Download Scalar Field Theories with Screening Mechanisms

Scalar Field Theories with Screening Mechanisms
Tom Charnock
If new scalar fields are included in theories then new forces can be found. A screening mechanism
dynamically suppresses theses forces without the need for fine tuning. This relies on the presence of
non-linearities in the equations of motion.
Most physics done is linear, so studying screening mechanisms can give insight into non-linear theories.
The screening mechanism can also be relevant in theories of dark energy. New large scale physics is
needed to explain dark energy, such as new fields, and for these to be relevant on large scales they need
to be light. This produces a problem since new light fields have not been seen.
Scalar fields are bosons, and bosons are force carriers. Scalar fields (with mass) in particular give rise to
to a Yukawa force,
g 2 1 −mϕ r
e
.
V (r) = −
4π r
Since new forces are not seen then this suggests that either g must be unnaturally small or that mϕ must
be very large. Clearly mϕ is not allowed to be large since the fields need to be light to affect large scales.
This leaves only the option of small g. Tests on gravity show that g 2 ≪ G, where G is Newton’s constant,
which would reveal that Mg ≫ Mp , i.e. a mass outside of where our current theories can probe.
1
Scalar Forces
A Lagrangian describing a scalar field and a fermion field with a coupling is given by
L=
1
1
(∂µ ϕ)2 − m2ϕ ϕ2 + ψ̄(iγ µ ∂µ − mψ )ψ + g ψ̄ψϕ,
2
2
where the signature is taken to be (+, −, −, −). The first two terms describe the scalar field with mass
mϕ , the third the kinetic energy of the fermion field and the last term is the coupling between the fermion
and the scalar field. 2-2 particle scattering is given by:
ψ
p′
k′
ϕ
iM =
q
p
k
ψ
The fermion propagator,
and the vertex,
p
is
i(p
/ + mψ )
, the scalar propagator,
p2 − m2ψ + iε
q
, is
i
,
q 2 − m2ϕ + iε
, is −ig.
In the non-relativistic limit then the macroscopic forces are most relevant and the four-momentum can
be written p = (m, p) and k = (m, k ) where p and k are small such that
e
e
e
e
′
2
(p − p) = −|p − p′ |2 + O(p4 ).
e e
1
The diagram written algebraically is
2
iM = (−ig)
µ
′
Ū (p )U (p)
µ
¶
i
(p′ − p)2 − mϕ
¶
Ū (k )U (k) ,
′
where iε is dropped since it is not important to the calculation and Ū (p) and U (p) are the fermion states.
In the non-relativistic limit this becomes
!
Ã
i
2
iM = −g 2mψ
2mψ .
−|p′ − p|2 − m2ϕ
e e
This is true of a quantum field theory calculation, but in quantum mechanics, fermions are governed by
the Scrödinger equation,
~2 2
−
∇ ψ + V (x)ψ = Eψ,
2m
where E can be absorbed into V . The gradient of the potential provides the force which causes the
fermion to scatter. A sketch of the scattering could be drawn as
ψ
ψ
V (x)
e
If the scattering is weak then the Born approximation can be used,
hin|iM|outi = −iVe (q)(2π)δ(Ein − Eout ),
where the δ(E) defines the conservation of energy and Ve (q) is the Fourier transform of the potential. The
Born approximation can be equated to the quantum field theory calculation once it has been renormalised
such that,
−iVe (q)(2π)δ(Ein − Eout ) = (2π)4 δ (4) (pin − pout )iM,
which means that,
g2
Ve (q ) = − 2
.
|q| + m2ϕ
e
The inverse Fourier transform of the potential is
Z
−g 2
d3 q
iq ·x
e .
V (x) =
3
2
(2π) |q| + m2ϕ e e
e
Ve (q ) does not depend on the direction of q so V (x) does not depend on the direction of x. This means
e
e
e x can arbitrarily be allowed to point e
that
along the z axis such that
e
µ
¶
Z ∞
Z π
Z 2π
−g 2
1
2
q dq
sin ϑdϑ
dφ
eirq cos ϑ
V (x) =
(2π)3 0
q 2 + m2ϕ
e
0
0
The φ and the ϑ integrals are easily solvable so
Z ∞
1
1
i(e−iqr − eiqr )
2
V (x)
q 2 dq 2
(−g
)
.
2
2
q + mϕ
qr
e (2π)
0
Since
Z
0
∞
e−A −
Z
∞
eA =
0
Z
0
eA +
0
−∞
∞
=
Z
−∞
2
Z
A
e ,
∞
eA
then the potential can be written,
V (x) =
ig 2
(2π)2
Z
∞
dq
−∞
qeiqr
.
+ m2ϕ
q2
This can be solved by contour integration
Contribution Vanishes as ℑ[E] → ∞
Contributions Cancel Out
E Complex Plane
m
−q
q
−m
Contribution Vanishes as ℑ[E] → −∞
The residue theorem gives,
Z
∞
dq
−∞
so that
qeiqr
= iπe−rmϕ ,
(q + imϕ )(q − imϕ )
ig 2 1
V (x) =
πie−mϕ r
(2π)2 r
e
g 2 e−mϕ r
=−
.
4π r
This has the same form as Coulomb’s law or the gravitational potential. The scattering of fermions which
couple to a scalar field reveals a classical looking force.
2
Universally Coupled Scalars
Dark energy theories often have scalars which couple identically to all other fields. This can be restated
by saying that matter fields/particles move on geodesics of
geµν = A(ϕ)gµν ,
where gµν is the spacetime metric and geµν is a conformal rescaling. A simple form of A(ϕ) can be assumed
here for calculational purposes, such as, A(ϕ) = 1 + aϕ, where a is a constant. The geodesic equation is
e µ∇
e µU
e ν = 0 where U
e µ is the velocity of particles normalised with respect to the conformally rescaled
U
µ
ν
e U
e = −1. Positions of particles along the geodesic are given by X where the geodesics
metric, geµν U
are parameterised by λ where ˙ = ∂/∂λ. As another simplification the spacetime metric will be taken
to be flat, gµν = ηµν . Another velocity can be defined such that ηµν U µ U ν = −1 and an acceleration is
aµ = U ν ∇ν U µ . The geodesic equation can also be written as
∂ 2 X ν (∼) ν ∂X µ ∂X ρ
+ Γ µρ
= 0.
∂λ2
∂λ ∂λ
The Christoffel symbols are
Γα
βγ =
1 αδ
g (gδβ,γ + gδγ,β − gβγ,δ ),
2
3
where the derivative of the conformally rescaled metric is
∂
∂
geδβ =
[(1 + aϕ)ηαβ ]
∂xγ
∂xγ
∂ϕ
= a γ ηδβ .
∂x
e µU
e ν = −1 and ηµν U µ U ν = −1 then U µ = √1 + aϕU
e µ and hence
Since (1 + aϕ)ηµν U
aµ = U ν ∇ν U µ
hp
i
p
e ν ∇ν
eµ
1 + aϕU
= 1 + aϕU
³
h³
a ´ eν
a ´ e µi
≈ 1+ ϕ U
∇ν 1 + ϕ U
2
2
= −a∂ν ϕ (3U µ U ν + η µν ) ,
where in the third line it has been assumed that aϕ ≪ 1. According to the spacetime metric there is
an acceleration since particles are not travelling on the spacetime geodesics. If the system is static and
spherically symmetric then U µ = (1, 0), ϕ = ϕ(r) and ar = −aϕ(r)′ . ar shows that the radial acceleration
is due to the derivative of the scalar efield.
Now using signature (−, +, +, +), the Lagrangian describing a scalar field with some mass, mϕ and matter
travelling on geodesics of the conformally rescaled metric is
1
1
Lϕ = − (∂ϕ)2 − m2ϕ ϕ2 + Lm (ψ; (1 + aϕ)gµν ).
2
2
The equations of motion for ϕ are
1 δLm
(−ag µν ) = 0.
¤ϕ − m2ϕ ϕ + √
g δe
gµν
The energy-momentum tensor from the spacetime metric is
2 δLm
Tµν = − √
−g δg µν
2
δLm
− √ (1 − aϕ) µν ,
−g
δe
g
and as such, the equations of motion for the scalar field can now be rewritten
¶
µ
a
−Tµν
2
µν
= ¤ϕ − m2ϕ ϕ + T µµ
¤ϕ − mϕ ϕ − ag
2(1 − aϕ)
2
= 0.
In the flat, spherically symmetric, non-relativistic system then a source has T µµ = −̺(r) so that
¤ϕ − m2ϕ ϕ =
a̺(r)
.
2
If the source is a point mass, Mc , then ̺(r) = Mc δ (3) (r) so
ϕ′′ +
2ϕ′
a
− m2ϕ ϕ = Mc δ (3) (r).
r
2
Everywhere apart from r = 0 then the solutions are of the form
ϕ=
A −mϕ r B mϕ r
e
+ e
r
r
where B = 0 since ϕ needs to vanish as r → ∞. As r → 0 then a small sphere of radius ε around the
origin gives
µ
¶
¸
·
Z ε
aMc
1 ∂
2 ∂ϕ
2
2
r
− mϕ ϕ =
.
r dr 2
4π
r
∂r
∂r
2
0
4
Substituting in ϕ =
A −mϕ r
re
gives
−A =
aMc
.
8π
This means that
aMc −mϕ r
e
8πr
which has exactly the same form as the Yukawa term from the first section. Radial forces are given by
derivatives of the scalar field,
ϕ(r) = −
Fr = a∂r ϕ
=−
a2 Mc
(1 + mϕ r)e−mϕ r .
8πr2
A particle in a scalar field would therefore feel a force.
3
Chameleons
A scalar field which changes its mass depending on its environment is known as a chameleon. This is a
useful mechanism since the mass dictates the distance over which a force can be felt. The action for such
a chameleon theory is,
Ã
! Z
Z
Mp2
1
4 √
2
S = d x −g
R − (∂ϕ) − V (ϕ) + d4 xLm (ψ; geµν (ϕ)),
2
2
i.e, the same form as in the previous section, but now the conformally rescaled metric will take the form
geµν (ϕ) = e2ϕ/M gµν .
2ϕ/M fixes the energy scale and is taken to be much less than one so that
¶
µ
2ϕ
gµν .
geµν = 1 +
M
Varying the action with respect to ϕ gives the equations of motion for the scalar field, where the varied
action is
¶
µ
Z
Z
δL
2δϕ
′
4
4 √
g µν .
−
δϕ S = d x −g (δϕ¤ϕ − δϕV (ϕ)) + d x
δe
gµν
M
The energy-momentum tensor is
2 δLm
Tµν = − √
−g δg µν
µ
¶
2ϕ δLm
2
1−
,
= −√
−g
M δe
g µν
so placing this into the varied action gives the equations of motion,
µ
¶
2ϕ
1
1+
T µµ = 0.
¤ϕ − V ′ (ϕ) +
M
M
The matter fields can be treated as a perfect fluid so

−̺

p
µ
T ν =


p
p

,

and in non-relativistic cases then p ≪ ̺ so T µµ = ̺. To first approximation the equation of motion is
¤ϕ = V ′ (ϕ) +
′
= Veff
(ϕ),
5
̺
M
where the effective potential is
Veff = V (ϕ) +
̺ϕ
.
M
If V (ϕ) has a power law form, for demonstration V (ϕ) = Λ4−n ϕn , then the effective potenial is
Veff = Λ4−n ϕn +
̺ϕ
,
M
′
where a minimum needs to exist for stability. To find this minimum then solutions to Veff
(ϕ) = 0 need
to be found. These solutions are
̺
ϕn−1
,
min = −
nM Λ4−n
where n < 0 or n > 0 and odd. The mass is the second derivative of the effective potential,
′′
m2min = Veff
(ϕmin )
³
= n(n − 1)Λ4−n −
´(n−2)/(n−1)
̺
,
nM Λ4−n
where, as long as n 6= 1 and n 6= 2, the mass in the minimum depends on the density, ̺. In fact
dm2
> 0.
d̺
n
For n < 0 then the potential is of the form V = Λ4 (Λ/ϕ) which looks like
V
V
V
V
Veff
Veff
̺
M
̺
M
ϕmin
Low Density
i.e. Interstellar Medium
ϕmin
High Density
i.e. Water
As the density increases the minimum occurs at smaller field values and the mass scale also increases.
Take an object, such as a planet, with a radius r = Rc and density ̺c in a much less dense background,
with almost constant density ̺∞ . The object is taken to be spherical such that the system is spherically
symmetric and static and the mass is
4
Mc = πRc3 ̺c .
3
̺∞
̺c
4
Mc = πRc3 ̺c
3
Rc
6
The equations of motion can now be solved in a piecewise manner. Far from the object, the scalar field
sits in the minimum of the potential at ϕ∞ = ϕ(̺∞ ). If there is some small disturbance in the density
then a Taylor expansion of the potential can be made,
1
Veff (ϕ) = Veff (ϕ∞ ) + m2∞ (ϕ − ϕ∞ )2 ,
2
′′
(ϕ∞ ). In this static, spherically symmetric system the equation of motion is
where m2∞ = Veff
d2 ϕ 2 dϕ
+
= m2∞ (ϕ − ϕ∞ ),
dr2
r dr
with solutions of the form
A −m∞ r B m∞ r
e
+ e
.
r
r
Since these solutions must hold far from the object, then the r → ∞ boundary condition sets B = 0.
These solutions are assumed to be true all the way down to r = Rc even though this is a non-trivial
assumption. It turns out that it actually is a good approximation although the calculation is not done
here.
ϕ = ϕ∞ +
If the object is small, in size or density or both, then only small perturbations of the field about ϕ(̺∞ )
are made. This means that inside the object, r < Rc , then the effective potential is well approximated
by Veff (ϕ) = ̺c ϕ/M and so the equation of motion is
̺c
2
,
ϕ′′ + ϕ′ =
r
M
which has solutions
̺c r2
C
+ + D,
6M
r
where C and D are constants of integration. This is true all the way down to the origin and thus C = 0
is set as the r = 0 boundary condition. The field must be smooth at the boundary of the object and so
ϕ and ϕ′ are matched at r = Rc to get
¶
µ

r2
Mc

 ϕ∞ −
0 < r < Rc
2− 2
8πM Rc
Rc
.
ϕ=

 ϕ − Mc e−m∞ (r−Rc ) R < r
∞
c
4πM r
ϕ=
Since the force is the derivative of the field, then assuming m∞ r < 1
∂r ϕ
M
Mc
=
.
4πM 2 r2
|F | =
This is the Newtonian force law. There has been no change in the chameleon when the change in density
is only small.
When the object is large then the field reaches ϕc = ϕmin (̺c ) inside the source. The mass is large in this
environment and since the potential is steep then the field is trapped at ϕc . The object with radius Rc
has a surface inside at r = S so that when r < S, ϕ = ϕc .
Rc
S
7
The exterior solution remains the same outside of r = Rc and the field interpolates smoothly in the region
S < r < Rc such that Veff (ϕ) ≈ ̺ϕ/M . The field is therefore

0<r<S

 ϕc 2

 ̺c r
C
+ + D S < r < Rc .
ϕ=
6M
r



 ϕ∞ + A e−m∞ r Rc < r
r
′
Again, matching ϕ and ϕ across r = Rc and again at r = S then
µ
¶
̺c Rc3 S 3
−m∞ Rc
−1 ,
=
Ae
3M
Rc3
̺c S 3
,
C=
3M
̺c S 2
D = ϕc −
,
2M
2M
S2
(ϕ∞ − ϕc ).
1− 2 =
Rc
̺c Rc2
The last expression defines the condition for an object to be large or small. For an object to be large
then it must have
2M
(ϕ∞ − ϕc ) < 1.
̺c Rc2
To understand what this means then it makes sense to write this as
(ϕ∞ − ϕc )/M
< 1,
3(GMc /Rc )(Mp /M )2
where G is Newtons constant and Mp is the Planck mass. GMc /Rc is the Newtonian potential, Mp /M is
the ratio of the gravitational coupling to the scalar coupling and (ϕ∞ − ϕc )/M is the depth of the scalars
potential well. If M ∼ Mp then this expression is just the ratio of the force of the scalar field to the force
of gravity and so, for large objects, the scalar force must be weaker than gravity. This is called the thin
shell condition. Typically the thin shell condition is much less than one, which implies that S lies close
to Rc and hence there is a thin shell. If S = Rc (1 − ε) where ε is small then
−2ε =
2M
(ϕ∞ − ϕc ),
̺c Rc
which leads to
So outside of the object, r > Rc , then
e−m∞ Rc A = −(ϕ∞ − ϕc )Rc .
ϕ = ϕ∞
µ
¶
Rc −m∞ r
,
e
1−
r
where the assumptions are ϕ∞ ≫ ϕc and m∞ Rc ≪ 1. The first condition arises because ̺c ≫ ̺∞ . The
scalar force is
∇ϕ
Fϕ =
M
ϕ∞ R c
=
(1 + m∞ r)e−m∞ r
M r2
ϕ∞ R c
.
=
M r2
Compared to the gravitational force (per unit mass) then
Fϕ
FNewton
ϕ∞ R c r 2
M r2 GMc
ϕ∞ /M
=
,
(GMc /Rc )
=
which for a gravitational strength scalar force, M ∼ Mp , the thin shell condition shows that the scalar
force must be much weaker than gravity.
8
Quantum Chameleons
Quantum chameleons are a problem. Take the potential to be V = Λ5 /ϕ so that Veff = Λ5 /ϕ + ̺ϕ/M
with a minimum at ϕ2 = Λ5 M/̺ then Taylor expanding the potential gives
1
Veff = Veff (ϕmin ) + V ′′ (ϕmin )(ϕ − ϕmin )2 + · · ·
2
µ 2 ¶
µ 5 ¶1/2 µ 3 ¶1/2
̺
̺
Λ ̺
2
(ϕ − ϕmin )3
+
(ϕ
−
ϕ
)
−
=2
min
5
3
5
M
Λ M
Λ M2
5 ³ ̺ ´5/2
+
(ϕ − ϕmin )4 + · · · .
6 Λ3 M
Truncating this Taylor expansion is fine as long as the coefficients are small. The coefficient of the ϕ4
term is dimensionless,
5 ³ ̺ ´5/2
λ=
.
6 Λ3 M
This becomes larger than one when ̺ > Λ3 M . The typical values of these are Λ ∼ 10−3 eV for the dark
energy scale and M ∼ Mp ∼ 1018 GeV. The truncation of the ϕ4 term is there for only allowed when
̺ < 10−1 gcm−3 . This is problematical since the density of water is ̺ ∼ 1gcm−3 and hence the ϕ4 term
would need to be calculated for most reasonable matter. Other orders in the expansion would also need
to be calculated and kept and so the truncation of the Taylor expansion does not really work in this
regime.
The sum of the one loop chameleon corrections is
m4ϕ (ϕ)
ln
∆V1-loop (ϕ) =
64π 2
Ã
m2ϕ (ϕ)
M2
!
,
where M is an arbitrary energy scale. For these corrections to be negligible then again ̺ . Λ3 M . The
quantum corrections therefore become large for any normal sized densities.
Screening Mechanisms
The chameleon relies on the non-linear terms in the potential at the level of the equations of motion,
i.e. the potential is not just a mass term. This allows the properties to vary with the environment. The
chameleon Lagrangian for the potential used in the previous section is
Λ5
̺ϕ
1
−
,
Lϕ = − (∂ϕ)2 −
2
ϕ
M
where ̺ is the background matter with solutions ̺(x) and ϕ0 (x). The fluctutations on top of the background are δ̺ and δϕ which, when fed back into the Lagrangian, gives
·
¸
1
Λ5 δϕ2
δ̺δϕ
2
3
Lδϕ = − (∂δϕ) −
+ O(δϕ ) −
.
2
ϕ0 ϕ20
M
The mass of the scalar field is
′′
Veff
(ϕ) = V ′′ (ϕ)
2Λ5
ϕ3
= m2 (ϕ),
=
which leaves the Lagrangian
1
δ̺δϕ
1
+ O(δϕ3 ).
Lδϕ = − (∂δϕ)2 − m2 (ϕ0 )δϕ2 −
2
2
M
More generally, a similar Lagrangian could be constructed as
1
β(ϕ0 )
1
δ̺δϕ + O(δϕ3 ).
Lδϕ = − Z µν (ϕ0 )∂µ δϕ∂ν δϕ − m2 (ϕ0 )δϕ2 −
2
2
M
9
So now each term in the Lagrangian depends on the background field and not just the mass term as for
the chameleon.
If β(ϕ0 ) becomes small in experimental environments then any force would be screened. The coupling to
matter would become weak. This is known as the symmetron model as introducted by Hinterbichler and
Khoury, but was used earlier by Damour and Polyakov for the string dilaton.
If Z µν (ϕ0 ) is large then it makes it difficult for the field to propagate and hence hard to transmit a force.
c = Z 1/2 δϕ which reveals
Another way to see this is to canonically normalise, Z µν ∼ Zg µν , so that δϕ
β
1 c 2 1 m2 c 2
c
δϕδ̺.
δϕ −
Lδϕ
c = − (∂ δϕ) −
2
2 Z(ϕ0 )
M Z 1/2 (ϕ0 )
Large Z suppresses the couplings and hence the force is weak. This mechanism is used explicitly in
Nicolis, Rattazzi and Trincherini’s Galileon model which is related to the DGP model. In general this
type of screening is known as Vainshtein screening and was used by Vainshtein to hide the scalar mode
of the massive graviton.
4
Galileons
Working in flat space, there is a symmetry where ϕ → ϕ + bµ xµ + c. The name of the model comes from
the similarity to the Galilean spacetime symetry, xi → xi − vi t and t → t.
In four dimensions the most general Lagrangian which respects the symmetry, up to total derivatives,
and has second order equations of motion is
L=
5
X
n=1
Λ3(2−n) cn Ln (ϕ),
where Λ is a dimensionful energy scale and cn are dimensionless coefficients. The values of Ln are
L1 = ϕ,
L2 = (∂ϕ)2 ,
L3 = ¤ϕ(∂ϕ)2 ,
¡
¢
L4 = (∂ϕ)2 (¤ϕ)2 − ∂µ ∂ν ϕ∂ µ ∂ ν ϕ ,
¡
¢
L5 = (∂ϕ)2 (¤ϕ)3 − 3(¤ϕ)∂µ ∂ν ϕ∂ µ ∂ ν ϕ + 2∂µ ∂ν ϕ∂ ν ∂ ̺ ϕ∂̺ ∂ µ ϕ .
A coupling to matter can also be included,
Lcoupling =
ϕ µ
T .
M µ
This does break the symmetry, but only in a soft, controllable way. When studying the Galileon model
then L1 tends to be ignored so c1 = 0 and the rest is then normalised such that c2 = 1. The equation of
motion is extremely long and complicated in general but there is never more than two derivatives acting
on the scalar. This means there are no Ostrogradski ghosts. For a static, spherically symmetric system,
then the equation of motion reduces to
à (µ ¶
µ ¶2
µ ¶3 )!
c3 ϕ′
Mc δ (3) (r)
ϕ′
c4 ϕ′
1 ∂
3
r
+
=
+
,
r2 ∂r
r
Λ3 r
Λ6 r
M
with a point source of mass Mc . Integrating this once gives
µ ¶2
µ ¶3
ϕ′
c4 ϕ′
Mc
c3 ϕ′
+ 6
=
+ 3
.
r
Λ
r
Λ
r
4πM r3
For c3 = c4 = 0 then a canonical scalar field is found and
ϕ′ =
Mc
,
4πM r2
10
which is simply the force
F =
=
ϕ′
M
Mc
,
4πM 2 r2
i.e. exactly the gravitational force on a test mass when M = Mp . When c3 , c4 6= 0 then, writing
ϕ′ =
Mc
g(r),
4πM r2
it can be seen that if g < 1 then the force due to the scalar field is weaker than gravity and vice versa
for g > 1. Placing ϕ′ into the equation of motion gives
µ ¶6
µ ¶3
R2
R1
2
g +
g 3 = 1,
g+
r
r
with
1
R1 =
Λ
µ
c3 Mc
4πM
¶1/3
1
R2 =
Λ
Ã
c4 Mc
4πM
and
1/2
!1/2
,
which are known as the Vainshtein radii. As r → 0 then g → 0 also. This means that the force must
become weaker than gravity for the equation of motion to be satisfied. This model therefore does not
depend on the size of the density of the object unlike when using the chameleon screening mechanism.
The solutions to the equations of motion are stable under small perturbations if c2 > 0, c3 ≥
c4 ≥ 0 and c5 < 3c24 /4c3 . Since c3 and c4 are related then it can be seen that R1 > R2 :
p
3c2 c4 /2,
Mc
R2
R1
A long way from the source, r ≫ R1 , R2 , the non-linear terms in the equation of motion become negligible
and thus
Mc
.
ϕ′ =
4πM r2
When (R2 /R1 )3 R2 ≪ r ≪ R1 the solution is
s
Mc Λ3 −1/2
′
ϕ =
r
,
4πc3 M
and so the force is weaker than it is further from the source. Finally, from 0 < r < (R2 /R1 )3 R2 the
solution is constant in r,
¶1/3
µ
Mc Λ 6
′
.
ϕ =
4πc4 M
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Comparing these solutions to the Newtonian force, remembering that Fϕ = ϕ′ /M , it can be seen that
 µ
¶2

Mp


R1 ≪ r


M


¶2 µ ¶3/2 µ ¶3
µ

Fϕ
r
R2
Mp
=
R1 ≪ r ≪ R1 .

FN
M
R
R1

1

µ
¶
µ
¶
µ ¶3
2
2


Mp
r
R2


0<r<
R2

M
R2
R1
These properties are true for any source. Typical parameter values are M ∼ Mp and Λ3 ∼ H02 Mp where
H02 is used to make a connection to dark energy. Taking c3 ∼ O(1) then, for the earth, R1 ≈ 1018 cm≈ 1pc.
The screening mechanism for the earth is effective over gigantic distances. For the sun then R1 ≈ 10Mpc!
Only galaxy clusters have sizes comparable to their screening radius.
Because the equations of motion are non-linear then the forces cannot be superimposed as they can in
Newtonian systems. This makes it difficult to solve systems where one object may be inside anothers
Vainshtein radius.
Quantum Corrections
Flat space Galileons respect the symmetry
ϕ → ϕ + bµ xµ + c.
The field theory was constructed using this symmetry, under the condition that the equations of motion
were second order. The most general theory with second order equations of motion, which does not
respect the symmetry is
5
X
fn (ϕ, X)Ln ,
L=
n=1
where fn are arbitrary functions of the field and its kinetic energy, X = (∂ϕ)2 /2. This is Horndeski’s 1974
theory, which was largely forgotten until 2009 when Deffayet, Deser and Esposito-Farese approched it
from the Galileon point of view. The symmetry of the Galileon theory restricts the quantum corrections
from generating terms such as L ≥ (ϕ/M )10 (∂ϕ)2 , · · · .
Λ is the scale in the Galileon theory which controls when non-linearities enter. This means that it controls
when loop corrections are necessary. For a typical Λ ∼ (H02 Mp )1/3 ∼ 1.7 × 10−22 GeV then the theory
cannot be trusted classically at energies higher than this. Equivalently, Λ−1 ≈ 1.1 × 103 km so quantum
corrections need to be calculated when the distance scale is less than r ≈ 1000km. This is only true in
vacuum. If the background is sourced, then the region for the necessity of quantum corrections when
perturbations are small can be calculated, i.e. are predictions in a room on the surface of earth possible
given that the vacuum solution requires quantum corrections for distances r < 1000km? For the earth,
small perturbations in the scalar field are
ϕ = ϕ⊕ (r) + δϕ,
with
ϕ′⊕
=
s
M⊕ Λ3 −1/2
r
.
4πc3 M
The Lagrangian for quantum fluctuations is
Lδϕ = Zµν (ϕ⊕ )∂ µ δϕ∂ ν δϕ +
c3
¤δϕ(∂δϕ)2 ,
Λ3
where
1
c3
Zµν = − ηµν +
(¤ϕ⊕ ηµν − ∂µ ∂ν ϕ⊕ ) .
2
Λ
Inside the screening radius the non-linear terms become important. When r ≪ R1 = R⊕ then this
simplifies to Zµν ≈ diag where all the energies have a common magnitude, Z ∼ (R1 /r)3/2 , i.e. really
c = Z 1/2 δϕ reveals the Lagrangian
large. Canonically normalising the fluctuations as δϕ
c 2
Lδϕ
c ≈ −(∂ δϕ) +
12
c δϕ)
c 2
c3 ¤δϕ(∂
.
3
3/2
Λ
Z
e = ΛZ 1/2 and so Λ
e is much larger than Λ, therefore quanThe strong coupling scale now becomes Λ
tum corrections needed to be calculated for much higher energies. For the surface of the earth then
e ≈ 5.1 × 10−15 GeV which corresponds to a distance scale Λ
e −1 ≈ 4cm. This is still not brilliant,
Λ
but better than only being able to calculate classical solutions above 1000km. If higher order terms in
the Galileon theory are kept then the background dependence makes the way the strong coupling scale
changes less effective.
Unlike for the chameleon, where quantum corrections become important as soon as the screening mechanism takes affect, the Galileon theory has a region of space where the classical non-linearities are important, and so screening occurs, but the quantum corrections are small.
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