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45
Ferrer E.J., Incera V.
PHOTONS AND FERMIONS IN SPACETIME WITH ,4 COMPACTIFIED SPATIAL DIMENSION
Physics Department, State University of New York at Fredonia, Fredonia, NY 14063, USA
I. Introduction
A significant effort in the recent history of physics bears on the quest to unify all the known fundamental
interactions. This old Einstein's dream found a partial realization in the Standard Model, where the electromagnetic, weak and strong nuclear forces were unified in a single one. The fourth fundamental interaction, gravity, has been an obstacle, in that path. Quantum physics in the realm of the Riemann space, which is the natural habitat, for gravity, gives rise to an anomalous behavior for the quantum processes where the gravitational quantum particle, the graviton, participates. An incomplete unified scenario, where the four fundamental forces cannot be single out to one, limits the possibility to understand the physics of our universe at the earliest time.
An innovative approach in the search for a
CC)rp^'H;:h1" '”"y tO f">•<’-—
wa 1 in til " . 1
[ I ] ' posr 1 ■ , ' 1 i ; .
extra wj............... ,t.
extra d " ■ is w • ’■! ■ 1 ;s
that ar v i to low - -/sics 1 ■■ *e
..tri r~ fietcts in the remnant flat four-
■ time. More recently, the idea of
1 ■ . dimensi,: ' : . xtensively
a supergraviiy, super ■ • e theones
• a different developr. ~ - 1 ; •r ■ intensions have been also applied at lower energy scale in an attempt to understand hierarchical scales existing between the weak and Planck energies. The main new ingredient of this approach is that only gravitons can propagate in the bulk corresponding to the extra dimensions, while the other gauge fields of the Standard Model are constrained to the four dimensional wall.
Motivated by the importance of the Kaluza-Klein scenario, the study of QFT in nontrivial space-time has been the focus of attention of many investigators in recent years. It is well known that the global properties of the spacetime, even if it is locally flat, can give rise to new physics. A seminal discovery in this direction is the so called Casimir effect [4]. In this phenomenon, an attractive force appears between neutral parallel perfectly conducting plates. The materialized attractive force is mediated by the zero-point fluctuations of the electromagnetic field in vacuum. Hence, the Casimir force is interpreted as a macroscopic manifestation of the vacuum structure of the quantized fields in the presence of domains restricted by boundaries or nontrivial topologies [5].
As it is known, QFT in spacetime with non-trivial topology has nonequivalent types of fields with the same spin [6]. The allowed number of distinct field configurations is determined by the topological structure of the spacetime; generally being more than one in non-simply connected spaces. In particular, for a fermion system in a space-time which is locally flat but with topology represented by the domain S1 x.K3 (i.e. a Minkowskian space with one of the spatial dimensions compactified in a circle Sl of finite length a), the non-trivial topology is transferred into periodic boundary conditions for untwisted fermions or antiperiodic boundary conditions for twisted fermions tjf(t, x, y, z — a / 2) = ±v/(t, x,y,z +a! 2), (1)
but for vector fields, only untwisted configurations are realized. In (1) the compactified dimension with length a has been taken along the OZ -direction.
Quantum electrodynamics (QED) with photons coupled to untwisted fermions or to a combination of twisted and untwisted fermions is an unstable theory [7j, The instability arises due to polarization effects of untwisted electrons which produce tachyonic electromagnetic modes [7]. For self-interacting calar fields the space periodicity cart also produce o-.ciaKoii-o«, -»..«png a symmetry breaking that makes s i 'd to become massive [8]. The acquired on the periodicity length, so the jpilCilUUJCIiU'fJ 13 called topological mass generation.
To understand in qualitative terms how the fermion boundary conditions in a non-trivial topology can produce instabilities in QED, we should have in mind that, thanks to the vacuum polarization, the photon exists during part of the time as a virtual e*e~ — pair . The virtual pairs can then transfer to the real photon the properties of the quantum vacuum which, as known, depend on the non-trivial topology and boundary conditions of the space under consideration. Thus, to study the photon in the compactified space, one needs to investigate the effects connected to vacuum polarization in the S1 x/?3 space-time.
Our main goal in the present report is to analyze the consequences of the non-trivial topology for photon propagation in QED and for fermion condensation in a gauged-NJL theory. We will show that the non-simply connected character of the spacetime may give rise to different photon modes of propagation, which are normally absent in QED in a flat space with trivial topology. Another topic that we will discuss is the chiral symmetry restoration effect of a compactified dimension in a gauged-NJL theory.
If. Non-trivial Vacuum Solutions in Compactified QED
The vacuum polarization in the non-trivial spatial topology S1 xi?3 can be influenced by both virtual untwisted and twisted e+e~ - pairs. The results for twisted fermions can be easily read off the results at finite temperature, since in the Euclidean space the two theories are basically the same after the interchange of the four-space subindexes 3 <-» 4. Nevertheless, a different situation occurs with untwisted fermions that has no analogy in the statistical case. Henceforth, we concentrate our attention in the untwisted-fermion case.
Let us consider the QED action in a spacetime
domain with compactified dimension of length a in
the OZ -direction /2
5= | dx3 ¡dx()d2
(2)
When this compactified QED action is considered for untwisted fermions, the effect of vacuum polarization upon photon, propagation yields a tachyonic mass for the third component of the photon field [7]. In a quantum theory the existence of tachyonic modes are an indication that the considered vacuum is not the physical one, and that a symmetry breaking mechanism is in order. Indeed, in compactified QED with untwisted fermions it has been shown [9] that a constant expectation value of the electromagnetic potential component along the compactified direction minimizes the effective potential, thereby stabilizing the theory. The same stable vacuum solution obtained in QED with S1 x R* topology in Ref. [8], is also present in the case of massless QED with periodic fermions on a circle [10] (QED with S‘xJ?3 topology). Notice that, even though a constant vacuum configuration has F = 0, it
cannot be gauged to zero, because the gauge transformation that would be needed does not respect the periodicity of the function space in the S1 x R3 domain. This is a sort of Ahatonov-Bohm effect which makes ./l(l a dynamical variable due to the non-simply
connected topology of the considered space-time. The lack of gauge equivalence between a constant component of the gauge potential (4, in this case) and zero is also manifested in QED at finite temperature and/or density, due to the compactification of the time coordinate [11],
In the statistical case however, the minimum of the potential is at ^ =0, since only twisted fermions are allowed. On the other hand, in the electroweak theory with a finite density of fermions, a non-trivial constant, vacuum A0 is induced by the fermion density and cannot be gauged away [12]. There, in contrast to the system considered in the present paper, an additional
parameter (a leptonic and/or baryonic chemical potential) is needed to trigger the non-trivial constant minimum for .
To find the physical vacuum that stabilizes the untwisted fermion theory, we propose, following Ref. [9], the following ansatz [13]
4=A8v3 (3)
for the vacuum solution, with A an arbitrary constant that will be determined from the minimum equation of the effective potential. Due to the periodicity of the
fields in the S1 x i?3 space, the gauge transformations
AA—3 a are restricted to those satisfying e
ot(x, +a) = a(x3)2/ii ,/e Z [11]. Thus, the gauge transformation a(x) = (x • n)eA , which connects the constant field configuration (3) with zero, does not satisfy the required periodicity condition unless A will be given by
2lK ■ - <4,
A = •
ieZ
ea
Let us consider then the one-loop effective potential of the theory (2) around the vacuum configuration (3)
V = -~a-]ln(DetG~l). (5)
Here DetG~‘ - > < 1 ,d‘ pda
with p} = Innla (« = 0,±1,±2,...) being the discrete frequencies associated with periodic fermions. In (5) CT1 = j-p-\-rn is the fermion inverse Green’s function in the background A, with "pft = (pu,pd, p3 - eA).
After the Wick rotation to Euclidean space and summing in p} we obtain
V(A) = - f-li-k + 2a""' Retail —e’'"<E,’WeA)
2 (2k) L ^ \
). (6)
where d3p = idp4d2pt and tp = \jp2 + m2 .
The extremum of the effective potential (6) satisfies dV(A)
3A “r dip
A=Am. 3 -
2e a€f sin(aeA)
(7)
si
(2k) l + e " — 2e~œ" cos(aeA) In
- = 0.
The solution to (7) is A = — , I e Z . Nevertheless,
ea
the minimum condition 3V(AIIIin)/32A > 0 is only
satisfied by the subset (21 + l)tt
A mid ~
IeZ
(8)
ea
The elements in the set of minimum solutions (8), are gauge equivalent since they are all connected by allowed gauge transformations
(a(x3 +a) = a(xi) + 2ln). It should be pointed out, however, that the solutions (8) are not gauge equivalent to the trivial vacuum A = 0, since none of them satisfies (4). That is, the trivial vacuum belongs to a different gauge class.
Substituting with the minimum solution (8) in Eq. (6) we obtain
Vr(Amin ) - - J”~“r[£p + 2a-1 Retail + )].
,(2n)
(9)
The expression (9) coincides with the one-loop effective potential of the theory (2) around the trivial vacuum for twisted fermions. As expected, in the
am«1 approximation, the effective potential (9)
reduces to
V(A-)-^- (,0)
which is the result .reported for twisted fermions in Ref. f7j. Thus, the vacuum energy of both classes of fermions coincides if the corresponding correct vacuum solution is used.
Notice that since for twisted fermions the zero vacuum solution enters as a shift in the discrete momentum component: p., —> p^-eA: the parameter A appears only in the distribution 'functions associated to the sums in /?,. Hence, when A is evaluated in the minimum solution (8) it turn rf the
untwisted fermions into that oft', - > t • -
If, Photon Proptaiiort in S] xR'' Space-Time
To study the photon propagation in the S'xJ?3 domain, we should solve the dispersion relations of the electromagnetic modes, which in the low-frequency limit has the general form
kl - It2 +ll(k2) = 0, (11)
where ri(k2) accounts for vacuum polarization effects. Different external conditions, as external fields, geometric boundary conditions, temperature, etc., may modify the vacuum and produce, through ri(k2), a variation in the spectrum of the photon, modes.
Tire solution of the photon dispersion equations (11) can be obtained as the poles of the photon Green’s function. Due to the explicit Lorentz symmetry breaking in the 51 x J?3 topology, we must consider, in addition to the usual tensor structures k and g, a spacelike unit vector pointing along the compactified direction n11 = (0,0,0,1). Then, the general structure of the electromagnetic field Green's function is
Auv (k) = P
k k \L V
J Vv VV+«A
H. *2~ (k-n)
k\n^
(k • nf
a , , + -rkjL
(12)
where a is a gauge fixing parameter corresponding to the covariant gauge condition —3^ = 0, and P and
a
Q are defined as 1
P = -
*2+nn
6 = -
n,
(13)
(k2+n0){k2 +n0-ni[*i/(*•«)* +1]}
The parameters FI0 and II, are the coefficients of
the polarization operator
space can be written as LL
ïl(k) = Tï0 -
which in the S' xR
+n,
k k
V*v +n»K
k2n„n
(14)
^ k2 (k ■ n) (k ■ nf
From (12)-(13) the photon dispersion relations are 0, (15)
\
r
K ~ k ' + rio
K ~k" +1 IX,
= 0,
(1.6)
J
with k1 - kl -k\ and k: = k{ +k; . We should point out that in addition to the transverse inode associated to Eq. (15) (normally present in Minkowski space-time with trivial topology), a longitudinal mode, Eq. (16), arises here due to the presence of the extra coefficient 11,. The situation resembles the finite temperature case. Nevertheless, as discussed below, the physical consequences of the spatial compactification are radically different from those already known at finite temperature.
Since the compactified OZ -direction distinguishes itself from the other spatial directions, we should separate the analysis for photons propagating along OZ (k±=0 ), from those propagating perpendicularly to it ( k3 =0).
The dispersion relations (15) and (16) for photons propagating perpendicularly to the compactified direction (¿3=0) are found, from Eqs.(14)-(16), to reduce respectively to
kl
k2
.k1 -T1 =0
-L . 2 00 U*
(17)
kl-kl- n33=0. (18)
To find the solutions of Eqs. (17)-(18) at the one-loop level, we need to calculate the one-loop polarization operator components H00 and 11,3 for untwisted fermions. Considering the free propagator of untwisted fermions on the minimum solution (2)
G(x- x) ----— ^d4pexp[ip(x- x')]G(p), (19)
where G(p) = -
(2n) a
p-m
. —. h =(Po>P±,Pi-eA), (20)
p - m~ + it
and ^d4p = Jd3p, p3 = 2nnla, (n = 0,±l,±2,...)
ft
being the discrete frequencies associated to periodic fermions, the corresponding one-loop polarization operator is given by 11^ (*) =
Aie
(2k)~ a
PuiPv ~K)--[p(p-k)-m~]g
(p - m ){(p - k) -m
+p <-> v}.
In the a | k |« am 1 limit, we obtain
ntw (¿3 = 0, iu = 0, kL
-k
.2
3k
11-, 3(k3 =0 ,Jc0 =0,A:a
0)= '
+ 0(kV),
(21)
(22)
°>-7
+ 0(1:*)
(23)
where am/2nd. Using the results (22) and (23) in the dispersion equations (17), (18), and taking into account that the photon velocity for each propagation mode can be obtained from v(k) = /»’ j >.;. we find
that within the considered approximation the transverse and longitudinal modes propagate perpendicularly to the compactified direction with velocities
d-
-K,
12ic
vi:,l-[(Mxt)2/2il](
(24)
(25)
a < Il m ~ 103 fm , the transverse velocity (24) is about 0.1% larger than the light velocity in trivial space-time. Here we should point out that albeit v* > c, in this problem there is no causality violation. To understand this, let us recall that the velocity (24) is a low-frequency mode velocity. On the other hand, the velocity of interest for signal propagation, and hence the relevant one for causality violation problems, is the high-frequency velocity (q0 —>°°). To determine the difference between the two, one would need to investigate the absorption coefficient, Irn[n.(c|0)], with n(q0) being the refraction index as a function of the
frequency in the space with SlxR3 topology. However, aside from any needed calculation, we agree with the analysis of Reft. [16] about the lack of causality violations in similar systems. We believe that in the case under study no (micro-)causality should be violated, because the events taking place in the S1 x R3 space are not constrained by the null cone of a Minkowskian system, as Lorentz symmetry is explicitly broken in the present situation.
The low-frequency limit ( <c0 = 0, | k [-4 0 ) used to
obtain the longitudinal-mode mass ML is essential to
study the static properties of the el
in this space. The mass obtained in
role of a magnetic mass of me longituontai
electromagnetic inode [17]. As showed in Ref. 191, this
topological mass affects the magnetic response of the
system.
Considering now photons propagating along the OZ -direction (kl=0), we have that the dispersion relations (15) and (16) can be written respectively as
kl
kl
¿Va
- Arp - n,, =0
—FIjj = 0.
(26)
(27)
respectively, where (M¡) — TI33 = e ¡3a' >0 plays the role of an effective topological mass for the longitudinal mode.
We should notice that the modifications found for the two velocities, v* and Vj_, have different origins. The modification of the longitudinal velocity v^ is due to the appearance of the topological mass Ml± ; while the transverse superluminal velocity v[ (note that VÎ > c because £; < 1 in the used approximation) appears as a consequence of a genuine variation of the refraction index in the considered space-time. Modifications of the photon speed in non-trivial vacua have been previously reported in the literature [15].
It is easy to corroborate that for compactification lengths in agreement with the used approximation,
Assuming ak0 arn«l in (21), the components of the polarization operator appearing in (26) and (27) become
nn(kL=0,k„k() ~ 0) :
I + 0(|f)j + 0(li,2) (28)
FI33 (k^ 0, k^, ky
I
+o(i4)
(29)
From (26), (27), (28) and (29) we can straightforwardly find the low-frequency limit of the photon velocities for transverse and longitudinal modes propagating along the OZ -direction
(30)
where both transverse and longitudinal mode masses coincide and are given by Mj, =M^ —e/3a. We
stress that in this case both velocities are smaller than the light velocity in trivial Minkowski space c, and that the modification is due, as in (25), to the appearance of a topological photon mass for each mode.
IV. Effect of Compactification on Fermion
Condensation
Dynamical chiral symmetry breaking in phenomenological models with fermion interactions of Nanibu-Jona-Lasinio (NJL) type [18] have attracted great attention in recent years [19], [20]. Our interest now is to consider dynamical chiral symmetry breaking in a non-trivial topological space taking into account the results of Sec. II. With this aim, let us add a NJL four-fermion tenu with coupling G to the Lagrangian density (2), so that it becomes
£ _ __ F* + _ myy
4
G
(32)
+-
2 N
[(VY)2 + (fiT¥)2]-
The terrnions in (32) are assumed to carry out a flavor index a = 1,2,..., A’. Introducing the composite
(1 ^
fy), rc=.-~(ij?iYV) gauged-
NJL lagrangian density (32) can be rewritten as CSJL = -[? .if1 Dy) - f (0 + if ii)\j/ -
(33)
and 0 constant, since the effective potential V only depends on the chiral invariant p2 = a2 + it2.
The vacuum solution is determined by the stationary point of the effective potential (34). At G > Gc = 4n21 A2, the stationary equation dV(o)/da = 0 has a non-trivial solution o, which corresponds to a global minimum of (34).
If the third spatial dimension is compactified in a circle of radius a, we have that, for antiperiodic fermions, the potential (34) becomes
F (0) = É--2 tÎJ^inia2 + p\)-A> 2G J (2tz) e
(35)
N
-..—(cr + ir).
2G
The Lagrangian (32) has a continuous chiral symmetry f —> e№f y, but it is clear from Eq. (33) that if a gets a different from zero vacuum expectation, value (vev) 0, this chiral symmetry is broken and the fermions acquire mass. In the large N limit, assuming e <g. G, the interaction of the fermions with the ^
field can be neglected to investigate the fermion condensation in the strong- G regime. The effective potential for the composite fields a and \|/ is obtained by integrating out all the fields in the path integral In a flat and topologically trivial spacetime, the effective potential in leading order at large-N, is given, after performing the Wick rotation to Euclidean space, by
<*>
where A is a large momentum cutoff. In expression
(34) we dropped all the O' -independent terms, as they will not contribute to the stationary solution of the potential. We considered a configuration with ji = 0
a J (2k)
where £p -\Jp2 + cr . The last term in the RHS of Eq.
(35) is obtained after summing in the discrete momentum p3. The appearance of this new term gives
rise to the critical coupling Gac -6a2Gcl(6a2 -Gc), which depends on the compactification radius a.
Notice that G‘‘ > Gc. Hence, for twisted fermions the compactification tends to restore the symmetry, in agreement with results previously found in Refs. |20j, [2.1] within a pure NJL theory (without gauge fields). When periodic fermions are considered in the trivial vacuum, the corresponding effective potential Vp, is similar to (35), with the only change of a negative sign in front of the exponential in the RHS of Eq. (35). This ease was also studied in Ref. [20] in the context of a
pure NJL model There, the analysis of the minimum
of the potential revealed that the effect of the compactified dimension is to enhance the chiral condensate, i.e., to decrease the critical value of the coupling.
The situation is different however for the gauged-NJL theory with untwisted fermions. Here, in analogy with QED, the stable vacuum is given by the constant vector potential (8). Consequently, the effective potential must depend on a nontrivial vacuum solution explicitly. Summing in the discrete momentum p3 and taking into account that the constant vacuum (3) enters in the calculation of the effective potential as a shift in P-i Pi + A, the effective potential is given by
V, (0, A) = — - 2 f In (a2 + pi ) -
2G •> (2jt) "
4 f d*p
(36)
“J;
a J(2n)3 L J
Notice that when the minimum solution (8) is substituted on (36), it is obtained that Vp(o,AIIlin) = VA/,((j). As a consequence, the critical coupling for the chiral condensation with untwisted
fermions reduces to the same one ( Gac ) already found for twisted fermions in the trivial vacuum. Thus, we conclude that independently of the fermion boundary condition, the effect of the compactification in the theory (32) is to decrease the condensate, and eventually, to reinstate the chiral symmetry at some critical value of the compactification radius.
V. Concluding Remarks
In this paper we have shown that in a nonsimply connected spacetime with topology Sl xR3, the stable vacuum solution for QED with untwisted fermions is given by constant field configurations that are gauge _________________
equivalent to A3 = —, while for twisted fermions the ea
stable solutions correspond to constant gauge configurations equivalent to the trivial vacuum. As a consequence, the one-loop effective potentials for twisted and untwisted fermions coincide when’ the corresponding stable vacuum solutions are considered, A direct implication of the relation between fermion boundary conditions and QED vacua, is that the vacuum polarization cannot distinguish between the two classes of fermions when the corresponding true vacuum is taking into account.
Another interesting outcome of this investigation is the anisotropy in the photon propagation due to the nontrivial topology of the spacetime. In the S'xj?3 domain, the photons have several massive modes and a transverse superluminal one. The masses of the photon modes increase as the inverse of the compactification radius, while the superluminal velocity of the massless
mode increases logarithmically .with that topological distance. The existence of massive modes implies that at very small radius of compactification, the photon propagation at low energies is effectively confined to a Min.kowsk.ian (2+l)-dimensionai manifold, on which only superluminal photons propagate. Therefore, photons moving in such a lower dimensional space experience the lack of Lorentz symmetry of the general manifold (S'xj?3) on which the lower-dimensional space is embedded, allowing them to have a group velocity larger than the usual Minkowskian velocity c .
We also considered how the non-trivial topology affects the condensation of fermion-antifermion pairs. This was done in the framework of QED with an additional four-fermion interaction (gauged-NJL theory). In this model we found that the smaller the compactification radius, the larger the critical four-fermion coupling needed to generate a fermion-antifermion chiral symmetry breaking condensate. Contrary to what occurs in a pure NIL model [20], this result is obtained for both twisted and untwisted fermions, once the corresponding stable vacuum is considered. Thus, we conclude that in the gauged-NJL theory the tendency of the compactification is to help to reinstate the chiral symmetry.
The results we are reporting here can be of interest for condensed matter quasi-planar systems, as well as for theories with extra dimensions.
Acknowledgments
This research was supported by the National Science Foundation under Grant No. PHY-0070986.
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Neves I',1
ON THE STAB! RANDALL-SUNDRUM BRANEWGRLDS WITH CONFORMAL BULK FIELDS
Departamento de Física, Faeuiclacle cíe Ciencias e Tecnología, Universidade do Algarve Campus de Gambeias, 8000-
117 Faro, Portugal
11ntroduction
In the Randall-Sundrum (RS) scenario [1,2] the observable Universe is a 3-brane world of a Z, symmetric 5-dimensional anti-de Sitter (AdS) space. In the RSI model [1] the AdS orbifold has a compactified fifth dimension and two brane boundaries. The gravitational field is bound to the hidden positive tension brane and decays towards the visible negative tension brane. In this model the hierarchy problem is reformulated as an exponential hierarchy between the weak and Planck scales [1]. In the RS2 model [2] the AdS orbifold is non-compact with an infinite fifth dimension and a single positive tension brane. Gravity is localized on the positive tension brane now interpreted as the visible brane.
At low energies the theory of gravity on the observable brane is 4-dimensional general relativity and the cosmology may be Friedmann-Robertson-Walker [1]-[10]. In the RSI model this is only possible if the radion mode is stabilized and this as been achieved using a scalar field in the bulk [3,6,9,10]. The
gravitational collapse of matter was also analyzed in the RS scenario [11]-[16). Using an extended black string solution (first discussed in a different context by Myers and Perry [1?]) Chamblin, Hawking and Reall showed that it was possible to induce on the brane the Schwarzschild black hole metric [11]. However, this solution is divergent at the AdS horizon and at the black string singularity. Consequently, it is expected to be unstable [11,18]. A black cylinder localized near the brane which is free from naked singularities was conjectured to be its stable decay product. This exact 5-dimensional solution has not yet been found. The only known static black holes localized on a brane remain to be those found for a 2-brane in a 4-dimensional AdS space [12]. The problem lies in the simultaneous nonsingular localization of gravity and matter in the vicinity of the brane [11], [13]-[16], This has lead to another conjecture stating that D + l -dimensional black hole solutions localized on a D-1 -brane should correspond to quantum corrected D -dimensional black holes on the brane [15]. This is an extra motivation to look for 5-dimensional collapse solutions localized on
