Compactification Length Research Articles

Lightcone fluctuations in a five dimensional Kaluza-Klein model and an estimation on the size of the extra dimension

In this work we consider lightcone fluctuation effects in a five-dimensional spacetime arising as a consequence of the compactification of the extra dimension via a quasiperiodic condition, characterized by a phase angle 2πα. By considering light propagating in a non-compactified direction we are able to compute both the renormalized graviton two-point function and the flight time of a photon caused by the fluctuations. We show that the resulting expressions depend on the quasiperiodic parameter, the compactification length and also on the distance traveled by the photon. Based on the Near-Infrared Spectrograph (NIRSpec) sensitivity built on the James Webb Space Telescope, we discuss the possibility of making estimations on the size of the extra dimension if one assumes that the mean deviation on the flight time of photons can be observed through the NIRSpec. We also analyze the differences between the estimations in the periodic α = 0, antiperiodic α = 1/2 and other condition cases for α.

Feynman amplitudes in periodically compactified spaces: Spin 0

We propose an extension of the Schwinger parametric representation for Feynman amplitudes in $D$ euclidean dimensions to a scenario where $d$ dimensions are compactified ($d<D$) through the introduction of periodic boundary conditions in space. We obtain two valid representations, one useful near the bulk (large compactification length) and another useful near the dimensional reduction (small compactification length). Also, to illustrate, we exhibit some Feynman amplitudes up to three loops in a compactified scalar field theory.

Inquiring the Unruh–DeWitt detector about the global property of spacetime

We investigate the effect of spatial compactification on the transition rate of an inertial or accelerated Unruh–Dewitt detector coupled to an untwisted or twisted massless quantum scalar field. Four typical cases of the detector’s motion in the compactified Minkowski spacetime are considered respectively. Our results indicate that the detector’s transition rates are crucially dependent on the spatial compactification length, the magnitude and direction of detector’s velocity, the detector’s acceleration, and the field structure. As these factors change, the detector’s spontaneous emission and absorption processes can be enhanced or weakened at different degrees. In particular, when the compact length is small, the behavior of transition rate in the case of the twisted field is quite distinct from that of the untwisted field. Notably, when the detector moves at a constant velocity with nonzero components along the compact and non-compact directions, our study finds that the spontaneous emission rate depends on the component of velocity along the non-compact direction besides that along the compact direction. This is in sharp contrast with the case of the free Minkowski spacetime. Moreover, for the uniformly accelerated detector along the non-compact direction, the spatial compactification clearly modifies both the spontaneous emission and absorption rates, as the modifying factors depend on the compact length and the detector’s acceleration. Our work indirectly but comprehensively examines the nonequivalence of inertial frames in the compactified Minkowski spacetime, and meanwhile provides a theoretical way to identify the orientation and the size of the compact spatial dimension.

Dimensional crossover in non-relativistic effective field theory

Isotropic scattering in various spatial dimensions is considered for arbitrary finite-range potentials using non-relativistic effective field theory. With periodic boundary conditions, compactifications from a box to a plane and to a wire, and from a plane to a wire, are considered by matching S-matrix elements. The problem is greatly simplified by regulating the ultraviolet divergences using dimensional regularization with minimal subtraction. General relations among (all) effective-range parameters in the various dimensions are derived, and the dependence of bound states on changing dimensionality are considered. Generally, it is found that compactification binds the two-body system, even if the uncompactified system is unbound. For instance, compactification from a box to a plane gives rise to a bound state with binding momentum given by in units of the inverse compactification length. This binding momentum is universal in the sense that it does not depend on the two-body interaction in the box. When the two-body system in the box is at unitarity, the S-matrices of the compactified two-body system on the plane and on the wire are given exactly as universal functions of the compactification length.

Proof That Spatial Transitions Release/Absorb Energy That Compactification Necessarily Leads to Changes in Volume, Energy and Entropy

Using simple box quantization, we demonstrate explicitly that a spatial transition will release or absorb energy, and that compactification releases latent heat with an attendant change in volume and entropy. Increasing spatial dimension for a given number of particles costs energy while decreasing dimensions supplies energy, which can be quantified, using a generalized version of the Clausius-Clapyeron relation. We show this explicitly for massive particles trapped in a box. Compactification from N -dimensional space to (N - 1) spatial dimensions is also simply demonstrated and the correct limit to achieve a lower energy result is to take the limit, Lw → 0, where Lw is the compactification length parameter. Higher dimensional space has more energy and more entropy, all other things being equal, for a given cutoff in energy.

Vector Boson Mass Spectrum from Extra Dimension

Based on the mechanism for mass creation in the space-time with extradimensions proposed in our previous work, (arXiv: 1301.1405 [hep-th] 2013) we consider now the mass spectrum of vector bosons in extradimensions. It is shown that this spectrum is completely determined by some function of compactification length and closely related to the metric of extradimensions.

Phase transition in the massive Gross-Neveu model in toroidal topologies

We use methods of quantum field theory in toroidal topologies to study the $N$-component $D$-dimensional massive Gross-Neveu model, at zero and finite temperature, with compactified spatial coordinates. We discuss the behavior of the large-$N$ coupling constant ($g$), investigating its dependence on the compactification length ($L$) and the temperature ($T$). For all values of the fixed coupling constant ($\ensuremath{\lambda}$), we find an asymptotic-freedom type of behavior, with $g\ensuremath{\rightarrow}0$ as $L\ensuremath{\rightarrow}0$ and/or $T\ensuremath{\rightarrow}\ensuremath{\infty}$. At $T=0$, and for $\ensuremath{\lambda}\ensuremath{\ge}{\ensuremath{\lambda}}_{c}^{(D)}$ (the strong-coupling regime), we show that, starting in the region of asymptotic freedom and increasing $L$, a divergence of $g$ appears at a finite value of $L$, signaling the existence of a phase transition with the system getting spatially confined. Such a spatial confinement is destroyed by raising the temperature. The confining length, ${L}_{c}^{(D)}$, and the deconfining temperature, ${T}_{d}^{(D)}$, are determined as functions of $\ensuremath{\lambda}$ and the mass ($m$) of the fermions, in the case of $D=2,3,4$. Taking $m$ as the constituent quark mass ($\ensuremath{\approx}350\text{ }\text{ }\mathrm{MeV}$), the results obtained are of the same order of magnitude as the diameter ($\ensuremath{\approx}1.7\text{ }\text{ }\mathrm{fm}$) and the estimated deconfining temperature ($\ensuremath{\approx}200\text{ }\text{ }\mathrm{MeV}$) of hadrons.

Circles-in-the-sky searches and observable cosmic topology in a flat universe

[Abridged] In a Universe with a detectable nontrivial spatial topology the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations - the so-called circles-in-the-sky. Searches for nearly antipodal circles in maps of cosmic microwave background have so far been unsuccessful. This negative outcome along with recent theoretical results concerning the detectability of nearly flat compact topologies is sufficient to exclude a detectable nontrivial topology for most observers in very nearly flat positively and negatively curved Universes ($0<|\Omega_{tot}-1| \lesssim 10^{-5}$). Here we investigate the consequences of these searches for observable nontrivial topologies if the Universe turns out to be exactly flat ($\Omega_{tot}=1$). We demonstrate that in this case the conclusions deduced from such searches can be radically different. We show that for all multiply-connected orientable flat manifolds it is possible to directly study the action of the holonomies in order to obtain a general upper bound on the angle that characterizes the deviation from antipodicity of pairs of matching circles associated with the shortest closed geodesic. This bound is valid for all observers and all possible values of the compactification length parameters. We also show that in a flat Universe there are observers for whom the circles-in-the-sky searches already undertaken are insufficient to exclude the possibility of a detectable nontrivial spatial topology. It is remarkable how such small variations in the spatial curvature of the Universe, which are effectively indistinguishable geometrically, can have such a drastic effect on the detectability of cosmic topology.

Where in the string landscape is quintessence?

We argue that quintessence may reside in certain corners of the string landscape. It arises as a linear combination of internal space components of higher-rank forms, which are axionlike at low energies, and may mix with 4-forms after compactification of the Chern-Simons terms to 4D due to internal space fluxes. The mixing induces an effective mass term, with an action which preserves the axion shift symmetry, breaking it spontaneously after the background selection. With several axions, several 4-forms, and a low string scale, as in one of the setups already invoked for dynamically explaining a tiny residual vacuum energy in string theory, the 4D mass matrix generated by random fluxes may have ultralight eigenmodes over the landscape, which are quintessence. We illustrate how this works in the simplest cases, and outline how to get the lightest mass to be comparable to the Hubble scale now, ${H}_{0}\ensuremath{\sim}{10}^{\ensuremath{-}33}\text{ }\text{ }\mathrm{eV}$. The shift symmetry protects the smallest mass from perturbative corrections in field theory. Further, if the ultralight eigenmode does not couple directly to any sector strongly coupled at a high scale, the nonperturbative field theory corrections to its potential will also be suppressed. Finally, if the compactification length is larger than the string length by more than an order of magnitude, the gravitational corrections may remain small too, even when the field value approaches ${M}_{\mathrm{Pl}}$.

Thermal behavior of the compactified 3-D gross-neveu model

We consider the N-component tri-dimensional massive Gross-Neveu model at finite temperature and with compactified spatial coordinates. We study the behavior of the renormalized large-N effective coupling constant, investigating its dependence on the compactification length and the temperature. We show that spatial confinement exists for the model at T = 0, which is destroyed by raising the temperature.

Lightcone fluctuations in quantum gravity and extra dimensions

We discuss how compactified extra dimensions may have potentially observable effects which grow as the compactification scale decreases. This arises because of lightcone fluctuations in the uncompactified dimensions which can result in the broadening of the spectral lines from distant sources. We analyze this effect in a five dimensional model, and argue that data from gamma ray burst sources require the compactification length to be greater than about 10 5 cm in this model.

Bulk U(1) messenger

We propose a new U(1) gauge interaction in the bulk in higher dimensional spacetime, which transmits supersymmetry-breaking effects on the hidden brane to the observable our brane. We find that rather small gauge coupling constant of U(1) bulk, α bulk≃5×10 −4, is required for a successful phenomenology. This result implies the compactification length L of the extra dimension to be L −1≃2×10 15 GeV for (4+1)-dimensional spacetime. This large compactification length L is a crucial ingredient to suppress unwanted flavor-changing neutral currents and hence our proposal is very consistent with the Randall–Sundrum brane-world scenario.

Geometric bounds on Kaluza–Klein masses

We point out geometric upper and lower bounds on the masses of bosonic and fermionic Kaluza-Klein excitations in the context of theories with large extra dimensions. The characteristic compactification length scale is set by the diameter of the internal manifold. Based on geometrical and topological considerations, we find that certain choices of compactification manifolds are more favoured for phenomenological purposes.

Renormalization group and spontaneous compactification of a higher-dimensional scalar field theory in curved spacetime.

The renormalization group (RG) is used to study the asymptotically free $\phi_6^3$-theory in curved spacetime. Several forms of the RG equations for the effective potential are formulated. By solving these equations we obtain the one-loop effective potential as well as its explicit forms in the case of strong gravitational fields and strong scalar fields. Using zeta function techniques, the one-loop and corresponding RG improved vacuum energies are found for the Kaluza-Klein backgrounds $R^4\times S^1\times S^1$ and $R^4\times S^2$. They are given in terms of exponentially convergent series, appropriate for numerical calculations. A study of these vacuum energies as a function of compactification lengths and other couplings shows that spontaneous compactification can be qualitatively different when the RG improved energy is used.

ON THE CREATION OF DENSITY FLUCTUATIONS FROM THE HETEROTIC COSMIC SUPERSTRING

It has been proved by Witten that the heterotic superstring theory appears phenomeno-logically as a closed, superconducting, axionic cosmic string attached to one axion domain wall. After reduction from ten dimensions to four dimensions, the effective mass per unit length of the string is [Formula: see text], where α′ = 1/2πμ is the Regge slope parameter and bc is the compactification length. The string-wall system is present ab initio in the expanding Friedmann universe, and is dominated energetically by the string. Consequently, a scale-free spectrum of density (metric) fluctuations is impressed upon the space-time at its classical inception, whose magnitude, according to Zel'dovich, is ξ ≡ δρ/ρ ≈ 4πGμc ≈ 4 × 10−2g2, where [Formula: see text] is the Newton gravitational constant, M P is the Planck mass and g is the string gauge coupling parameter. The string disappears via gravitational or electromagnetic radiation on the time scale τ ≲ 2M P /μc, thus removing all conflict with the non-observation of timing residuals in the millisecond pulsars. The value of ξ should be commensurate with that previously obtained from the effective action, illustrating the notion of reciprocity between the two-dimensional, world-sheet and four-dimensional, space-time formulations of the theory. It therefore suggests that bc ≈ b H ≡ 1/2πT H , where [Formula: see text] is the Hagedorn temperature, since the two-loop calculation of the free energy by Hellmund and Kripfganz indicates that g2 → 0 as T → T H .

Topological symmetry breaking in self-interacting theories on toroidal space–time

The possibility of topological mass generation through symmetry breaking in a simple model consisting of a self-interacting (massive or massless) λφ4 scalar field on the space–time TN×Rn, n, N∈N0−TN being a general torus—is investigated. The nonrenormalized effective potential is calculated and the specific dependences of the generated mass on the compactification lengths and on the initial mass of the field are determined. Later, in order to obtain the renormalized topologically generated mass, the analysis is restricted to n+N=4 dimensions. It is shown that if the field is massive no symmetry breaking can occur. On the contrary, when it is massless, for n=1 and n=0 and for values of the vector of compactification lengths belonging to some specific domain of RN, symmetry breaking does actually take place. Explicit values of the mass topologically generated in this way are obtained.

Gauge field mass generation by toroidal spacetime

We consider an Abelian gauge field theory on the partially compactified spacetime , where is the N-dimensional torus and is a n-dimensional Riemannian manifold. The mass of the gauge field generated by quantum fluctuations of a massive or massless scalar field, minimally coupled to a constant background gauge potential, is the quantity of our main interest. As long as the eigenvalue spectrum of is positive (where is the Laplace--Beltrami operator of the manifold , R the Riemann scalar curvature and m the mass of the quantum field), it is found that the topologically generated mass is real for arbitrary N. If, however, the spectrum has zero eigenvalues we found that, depending on the compactification lengths of the torus, the generated mass may also be imaginary.

Topological gauge field mass generation by toroidal spacetime

The author considers an Abelian gauge field theory on Euclidean partially compactified spacetime TN*Rn for arbitrary N and n. The one-loop effective potential generated by quantum fluctuations of a massive scalar field, minimally coupled to a constant background Abelian gauge potential is calculated for arbitrary compactification lengths L1,. . .,LN of the multi-dimensional torus. In particular the topologically generated mass of the gauge field is obtained and its complicated dependence on the parameters involved (compactification lengths, mass M of the scalar field) is given explicitly. It is found that the topologically generated mass is positive for arbitrary N and n and that it does not depend on a renormalization parameter. For n>2 the limit M to 0 is smooth, but for n=0,1,2 zero modes play a crucial role and the generation of real or imaginary gauge field masses is possible.

Duality in nontrivially compactified heterotic strings.

We study the implications of duality symmetry on the analyticity properties of the partition function as it depends upon the compactification length. In order to obtain nontrivial compactifications, we give a physical prescription to get the Helmholtz free energy for any heterotic string, supersymmetric or not. After proving that the free energy is always invariant under the duality transformation R\ensuremath{\rightarrow}\ensuremath{\alpha}'/(2R) and getting the zero-temperature theory whose partition function corresponds to the Helmholtz potential, we show that the self-dual point ${\mathit{R}}_{0}$= \ensuremath{\surd}\ensuremath{\alpha}'/2 is a generic singularity like the Hagedorn one. The main difference between these two critical compactification radii is that the term producing the singularity at the self-dual point is finite for any R\ensuremath{\ne}${\mathit{R}}_{0}$. We see that this behavior at ${\mathit{R}}_{0}$ actually implies a loss of degrees of freedom below that point.

Upper bounds to 1-loop symmetry restoration for the Goldstone model in R q+1 × T p in the low-mass and weak-coupling limit

We obtain critical values for the compactification lengths relevant for the restoration of spontaneously broken symmetries in the Goldstone model. We find them by varying both the space dimension and the number of compactified directions. More precisely, our approach consists in applying Epstein zeta-functions to evaluate the first corrections to the potential, and is used for untwisted and also for twisted scalar fields.