One loop effective potential in heterotic M-theory

0.27 MB·English

by  Ian G. Moss

One loop effective potential in heterotic M-theory
Ian G. Moss and James P. Norman
∗ †
School of Mathematics and Statistics, University of Newcastle upon Tyne,
Newcastle upon Tyne, NE1 7RU, United Kingdom
(Dated: February 1, 2008)
We have calculated the one loop effective potential of the vector multiplets arising
4
from the compactification to five dimensions of heterotic M-theory on a Calabi-Yau
0
0
2 manifold with h1,1 > 1. We find that extensive cancellations between the fermionic

n
and bosonic sectors of the theory cause the effective potential to vanish, with the
a
J
exception of a higher order curvature term of the type which might arise from string
4
2
corrections.

1
v
1
8
1
I. INTRODUCTION
1
0
4
0 The weakly coupled E E heterotic string is one of the most phenomenologically
8 8
/ ×
h
viable of the five consistent string theories. Unfortunately, the predicted value for Newton’s
t
-
p
constant in this theory is too large. Witten [1] has shown that this situation can be resolved
e
h
in the strong coupling limit, which is believed to be eleven-dimensional supergravity on
:
v
i the orbifold 10 S1/Z , with E Super Yang-Mills gauge theories on the orbifold fixed
X 2 8
M ×
r points [2]. This theory can be compactified on a Calabi-Yau three-fold to obtain six internal
a
dimensions. It is known that in order for the theory to predict the correct known values of
Newton’s constant and grand unification gauge couplings, the orbifold radius must be an
order of magnitude or so larger than the Calabi-Yau compactification scale. Hence, at some
intermediate energy scale, the theory has a consistent five-dimensional description.
Lukas et al. [3, 4] have derived the five-dimensional effective action by reducing Hoˇrava-
WittentheoryonaCalabi-Yauspace. They have shownthat theresulting theoryisagauged
version of N = 1 supergravity in five dimensions, with a non-abelian set of E gauge fields
8
on one orbifold plane, spontaneously broken to E , and an unbroken E gauge group on the
6 8
∗Electronic address: [email protected]
†Electronic address: [email protected]
2
other. The vacuum solution for this theory has a domain wall structure with a curved bulk
metric. This was the predecessor of the “brane-world” scenarios [5].
In addition to a gravitational and universal hypermultiplet present for every Calabi-
Yau compactification, the five-dimensional theory contains a number of vector multiplets,
depending on the topological properties of the Calabi-Yau. More precisely, there are n =
V
h1,1 1 of these vector multiplets. The background solution is a multi-charged BPS domain
−
wall with h1,1 charges. Lukas et al. show that there always exists a background solution
with h1,1 1 with two moduli parameters representing the separation of the domain walls
≥
and the dilaton field.
In this paper, we will calculate the one loop effective potential of these vector multiplets
about this simple background solution as a function of the moduli.
Thevector multiplets decouple fromthegravitationalanduniversal hypermultiplets mak-
ing them the simplest multiplets for the one loop calculation. Furthermore, it is possible
to change the number of vector hypermultiplets by changing h1,1 without altering the back-
ground solution or any of the other multiplets at one loop order. Therefore, the vector
multiplets tell us how the total vacuum energy depends on the topology of the Calabi-Yau
manifold.
The five dimensional domain wall solution gives rise to an effective four dimensional
theory in which the separation of the domain walls becomes one of the moduli fields. It
is important to identify effects which can provide a potential for the brane separation and
fix this particular modulus. One possible mechanism is that quantum fluctuations of the
bulk fields stabilise the branes at phenomenologically acceptable positions. This has been
discussed extensively in the context of the Randall-Sundrum brane world scenario [6, 7, 8,
9, 10]. Previous work of this kind in five-dimensional heterotic M-theory has been done for
scalar fields by Garriga et al. [11].
The BPS solution described above preserves half of the supersymmetry, hence one would
expect cancellation of the contributions to the effective action of the bosonic and fermionic
sectors. In a related calculation in an eleven dimensional setting, this expectation was con-
firmed when the vacuum energy was shown to be zero [12]. However, the Hoˇrava-Witten
theory has only been constructed so far as an expansion in powers of the eleven dimen-
sional gravitational coupling constant κ, and it is not known whether there exists a fully
supersymmetric formulation valid to all orders in κ.
3
We evaluate the one-loop effective action of the various fields using ζ-function regulari-
sation. We have extended the conformal transformation technique [13] used in [11] to apply
to spin 1/2 and spin 1 fields.
In the following, rather than working in the “upstairs” picture of the orbifold S1/Z , we
2
work in the equivalent “downstairs” description of an interval I. That is, we consider the
space to be a manifold with a boundary, rather than a manifold with singular delta-function
sources. In this way, the domain walls in the above are replaced by boundary “branes”.
Our conventions are as follows: Tensor indices are α,β,γ, ... = 0,1,2,3,11 in the bulk,
and µ,ν,ρ, ... = 0,1,2,3 on the boundary. The coordinate in the direction of the interval
is x11. Capital letters A,B,C,... = 0,1,2,3,11 index a local orthonormal frame in the bulk,
while I,J,K,... = 0,1,2,3 label a local orthonormal frame on the boundary. Conventions
for the Riemann and Ricci tensors are as in Misner, Thorne and Wheeler [14]. The extrinsic
curvature is defined as K = N where N is the unit outward pointing normal. Gamma
µν µ ν α
∇
matrices are defined by Γα,Γβ = 2gαβ. The index N denotes contraction with N .
α
(cid:8) (cid:9)
II. CLASSICAL BACKGROUND SOLUTION
The five dimensional action of heterotic M-theory including the (1,1) moduli was derived
in [4]. Here we will briefly review the field content and background solution.
The background solution consists of setting as many of the fields as possible to zero. The
minimum field content consistent for describing the vacuum solution is given by the action
S0 = S0( )+S0(∂ ), where the bulk and boundary actions are
M M
1 1 1
S0( ) = dµ R+G ∂ bi∂αbj + ∂ Φ∂αΦ+ e 2ΦGijα α . (1)
M −κ2 − ij α 2 α 2 − i j
5 Z (cid:18) (cid:19)
M
2 √2 2 √2
S0(∂ ) = dµ K + α bie Φ + dµ K α bie Φ . (2)
M κ2 2 i − κ2 − 2 i −
5 Z∂ (1) ! 5 Z∂ (2) !
M M
Let us first remind the reader of the origin of some of these fields. The scalar field Φ is the
moduli field describing the volume of Calabi-Yau and will be referred to as the dilaton. The
fields bi are the Calabi-Yau shape moduli. Indices i,j,... run from 1,...,h1,1 and are raised
and lowered by the Kahler metric G = 1∂ ∂ ln , where = d bibjbk is the Kahler
ij −2 i j K K ijk
potential. The constants d are the Calabi-Yau intersection numbers. The charges α are
ijk i
constants. The shape moduli are constrained by = 6 (after the differentiation) and hence
K
only represent h1,1 1 degrees of freedom.
−
4
The existence of the bulk and boundary potentials leads to a curved bulk metric and to
non-trivial profiles of Φ and bi across the orbifold. We denote y = x11. The position of the
visible brane is taken to be at y = 0 and the hidden brane is at y = π. Upon substitution
of the ansatz
ds2 = a(y)2η dxµdxν +b(y)2dy2 (3)
µν
Φ = Φ(y) (4)
bi = bi(y), (5)
Lukas et al. have found an implicit solution to the equations of motion in terms of h(1,1)
functions fi(y). However, they also show that by defining new constants α¯i and α by
2 1
d α¯iα¯j = αi, α = 9 d α¯iα¯jα¯k , (6)
ijk ijk
3 6
(cid:18) (cid:19)
andbyusinganappropriatechoiceofintegrationconstants, onecanfindtheexplicitsolution,
a = a H1/2 (7)
0
b = b H2 (8)
0
Φ = ln b H3 (9)
0
3αi
bi = (cid:0) (cid:1) (10)
2 α
where
√2
H = αy +c , (11)
0
3
which is just the “universal” solution obtained when h1,1 = 1. In this classical solution, the
fields bi are constant across the orbifold.
III. ONE-LOOP EFFECTIVE ACTION
To evaluate the one-loop correction to the classical action for a field φ , we perform a
a
continuation of the Lorentzian metric to a positive definite Euclidean metric. We expand
the Euclidean action I around the background fields φa, i.e., I = I(φ )+I,a(φ )(φ φ ) +
0 0 0 − 0 a
I,a (φ )(φ φ ) (φ φ )b +.... The one-loop correction to the classical action is
b 0 − 0 a − 0
1
W = ( 1)f logdet µ 2∆a (12)
− 2 −R b
(cid:0) (cid:1)
5
where we use comma (,) to denote functional differentiation with respect to the field φ .
a
f = 0 for bosons and f = 1 for fermions. ∆a = I,a is an operator. We define the
b b
determinant of an operator using generalised ζ-functions. The ζ-function is defined by a
generalised trace,
ζ(s) = tr ∆ s . (13)
−
(cid:0) (cid:1)
for some range of s in the complex plane where the trace converges. We define the determi-
nant of the operator by
logdet µ 2∆a = ζ (0) ζ(0)logµ2 (14)
−R b − ′ − R
(cid:0) (cid:1)
where we have used the analytic continuation of the ζ-function at s = 0.
In curved space, the trace of an operator is often difficult to evaluate. However, we can
use the properties of the effective action under conformal transformations of the metric
g g˜ = Ω2g , (15)
µν µν µν
→
to relate the effective actions of an operator in conformally related spacetimes. We use a
tilde to denote quantities calculated in the metric g˜. We restrict attention here to second
order operators of Laplace type – that is, operators which can be written in the form
∆ = D2 +X, (16)
−
where D = + iA is a covariant gauge derivative. We use the n-bein formalism to
α α α
∇
treat fields of general spin. The covariant derivative is = ∂ +iω ABΣ where Σ is
∇α α α AB AB
the generator of Lorentz transformations for the appropriate representation of the Lorentz
group. Under a conformal rescaling (15), the n-bein and its dual rescale as
e˜A = ΩeA, e˜α = Ω 1eα, (17)
α α A − A
while the operator (16) rescales to ∆˜, where
2
∆˜ = ˜ +iA˜ +X˜. (18)
− ∇
(cid:16) (cid:17)
If we define the conformal rescaling properties of X˜ and A˜ as
3
X˜ = Ω 2X Ω 2R R˜ , A˜ = A 2Ω 1Ω Σ β, (19)
− − 16 − − α α − − ;β α
(cid:16) (cid:17)
6
then the rescaled operator ∆˜ is given by
∆˜ = Ω 7/2∆Ω3/2. (20)
−
Boundary conditions will also be affected by conformal transformations. If the field
satisfies Robin boundary conditions
(D )φ = 0 on ∂ , (21)
N
−S M
then under a conformal rescaling, this will become
˜ ˜ ˜
D φ = 0 on ∂ . (22)
N
−S M
(cid:16) (cid:17)
since the field transforms as φ φ˜ = Ω 3/2φ. The boundary conditions retain the same
−
→
form if we define
3
˜= Ω 1 + Ω 1K K˜ . (23)
− −
S S 8 −
(cid:16) (cid:17)
Dirichlet boundary conditions φ = 0 on ∂ are unchanged under conformal rescalings.
M
Following Dowker [13], we introduce a one parameter family of metrics which interpolate
between two conformally related spacetimes. We take Ω = Ω(σ). One can then show that,
for operators which transform covariantly under conformal transformations,
W [σ = σ ] = W [σ = σ ]+C[Ω], (24)
2 1
where the cocycle function C[Ω] is given (in five dimensions) in terms of the generalised
heat kernel coefficient a (p,∆) as
5/2
σ2
C[Ω] = ( 1)f dσ a ∂ lnΩ(σ),∆˜ . (25)
5/2 σ
− −
Zσ1 (cid:16) (cid:17)
The a (p,∆) coefficient is known for general Laplace type operators with mixed boundary
5/2
conditions [15]. It is composed of geometric invariants in the metric evaluated only on the
boundary of the spacetime. Hence, we can relate the effective actions of the conformally
related operators in two conformally related spacetimes.
The classical background in Section II is conformally flat in the coordinate z defined by
dz = a(y)/b(y)dy. In this new coordinate, the metric is
2/5
z
ds2 = η dxµdxν +dz2 , (26)
µν
z
(cid:18) 1(cid:19)
(cid:0) (cid:1)
7
where
3√2b
0
z = . (27)
1
5αa
0
The dilaton field,
6 z 5αz
1
Φ = log +log . (28)
5 z 3√2
(cid:18) 1(cid:19) (cid:18) (cid:19)
The values of z on the two branes, z and z can be used as the moduli parameters of
1 2
the background solution. We can use the technique described above to relate the one-loop
effective action in the warped background spacetime to one in flat space.
For definiteness, we take the function Ω(σ) to be
Ω(σ) = e(1 σ)ω(z) (29)
−
so that flat space is at σ = 0 and the physical metric is at σ = 1. Then, the effective action
is related to the flat space effective action W by
0
1
W = W +C[Ω], C[Ω] = ( 1)f dσ a ω,∆˜ . (30)
0 5/2
−
Z0
(cid:16) (cid:17)
From now on we use the subscript “0” to indicate a quantity calculated in the conformally
transformed flat space.
IV. THE ONE LOOP EFFECTIVE ACTION OF THE VECTOR MULTIPLET
A. The scalar field
There are h1,1 scalar fields bi in the vector multiplet. However, these are constrained,
and as such only represent n = h1,1 1 degrees of freedom. We can, however, write
V
−
the action in terms of n unconstrained fields ϕ . The indices x, y, etc are raised and
V x
lowered using the metric g = bibjG . The action contains both bulk and boundary terms
xy x y ij
S = S( )+S(∂ ), which are [26]
M M
1
S( ) = dµ g ∂ ϕx∂αϕy +V(ϕ) (31)
xy α
M − 2
Z (cid:18) (cid:19)
M
S(∂ ) = dµ U(ϕ) dµ U(ϕ). (32)
M −
Z∂ (1) Z∂ (2)
M M
The bulk potential V(ϕ) and the boundary potential U(ϕ) are only expressible implicitly in
terms of the constrained fields bi. They are
1 √2
V(ϕ) = e 2ΦGijα α , U(ϕ) = e Φα bi. (33)
− i j − i
4 2
8
The second variationofthe classical actionabout the background solution gives the operator
∆x = 2δx +V(ϕ),x (34)
y −∇ y y
together with the boundary conditions
δx U(ϕ),x ϕy = 0 on ∂ (1), δx +U(ϕ),x ϕy = 0 on ∂ (2). (35)
∇N y − y M ∇N y y M
(cid:0) (cid:1) (cid:0) (cid:1)
Although the bulk andboundary potentials are only known interms of theconstrained fields
bi, the second derivatives of the bulk and boundary potentials with respect to the fields ϕx
can be calculated in terms of the background scalar field Φ.
4 √2
V(ϕ) = α2e 2Φg , U(ϕ) = αe Φg (36)
,xy − xy ,xy − xy
9 3
Since the operator has a trivial index structure in the indices x,y, from this point we
drop this index to avoid clutter. The operator (34) is of Laplace type (16). We now
perform a conformal transformation (15) with Ω = eω(z). The one-loop effective action in
the conformally transformed space requires the eigenvalues of the conformally transformed
operator ∆ ϕ˜ = λ ϕ˜ . The general solution can be found in terms of Bessel functions;
0 n n n
ϕ˜ (z) = √zeikµxµ AJ (m z)+BY (m z) , (37)
n 3/5 n 3/5 n
(cid:0) (cid:1)
where λ = m2 +k2. We take the position of the visible brane to be at z and the hidden
n n 1
brane to be at z (z > z ). We also introduce τ = z /z . Applying the conformally
2 2 1 1 2
transformed boundary condition, we obtain an implicit equation for µ = z m as the roots
n 2 n
of
F(m ) = J (µ τ)Y (µ ) J (µ )Y (µ τ) = 0. (38)
n 2/5 n 2/5 n 2/5 n 2/5 n
−
The ζ-function is
d4k k2 +m2 −s
ζ(s) = n d4x n . (39)
V (2π)4 µ2
Z Z n (cid:18) R (cid:19)
X
The factor of n is included since we have n scalar fields. Performing the k integrals first
V V
gives us
Γ(s 2)
ζ(s) = n µ2s d4x − µ4 2sz2s 4. (40)
V R (4π)2Γ(s) n− 2 −
Z n
X
The evaluation of the sum over µ is complicated since we only know them through an
n
implicit equation. We leave the details of the evaluation of the sum in (40) to Appendix A.
9
Here, we simply quote the result for the effective action of the conformally transformed
operator;
n A B G (τ) 2781 ln(µ z ) ln(µ z )
Wscalar = V d4x 2/5 + 2/5 + 2/5 + R 1 + R 2 . (41)
0 (4π)2 z4 z4 z4 80000 z4 z4
Z (cid:20) 1 2 2 (cid:18) 1 2 (cid:19)(cid:21)
The function G (τ) is
ν
K (x)I (τx)
G (τ) = ∞dxx3ln 1 ν ν , (42)
ν
− I (x)K (τx)
Z0 (cid:18) ν ν (cid:19)
and the constants A and B are defined in Appendix A. To obtain the full expression for
ν ν
the effective action, we must add the cocycle function to this result. After a simple but
lengthy calculation, the details of which can be found in Appendix B, the cocycle function
for this operator is found to be
n 2781 ln(cz ) ln(cz ) 4601 1 1
Cscalar[Ω] = V d4x 1 + 2 + + . (43)
(4π)2 400000 z4 z4 9600000 z4 z4
Z (cid:20) (cid:18) 1 2 (cid:19) (cid:18) 1 2(cid:19)(cid:21)
The constant c = 6b /(5√2αa ) can be absorbed into the renormalization scale µ . After
0 0 R
this redefinition we find the total effective action for the scalar field
n A B G (τ)
Wscalar = V d4x 2/5 + 2/5 + 2/5
(4π)2 z4 z4 z4
Z (cid:20) 1 2 2
8343 ln(µ z ) ln(µ z ) 4601 1 1
R 1 R 2
+ + + + .(44)
200000 z4 z4 9600000 z4 z4
(cid:18) 1 2 (cid:19) (cid:18) 1 2(cid:19)(cid:21)
B. The vector field
The five dimensional M-theory action contains h1,1 U(1) gauge fields i. One of these
Aα
gauge fields is the graviphoton of the gravity multiplet. This leaves n gauge fields
α V
A
for the n vector multiplets. After decomposing the i into the graviphoton and vector
V Aα
multiplet gauge field by x = bx i and = 2b i, the action for the vector multiplet
Aα iAα Aα 3 iAα
gauge field to quadratic order is
1
S = dµ x AB (45)
gauge 4FABFx
Z
M
Integration by parts yields
1
S = dµ A 2δB +R B + B x (46)
gauge 2Ax −∇ A A ∇A∇ AB
Z
M
(cid:0) (cid:1)
10
werewehaveneglectedboundaryterms. Weworkinatetradframe. Thecovariantderivative
contains the connection = ∂ +iω CDΣ where Σ D B = i η ηDB δBδD is the
∇α α α CD C A −2 CA − C A
generator of Lorenz transformations appropriate for(cid:0)a ve(cid:1)ctor repre(cid:0)sentation of the(cid:1)Lorenz
group. Again, since the index structure in the index x is trivial, we drop this index from
now on.
We can add to the action a gauge fixing term S and anticommuting ghost fields c,c¯so
gf
that S = S +S +S where
total gauge gf ghost
1
S = dµ F2, (47)
gf
2
Z
M
δF
S = dµ c¯ c. (48)
ghost
δΛ
Z
M
A convenient choice of gauge fixing term will turn out to be
F = +2ω α. (49)
;α
∇·A A
Since φ = 6ω , this gauge fixing term can also be expressed in terms of the background
;α ;α
−
dilaton field. The gauge fixed action for the gauge field S = S + S can then be
g+gf gauge gf
written, after an integration by parts, as
1
S = dµ A 2δB +R B +2ω B 2ω;B 2ω B +4ω ω;B . (50)
g+gf 2A −∇ A A ;A∇ − ∇A − ;A ;A AB
Z
M
(cid:0) (cid:1)
The action can be simplified by introducing the derivative (D ) C = ( ) C + i(A ) C,
A B ∇A B A B
where the connection A is chosen to be
A
(A ) C = i ω;Cη ω δC = 2ω (Σ D) C (51)
A B − AB − ;B A ;D A B
(cid:0) (cid:1)
The gauge fixed operator can now be written in the form of a Laplace-type operator, i.e.,
∆ B = (D2) B +X B, (52)
A − A A
where
X B = R B 2ω B +ω ω;B ω ω;CδB. (53)
A A − ;A ;A − ;C A
Boundary conditions can be found from the requirement of invariance under BRST trans-
formations [16]. This ensures that the path integral is gauge independent. The two admis-
sible sets of boundary conditions are known as absolute and relative. Absolute boundary
conditions have
= 0, D δJ +K J = 0, c = c¯= 0 on ∂ (54)
AN N I I AJ ∇N ∇N M
(cid:0) (cid:1)