Coleman meets Schwinger

0.15 MB·German

by  Lorenzo Cornalba

CFP–05–01
ROM2F/2005/01
Coleman meets Schwinger!
Lorenzo Cornalba,1 Miguel S. Costa,2 and Jo˜ao Penedones2
1Dip. di Fisica e Sez. INFN, Universit´a di Roma “Tor Vergata”, Roma, Italy
2Centro de F´ısica do Porto e Departamento de F´ısica,
Faculdade de Ciˆencias da Universidade do Porto, Portugal
It iswell knownthat spherical D–branesarenucleated in thepresenceof an externalRRelectric
field. Using the description of D–branes as solitons of the tachyon field on non–BPS D–branes, we
showthat thebranenucleation process can beseen as thedecay ofthetachyonfalse vacuum. This
processcandescribethedecayofflux–branesinstringtheoryorthedecayofquintessencepotentials
arising in flux compactifications.
5
PACSnumbers: 11.25.-w,11.25.Uv
0
0
2
One of the most beautiful results in Quantum electric field self–gravitates on the length scale 1/E

n
Field Theory,derivedby Schwinger[1], is thatpairs andthenon–BPSbraneisplacedatthecenterofthe
a
ofchargedparticlesareproducedinanexternalelec- flux–brane, which is a gravitationally stable point.
J
tric field. This mechanism can be generalized to In this language the decay of the flux–brane to-
9
1 the nucleation of spherical branes in theories with wards flat space is explicitly described as the decay
p–form gauge potentials and was studied in [2] us- of the open string false vacuum of many non–BPS

1 ing semiclassical instanton methods, starting from D-branes. Secondly, whenever the transverse direc-
v
theNambu–Gotobraneactionminimallycoupledto tions are compact, the electric field will give rise to
1
the gauge field. String theory has many extended a quintessence potential in the compactified theory
5
1 charged objects associated to massless gauge fields [8]. Thedecayofthetachyonfieldwillthendescribe
1 and therefore analogous computations of brane nu- thedecayofthequintessencepotential. Thisprocess
0 cleation rates can be performed. generalizes the dynamical decay of the cosmological
5 According to Sen [3], D–branes can be thought of constant of [9]. We shall neglect the effect of the
0
as tachyon solitons. It is therefore natural to ask if expansion of the universe in the tachyon dynamics.
/
h thebranenucleationprocesscanbedescribedinthis Throughout this paper we neglect closed string
-t language. Consider, in particular, the tachyon kink effects by taking α′E2 1 and gs 1. Therefore
p ≪ ≪
solution on a non–BPS Dp–brane that interpolates we can consider the open string dynamics indepen-
e
betweenthe T = vacuaofthetachyonfieldand dently,ignoringthebackreactionontheclosedstring
h
±∞
: which describes a BPS D(p 1)–brane. By turning fields. The effects in the closed string fields due to
v −
on an external RR electric field thevacuumdecaycanbecomputedtoleadingorder
i
X anddescribe a changeoforderg . We workin units
dC =E dt dx1 dxp s
r p ∧ ∧···∧ such that α′ =1.
a
Recall the form of the classical action for the
paralleltotheworldvolumeofanon–BPSDp–brane,
tachyon field on a non–BPS Dp–brane [3]
we shall see that the tachyon vacuum degeneracy
is lifted and that the brane nucleation corresponds
to the decay of the tachyon false vacuum. Hence S = dp+1xV(T) 1+ηµν∂µT∂νT
−
Coleman’s analysis of the decay of the false vacuum Z
p
[4] describes Schwinger’s nucleation process. Simi- + W(T)dT Cp , (1)
∧
larmethods wereusedin[5]todescribebrane/anti– Z
brane decay by creation of a throat [6]. where the massless fields on the brane are consis-
We shall take as closed string backgroundfor our tentlysettozeroandµ,ν =0,1, ,p. Weconsider
···
computations flat space with a small RR electric flat space–time with a constant dilaton, but we al-
field and we shall neglect the backreaction on the low for the presence of a RR p–form potential. Sev-
closed string geometry. The electric field E will in eral properties of the functions V(T) and W(T) are
general distort the geometry on length scales larger known. Both are evenfunctions and behave asymp-
than 1/E, therefore our analysis will be valid only totically as e T /√2. Moreover, V(0) = T˜ is the
−| | p
within this scale. There are two simple settings to tension of the non–BPS Dp–brane, which is related
keepinmind. Firstly,wheneverthedirectionstrans- tothetensionT ofaBPSDp–branebyT˜ =√2T .
p p p
verseto the electric field arenotcompact,the back- Finally, the fact that the tachyonkink solution rep-
ground geometry is that of a flux–brane [7]. The resents a BPS D(p 1)–brane gives the additional
−
2
requirements
~
T
p
∞ ∞ V
dT V(T)= dT W(T)=Tp 1 , U
− E Z
Z−∞ Z−∞
sothatthetensionandchargeofthesolitonarecor-
rectly normalized. The qualitative results in this
notewillnotdependonthespecificformofthefunc-
tions V(T) and W(T). However, for specific exam-
ples we shall use the explicit form [3]
T˜ T 0 T
p 0
V(T)=W(T)= . (2)
cosh T/√2
-ET
To analyze the tachyon dynam(cid:0)ics in(cid:1)the presence p-1
of an external RR electric field it is convenient to
FIG. 1: Tachyon potential U(T) and functions V(T)
integrate by parts the Wess–Zumino termin the ac-
and EZ(T).
tion. Defining
∞
Z(T)= dT′W(T′) ,
− Then,underatranslationxp xp+δxp,thechange
ZT →
in the energy density is δxpEU( ), which gives
anddroppingtheboundaryterm,oneobtainssimply −∞
the constant pressure ETp 1.
−
Since T = + is metastable one expects that
S = Z(T)dC .
WZ p ∞
− there is a finite rate per unit time and volume for
Z
the nucleation of bubbles of the true vacuum in-
The function Z(T) was defined so that Z( ) = 0
∞ side the false vacuum. Furthermore, we see that
and Z( ) = Tp 1. From the asymptotics of
−∞ − − the surface of the bubble is the locus where the
W(T) it follows that Z(T) approaches its asymp-
tachyon field interpolates between the two minima,
totic values at T = as e T /√2. The tachyon
±∞ ∓ −| | and should therefore be interpreted as a spherical
effective action in the presence of an electric field
D(p 1)–brane. Wewillestimatethenucleationrate
can then be written as −
using the standard semiclassicalmethods developed
by Coleman. This amounts to finding the bounce:
S = dp+1x V(T) 1+ηµν∂ T∂ T +EZ(T) ,
− µ ν the non–trivial classical solution of the Euclidean
Z h p i equationsofmotion,whichapproachesthefalsevac-
and the corresponding classicaleffective potential is
uum at infinity and has minimal Euclidean action
U(T)=V(T)+EZ(T) . S0. Then Γ/V ≃ ∆e−S0 is the nucleation rate per
unit time and p–volume, with ∆ a pre–factor. This
The constant of integration in the function Z(T) pre–factor is related to the determinant of the open
abovewasfixedsothatU( )=0. Fromtheasymp- string field fluctuations around the bounce. Since
∞
totics of V and Z it is clear that, for small electric the tachyonclassical actionis obtained by eliminat-
field, both T = are still vacua of the theory. ingthemassiveopenstringfieldsviatheirequations
±∞
However, since U( ) = ETp 1, T = + is a of motion, the naive pre-factor computed from the
metastable vacuum−a∞nd wil−l tunn−el to the tru∞e vac- action (1) would neglect fluctuations of the massive
uum at T = (see FIG. 1). On the other hand, string modes.
−∞
fromtheasymptoticsofV andZ itisalsoclearthat, Since the kinetic term inthe actionis notcanoni-
as one increases E above a critical value, T = + cal,oneshouldverifythevalidityofstandardinstan-
∞
ceases to be a metastable vacuum. The criticalfield ton methods for the Born-Infeld Euclidean action
dependsonthespecificformofV andW,butitwill
beoforderthestringscale. Fortheparticularchoice
(2) the critical value is 1/√2. S = dp+1x V(T) 1+(∂T)2+EZ(T) .
E
In the above language it is easy to check that the Z h p i
kink representing a D(p 1)–branefeels a force due
to the external field. C−onsider a configuration de- Let us call the bounce solution T¯ and define a one–
pending onasinglevariablexp withT( )= . parameterfamily offunctions T (x) T¯(x/λ). One
λ
±∞ ±∞ ≡
3
Π
can easily show that
1
d
S [T ] = (p+1)S T¯
E λ E
dλ
(cid:12)λ=1 B
(cid:12)(cid:12)(cid:12) dp+1x(cid:2)V(cid:3)(T¯)(∂T¯)2 .
− 1+(∂T¯)2
Z
Since λ = 1 is a stationary pointpof the action it
follows that S T¯ > 0. One may also compute
E
the second derivative at λ=1 and easily show that -1 Φ Φ A 1 Φ
(cid:2) (cid:3) c 0
it is negative. This fact guarantees the existence
of a negative eigenmode in the quadratic fluctua-
tions around the bounce. One also expects that the
K > 0
bounce issphericallysymmetricandthatithaspre-
cisely one negative eigenmode. C
K < 0
To simplify the analysis of the dynamics of the
bounce, we shall work with potential functions -1
, , ,whichareobtainedfromtheoriginalV,U,Z
V U Z
by rescaling them by 2/Tp 1. Start by defining a FIG. 2: Contour plot of K and typical trajectories for
new field − E > Ec. The thick line is the K = 0 contour and the
dashed lines are other K contours. The line starting at
T
Φ= (T′)dT′ , Φc is the bounce for p > 0 and the other two lines are
0 V typical trajectories. When E ≤Ec the bounce degener-
Z
ates to theboundary of phase space.
such that T = corresponds to Φ= 1. For the
±∞ ±
particular choice (2), one has
Using (4) it is easy to show that
√2 π
= cos Φ , =Φ 1 . (3)
V π 2 Z − p (Φ)Π2
(cid:16) (cid:17) K˙ = V 0 .
For a Euclidean solution with maximal symmetry r√1 Π2 ≥
Φ = Φ(r) where r2 = x xµ, the effective one– −
µ
dimensional action reads Thus,itisinstructivetoconsidertheKcontourplot,
since K never decreases along the (Φ(r),Π(r)) tra-
I = ∞drrp 2+Φ˙2+E , jectories(seeFIG.2). Sphericalsymmetrydemands
E
V Z
Z0 hp i Π(0)=0andthereforealltrajectoriesstartfromthe
where dot denotes the radial derivative. The Eu- Π=0 horizontal axis.
clidean action is SE = IEApTp 1/2, with Ap the Consider first the conservative case p = 0, where
volume of the p–sphere. − thetrajectoriesfollowlinesofconstantK clockwise.
The equations of motion turn out to be simpler The bounce solution is the trajectory with K = 0,
in the Hamiltonian formulation. The conjugate mo- with the initial value Φ0 defined by the equation
mentum to the field Φ is related to K(Φ0,0) = (Φ0) = 0. This trajectory approaches
U
1 ∂L Φ˙ the vertical line Φ = 1 as r → ∞ (point B in FIG.
= Π . 2). The corresponding Euclidean action is given by
rp ∂Φ˙ 2+Φ˙2 ≡ the integral
V
Thus,since isnon–npegative,the variableΠ isalso
bounded betVween 1 and 1. Finally, the equations 1 E 2
of motion reduce t−o the following dynamical system S0 =Tp−1ZΦ0dΦs1−(cid:18) VZ(cid:19) .
p
Π˙ = ′(Φ) 1 Π2+E ′(Φ) Π , In the limit E 0 the above action remains finite
V − Z − r and describes a→D/D¯–instanton pair. Trajectories
(Φ)pΠ
Φ˙ = V . (4) that start with Φ(0) between Φ0 and 1 are periodic
√1 Π2
− andthosethatstartbetween 1andΦ0 endinfinite
−
Although for p > 0 the system is not conservative, r at the horizontal line Π=1.
it is still useful to consider the Hamiltonian Nextweconsiderthecasep>0. Nowalltrajecto-
1 ries that enter the K > 0 region will inevitably ap-
H = (Φ) 1 Π2+E (Φ) K . proachpointAinFIG.2atr = . Thiswillhappen
−rp V − Z ≡ ∞
p
4
for Φ(0) greater than some critical value Φ satisfy- After nucleationthe brane will actas a source for
c
ing 1 Φc < Φ0. On the other hand, trajectories closed strings. The tadpoles for the closed string
− ≤
with Φ(0)<Φ reach Π= 1 in finite r and can not fieldscanbe computedbycouplingthe action(1)to
c
becontinuedfurther. Thebounceispreciselytheso- these fields. Since the transverse coordinates of the
lution with Φ(0)=Φ . To understand the behavior original non–BPS brane vanish at the bounce, the
c
ofΦ asafunctionofthe electric fieldE, letuslook only non–vanishing components of the sources for
c
at the extreme solution with Φ(0) = 1. This tra- the metric and RR p–form field lie along the non–
−
jectory staysin the verticalline Φ= 1until it hits BPS brane world–volume. A simple computation
−
Π = 1 at a finite radius r¯. Expanding the solution shows that the metric, RR p–form and dilaton tad-
around this point, one can show that, for poles are
x x
E >Ec = p 1 , Tµν = Tp−1 Rµ2ν −ηµν δ(r−R)δ⊥(x) ,
8 ( 1)
r Z′ − (cid:16)xν (cid:17)
we necessarily have Π˙(r¯) > 0 and therefore the tra- Jµ1···µp = Tp−1 R ǫνµ1···µpδ(r−R)δ⊥(x) ,
jectory ends. We conclude that for E > Ec the Q = Tp−1δ(r−R)δ⊥(x) ,
bounce starts with Φ > 1 and approaches point
c − where r2 =xµxµ is the analytic continuation of the
B in FIG. 2 at r = . This behavior is analogous
∞ Euclidean radial coordinate and δ (x) is a delta
to the case p = 0, where the electric field is always ⊥
function on the transverse space. These are the
above E . Although this solution is interesting,it is
c sourcesthatdescribethedecayofaRRfluxp–brane.
not clear if it survives string corrections. As E de-
The linearized solution for the closed fields can be
creases to E the bounce trajectory degenerates to
c solved in terms of retarded Green functions and de-
theboundaryofthephasespace. ForE E theso-
≤ c scribes the radiation emitted by the expanding nu-
lution starts at Φ = 1 and hits Π=1 at a radius
R = p/E, wherecΠ˙ v−anishes. Then the trajectory cleatedbrane. This processcanalsobe describedin
the framework of Euclidean quantum gravity using
jumps discontinuously to Φ = Π = 1 and descends
generalizations of the Ernst metric [10, 11]. Here,
vertically to point B in Fig. 2. As a function of
instead of an effective closed string description, we
r, the tachyon field Φ jumps discontinuously from
used open strings to describe the decay process and
the true to the false vacuum at r = R, while Π(r)
the sources for the emitted radiation.
is smooth. We conclude that in this case the thin
When the transverse directions to the non–BPS
wall approximation is exact. This is a consequence
brane are compact, the delta functions δ (x) in the
of the kinetic term of Born–Infeld type. The Eu- ⊥
above sources should be replaced by the inverse of
clidean action of the bounce can then be explicitly
the volume of the transverse space. The sources in-
evaluated
duce a jump in the derivative of the dilaton, gauge
A potential and metric. Moreover, the dilaton will ra-
S0 = p+p1Tp−1Rp . diate. Inthedimensionallyreducedtheorythejump
in the electric field corresponds to the decay of a
This result was obtained by Teitelboim almost 20 quintessence potential. It would be interesting to
years ago using first quantized p-brane mechanics consider the explicit form of the background geom-
[2]. Here we have derived it as the bubble decay etry and analyze the decay process throughout the
of the tachyon false vacuum for the case of D-brane cosmologicalevolution.
nucleation in string theory. The techniques exploredin this paper canalso be
The time evolution of the nucleated branes is de- applied to black hole physics. In particular, one
terminedby theanalyticcontinuationofthebounce could try to make predictions regarding the emis-
solution. For p > 0 a D(p 1)–brane appears as a sionofbranesby non–extremalblackholes in string
−
(p 1)–sphere of radius R and then expands with theory.
−
constantradialacceleration1/Rdescribingahyper-
boloid in Minkowski space. In this case all the en- This work was supported in part by INFN,
ergy arising from the decay of the false vacuum is by the MIUR–COFIN contract 2003–023852,
transfered to the brane. For p=0 there is no brane by the EU contracts MRTN–CT–2004–503369,
to absorbthe energy andthereforethe tachyonfield MRTN–CT–2004–512194 and MERG–CT–2004–
can not appear exactly in the true vacuum. In this 511309, by the INTAS contract 03–51–6346,
case the tachyon field rolls down classically from T0 by the NATO grant PST.CLG.978785 and by
(see FIG. 1) to the true vacuum. the FCT contracts POCTI/FNU/38004/2001
5
and POCTI/FP/FNU/50161/2003. L.C. is [6] C.G.CallanandJ.M.Maldacena,Nucl.Phys.B513
supported by the MIUR contract “Rientro dei (1998) 198; K.G. Savvidy,[arXiv:hep-th/9810163].
cervelli” part VII and J.P. by the FCT fellowship [7] P.M. Saffin, Phys. Rev. D 64 (2001) 024014; M.
GutperleandA.Strominger,JHEP0106035;M.S.
SFRH/BD/9248/2002.
Costa, C.A.R. Herdeiro and L. Cornalba, Nucl.
Phys. B 619 (2001) 155.
[8] L. Cornalba, M.S. Costa and C. Kounnas, Nucl.
Phys. B 637 (2002) 378.
[9] J.D. Brown and C. Teitelboim, Phys. Lett. B 195
[1] J.S. Schwinger, Phys. Rev. 82 (1951) 664. (1987) 177.
[2] C. Teitelboim, Phys. Lett. 167B (1986) 63. [10] F.Dowker,J.P.Gauntlett,G.W.GibbonsandG.T.
[3] A.Sen,[arXiv:hep-th/0410103]. Horowitz, Phys. Rev.D 53 (1996) 7115.
[4] S.Coleman, Phys.Rev D15 (1977) 2929. [11] M.S. Costa and M. Gutperle, JHEP 0103 027.
[5] K.Hashimoto, JHEP 0207 035.