Embedding Brans-Dicke gravity into electroweak theory

Abstract

We argue that a version of the four dimensional Brans-Dicke theory can be embedded in the standard flat spacetime electroweak theory. The embedding involves a change of variables that separates the isospin from the hypercharge in the electroweak theory.

Full text

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arXiv:0709.0586v2 [hep-ph] 27 Jun 2008
Embedding Brans-Dicke gravity into electroweak theory
M. N. Chernodub1, ∗and Antti J. Niemi2,3,4, †
1Institute of Theoretical and Experimental Physics,
B. Cheremushkinskaya 25, Moscow 117218, Russia
2Laboratoire de Mathematiques et Physique Theorique CNRS UMR 6083,
F´ ed´ eration Denis Poisson, Universit´ e de Tours, Parc de Grandmont, F37200, Tours, France
3Department of Theoretical Physics, Uppsala University, P.O. Box 803, S-75108, Uppsala, Sweden
4Chern Institute of Mathematics, Tianjin 300071, P.R. China
(Dated: June 27, 2008)
We argue that a version of the four dimensional Brans-Dicke theory can be embedded in the
standard flat spacetime electroweak theory. The embedding involves a change of variables that
separates the isospin from the hypercharge in the electroweak theory.
PACS numbers:04.20.Cv, 11.15.Ex, 12.15.-y, 98.80.Cq
The construction of a consistent four dimensional
quantum theory of gravity remains a challenge. Super-
string theory with its pledge to unify all known interac-
tions is the most attractive candidate for resolving this
conundrum [1]. Some colleagues have also argued that
conformal Weyl gravity is a renormalizable albeit not
apparently unitary four dimensional quantum theory of
gravity [2]. Finally, there are indications that N = 8 su-
pergravity theory might be ultraviolet finite, for reasons
that resemble those ensuring the finiteness of the N = 4
supersymmetric Yang-Mills theory [3]. The spectrum of
the latter appears to coincide with that of the AdS5×S5
solution of ten dimensional IIB supergravity theory [4].
This duality has led [5] to propose that even within the
strong and the electroweak components of the Standard
Model there is an embedded quantum theory of gravity
that remains to be discovered. (See also [6].)
In the present paper we inspect how a gravity the-
ory could be embedded in the bosonic sector of the (Eu-
clidean signature) Weinberg-Salam Lagrangian [7],
LWS=1
4
?F2
µν+1
4B2
µν+ |Dµφ|2+ λ(φ†φ)2+ µ2φ†φ (1)
All our notations are exactly as in [7]. We decompose the
Higgs field φ and the SUL(2) gauge field Aa
µas follows,
φ =
?σeiα?· S ,
??
+?W+
S =
eiγ
?2(1 − n3)
?
n1− in2
1 − n3
?
(2)
? Aµ =
W3
µ+2i
g
? m+· ∂µ? m−
?
ˆ n −
i
2g[∂µˆ n, ˆ n]
ˆ Aµ +ˆ Xµ
?
(3)
µ· ˆ m++ W−
µ· ˆ m−?
≡
Here ˆ n is the isospin projection operator
ˆ n = ? n · ? τ = −φ†? τφ
φ†φ
· ? τ
and ? m±= ? m1±i? m2so that (? m1, ? m2,? n) is a right-handed
orthonormal triplet. If the phases α and γ are combined,
(2), (3) involves sixteen independent fields as a complete
decomposition should. But for future reference we prefer
to keep α and γ separate. Notice that ? m±is defined up
to a phase that we may identify with γ. We have se-
lected (2) and (3) so that in the unitary gauge where ˆ n
becomes equal to the diagonal Pauli matrix τ3, the diag-
onal component of the gauge field coincides with W3
the off-diagonal components coincide with the W±
the low-temperature Higgs phase W3
the UY(1) hypergauge field Bµinto the massive neutral
Z-boson and the massless photon, and the off-diagonal
gauge fields W±
charged W-bosons [7].
In the present paper we shall argue that a gravity the-
ory emerges from (1) in terms of spin-charge decomposed
variables. For this we start by noting that the Higgs field
is a Lorentz scalar but with a nontrivial isospin and a
nontrivial hypercharge. Thus we decompose it accord-
ingly, and the result is displayed in (2). Since the phys-
ical Z-boson and photon are both charge neutral, there
is no room for any spin-charge decomposition in W3
Bµ. But the W-bosons are charged and they also have
a nontrivial Lorentz spin, and following [8] we separate
their spin from their charge by decomposing
µand
µ. In
µthen combines with
µ=
1
√2
?
W1
µ∓ iW2
µ
?
become the massive
µand
W+
µ= ψ1
?µ+ ψ2¯
?µ
(4)
The ψ1,2are two complex scalars, they carry the charge
of W±
normalized according to
µ[8]. The complex four-vector
?µcarries spin, it is
?µ
?µ= 0&
?µ¯
?µ= 1(5)
The decomposition (2) admits an internal Uφ(1) gauge
symmetry: If we send α → α+δ and γ → γ−δ the Higgs
field φ remains intact. Similarly, if we multiply ψ1and ψ2
by a phase and
and (4) do not change under this internal UW(1) gauge
transformation. The gauge fields for these internal sym-
metries are composite vector fields. For Uφ(1) we have
Λµ= −iS†∂µS and for UW(1) we have Cµ= i¯
?µby the complex conjugate phase, (3)
? · ∂µ
?.

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We now define a number of auxiliary quantities. We
start by introducing the three component unit vector
?t =
1
ρ2·?¯ψ1
¯ψ2
?? σ
?ψ1
ψ2
?
where ρ2= |ψ1|2+|ψ2|2. With gYµ= g? n ·?Aµ−2Λµwe
define a Uφ(1) × UW(1) covariant derivative as follows,
DC
µψ1,2= (∂µ+ igYµ∓ iCµ)ψ1,2,
DC
µ
?ν= (∂µ+ iCµ) ?ν
and a Uφ(1) × UY(1) covariant derivative as follows,
Dµ= ∂µ− ig
2ig · ρ2t3· ( ?µ¯
2Yµ− ig′
2Bµ
We then introduce the gauge invariant supercurrents
J±
µ
=
i
2ρ2{?¯ψ1DC
−
µψ1− ψ1¯DC
µ¯ψ1
?± (1 → 2)}
Tµ
=
i
2σ2{φ∗Dµφ − φ¯Dµφ∗}
(6)
Finally,
Pµν=1
?ν−
?ν¯
?µ) ≡ gρ2t3Hµν.
Following [8] we interpret ρ2as the conformal scale of a
locally conformally flat metric tensor,
?ρ
and from now on all the Greek indices µ,ν,λ,... refer to
the ensuing locally conformally flat spacetime. Note that
in a coordinate basis the metric tensor is dimensionless
while ρ has the dimensions of mass. Dimension analysis
then tells us to introduce the a priori arbitrary mass pa-
rameter κ. With the metric tensor we have the vierbein
Eaµ that relates a coordinate basis (µ) to a local or-
thogonal frame (a), the Christoffel symbol Γλ
connection ωλ
µ νand all other geometric quantities that
are defined in the usual, standard fashion. The covariant
derivative of the zweibein field
Gµν =
κ
?2
δµν
(7)
µν, the spin
?µis [8]
∇µ
?ν+ ωλ
µ ν
?λ= ∂µ
?ν− Γλ
µν
?λ+ ωλ
µ ν
?λ= ρ · ∂µ(
?ν
ρ)
The covariantized UW(1) gauge field is
Cµ= i¯
?σ∇µ
?σ+ i¯
?λωσ
µ λ
?σ
and we also introduce the following twisted covariant
derivative operator
Dν
µ λ= δν
λ∇C
µ+ ων
µ λ= δν
λ(∇µ+ iCµ) + ων
µ λ
Finally, in 4D the Ricci scalar for our metric tensor is
?κ
R = −6
ρ
?2?1
ρ2(∂µρ)2+ ∂µ(1
ρ∂µρ)
?
In order to relate (1) to a gravity theory, we first em-
ploy the present geometrical structure to convert it into
a generally covariant form. The result is a sum of two
terms LWS= L(1)
rately. We start with L(1)
the first one is
WS+ L(2)
WSthat we now inspect sepa-
WS. It admits two contributions,
L(11)
+1
4
√
WS=1
√
4
√GGµνGρσGµρGνσ
GGµνGρσTµρTνσ+ ς2√
GGµν∂µς∂νς +ς2
(8)
GGµνTµTν
(9)
+
6· R
√
G +
√
G{λς4+ rς2} (10)
We have defined
Gµν= Gµν(C) − (∂µJ+
where Gµν(C) is the ’t Hooft tensor [9]
Gµν(C) = ∂µ[t3Cν] − ∂ν[t3Cµ] −1
ν− ∂νJ+
µ) − 2g2κ2t3Hµν
(11)
2?t · ∂µ?t × ∂µ?t
(12)
and
Tµν=1
g′
?Gµν(C) − (∂µJ+
ν− ∂νJ+
µ) − 2(∂µTν− ∂νTµ)?
and ς = G−1/8σ and r = G−1/4µ2. Note in particular
that (8)-(10) have no explicit κ dependence except for
the last term in Gµν.
The second contribution to L(1)
?1
+(¯Dσ
+1
12· R
WSis
L(12)
WS= κ2·
2
√
GGµν
?
J+
µJ+
ν+1
4∇C
µ?t · ∇C
ν?t
(13)
?
µ λ¯
?σ)(Dτ
√
G +1
ν η
?τ) +1
2t−(Dσ
µ λ
?σ)(Dτ
√
G
ν η
?
?τ) + c.c.
(14)
4g2(ς2−3
8t2
3κ2} ·
(15)
Notice that the entire (13)-(15) is proportional to κ2.
Before we proceed to the final term
L(2)
WS= −1
2(Dab
µ[A]Xµb)2
(16)
we first point out some salient features in the structure
of (8)-(10), (13)-(15):
We start by observing that L(1)
SUL(2) × UY(1) gauge independent variables.
are sixteen independent fields, in addition to the UW(1)
phase.
Paramount to our geometric interpretation of L(1)
that the density ρ is a nonvanishing quantity, ?ρ? = ∆ ?=
0. A priori it could be natural to identify ∆ ≡ κ but we
keep them separate. Arguments have been given [10],
[11] that in a SU(2) Yang-Mills theory ∆ is nonvanish-
ing. Assuming that this persists in the Weinberg-Salam
WS
involves only
There
WSis

Page 3

3
model, the Lagrangian (8)-(10), (13)-(15) is defined in a
locally conformally flat spacetime which is different from
the flat R4of perturbation theory. This emergence of a
novel spacetime is in line with the no-go theorem [12], [13]
that forbids an embedded theory of gravity from residing
in the same spacetime with the underlying nongravity
theory; see also [5].
The contribution (15) has the standard Einstein-
Hilbert form with a cosmological “constant” term. Simi-
larly the two first terms in (10) constitute a Brans-Dicke
Lagrangian: With ψ = ς2we arrive at the standard
Brans-Dicke form [14] with the conformally invariant pa-
rameter value ω = −3
The contribution (8) together with the first term in
(13) describe the embedded dynamics of the supercurrent
J+
in the first term of (9). Similarly, the two terms in (9)
describe the embedded dynamics of the supercurrent Tµ.
It becomes massive whenever we are in a Higgs phase
where ς acquires a nontrivial expectation value.
It is notable that when κ ?= 0 the vector J+
mass even in the absence of the conventional Higgs effect.
Both Gµνand Tµνcontain the ’t Hooft tensor (12). To-
gether with the kinetic term in (13) for the vector field
?t this gives us an embedded, unitary gauge version of
the spontaneously broken SO(3) Georgi-Glashow model.
The unbroken symmetry group is the compact UW(1)
that has the capacity of supporting embedded magnetic
monopoles.
The second term in (12) in combination with the?t con-
tribution in (13) describes the embedding of the Faddeev
model [15]. Consequently we expect embedded knotted
solitons [16] to be present. Furthermore, in a Lorentz
invariant ground state we must have t3= ±1 [8] and this
prevents?t from supporting any massless modes.
The contribution (14) and the last term in Gµν de-
fines a (gauged) Grassmannian nonlinear sigma-model.
Its properties are detailed in [8], [17]. Together the unit
vector?t and the complex vector
sional internal space with the structure of S2× S2× S2.
The last term in (10) and the second term in (15) com-
bine into a cosmological “constant” contribution. The
original constant parameter µ2has become a spacetime
dependent variable r.We can interpret it as a back-
ground scalar curvature and we can combine it with the
middle term in (10). In addition, the Brans-Dicke-Higgs
field ς has a mass term which is proportional to κ. This
mass together with the scalar curvature R in (10) influ-
ence how symmetry becomes broken by the Higgs poten-
tial. In particular, there can be regions in the spacetime
where the symmetry is broken while in other spacetime
regions symmetry remains unbroken [18].
In the vicinity of the (Lorentz invariant [8]) t3 = ±1
ground state and when the field variables are slowly vary-
ing, we may delete all derivative contributions to the
Lagrangian (8)-(10), (13)-(15). We also assume that r
2.
µwith mass κ. The kinetic term of J+
µis also embedded
µacquires a
?µdescribe a six dimen-
describes the entire ground state scalar curvature ?R?.
When we minimize the ensuing potential for ς in (15) and
account for the Hµνin the first term of (8), we conclude
that the (classical level) cosmological constant becomes
vanishingly small when the background scalar curvature
r ∼ ?R? is
r · ∆ = µ2≈ −
??
λ
2g +g2
4
?
∆(17)
This gives
?ς2? ≈
1
2√2
g
√λ· κ2
(18)
Suppose now that (17), (18) hold and that we are near
the BPS limit so that λ is vanishingly small, and that
κ is finite (e.g. of the order of the electroweak scale).
The cosmological constant then vanishes and the effective
Planck’s mass in the second term of (10) can become very
large. The vector fields Jµand?t both have a mass which
is of the order of the electroweak scale, but the vector
field Tµbecomes very massive.
Finally, in the London limit where ρ is constant, L(1)
describes the interactions between J±
a flat spacetime which is different from the flat R4where
(1) is defined.
We now proceed to the remaining contribution (16).
Notably it is independently SUL(2) × UY(1) gauge in-
variant. It describes the interactive dynamics between
the Grassmannian vector field
currents
WS
µ, Tµ,?t,
?µand ς in
?µand the two complex
J(±)
µ
=
1
4Γν
νµ+1
2∂µln(1 ± t3) −
i
t3± 1· (J+
µ± J−
µ)
It also acquires a form which is generally covariant w.r.t.
the metric tensor (7). In particular, in parallel with (13)-
(15) the entire contribution (16) is proportional to κ2.
Since J(±)
the invariance under four dimensional diffeomorphism
group Diff(4) into SDiff(4), its volume preserving sub-
group. But we have also observed that when κ ?= 0 both
vector fields J±
As a consequence whenever κ ?= 0 the physical photon
field becomes subject to the Meißner effect and acquires
a mass in the Higgs phase. But since the photon mass
(if there is any!) is tiny [19], in order for us to reconcile
with the observed Physics we must take the limit κ → 0.
Since both (13)-(15) and (16) are proportional to κ2this
truncates the entire Lagrangian into (8)-(10).
We are now in the position to state the main proposal
of the present paper: When the metric tensor Gµν in
the Lagrangian (8)-(10) is taken to be arbitrary, this La-
grangian describes the gravity theory that emerges from
the electroweak Lagrangian (1).
Since the ’t Hooft tensor (12) is closed it can be writ-
ten as the exterior derivative of a (generally singular)
µ
contains Γν
νµ, the presence of (16) breaks
µand Tµare massive in the Higgs phase.

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4
vector field, and this vector field can be combined with
J+
when κ → 0. When the metric tensor is arbitrary but
Diff(4) symmetry remains broken into volume preserv-
ing SDiff(4) as κ → 0, the Lagrangian (8)-(10) with a
a priori arbitrary metric tensor engages sixteen SDiff(4)
invariant fields and this coincides exactly with the num-
ber of SUL(2) × UY(1) invariant fields in (1): The grav-
ity Lagrangian describes the interactive dynamics of the
conformal Brans-Dicke theory with a massless J+
with a Tµthat acquires a mass in the Higgs phase of the
Brans-Dicke-Higgs scalar field ς2.
The κ → 0 limit is like a Wigner-In¨ on¨ u contraction:
The identification (7) becomes singular but at the level
of the Lagrangian (8)-(10), (13)-(15), (16) with the arbi-
trary and in particular κ-independent metric, the κ → 0
limit is well defined.
Finally, we propose the following interpretation for the
appearance of a locally conformally flat metric tensor
in the original Lagrangian (8)-(10): We view this La-
grangian as the short distance limit of a higher deriva-
tive (one loop) renormalizable gravity theory [2] with the
additional term
µ. Thus both?t and
?µentirely disappear from (8)-(10)
µand
∆L ∼
1
4γ2· W2
µνρσ
where Wµνρσ is the Weyl tensor. Since the coupling γ
is asymptotically free, the β-function for γ enforces the
Weyl tensor to vanish at the short distance limit [2]. This
reduces the general metric tensor into its locally confor-
mally flat form in the short distance limit; see also [8].
In conclusion, we have constructed a change of vari-
ables that converts the bosonic Weinberg-Salam La-
grangian into a variant of the Brans-Dicke Lagrangian
in a locally conformally flat spacetime. We have argued
that when the metric tensor becomes arbitrary and we
take the limit where the physical photon becomes mass-
less, this Brans-Dicke Lagrangian determines the gravity
theory which is embedded in the Weinberg-Salam La-
grangian. We expect that one can similarly relate a grav-
ity theory to the strong sector of the Standard Model. It
would be interesting to work out the details in particular
since the enlarged structure of the SU(3) gauge group
may directly engage the remaining components of a full
metric tensor. We leave it as a puzzler to physically inter-
pret the possibility that within the Standard Model there
may be two distinct embedded gravity theories with their
own distinct spacetimes.
This work has been supported by a STINT Institu-
tional grant IG2004-2 025. The work by M.N.Ch. is also
supported by the grants RFBR 05-02-16306aand RFBR-
DFG 06-02-04010. The work by A.J.N is also supported
by a VR Grant 2006-3376 and by the Project Grant
ANR NT05 − 142856. The authors are most grateful to
L. Faddeev for numerous discussions, valuable comments
and criticism. We also thank S. Slizovskiy, F. Wilczek,
M. Zabzine and K. Zarembo for discussions. M.N.Ch.
is thankful to the members of Department of Theoreti-
cal Physics of Uppsala University, Institute for Theoret-
ical Physics of Kanazawa University, and Laboratoire de
Mathematiques et Physique Theorique of Tours Univer-
sity for hospitality and stimulating environment. A.J.N.
thanks the Aspen Center for Physics for hospitality.
∗Electronic address: Maxim.Chernodub@itep.ru
†Electronicaddress:
URL: http://www.teorfys.uu.se/people/antti
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