Table Of ContentLocal Axion Cosmic Strings from
Superstrings
5 Stephen C. Davisa, Pierre Bin´etruyb,c and Anne-Christine Davisd
0 ∗ † ‡
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aInstitute of Theoretical Physics, E´cole Polytechnique F´ed´erale de
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a
Lausanne, CH–1015 Lausanne, Switzerland
J
5 bLPT, Universit´e de Paris-Sud, Baˆt. 210, 91405 Orsay Cedex, France
2
cAPC, Universit´e Paris VII, 2 place Jussieu, 75251 Cedex 05, France
1
dDepartment of Applied Mathematics and Theoretical Physics, Centre for
v
0
Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WA, UK
0
2
1
February 1, 2008
0
5
0
/
h
t
- Abstract
p
e
Axionic cosmic string solutions are investigated in a superstringmotivated model
h
: with a pseudo-anomalous U(1) gauge symmetry. The inclusion of a gauge field and
v
i spatially varyingdilatonallow localdefectsolutionswithfiniteenergyperunitlength
X
to be found. Fermion zero modes (whose presence is implied by supersymmetry)
r
a are also analysed. The corresponding fermion currents suggest strong cosmological
boundson the model. It is shown that the unusualform of the axion strings weakens
these bounds. Other cosmological constraints on the underlying theory are also
discussed.
1 Introduction
Spontaneous breaking of a pseudo-anomalous U(1) gauge symmetry occurs in a large
class of superstring compactifications [1]. The U(1) symmetry arises generically as a
remnant of the Green-Schwarz [2] mechanism of anomaly cancellation in the underlying
ten-dimensional supergravity.
∗Stephen.Davis@epfl.ch
†[email protected]
‡[email protected]
1
It is well known that line-like topological defects form during an ordinary U(1) symme-
trybreaking. Suchdefectsarecalledcosmicstrings, andhavemanyinterestingcosmological
properties [3]. Similar defects can also form if the symmetry is pseudo-anomalous. These
strings areexpected to beglobal, because of their coupling tothe axion [4]. Furthermore, it
seems that a cutoff must be introduced inside the string core in order to give a finite energy
per unit length. It has been shown that by coupling the axion to a gauge field, local axion
strings can be obtained (although the cutoff inside the core is still needed) [5]. The cou-
pling of the dilaton-axion field to the gauge fields plays a central role in the Green-Schwarz
mechanism, so local axion strings are natural in this context.
Recent work on the formation of topological defects in superstring theory has led to a
renewed interest in cosmic strings [6]. In contrast to other work, we will just be considering
a four-dimensional model, although we expect our results to be of general relevance.
In this paper we discuss axion string solutions which do not require a cutoff in the
core. In a supersymmetric theory, the dilaton forms part of the same supermultiplet as
the axion. By allowing the dilaton to vary inside the string, finite energy solutions can be
found. The axion field varies (in a topologically non-trivial way) around the string, and so
plays the role of the phase of the Higgs field in a conventional U(1) cosmic string model.
In a similar way the dilaton plays the role of the magnitude of the Higgs field. This type of
solution relies on particular couplings between the axion and dilaton fields, although they
are completely natural in a supersymmetric (or superstring) context.
Although we have referred to the dilaton in the above discussion, the same ideas ap-
ply for pseudo-anomalous U(1) breaking with an axion-moduli superfield. Similarly the
analysis and results of this paper will also apply to moduli, although the corresponding
couplings and mass scales coming from the underlying theory will be different.
We will investigate local, axion strings in a superstring motivated, supersymmetric
model. A simplified version of a ‘racetrack’ potential [7] will be used to stabilise the
dilaton. Such a potential could arise from gaugino condensation. It is quite natural for its
energy scale to be far lower than that associated with supergravity, perhaps even as low as
the QCD scale. For simplicity we model these lower energy effects with two effective scalar
fields. The model is outlined in section 2, and the cosmic string solutions themselves are
discussed in section 3.
Acommonfeatureofsupersymmetric cosmicstringmodelsistheexistence ofconserved,
massless fermion currents which flow along the strings [8]. Like the strings themselves,
these have interesting cosmological implications [9]. The massless fermion bound states
of our solutions are examined in section 4. The cosmological evolution of a network of
such strings is discussed in section 5. The bounds on the underlying theory coming from
the cosmological properties of the axion strings are derived in section 6. The possibility
of cosmic string loops which are stabilised by the fermion currents generally gives rise
to particularly strong bounds. However, as we will show, these bounds are significantly
weakened for our string solutions.
In section 7 we discuss an apparent analytic simplification of the field equations, and
in section 8 we summarise our results.
2
2 Axion cosmic string model
We will consider a supersymmetric model with a pseudo-anomalous U(1) gauge symmetry.
Working in units with m = 1, the Lagrangian is
Pl
1
˜= Φ†e2qiVΦ + + SWαW +W(Φ ,S) δ(θ¯2)+(h. c.) . (1)
L i i K 4 α i
(cid:18) (cid:19)
(cid:16) (cid:17)
−1/2
S(s,2s χ ,F ) is the axion-dilaton superfield, V(A ,s λ ,D) is the gauge vector su-
R α s µ R α
perfield, and Φ (φ ,ψ ,F ) are chiral superfields with charges q . The kinetic term for the
i i iα i i
superfield S is described by the modified K¨ahler function = log(S+S¯ 4δ V), where
K − − GS
δ is the Green-Schwarz parameter [1].
GS
If the dilaton is to have a finite vacuum expectation value, a non-trivial superpotential
(W) is required. We will consider a racetrack [7] style model with the effective superpo-
tential
W = Φ h (Φ/η)n1e−3S/(2b1) h (Φ/η)n2e−3S/(2b2) (2)
0 1 2
−
where bi = 3Ni/(16π2). The(cid:8)integers Ni are determined by the symm(cid:9)etry breaking of the
original theory. The above superpotential resembles that arising from an E SU(N )
8 1
→ ×
SU(N ) breaking, in which case N +N 10.
2 1 2
≤
The effective superfields, Φ and Φ , represent superstring and gaugino condensation
0
effects. We take q = 1 and q < 0. Gauge invariance implies that q = n 3δ /b , and
φ − 0 0 i− GS i
so for our model we require
N N (n n )
1 2 1 2
δ = − . (3)
GS 16π2(N N )
2 1
−
Typically we expect δ 1/10 [1], but we will consider more general values.
GS ∼
The Lagrangian (1) can be expanded in terms of the component fields. The auxiliary
fields can then be eliminated using their equations of motion. Working in Wess-Zumino
gauge, the bosonic part of the Lagrangian is
1 1 s a
= ∂ s ∂µs + (∂ a 2δ A )2 RFµνF + FµνF˜
LB 4s2 µ R R 4s2 µ − GS µ − 4 µν 4 µν
R R
1 s
+ φ 2 + φ 2 F 2 F 2 F 2 RD2 (4)
|Dµ | |Dµ 0| −| 0| −| | − 4s2| s| − 2
R
where s = s +ia, φ = (∂ iA )φ, and φ = (∂ +iq A )φ . The real part of the
R Dµ µ − µ Dµ 0 µ 0 µ 0
dilaton defines the gauge coupling 1/g2 = s . The auxiliary fields are given by
R
F† = h (φ/η)n1e−3s/(2b1) +h (φ/η)n2e−3s/(2b2) (5)
0 − 1 2
1 δ
D = q φ 2 φ 2 + GS . (6)
0 0
−s | | −| | s
R (cid:18) R (cid:19)
F and F are both proportional to φ . The scalar potential is minimised by s = or
s 0 R ∞
1 δ
φ = 0 , φ = η , s = = GS , (7)
0 | | R g2 η2
0
3
where g is the vacuum value of the coupling g. The superpotential is related to g by
0 0
2δ h 1
GS ln 1 = . (8)
n n h g2
1 − 2 2 0
We see that the model has two distinct mass scales. One,
√2η2
m = 1+η2 = 2δ g2 1+η2 , (9)
D √δ GS 0
GS
p p p
arises from the D term, and because of supersymmetry is also the mass of the gauge field
A . If δ 1/10 and g 1, as suggested by string theory, we will have m 1/√5.
µ GS ∼ 0 ∼ D ≈
The other mass scale,
h 3δ
m = n n | 1| exp GS 1+η2
F 2 1
| − | η2 −2b η2
(cid:18) 1 (cid:19)
h N N 8pπ2
= 16π2| 1|| 1 − 2| exp 1+η2 , (10)
g2 N N −N g2
0 1 2 (cid:18) 1 0(cid:19)
p
arises from the F term, and will generally be far smaller. If η is small, the fields s and φ
0
are approximate mass eigenstates. Their masses are respectively m and m .
F D
3 Cosmic Strings
We will look for cosmic string solutions of the form
φ = ηf(r)einϕ (11)
v(r)
A = n (12)
ϕ
r
δ
s = GS +2inδ ϕ (13)
η2γ(r)2 GS
φ = 0 . (14)
0
Note that q n must be an integer so that F is single valued. As r approaches infinity f,
0 0
γ and v all tend to 1. The boundary conditions at r = 0 are γ = f = v = 0. In contrast
to previous work [4, 5], we allow spatial variations of the dilaton.
The equations of motion for the various fields are
1 1 s ∂F∗
∂µ ∂ s + (∂ a 2δ A )2 + RFµνF +4s F 0
s µ R s2 µ − GS µ 2 µν R 0 ∂s∗
(cid:18) R (cid:19) R
1 δ δ
φ2 GS φ2 3 GS = 0 (15)
− s | |− s | |− s
R (cid:18) R (cid:19)(cid:18) R (cid:19)
∂F∗ 1 δ
µ φ+F 0 + φ2 GS φ = 0 (16)
D Dµ 0∂φ∗ s | |− s
R (cid:18) R (cid:19)
4
1
v
0.8
0.6 γ
0.4
0.2
f
0
−2 2 4 6
10 1 10 ρ 10 10
Figure 1: Plot of string profile functions
δ
∂µ(s F )+2Im[φ(∂ +iA )φ∗] GS[∂ a 2δ A ] = 0 . (17)
R µν ν ν − s2 ν − GS ν
R
The equations for the string profile function are found by substituting the string ansatz
(11–14) into the above field equations. It is useful to introduce ρ = rη2 2/δ . The
GS
equations (15–17) for the string solution are then
p
1 ργ′ ′ (1 v)2 v′ 2 η2
2n2η4 − γ4 +n2η2γ2 (f2 γ2)(f2 3γ2)γ2
ρ γ − ρ2 γ2ρ − 4 − −
(cid:18) (cid:19) (cid:18) (cid:19)
2η4
h˜ fn1e−1/(˜b1γ2) h˜ fn2e−1/(˜b2γ2) h˜ ˜b−1fn1e−1/(˜b1γ2) h˜ ˜b−1fn2e−1/(˜b2γ2) = 0 (18)
− γ2 1 − 2 1 1 − 2 2
h ih i
1 (1 v)2 γ2
(ρf′)′ n2 − f (f2 γ2)f
ρ − ρ2 − 2 −
η2 h˜ fn1e−1/(˜b1γ2) h˜ fn2e−1/(˜b2γ2) h˜ n fn1−1e−1/(˜b1γ2) h˜ n fn2−1e−1/(˜b2γ2) = 0(19)
1 2 1 1 2 2
− − −
h ih i
v′ ′ 1 v
+ − (f2 +η2γ4) = 0 (20)
γ2ρ ρ
(cid:18) (cid:19)
where h˜ = h η−4 δ /2, ˜b = 2b η2/(3δ ). Hence m /m = (n n )h˜ e−1/˜bi.
i i GS i i GS F D 1 − 2 i
There are two distinct length scales for this type of string solution. The string has an
p
inner core of radius r m−1 (ρ 1) in which v < 1 and f = γ, so D = 0. Around that
D ∼ D D ∼ 6 6
region there is an outer core in which v 1 and f γ but f,γ < 1 so F = 0. This region
0
≈ ≈ 6
5
is of radius r m−1, which can be far greater then r even for moderate values of the
F ∼ F D
parameters.
A numerical solution of (18–20) for n = 1 is shown in figure 1. The parameters used
were n = 1, n = 0, g = 1, m /m = 10−5 and b = 3N /(16π2) with N = 4, N = 6 (as
1 2 0 F D i i 1 2
suggested by the symmetry breaking discussed in section 2). The remaining parameters
are then q = 2, δ = η2 = 3/(4π2), h 111 and h 0.154.
0 − GS 1 ≈ 2 ≈
It is useful to have an approximate analytic solution to the field equations (18–20).
Inside the inner core (ρ < 1) of the string the kinetic terms dominate the field equations.
At ρ = 1 we take f = γ and v = 1. Assuming n η2 is small, the approximate solution for
| |
ρ < 1 is
A
γ (21)
≈ 1 2 n η2A2logρ
− | |
p f Aρ|n| (22)
≈
ρ2
v . (23)
≈ 1 2 n η2A2logρ
− | |
If m m then an approximate solution for the outer core is also needed. Here
D F
≫
D 0 so f γ. The potential terms in the field equations can still be neglected, and
≈ ≈
v 1 so some kinetic terms can also be dropped. Solving eq. (18) and imposing continuity
≈
of γ and γ′ at ρ = 1 implies
f γ AρA2|n|η2 (24)
≈ ≈
and v = 1 for 1 < ρ < m /m . For ρ > ρ we take f = γ = v = 1. The constant A is
D F F
determined by continuity at ρ = m /m . It satisfies ( logA)/A2 = n η2log(m /m ).
D F D F
− | |
For small n η2log(m /m ) this implies A 1 n η2log(m /m ).
D F D F
| | ≈ −| |
If m m , there is no outer core. φ and s take their vacuum expectation values
D ∼ F | | R
for ρ > 1, and A = 1.
In all cases the dilaton is thus s 2 n δ logρ + const. for small ρ. Substituting
R ≈ − | | GS
this into the Lagrangian (4) we see that 1/(rlogr)2 inside the string core. Thus the
L ∼
string energy, which is proportional to rdr, is finite for this type of solution. If we had
L
taken s = const. then 1/r2 and the energy would have diverged logarithmically at
R L ∼ R
the string core. We see that our solution is a local string with finite energy, in contrast the
usual axion strings which are global.
The energy per unit length of the string is equal to
γ′2 (1 v)2
µ = 2πη2 ρdρ +f′2 +n2 − (f2 +η2γ4)
η2γ2 ρ2
Z n
v′2 1 2
+n2 + (f2 γ2)2γ2 +η2 h˜ fn1e−1/(˜b1γ2) h˜ fn2e−1/(˜b2γ2) .(25)
γ2ρ2 4 − 1 − 2
(cid:16) (cid:17) o
Substituting in the above approximate solution, we find that for small η,
m
µ = 4πn2η2 2a + n η2b log D +O(η4) (26)
n n
| | m
(cid:18) F (cid:19)
with a ,b 1.
n n
∼
6
0.2
λ
iψ m−1
0.1 0 F
χδ−1/2
GS
0
−0.1
ψ
−0.2
−2 2 4 6
10 1 10 ρ 10 10
Figure 2: Plot of fermion fields for ǫ = 1.
1
4 Fermion Zero Modes
Wewillnowconsiderthefermionicsectorofthetheory. Thenon-zerotermsinthefermionic
part of the Lagrangian are
i i
= iχ¯σ¯µ ∂ (∂ a 2δ A ) χ iλ¯σ¯µ ∂ + ∂ a λ iψ¯σ¯µ ψ
LF − µ − s µ − GS µ − µ 2s µ − Dµ
(cid:20) R (cid:21) (cid:20) R (cid:21)
i δ s
iψ¯ σ¯µ ψ + 3 GS φ 2 (λ¯χ¯ λχ)+ R (χ¯σ¯νσµλ¯ +λσµσ¯νχ)F
0 µ 0 µν
− D √2s s −| | − 2√2
R (cid:18) R (cid:19)
i√2 3 ∂2W ∂2W
(φ¯λψ φλ¯ψ¯)+ χ2χ¯2 ψψ +2s χψ +(h. c.) . (27)
− √s − 2 − ∂φ ∂φ 0 R∂φ ∂s 0
R (cid:20) 0 0 (cid:21)
A common feature of cosmic string models is the existence of fermion bound states
which are confined to the string core. These occur in models in which fermion fields gain
their masses from the cosmic string Higgs bosons. This is the case for all supersymmetric
cosmic string models, including the one considered in this paper.
Ofparticularinterest arezeroenergyfermionboundstates, or‘zeromodes’. Excitations
of these are massless particles, with the unusual property that all excitations of a given
zero modemove in the same direction along the string. The resulting currents canradically
alter the model’s cosmology.
In supersymmetric models some of the zero modes can be found by applying a super-
symmetry transformation to the string solution [8]. Applying the transformation (52–54),
with parameter ξ = i√δ /(2η3)ǫ , to the string solution (11–14), and using a gauge
α GS α
7
transformation to return to Wess-Zumino gauge gives
h˜ fn2 h˜ fn1
δψ = iη 2 1 e−inq0ϕǫ (28)
0α e1/(˜b2γ2) − e1/(˜b1γ2)! α
1 v
δψ = f′ n − f ei(n∓1)ϕǫ∗ (29)
α − ± ρ α
(cid:18) (cid:19)
γ v′
δλ = γ2 f2 n ǫ (30)
α α
2 − ± γρ
(cid:20) (cid:21)
γ(cid:0)′ (cid:1)1 v
δχ = nηγ2 − e∓iϕǫ∗ (31)
α γη ∓ ρ α
(cid:18) (cid:19)
where the upper (lower) signs corresponds to the index α = 1 (2). Since (28–31) are non-
zero, supersymmetry is completely broken inside the string core. The above expressions
must be normalisable if they are to correspond to physical states. At large ρ they all decay
exponentially, so there is no problem there. Using (22–23) we see that (28–30) are all well
behaved as ρ 0. The small ρ behaviour of χ is less obvious. Solving (18) for small ρ
α
→
we find
γ′ 1 1 sgnn
nηγ2 = ∓ +O(ρ) (32)
γη ∓ ρ ηρlogρ
(cid:18) (cid:19) −
so χ (χ ) is well-behaved as ρ 0 if n > 0 (n < 0). Thus only one of the transformations
1 2
→
(ξ = 0 if n > 0) gives a normalisable fermion zero mode solution. A plot of this solution is
1
6
shown in figure 2. Excitations of it are massless currents, which only flow in one direction
along the string. The direction is determined by the sign of n. While the other transfor-
mation also gives a solution of the fermion field equations, it is unphysical since it is not
single valued as ρ 0.
→
The total number of zero modes can be determined by considering solutions to the
fermion field equations which are normalisable at large and small r. By trying to match
up these solutions it is possible to find a lower bound for the total number of physical zero
mode solutions. This was done for a general cosmic string solution in ref. [10]. There it
was assumed that all fermion fields had power law behaviour near the centre of the string.
The s−1∂ a 1/(rlogr) terms in (27) produce extra logarithmic factors in the small r
R µ ∼
solutions, but these do not affect their normalisability, so the expression derived in ref. [10]
still applies. We find (for q < 0) there are n (1 q ) zero modes, all of which are left
0 0
| | −
(right) movers if n > 0 (n < 0).
5 String Network Evolution
By considering the cosmological properties of the strings, we can obtain constraints on the
underlying theory. Before doing this, we need to consider the evolution of a network of our
cosmic strings, which may be non-standard. For simplicity we will assume that g 1.
0
∼
This implies that the energy per unit length of the string µ m2 , and that the only other
∼ D
mass scales in the model are m and m .
Pl F
8
When discussing the evolution of a cosmic string network, it is useful to introduce the
correlation length of the network, ξ. This gives the typical length scale of the network.
For example, in a volume ξ3 there will typically be length ξ of string.
At the phase transition (T = T ) where the strings form, ξ 1/T . Our strings are
c c
∼
essentially formed when the F-terms appear in the potential, at T = T m . This can
c F
∼
be far less than the energy scale of the strings, which is an unusual feature of our model.
After a brief period of acceleration, the network’s evolution is dominated by the fric-
tional force coming from its interaction with the plasma. During this friction domination
1/2 1/4
m m t
ξ Pl D g1/4t (33)
∼ g∗1/4T5/2 ∼ ∗ (cid:18)t∗(cid:19)
(we have used t m /(√g T2)). This lasts until temperature
Pl ∗
∼
m2
T D , (34)
∗
∼ m
Pl
after which the expansion of the universe (which simply stretches the strings) will be more
significant than the effects of friction. For our strings, if
m m
F . D , (35)
m m
D Pl
then the friction domination regime will be absent.
For T < T , the stretching of the strings would soon result in them dominating the
∗
energy density of the universe. This is clearly not true for our universe, and so some
mechanism is needed for the network to lose energy. In fact, as the strings evolve, loops
of string break off from the network. Typically these decay by radiating gravitons. As a
result of this energy loss the strings do not dominate the universe and instead the network
approaches a scaling solution, where its energy density scales like matter. The strings move
with relativistic velocities and ξ = γt with γ 1. The behaviour of the network in this
∼
scaling regime is practically independent of the network’s earlier evolution. We therefore
expect this part of the evolution to be standard for our strings.
If the friction regime is absent or very short, then the string density at early times
will be much higher than that of the scaling regime. In order for the network to reach a
scaling solution, it must undergo a period of intense loop production. Once the strings
have reached relativistic speeds (which we assume happens rapidly) we find that during
this period the correlation length is given by [3]
3/4
t t
c
ξ γt+ξ 1 γ , (36)
c
≈ − ξ t
(cid:18) c(cid:19)(cid:18) c(cid:19)
which approaches the scaling solution as t increases.
Most of the cosmological constraints on our model arise from loop production by the
string network. We will take the typical size of a loop produced by an evolving network to
9
be ℓ αξ. The rate at which loops are produced is [3, 15]
∼
dn 1 1 dξ
loop
. (37)
dt ∼ ξ3ℓ ∼ αξ4 dt
The parameter α depends on the behaviour of the network. In the friction domination
regime α 1. For strings moving with relativistic velocities (as in the scaling regime), the
∼
value of α is unclear. It is bounded below by α & ΓGµ, where the factor Γ is related to
how length scales in the network evolve with time. Numerical simulations suggest Γ 65.
∼
Simulations of conventional strings indicate α 10−3, so we expect large numbers of small
∼
loops will be produced during the scaling regime.
Clearly the scaling regime will not be achieved if the string loops do not decay. In
the previous section we have seen that there are fermion zero modes in the core of our
strings. Including the z and t dependence of the zero modes results in the string carrying
a conserved current [9]. Sufficiently large currents will allow stable string loops, called
‘vortons’, to exist. If these do not decay the string energy density will come to dominate
the universe after all. This is generically a potential problem for all current carrying
strings. If the zero modes move in both directions the net current will be smaller than the
total number of particles, weakening the bounds on the model. However in our model we
have either left or right movers, but not both. Thus the net current is maximal. Vorton
bounds can sometimes be avoided by using supersymmetry breaking to change the fermion
spectrum. However, since our zero modes are chiral, they will survive this [11], as they are
unable to mix with other modes and become bound states.
Not all loops will go on to become vortons. Only those with a sufficiently large number
of fermion charge carriers will survive. In the friction dominated regime, the majority of
the loops produced when T m satisfy this requirement. Calculating the typical loop
F
∼
length and corresponding current allows the vorton density to be estimated [12]
ρ m3
v ǫ F (38)
g T3 ∼ √g m m
∗ ∗F Pl D
where g = g (T = m ). For our model, anything in the range 10 . g . 103 is possible.
∗F ∗ F ∗F
The parameter ǫ measures the efficiency with which a network produces potential vortons.
It will be in the range 0 < ǫ . 1.
If m < T then our strings will form in the scaling regime, and the above expres-
F ∗
sion (38) will not apply. During the scaling regime, far fewer loops will possess enough
charge carriers for them to form vortons. The number density of vortons will therefore be
smaller, as will the typical current per vorton. Taking these issues into account gives a
smaller expression for the vorton density [12]
ρ m2m2 Γm ǫ/(3−ǫ)
v ǫα3/2 F Pl D . (39)
g T3 ∼ Γ2m3 √g m
∗ D (cid:18) ∗F Pl(cid:19)
The smaller value of ρ will weaken any vorton-related constraints on our model.
v
10
