A note on the stability of axionic D-term strings

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by  Ana Achucarro

A note on the stability of axionic D-term s-strings
Ana Achu´carro
Lorentz Institute of Theoretical Physics, University of Leiden, The Netherlands and
Department of Theoretical Physics, The University of the Basque Country UPV-EHU, Bilbao, Spain
Kepa Sousa
Lorentz Institute of Theoretical Physics, University of Leiden, The Netherlands
WeinvestigatethestabilityofanewclassofBPScosmicstringsinN=1supergravitywithD-terms
recently proposed by Blanco-Pillado, Dvali and Redi. These have been conjectured to be the low
energymanifestationofD-stringsthatmightformfromtachyoncondensationafterD-anti-D-brane
annihilation in typeIIBsuperstring theory. There are threeone-parameter families of cylindrically
symmetric one-vortex solutions to the BPS equations (tachyonic, axionic and hybrid). We find
evidencethatthezeromodeintheaxioniccase,or s−strings,canbeexcited. Itsevolutionleadsto
6
thedecompactification of four-dimensional spacetime at late times, with arate that decreases with
0
decreasing brane tension.
0
2
PACSnumbers: 11.27.+d,11.25.Mj,04.65.+e,12.60.Jv

n
a
J Recently Blanco-Pillado, Dvali and Redi have pro- Supergravity effects were considered in [1] and, as

0 posed a model to describe a D-brane anti-D-brane expected on general grounds [6, 7], the zero mode
2 unstable system after compactification to four dimen- survives the coupling to supergravity.
sions [1]. In the type IIB case, the tension of the
1
branes appears in the four-dimensional effective theory The s-strings are peculiar . The authors of [1] argued
v
as a constant Fayet Iliopoulos term which allows for that they should be associated with D anti-D bound
1
the existence of non-singular BPS axion-dilaton strings states that areunstable in ten dimensions, andtherefore
5
1 generalisingearlierworkby[2,3,4]inheteroticscenarios. only exist after compactification. They also noted that
1 they share some features with semilocal strings in the
0 It was shown that this model contains three different Bogomolnyi limit [8]. In the semilocal case any excita-
6 families of BPS cosmic string solutions: tachyonic or φ- tion of the zero mode [9] leads invariably to the spread
0
strings,withwindingn(seealso[5]),axionic1 ors-strings of the magnetic field and the eventual disappearance of
/
h withwindingmandathirdtypewhichisthespecialcase the strings [10].
t where the following relation is satisfied: q m = n. We
-
| | | |
p areusingthesamenotationof[1]. Wecallthethirdtype Wehavestudiednumericallythedynamicsofthiszero
e hybrid strings. Each of the families is parametrized by a mode and we find that also in the s-string case it can
h
real number, κ, defined by the relation: be excited and will lead to the dissolution of the strings.
:
v Since in this process the field s grows without bound,
i δ δ andthereforealsothe compactificationvolumemodulus,
X s=2 (n q m)logr 2 log φ +κ. (1)
q | |− | | − q | | our result would appear to imply the decompactification
r
a of spacetime from four to ten dimensions at late times.
In the first two families the parameter κ is associated to
azeromodethatconnectsvortexsolutionswithdifferent We start by briefly reviewing the model of [1] and the
core radius and equal magnetic flux, as can be seen by three families of BPS strings. We then analyse the zero
setting: mode numerically and conclude that it can be excited.
Forcomparison,weshowtheresultsofthesameanalysis
δ
k =2 (n q m)logR (2) on the φ-string, where a perturbation of the string leads
q | |− | | to some oscillations but no runaway behaviour.
In this way R gives the scale of the core radius. In the
third case the same parameter measures which of the
I. THE MODEL
fields, the dilaton or the tachyon, contributes more to
compensate the Fayet-Iliopoulos term.
The authors of [1] proposed a supersymmetric abelian
Higgsmodelcontaininga vectorsuperfieldV andN chi-
ral superfields. Here we only need to consider one chiral
superfield Φ with charge q, that represents the tachyon.
1 Thename axionicis somewhat misleadingforthe solutions dis- The model also involves an axion-dilaton superfield S
cussedheresincethereisnoaFF˜coupling,inparticulartheydo whichiscoupledtothe gaugemultipletintheusualway.
notsharethefeaturesusuallyassociatedwithaxionicstrings[2]. Its lowest component is s+ia where the axion a is the
2
four-dimensional dual of the Ramond-Ramond two-form leading to a bound on the energy per unit length
zero mode after compactification. Its scalar partner s is
some combination of the dilaton and the volume modu- E =R d2xE ≥(ξg)2R d2xFxy. (8)
lus. WetaketheK¨ahlerformtobeK = M2log(S+S¯)
− p TheboundisattainedbythesolutionsoftheBogomolnyi
and the gauge kinetic function are set to be constant,
equations
f(S) = 1/g2, where g is the gauge coupling. With this
choice,thereisnoaF F˜µν termintheaction. TheU(1)
µν (D iD )φ =0
x y
symmetry is not anomalous, it is the diagonal combina- ±
(D˜ iD˜ )S˜ =0
tion of the U(1)s from the D- anti D system. x y
±
In component notation the bosonic sector of the la- F (1 s−1 φ2) =0 (9)
xy
∓ − −| |
grangian, after eliminating the auxiliary field from the
vector multiplet, is: Wewillfocusoncylindricallysymmetricvortices,that
can be described by the following ansatz:
L = −|D1gµ2φ(ξ|2+−2KδKSS¯|DµqSφ|22−)241.g−2FµνFµν (3) φ=f(r)einθ s−1 =h2
−2 S− | | a=mθ A =v(r)/r . (10)
θ
KS and KSS¯ represent the derivatives of the Kahler po-
tential respect to the fields S and S¯. φ is the lowest (The ansatz for the dilaton makes comparison to the
component of the chiral field field, A is the U(1) gauge semilocal case easier. Also we shall see that h vanishes
µ
field,andF =∂ A ∂ A theassociatedabelianfield in various cases, and with this choice we avoidhaving to
µν µ ν ν µ
−
strength, deal with infinities).
With this ansatz the energy density becomes:
D φ=∂ φ iqA φ, D S =∂ S+i2δA . (4)
µ µ µ µ µ µ
− E =A(r)2+B(r)2/(αh)2+1C(r)2+D(r)′/r (11)
δ is the coupling of the axion to the gauge field. In the 2
case of anomalous U(1), where f(S) = S, it is called the and the Bogomolnyi equations
Green-Schwarz parameter and is fixed by the value of
′
the anomaly. Here δ is not determined. A(r) f f(n v)/r =0
≡ − | |−
B(r) h′ α2h3(m v)/r =0 (12)
It is convenient to write the bosonic part of the la- C(r) ≡ v′/−r (1 |f2|− h2)=0
grangian using the rescalings: ≡ − − −
D(r) v f2(v n) h2(v m)=0 (13)
φ=pξ/q φˆ s=δMp2/ξ sˆ a=2δ/q aˆ ≡ − −| | − −| |
The total energy is:
Aµ =gpξ/qAˆµ x=(g√ξq)−1 xˆ , (5)
With these definitions the axion aˆ is defined modulo E =2πR drrE(r)≥2πv∞ (14)
2π, and δ is rescaled away. After dropping the hats, the wherev∞istheasymptoticvalueofthegaugefieldprofile
bosonic sector of the lagrangianreads: function for large values of r.
L(ξg)−2 = D φ2 1(αs)−2(∂ s)2 (α/s)2(∂ a+A )2 The condition A(r) = B(r) = 0 implies the following
−| µ | −4 µ − µ µ relation between the tachyon and the dilaton:
1FµνF 1(1 s−1 φ2)2 , (6)
−4 µν −2 − −| | 1/(αh)2 =2(n m)logr 2logf +κ. (15)
withD φ=∂ φ iA φ,andα2 =ξ/(qM2). Notethat, | |−| | −
µ µ − µ p
since α is the symmetry breaking scale in Plank units, Depending on the asymptotic value of the profile
ignoring supergravity and superstring corrections would functions for large values of r, f∞, h∞ and v∞, we can
only be a consistent approximation for α 1. How- classify the solutions to the Bogomolnyi equations in
≪
ever, this limit is difficult to analyze numerically. We three different families. Each of them is parametrized
will present numerical results for α = 1 and argue sepa- by the integration constant κ.
rately on the effect of lowering α.
To study straight vortices along, say, the z direction, In the first case,
− •
we drop the z dependence and set A = 0. For time
z
independent configurations and defining S˜ = s+2iα2a, f∞ =1, h∞ =0, v∞ =n , (16)
and D˜ S˜=∂ S˜+2iα2A , the energy functional can be
µ µ µ
thetachyonacquiresanonvanishingvacuumexpectation
written in the Bogomolnyiform:
valuefarfromthecenterofthestring. Themagneticflux
E(ξg)−2 = (D iD )φ2+1(αs)−2 (D˜ iD˜ )S˜2 ofthesevorticesisinducedbythewindingofthetachyon,
| x± y | 4 | x± y |
+1(F (1 s−1 φ2))2 F n.
2 xy ∓ − −| | ± xy The profile function h(r) tends very slowly, (loga-
∗ ∗
i[∂ (φ D φ) ∂ (φ D φ)]
∓ x y − y x rithmically), to zero at large r. The details about the
i1α−2[∂ (s−1D˜ S˜) ∂ (s−1D˜ S˜)]. (7) asymptotics of the fields can be found in [1]. Following
± 2 x y − y x
3
to a Nielsen-Olesen string. In the case of s strings the
−
widthofthestringincreaseswithdecreasingα,whilethe
condensate does not vary much. In fact, the main effect
of decreasing α will be a slowing down of the dynamics.
II. DISCRETIZED EQUATIONS OF MOTION
The functions f(r), h(r) and v(r) are substituted by
the set of quantities f , h and v which are the profile
k k k
functions evaluated in the lattice points r = (k+1)∆,
k 2
where ∆ is the lattice spacing.
We want to analyze the response of the solutions of
the Bogomolnyi equations under perturbations. We
mustmakesurethattheconfigurationsthatwearegoing
to perturb are stationary solutions of the equations of
FIG. 1: The axion string profile functions, for m=1, n=0,
motion. This is automatic in the continuous case but
a(r),(upperdashedline),h(r),(middledashedline)andf(r),
not in an arbitrary discretization.
(lower solid line), with the size of thecondensate, f0 =0.51.
Following Leese [10] we construct a discrete versionof
the energy functional for static configurations givenby:
Blanco-Pillado et al. we call these vortices φ-strings.
4B2 C2 (D D )
spo•nIsnibtlehefosreccoonmdpceanssea,tFinigg.(t1h)e, tDh-etedrimla.tonThaelonfuenicstiroen- Ek∆2=A2k+ α2(hk+1k+hk)2 + 2∆k2 + k+k1+−21 k(19)
h(r), approaches a non vanishing constant far from the where
centre of the string, while the tachyon expectation value f +f
k+1 k
tendstozero. Inthis familythemagneticfluxisinduced Ak = fk+1 fk (n vk) ,
− − − 2k+1
by the winding of the axion, m.
(h +h )3
B = h h α2(m v ) k+1 k ,
k k+1 k k
f∞ =0 h∞ =1 v∞ =m (17) − − − 2k+1
v v
C = k+1− k ∆2(1 h2 f2 ),
These vortices have been denominated s-strings. They k k+1 − − k+1− k+1
2
are regular thanks to the constant Fayet-Iliopoulos term
D = v f2(v n) h2(v m). (20)
k k− k k− − k k−
In the previous two families the constant κ has the Theprofilefunctionsareobtainedminimizingthetotal
same interpretation. Each value of this parameter is energy for static configurations:
associated to a particular width of the strings.
∞
For strings of the third family both the tachyon and E =2πX∆2(k+21)Ek, (21)
• k=0
the dilaton contribute to compensate the D-term. This
happenswhentheaxionanddilatonhavethesamewind- for which the discretized Bogomolnyi equations have to
ing. be satisfied: Ak =Bk =Ck =0.
f∞2 +h2∞ =1 v∞ =n=m (18) The boundary conditions used to solve them are set
at the points r and r . At r we impose v =0, and at
0 1 0 0
In this case f∞ and h∞ can have any value as long as r we fix the values of f and h . One of these two has
1 1 1
the previous relation is satisfied (18). Each particular to be tuned in order to obtain the correct asymptotic
f∞ can be associated to a single κ, which means that behavior,and eachvalue ofthe other one correspondsto
the interpretation of this parameter is different to the a different solution within a family.
previous two cases, and cannot be related any more to
the width of the strings, so we will not discuss it any The following discretized version of the action is nat-
further. urally associated to the energy functional (21):
∞
As can be seen in (13) the derivatives of h scale as
α2. In the case of the φ-strings, varying α for a fixed S=−2πτ∆2 X(k+21)(Ekl−Tkl)
l,k=0
valueoftheintegrationconstantkdoesnotchangemuch
the width of the string, but the condensate flattens. In (hl+1 hl)2 (al+1 al)2
τ2Tl=(fl+1 fl)2+ k − k k − k (22)
the limit when α is very small the φ strings are similar k k − k α2h2 − (2k+1)∆4
− k
4
The superscriptl labels the time slices, which are sep-
arated by an interval τ.
Here Tl represents the density of energy associated to
k
the time derivatives.
The equations of motion can be derived from (22) by
setting to zero the partial derivatives of the action with
respect to the variables fl, hl, vl.
k k k
As the solutions of the Bogomolnyi equations are
static and minimize the discretized energy functional
they must be stationarysolutions of the discretizedtime
dependent equations of motion.
The boundary conditions have been implemented us-
ingthesamemethodof[10]. Allthequantitiesmeasured
take information from a region of radius rcal =4.5. The FIG. 2: Response of a φ−string (n = 1, m = 0), with
simulation is stopped at t = 2(r r ). In this
max max − cal α = 1 and condensate size h1 = 0.5, to a perturbation with
way the region from which we take data is not affected strength W˙0 = −0.261. The plotted lines are, from top to
by the presence of the boundary. A typical value for the bottom,E,F,W,ET and T (exceptfor W andET,therest
size of the lattice is rmax =21.5, but it varies depending oftheplotshavebeenrescaled byafactorof1/2 tofitinthe
on the profile. window). The core width, W, oscillates but is constant on
average showing that the zero mode is not excited.
During the simulation we keep track of the following
quantities:
Duringthe simulationwehaveset∆=0.1andτ =0.05,
kcal with τ/∆ smaller than the Courant number 1/√2.
E(l,kcal) = 2πX∆2(k+12)Ekl (23)
k=0
kcal A. Tachyonic Strings
T(l,kcal) = 2π∆2X(k+12)Tkl (24)
k=0 Inthiscasetoimplementtheperturbationwetakethe
E(l,k )+T(l,k ) second time slice to be :
cal cal
E (l,k ) = (25)
T cal
E(0,k )+T(0,k )
cal cal f1 =(1+p(r))f0, a1 =(1+p(r))a0
kcal(k+1)2El k k k k
W = 2π∆3Pk=0 2 k (26) h1 =(1/(h0)2+2log(1 p(r)))−1/2 (29)
E(l,k ) k k −
cal
F(l,k ) = 2π(Dl Dl) (27) with the perturbation p(r)=p [1 3(r/r )2+2(r/r )3]
cal kcal − 0 0 − 0 0
for r <r and zero otherwise.
0
Here k is defined by r =(1 +k )∆. The perturbation has been chosen in order to maximize
cal cal 2 cal
E is the static energy contained in r < r , and T thefractionofenergyabsorbedbythezeromode. Notice
cal
the energy due to the time derivatives in the same the relation between the perturbations of the tachyon
region. E is the total energy normalized to the initial field and the dilaton, and the equation (15) that gives h
T
value. W is a measure of the width of the string. In intermsoff,infactforsmallvaluesoftheparameterp
0
(26) the expression in the numerator is similar to the the perturbed profile approximately satisfies the Bogo-
energyfunctional,buttheextrafactor(k+1)givesmore molnyi equations. This perturbation gives a coordinate
2
weight to the energy far away from the core. Then, W dependence to the integration constant κ.
increases as the energy spreads. F gives the magnetic The results shown are for a value of p such that the
0
flux confined in the region r <r . perturbation initially reduced the width of the string,
cal
but the same results are obtained in the opposite case.
To implement the perturbation, we set a solution of The parameter r varies for different initial conditions.
0
the discrete Bogomolnyi equations in the first time slice In general it has a value close to r =2.
0
l=0, andin the second, l=1,we put the same solution
slightly deformed. We characterize the strength of the Fig.(2) shows the evolution of a φ-string, with n = 1,
perturbation by the fractional change of width between m = 0 and a core size h = 0.51. We show the case
1
the first, (l =0), and second, (l =1), time slices: α = 1, which is also the choice made in [1]. The
perturbationappliedhasastrengthW˙ = 0.261,which
0
W˙ (W(l =1) W(l =0))/W(l=0). (28) corresponds to a 0.4% perturbation in th−e energy. We
0
≡ −
5
B. Axionic Strings
The relevant perturbation that excites the zero mode
in this case is:
f1 =(1 p(r))f0, a1 =(1 p(r))a0
k − k k − k
h1 =(1/(h0)2+2log(1+p(r)))−1/2 (30)
k k
The result of applying this perturbation with a
strength of W˙ = 0.095 to an s-string, with α = 1,
0
−
can be seen in Fig.(3). In this case the perturbation
in energy is 2.4%. The string has windings n = 0 and
m = 1, and core size f = 0.51. The functions plotted
1
are the same ones that appear in Fig.(2). One of the
mostrelevantfeaturesoftheseplotsisthatthe magnetic
flux and the total energy are decreasing with time,
which implies that the energy is flowing out from the
FIG. 3: Response of an s−string (n=0, m=1), with α=1 region r r , and the magnetic flux is spreading.
cal
andcondensatesizef1 =0.49,toaperturbationwithstrength At the sam≤e time the width of the string, ignoring the
W˙0 =−0.095. The plotted lines are, from top to bottom, E, oscillatorybehavior, increasesat a constantrate. Before
F,W,ET andT (exceptforW andET,therest oftheplots t = 5 oscillations are noisy. In this period the shock
have been rescaled by a factor of 1/2 to fit in the window).
waveproducedbytheperturbationisstillinsider r .
Thecore width,W,after atransient, oscillates and increases ≤ cal
at a constant rate, thezero mode is excited in this case.
Although the perturbation was chosen to reduce the
corewidth,thetimeintervalwhenthecoreiscontracting
cannot be seen clearly in the figures. The reason is
that the contracting regime ends before the initial
have plotted the observables defined in the last section
burst of radiation comes out from the observed region.
as a function of time. Upper solid line represents E, the
As the system is not in a steady state yet, the data
dashed line just below is the magnetic flux in r < r ,
cal
are difficult to interpret. We have chosen to show this
whichremainsalmostconstant. Thekineticenergy,T,is
casebecausetheexpandingregimeisshownmoreclearly.
representedby the bottom solid line. The dashed line at
the center of the figure is the total energy. Although it
can not be clearly seen in the plot, the data tell us that
during the period before t=12, a fraction of the energy
is lost. This is the initial burst of radiation emitted
after the perturbation. After that the system reaches a
stationary state where all quantities oscillate except the
magnetic flux and the total energy.
The remaining solid line in the center is the string
width. As the energy contained in the region r r
cal
≤
remains constant and the width of the vortex oscillates
only around the initial value we conclude that this
kind of string is stable under this perturbation. The
experiment has been repeated in a wide range of the
parameters, p , r , and for different initial widths
0 0
(parametrized by h ), and windings, but the results
1
are similar to the ones presented here. Nielsen-Olesen
vortices react in the same way to a perturbation, as was
shown in [10]. FIG. 4: Response of an n=0, m=2, α=1 s-string with a
corecondensateofsizef1 =0.51toperturbationswithdiffer-
ent strengths. Dashed lines correspond to F, the strength is
We have repeated the evolution for different values of lowest for the top one. Solid lines represent W, the strength
α but no qualitative change has been observed. As was is highest for the top one. The strengths are: W˙0 =−0.004,
mentioned before, the smaller the value of α, the more −0.014 and −0.024. F is rescaled by a factor of 1/4. The
similartheφ stringistoaregularNielsen-Olesenstring, figure shows how the growth rate of the core increases with
which is know−n to be stable. theperturbation strength.
6
In Fig.(5) it can be seen how the rate of expansion of
an s string, (m = 1, n = 0), is affected by varying the
−
valueofα. Inthiscasethe perturbationhasbeenchosen
to initially increase the core size W˙ =0.233>0. As we
0
decrease α, keeping the perturbation strength fixed, the
rate of expansion of the string decreases. This can be
understood from equation (22). The energy associated
tothefieldhscalesastheinverseofα2. Deviationsfrom
the solution to the Bogomolnyi equations cost more
energy for smaller values of α, thus for a fixed perturba-
tion strength the evolution rates should decrease with
α. The values of alpha are: α=0.95, 0.94, 0.90.
The precision of the technique used here does not
allow to obtain reliable data for values of α lower than
FIG. 5: Response of an n = 0, m = 1 s-string with a core 0.7, where already the evolution is so slow that it can
condensate of size f1 = 0.51 to a perturbation with W˙0 = hardly be appreciatedduring the time of the simulation.
0.233. The curvesrepresent F,themagnetic fluxrescaled by However,notethatinthesesimulationstimeismeasured
a factor of 1/4. From bottom to top the values of α are are:
in units of the inverse of the Higgs mass, thus even for
α = 0.95, 0.94 and 0.90. The figure illustrates the slowing
lower values of α decompactification is still possible on
down of thecore expansion with decreasing α
cosmologicaltime scales.
Fig.(4) shows the effect of applying perturbations of WearegratefultoJoseBlanco-Pillado,StephenDavis,
differentstrengthstoan=0,m=2vortex. Inthis case Koen Kuijken and Jon Urrestilla for very useful discus-
the vortex has also a condensate size of f = 0.51. The sions. This work is supported by Basque Government
1
strengths applied are: W˙ = 0.004, W˙ = 0.014 and grant BF104.203, by the Spanish Ministry of Education
0 0
W˙ = 0.024. − − under project FPA 2005-04823 and by the Netherlands
0
−
Organization for Scientific Research (N.W.O.) under
Notice that the bigger the strength of the perturba- the VICI programme. We are also grateful to the ESF
tion,thelargerthefractionofmagneticfluxlostthrough COSLAB Programme for incidental support in the
the boundary r =r . The rate of growth of the radius initial stages of this work.
cal
also increases with the strength of the perturbation.
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