2079 lines
63 KiB
TeX
2079 lines
63 KiB
TeX
\subsection{Motivation}
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Unfortunately and puzzlingly the first attempts to consider space-like~\cite{Craps:2002:StringPropagationPresence} or light-like singularities~\cite{Liu:2002:StringsTimeDependentOrbifold,Liu:2002:StringsTimeDependentOrbifolds} by means of orbifold techniques yielded divergent four points \emph{closed string} amplitudes (see \cite{Cornalba:2004:TimedependentOrbifoldsString,Craps:2006:BigBangModels} for reviews).
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These singularities are commonly assumed to be connected to a large backreaction of the incoming matter into the singularity due to the exchange of a single graviton~\cite{Berkooz:2003:CommentsCosmologicalSingularities,Horowitz:2002:InstabilitySpacelikeNull}.
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This claim was already questioned in the literature where the $O$-plane orbifold was constructed.
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This orbifold should in fact be stable against the gravitational collapse but it exhibits divergences in the amplitudes (see the discussion in \cite{Cornalba:2004:TimedependentOrbifoldsString}).
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In what follows we show a direct computation showing that the presence of the divergence is not related to a gravitational response.
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What has gone unnoticed is that in the Null Boost Orbifold (\nbo) \cite{Liu:2002:StringsTimeDependentOrbifold} even the four \emph{open string} tachyons amplitude is divergent.
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Since we are working at tree level gravity is not an issue.
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In fact in Equation (6.16) of \cite{Liu:2002:StringsTimeDependentOrbifold} the four tachyons amplitude in the divergent region reads
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\begin{equation}
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A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \ccA( q )
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\end{equation}
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where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vec{p}_{\perp}}^2}$ and $\ccA_{\text{closed}}( q ) \sim q^{1 - \ap \norm{\vec{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices).
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Moreover divergences in string amplitudes are not limited to four points: interestingly we show that the open string three point amplitude with two tachyons and the first massive state may be divergent when some \emph{physical} polarisations are chosen.
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The true problem is therefore not related to a gravitational issue but to the non existence of the effective field theory.
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In fact when we express the theory using the eigenmodes of the kinetic terms some coefficients do not exist, not even as a distribution.
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This holds true for both open and closed string sectors since it manifests also in the four scalar contact term.
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The issue can be roughly traced back to the vanishing volume of a subspace and the existence of a discrete zero mode of the Laplacian on this subspace.
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As an introduction to the problem we first deal with singularities of the open string sector.
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We try to build a consistent scalar \qed and show that the vertex with four scalar fields is ill defined.
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Divergences in scalar QED are due to the behaviour of the eigenfunctions of the scalar d'Alembertian near the singularity but in a somehow unexpected way.
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Near the singularity $u = 0$ in lightcone coordinates almost all eigenfunctions behave as $\frac{1}{\sqrt{\abs{u}}} e^{i \frac{\cA}{u}}$ with $\cA \neq 0$.
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The product of $N$ eigenfunctions gives a singularity $\abs{u}^{-N/2}$ which is technically not integrable.
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However the exponential term $e^{i \frac{\cA}{u}}$ allows for an interpretation as distribution when $\cA = 0$ is not an isolated point.
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When $\cA = 0$ is isolated the singularity is definitely not integrable and there is no obvious interpretation as a distribution.
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Specifically in the \nbo we find $\cA \sim \frac{l^2}{k_+}$ where $l$ is the momentum along the compact direction.
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As a consequence we find the eigenfunction associated to the discrete momentum $l = 0$ along the orbifold compact direction with an isolated $\cA = 0$.
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It is the eigenfunction which is constant along that direction and it is the root of all divergences.
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We then check whether the most obvious ways of regularizing the theory by making $\cA$ not vanishing may work.
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The first regularisation we try is to use a Wilson line along the compact direction even though the diverging three point string amplitude involves an anti-commutator of the Chan-Paton factor therefore it is divergent also for a neutral string, i.e.\ for a string with both ends attached to the same D-brane.
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This kind of string does not feel Wilson lines.
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Moreover anti-commutators are present in amplitudes with massive states in unoriented and supersymmetric strings and therefore neither worldsheet parity nor supersymmetry can help.
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The second obvious regularisation is the introduction of higher derivatives couplings to the Ricci tensor which is the only non vanishing tensor associated to the (regularised) metric.
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In any case it seems that a sensible regularisation must couple to all open string in the same way and this suggests a gravitational coupling.
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We then give a cursory look to whether closed string winding modes could help~\cite{Berkooz:2003:StringsElectricField}, as already suggested in~\cite{Liu:2002:StringsTimeDependentOrbifolds,Craps:2002:StringPropagationPresence} in analogy to the resolution of static singularities.
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Twisted closed strings become massless near the singularity and they should in some way be included.
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They generate a background potential $B_{\mu\nu}$ which is equivalent to a electromagnetic background from the open string perspective.
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Under a plausible modification of the scalar action which is suggested by the two-tachyons---two-photons amplitude the problems seem to be solvable.
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In any case the origin of the string divergence seems to originate from the lack of contact terms in the effective field theory.
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Since these terms arise from string theory also through the exchange of massive string states we examine three point amplitudes with one massive state.
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A deeper understanding of the subject requires the study of the polarisations of the massive state on the orbifold as seen from the covering Minkowski space before the computation of the overlap of the wave functions.
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We then go back to string theory and we verify that in the \nbo the open string three points amplitude with two tachyons and one first level massive string state does indeed diverge when some physical polarisation are chosen.
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We then introduce the generalised Null Boost Orbifold (\gnbo) as a generalization of the \nbo which still has a light-like singularity and is generated by one Killing vector.
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However in this model there are two directions associated with $\cA$, one compact and one non compact.
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We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation~\cite{Estrada:2012:GeneralIntegral}.
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However if a second Killing vector is used to compactify the formerly non compact direction, the theory has again the same problems as in the \nbo.
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In the literature there are however also other attempts at regularizing the \nbo such as the Null Brane.
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This kind of orbifold was originally defined in \cite{Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes,Cornalba:2004:TimedependentOrbifoldsString} and studied in perturbation theory in \cite{Liu:2002:StringsTimeDependentOrbifolds}.
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The Null Brane shares with the \gnbo the existence of a non compact direction on the orbifold.
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In this case it is indeed possible to build single particle wave functions which leads to the convergence of the smeared amplitudes.
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We finally present also a brief examination of the Boost Orbifold (\bo) where the divergences are generally milder~\cite{Horowitz:1991:SingularStringSolutions,Khoury:2002:BigCrunchBig}.
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The scalar eigenfunctions behave in time $t$ as $\abs{t}^{\pm i\, \frac{l}{\Delta}}$ near the singularity but there is one eigenfunction which behaves as $\log(\abs{t})$ and again it is the constant eigenfunction along the compact direction which is the origin of all divergences.
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In particular the scalar \qed on the \bo can be defined and the first term which gives a divergent contribution is of the form $\abs{\phi~\dphi}^2$, i.e.\ divergences are hidden into the derivative expansion of the effective field theory.
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Again three points open string amplitudes with one massive state diverge.
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\subsection{Scalar QED on NBO and Divergences}
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\label{sect:NOscalarQED}
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As discussed the four open string tachyons amplitude diverges in the \nbo.
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The literature on the subject (see for instance~\cite{Cornalba:2004:TimedependentOrbifoldsString} and references therein) suggests that this can be cured by the eikonal resummation.
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We therefore consider the scalar \qed on the \nbo as a first approach.
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In this case all eigenmodes can be written using elementary functions thus making the issues even more evident.
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Its action is given by
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\begin{equation}
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\rS_{\text{s}\qed}
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=
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\int\limits_{\Omega} \dd[D]{x}\,
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\sqrt{- \det g}
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\qty(
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- \qty(D^{\mu} \phi)^*\, D_{\mu} \phi
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- M^2 \qty(\phi^*)\, \phi
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- \frac{1}{4} f^{\mu\nu}\, f_{\mu\nu}
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- \frac{g_4}{4} \abs{\phi}^4
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),
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\end{equation}
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with
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\begin{equation}
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D_{\mu} \phi
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=
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\qty(\ipd{\mu} -i\, e\, a_{\mu}) \phi,
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\qquad
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f_{\mu\nu}
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=
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\ipd{\mu} a_{\nu} - \ipd{\nu} a_{\mu}.
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\end{equation}
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We reserve small letters for quantities defined on the orbifold and capital letters for those defined in flat space.
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Moreover $\Omega$ denotes the orbifold.
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We will construct directly both the scalar and the spin-1 eigenfunctions which we can use as a starting point for the perturbative computations.
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\subsubsection{Geometric Preliminaries}
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\label{sec:geometric_preliminaries_nbo}
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In Minkowski spacetime $\ccM^{1,D-1}$ with coordinates $\qty(x^{\mu}) = \qty(x^+,\, x^-,\, x^2,\, \vec{x})$
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and metric
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\begin{equation}
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\dss[2]{s}
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=
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- 2 \dd{x^+} \dd{x^-}
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+ \qty(\dd{x^2})^2
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+ \eta_{ij} \dd{x}^i \dd{x}^j,
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\end{equation}
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we consider the following change of coordinates to $\qty(x^{\alpha}) = (u,\, v,\, z,\, \vec{x})$
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\begin{equation}
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\begin{cases}
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x^- & = u
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\\
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x^2 & = \Delta u z
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\\
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x^+ & = v + \frac{1}{2} \Delta^2 u z^2
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\end{cases}
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\qquad
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\Leftrightarrow
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\qquad
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\begin{cases}
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u & = x^-
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\\
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z & = \frac{x^2}{\Delta\, x^-}
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\\
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v & = x^+ - \frac{1}{2} \frac{(x^2)^2}{x^-}
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\end{cases}.
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\label{eq:NBO_coordinates}
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\end{equation}
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Then the metric becomes:
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\begin{equation}
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\dss[2]{s}
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=
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- 2\, \dd{u}\, \dd{v}
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+ \qty(\Delta u )^2 (\dd{z})^2
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+ \eta_{ij} \dd{x}^i \dd{x}^j,
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\end{equation}
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along with the non vanishing geometrical quantities
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\begin{equation}
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-\det g = \qty( \Delta u )^2,
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\end{equation}
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and
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\begin{equation}
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\tensor{\Gamma}{_z^v_z} = \Delta^2 u,
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\qquad
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\tensor{\Gamma}{_u^z_z} = u^{-1}.
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\end{equation}
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Riemann and Ricci tensor components however vanish since at this stage we only performed a change of coordinates from the original Minkowski spacetime.
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Locally it is the same as the \nbo and they must have the same local differential geometry.
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The \nbo is introduced by identifying points along the orbits of the Killing vector:
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\begin{equation}
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\begin{split}
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\kappa
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& =
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- i \qty(2 \pi \Delta) J_{+2}
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\\
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& =
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\qty(2 \pi \Delta)\, (x^2 \ipd{+} + x^- \ipd{2})
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\\
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& =
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2 \pi \ipd{z},
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\end{split}
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\label{eq:nbo_killing_vector}
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\end{equation}
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in such a way that
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\begin{equation}
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x^{\mu} \equiv \cK^{n}\, x^{\mu},
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\qquad
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n \in \Z,
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\end{equation}
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where $\cK^{n}= e^{n\kappa}$, leads to the identifications
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\begin{equation}
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x=
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\mqty( x^- \\ x^2 \\ x^+ \\ \vec{x} )
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\equiv
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\cK^{n} x
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=
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\mqty(%
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x^- \\
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x^2 + n \qty(2 \pi \Delta) x^- \\
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x^+ + n \qty(2 \pi \Delta) x^2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 x^- \\
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\vec{x}
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)
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\end{equation}
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or to
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\begin{equation}
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\qty( u,\, v,\, z,\, \vec{x} )
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\equiv
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\qty( u,\, v,\, z + 2 \pi n,\, \vec{x} )
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\end{equation}
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in coordinates $\qty(x^{\alpha})$ where $\kappa = 2 \pi \ipd{z}$ is a global Killing vector.
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As a reference for the future, we notice that we could regularise the metric as
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\begin{equation}
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\dss[2]{s}
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=
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- 2\, \dd{u}\, \dd{v}
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+ \Delta^2 \qty(u^2 + \epsilon^2) (\dd{z})^2
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+ \eta_{ij} \dd{x}^i \dd{x}^j.
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\end{equation}
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The non vanishing geometrical quantities are then:
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\begin{equation}
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-\det g = \Delta^2 \qty(u^2 + \epsilon^2),
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\end{equation}
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and
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\begin{equation}
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\tensor{\Gamma}{_z^v_z} = \Delta^2 u,
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\qquad
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\tensor{\Gamma}{_u^z_z} = \frac{u}{u^2 + \epsilon^2},
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\end{equation}
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which lead to the following Riemann and Ricci tensor components:
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\begin{equation}
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\tensor{R}{^z_u_z_u} = - \frac{\epsilon^2}{\qty(u^2+ \epsilon^2)^2},
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\quad
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\tensor{R}{^v_z_z_u} = - \frac{\Delta^2 \epsilon^2}{u^2 + \epsilon^2},
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\quad
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\tensor{Ric}{_u_u} = -\frac{\epsilon^2}{\qty(u^2+ \epsilon^2)^2}.
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\end{equation}
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Since $\delta_{\text{reg}}(u) = \frac{1}{\pi} \frac{\epsilon}{u^2+ \epsilon^2}$ then $\tensor{R}{^z_u_z_u} = - \pi^2 \qty[ \delta_{\text{reg}}(u) ]^2$.
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\subsubsection{Free Scalar Action}
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We study the eigenmodes of the Laplacian operator to diagonalize the scalar kinetic term given by:\footnotemark{}
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\footnotetext{%
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The factor $-g^{\alpha\beta}$ is due to the choice of the East coast convention for the metric, namely:
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\begin{equation*}
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- g^{\alpha\beta}
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\ipd{\alpha} \phi^*\, \ipd{\beta} \phi
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-
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M^2 \phi^*\, \phi
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\sim
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\abs{\dot{\phi}}^2 - M^2 \abs{\phi}^2
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\sim
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\rE^2 - M^2.
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\end{equation*}
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}
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\begin{equation}
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\begin{split}
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\rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \phi ]
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& =
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\int\limits_{\Omega} \dd[D]{x}\,
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\sqrt{- \det g}~
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\qty(%
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- g^{\alpha\beta} \ipd{\alpha} \phi^*\, \ipd{\beta} \phi
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- M^2 \phi^*\, \phi
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)
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\\
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& =
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\int \dd[D-3]{\vec{x}}\,
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\int \dd{u}\,
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\int \dd{v}\,
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\finiteint{z}{0}{2\pi}
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\abs{\Delta u}
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\\
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& \times
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\qty(%
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\ipd{u} \phi^*\, \ipd{v} \phi\,
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+
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\ipd{v} \phi^*\, \ipd{u} \phi\,
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-
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\frac{1}{\qty(\Delta u)^2} \ipd{z} \phi^*\, \ipd{z} \phi\,
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-
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\ipd{i} \phi^*\, \ipd{i} \phi
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-
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M^2 \phi^*\, \phi
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).
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\end{split}
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\end{equation}
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The solution to the equation of motion is enough when we want to perform the canonical quantization.
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Since we use Feynman diagrams we consider the path integral approach: we take off-shell modes and solve the eigenvalue problem $\square \phi_r = r \phi_r$.
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Comparing with the flat case we see that $r$ is $2\, k_-\, k_+ - \norm{\vec{k}}^2$ when $k$ is the impulse in flat coordinates.
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We therefore have
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\begin{equation}
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-2 \ipd{u} \ipd{v} \phi_r
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-
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\frac{1}{u} \ipd{v} \phi_r
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+
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\frac{1}{\qty(\Delta u)^2} \ipd{z}^2 \phi_r
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+
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\ipd{i}^2 \phi_r
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=
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r \phi_r.
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\label{eq:nbo_eom}
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\end{equation}
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Using Fourier transforms it follows that the eigenmodes are
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\begin{equation}
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\phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x})
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=
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e^{i k_+ v + i l z + i \vec{k} \cdot \vec{x}}\,
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\tphi_{\kmkr}(u),
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\end{equation}
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with
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\begin{equation}
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\tphi_{\kmkr}(u)
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=
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\frac{1}{\sqrt{\qty( 2 \pi )^D~ \abs{2 \Delta k_+\, u}}}
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e^{
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- i \frac{l^2}{2 \Delta^2 k_+} \frac{1}{u}
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+ i \frac{\norm{\vec{k}}^2 + r}{2 k_+} u
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},
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\end{equation}
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and
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\begin{equation}
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\phi^*_{\kmkr}\qty(u,\, v,\, z,\, \vec{x})
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=
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\phi_{\mkmkr}\qty(u,\, v,\, z,\, \vec{x}).
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\end{equation}
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We chose the numeric factor in order to get a canonical normalisation:
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\begin{equation}
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\begin{split}
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&
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\qty( \phi_{\kmkrN{1}},\, \phi_{\kmkrN{2}} )
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\\
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= &
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\int \dd[D-1]{\vec{x}}\,
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\int \dd{u}\, \int \dd{v}\,
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\finiteint{z}{0}{2\pi}
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\abs{\Delta u}\,
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\phi_{\kmkrN{1}}\, \phi_{\kmkrN{2}}
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\\
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= &
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\delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\,
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\delta( r_{(1)} - r_{(2)})\,
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\delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\,
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\delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}.
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\end{split}
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\end{equation}
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We can then perform the off-shell expansion
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\begin{equation}
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\phi\qty(u,\, v,\, z,\, \vec{x})
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=
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\int \dd[D-3]{\vec{k}}
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\int \dd{k_+}
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\int \dd{r}
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\infinfsum{l}
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\cA_{\kmkr}\,
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\phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}),
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\end{equation}
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such that the scalar kinetic term becomes
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\begin{equation}
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\rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cA ]
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=
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\int \dd[D-3]{\vec{k}}\,
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\int \dd{k_+}
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\int \dd{r}
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\infinfsum{l}
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\qty(r - M^2)\,
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\cA_{\kmkr}\,
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\cA_{\kmkr}^*.
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\end{equation}
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\subsubsection{Free Photon Action}
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The action of the free photon can be written as
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\begin{align}
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\rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ a ]
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=
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\int\limits_{\Omega} \dd[D]{x}\,
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\sqrt{-\det g}\,
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\qty(%
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- \frac{1}{2} g^{\alpha\beta} g^{\gamma\delta}
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D_{\alpha} a_{\gamma} \qty( D_{\beta} a_{\delta} - D_{\delta} a_{\beta})
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).
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\end{align}
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We choose to enforce the Lorenz gauge:\footnotemark{}
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\footnotetext{%
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Indeed it is exactly the usual Lorenz gauge since locally the space is Minkowski.
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}
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\begin{equation}
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D^{\alpha} a_{\alpha}
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=
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- \frac{1}{u} a_{v}
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- \ipd{u} a_{v}
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- \ipd{v} a_{u}
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+ \frac{1}{\Delta^2 u^2} \ipd{z} a_z
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+ \eta^{ij} \ipd{i} a_j
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=
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0.
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\label{eq:Lorenz_gauge}
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\end{equation}
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As covariant derivatives commute since we are locally flat, the \eom read $\qty(\square a)_{\alpha} = 0$.
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Explicitly we have:
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\begin{equation}
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\begin{split}
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\qty( \square a )_u
|
|
& =
|
|
\frac{1}{u^2} a_{v}
|
|
-
|
|
\frac{2}{\Delta^2 u^3} \ipd{z} a_z
|
|
+
|
|
\qty[
|
|
-
|
|
2 \ipd{u} \ipd{v}
|
|
-
|
|
\frac{1}{u} \ipd{v}
|
|
+
|
|
\frac{1}{\Delta^2 u^2} \ipd{z}^2
|
|
+
|
|
\eta^{ij} \ipd{i} \partial_j
|
|
]
|
|
a_u,
|
|
\\
|
|
\qty( \square a )_v
|
|
& =
|
|
\qty[
|
|
-
|
|
2 \ipd{u} \ipd{v}
|
|
-
|
|
\frac{1}{u} \ipd{v}
|
|
+
|
|
\frac{1}{\Delta^2 u^2} \ipd{z}^2
|
|
+
|
|
\eta^{ij} \ipd{i} \partial_j
|
|
]
|
|
a_v,
|
|
\\
|
|
\qty( \square a )_z
|
|
& =
|
|
-
|
|
\frac{2}{u} \ipd{z} a_v
|
|
+
|
|
\qty[
|
|
-
|
|
2 \ipd{u} \ipd{v}
|
|
+
|
|
\frac{1}{u} \ipd{v}
|
|
+
|
|
\frac{1}{\Delta^2 u^2} \ipd{z}^2
|
|
+
|
|
\eta^{ij} \ipd{i} \partial_j
|
|
]
|
|
a_z,
|
|
\\
|
|
\qty( \square a )_i
|
|
& =
|
|
\qty[
|
|
-
|
|
2 \ipd{u} \ipd{v}
|
|
-
|
|
\frac{1}{u} \ipd{v}
|
|
+
|
|
\frac{1}{\Delta^2 u^2} \ipd{z}^2
|
|
+
|
|
\eta^{ij} \ipd{i} \partial_j
|
|
]
|
|
a_i.
|
|
\end{split}
|
|
\end{equation}
|
|
|
|
As in the previous scalar case we are actually interested in solving
|
|
the eigenmodes problem $\qty(\square a)_\alpha= r \,a_\alpha$.
|
|
We proceed hierarchically: first we solve for $a_v$ and $a_i$ whose equations are the same as in the scalar field, then we insert the solutions as a source in the equation for $a_z$ and eventually we solve for $a_u$.\footnotemark{}
|
|
\footnotetext{%
|
|
Notice that inside the square brackets of the differential equation for $a_z$ there is a different sign for the term $\frac{1}{u} \ipd{v}$ with respect to the equation for the scalar field.
|
|
}
|
|
We get the solutions:
|
|
\begin{equation}
|
|
\begin{split}
|
|
\norm{\tildea_{\kmkr\, \alpha}(u)}
|
|
\,=
|
|
\mqty(%
|
|
\tildea_u
|
|
\\
|
|
\tildea_v
|
|
\\
|
|
\tildea_z
|
|
\\
|
|
\tildea_i
|
|
)
|
|
& =
|
|
\sum\limits_{%
|
|
\underline{\alpha}
|
|
\in
|
|
\qty{ \underline{u}, \underline{v}, \underline{z},\underline{i} }
|
|
}
|
|
\cE_{\kmkr\, \underline{\alpha}}
|
|
\norm{\tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)}
|
|
\\
|
|
& =
|
|
\cE_{\kmkr\, \underline{u}}
|
|
\mqty(
|
|
1
|
|
\\
|
|
0
|
|
\\
|
|
0
|
|
\\
|
|
0
|
|
)\,
|
|
\tphi_{\kmkr}(u)
|
|
\\
|
|
& +
|
|
\cE_{\kmkr\, \underline{v}}
|
|
\mqty(
|
|
\frac{i}{2 k_+ u}
|
|
+
|
|
\frac{1}{2} \qty( \frac{l}{\Delta k_+} )^2 \frac{1}{u^2}
|
|
\\
|
|
1
|
|
\\
|
|
\frac{l}{k_+}
|
|
\\
|
|
0
|
|
)\,
|
|
\tphi_{\kmkr}(u)
|
|
\\
|
|
& +
|
|
\cE_{\kmkr\, \underline{z}}
|
|
\mqty(
|
|
\frac{l}{\Delta k_+ \abs{u}}
|
|
\\
|
|
0
|
|
\\
|
|
\Delta \abs{u}
|
|
\\
|
|
0
|
|
)\,
|
|
\tphi_{\kmkr}(u)
|
|
\\
|
|
& +
|
|
\cE_{\kmkr\, \underline{j}}
|
|
\mqty(
|
|
0
|
|
\\
|
|
0
|
|
\\
|
|
0
|
|
\\
|
|
\delta_{\underline{ij}}
|
|
)\,
|
|
\tphi_{\kmkr}(u),
|
|
\label{eq:Orbifold_spin1_pol}
|
|
\end{split}
|
|
\end{equation}
|
|
then we can expand the off-shell fields as
|
|
\begin{equation}
|
|
a_{\alpha}\qty(u,\, v,\, z,\, \vec{x} )
|
|
=
|
|
\int \ccD k
|
|
\sum\limits_{%
|
|
\underline{\alpha}
|
|
\in
|
|
\qty{ \underline{u}, \underline{v}, \underline{z},\underline{i} }
|
|
}
|
|
\infinfsum{l}
|
|
\cE_{\kmkr\, \underline{\alpha}}\,
|
|
{a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x} ),
|
|
\end{equation}
|
|
where ${a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x}) = \tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)\, e^{i\, \qty( k_+ v + l z + \vec{k} \cdot \vec{x})}$ and $\int \ccD k = \int \dd[D-3]{\vec{k}} \int \dd{k_+} \int \dd{r}$.
|
|
|
|
We can also compute the normalisation as
|
|
\begin{equation}
|
|
\begin{split}
|
|
\qty(a_{(1)},\, a_{(2)})
|
|
& =
|
|
\int \dd[D-3]{\vec{x}}
|
|
\int \dd{u}
|
|
\int \dd{v}
|
|
\finiteint{z}{0}{2\pi}
|
|
\abs{\Delta u}
|
|
\\
|
|
& \times
|
|
g^{\alpha\beta}\,
|
|
a_{\kmkrN{1}\, \alpha}\, a_{\kmkrN{2}\, \beta}
|
|
\\
|
|
& =
|
|
\cE_{\kmkrN{1}} \circ \cE_{\kmkrN{2}}
|
|
\\
|
|
& \times
|
|
\delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\,
|
|
\delta( r_{(1)} - r_{(2)})\,
|
|
\delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\,
|
|
\delta_{l_{\qty(1)} + l_{\qty(2)},\, 0},
|
|
\end{split}
|
|
\end{equation}
|
|
where:\footnotemark{}
|
|
\footnotetext{%
|
|
We use a shortened version of the polarizations $\cE$ for the sake of readability.
|
|
We write $\cE_{(n)\, \underline{\alpha}} = \cE_{\kmkrN{n}\, \underline{\alpha}}$ thus hiding the understood dependence of the components of $\cE_{(n)}$ on the momenta.
|
|
}
|
|
\begin{equation}
|
|
\begin{split}
|
|
\cE_{(1)} \circ \cE_{(2)}
|
|
=
|
|
- \cE_{(1)\, \underline{u}}\, \cE_{(2)\, \underline{v}}
|
|
- \cE_{(1)\, \underline{v}}\, \cE_{(2)\, \underline{u}}
|
|
+ \cE_{(1)\, \underline{z}}\, \cE_{(2)\, \underline{z}}
|
|
+
|
|
\eta^{\underline{ij}}\,
|
|
\cE_{(1)\, \underline{i}}\, \cE_{(2)\, \underline{j}}.
|
|
\end{split}
|
|
\end{equation}
|
|
Finally the Lorenz gauge reads
|
|
\begin{equation}
|
|
\eta^{i \underline{j}}\, k_i\, \cE_{\kmkr\, \underline{j}}
|
|
-
|
|
k_+\, \cE_{\kmkr\, \underline{u}}
|
|
-
|
|
\frac{\vec{k}^2 + r}{2\, k_+} \cE_{\kmkr\, \underline{v}}
|
|
=
|
|
0,
|
|
\label{eq:explicit_orbifold_Lorenz}
|
|
\end{equation}
|
|
which does not impose any constraint on the transverse polarization
|
|
$\cE_{\kmkr\, \underline{z}}$.
|
|
The photon kinetic term becomes
|
|
\begin{equation}
|
|
\rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cE ]
|
|
=
|
|
\int \dd[D-3]{\vec{k}}
|
|
\int \dd{k_+}
|
|
\int \dd{r}
|
|
\infinfsum{l}\,
|
|
\frac{r}{2}\,
|
|
\cE_{\kmkr}\,
|
|
\circ
|
|
\cE_{\kmkr}^*.
|
|
\end{equation}
|
|
|
|
|
|
\subsubsection{Cubic Interaction}
|
|
|
|
With the definition of the d'Alembertian eigenmodes we can now examine the cubic vertex which reads
|
|
\begin{equation}
|
|
\rS_{\text{s}\qed}^{(\text{cubic})}\qty[\phi,\, a]
|
|
=
|
|
\int\limits_{\Omega} \dd[D]{x}\,
|
|
\sqrt{- \det g}\,
|
|
\qty(%
|
|
-i\, e\,
|
|
g^{\alpha\beta}
|
|
a_{\alpha}
|
|
\qty(%
|
|
\phi^*\, \ipd{\beta} \phi
|
|
-
|
|
\ipd{\beta} \phi^*\, \phi
|
|
)
|
|
).
|
|
\end{equation}
|
|
Its computation involves integrals such as
|
|
\begin{equation}
|
|
\int \dd{u}\,
|
|
\abs{\Delta u}\,
|
|
\qty(\frac{l}{u})^2
|
|
\finiteprod{i}{1}{3} \tphi_{\kmkrN{i}}
|
|
\sim
|
|
\int_{u \sim 0} \dd{u}\,
|
|
\qty(\frac{l^2}{\abs{u}^{\frac{5}{2}}})
|
|
e^{%
|
|
-i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +)}}
|
|
\frac{1}{u}
|
|
},
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\int \dd{u}\,
|
|
\abs{\Delta u}\,
|
|
\qty(\frac{1}{u})
|
|
\finiteprod{i}{1}{3} \tphi_{\kmkrN{i}}
|
|
\sim
|
|
\int_{u \sim 0} \dd{u}\,
|
|
\qty(\frac{1}{u\, \abs{u}^{\frac{1}{2}}})
|
|
e^{%
|
|
-i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +}}
|
|
\frac{1}{u}
|
|
},
|
|
\label{eq:nbo_div_integral}
|
|
\end{equation}
|
|
which can be interpreted as hints that the theory is troublesome.
|
|
The first integral diverges if the exponential functions are all equal to unity.
|
|
Fortunately it happens when all factors $l_{\qty(i)}$ (where $i = 1,\, 2,\, 3$) vanish.
|
|
In this case however the integral vanishes if we set $l_{\qty(i)} = 0$ before its evaluation.
|
|
This however suggests that when all $l_{\qty(i)} = 0$, i.e.\ when the eigenfunctions are constant along the compact direction $z$, something suspicious is happening.
|
|
On the other side when at least one $l$ is different from zero we have an integral such as:
|
|
\begin{equation}
|
|
\int_{u \sim 0} \dd{u}\,
|
|
\abs{u}^{-\nu}\, e^{i \frac{\cA}{u}}
|
|
\sim
|
|
\int_{t \sim \infty} \dd{t}\,
|
|
t^{\nu-2}\, e^{i \cA t}.
|
|
\end{equation}
|
|
All $l_{\qty(i)}$ are discrete but $k_{\qty(i)\, +}$ are not thus $\cA$ has an isolated zero.
|
|
Otherwise it has continuous value and may be given a distributional meaning, similar to a derivative of the Dirac delta function.
|
|
The second integral has the same issues when all $l_{\qty(*)} = 0$ but, since it is not proportional to any $l$ as it stands, it is divergent unless we consider a principal part regularization.
|
|
|
|
We can give in any case meaning to the cubic terms
|
|
and we get:\footnotemark{}
|
|
\footnotetext{%
|
|
The notation $(2) \rightarrow (3)$ meaning is that all previous terms inside the curly brackets appear again in exactly the same structure but with momenta of particle $(3)$ in place of those of particle $(2)$.
|
|
}
|
|
\begin{equation}
|
|
\begin{split}
|
|
\rS_{\text{s}\qed}^{(\text{cubic})}\qty[ \cA,\, \cE ]
|
|
& =
|
|
\finiteprod{i}{1}{3}
|
|
\qty[%
|
|
\int \dd[D-3]{\vec{k}_{\qty(i)}}\,
|
|
\dd{r_{\qty(i)}}\,
|
|
\dd{k_{\qty(i)\, +}}
|
|
\sum_{l_{\qty(i)}}
|
|
]\,
|
|
\qty(2\pi)^{D-1}
|
|
\delta\qty(\finitesum{i}{1}{3} \vec{k}_{\qty(i)})\,
|
|
\delta\qty(\finitesum{i}{1}{3} k_{\qty(i)\, +})\,
|
|
\\
|
|
& \times
|
|
e~
|
|
\delta_{l_{\qty(1)} + l_{\qty(2)} + l_{\qty(3)},\, 0}\,
|
|
\qty(\cA_{\mkmkrN{2}})^*\, \cA_{\kmkrN{3}}
|
|
\\
|
|
& \times
|
|
\left\lbrace
|
|
\cE_{\kmkrN{1}\, \underline{u}}\,
|
|
k_{\qty(2)\, +}\,
|
|
\cI_{\qty{3}}^{\qty[0]}
|
|
\right.
|
|
\\
|
|
& +
|
|
\cE_{\kmkrN{1}\, \underline{z}}\,
|
|
\frac{%
|
|
k_{\qty(2)\, +} l_{\qty(1)}
|
|
-
|
|
l_{\qty(2)} k_{\qty(1)\, +}
|
|
}{\Delta k_{\qty(1)\, +}}\,
|
|
\cJ_{\qty{3}}^{\qty[-1]}
|
|
\\
|
|
& +
|
|
\cE_{\kmkrN{1}\, \underline{v}}\,
|
|
\ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)})
|
|
\\
|
|
& -
|
|
\left.
|
|
\eta^{\underline{i}\, j}\,
|
|
\cE_{\kmkrN{1}\, \underline{i}}\,
|
|
k_{(2)_j}\,
|
|
\cI_{\qty{3}}^{\qty[0]}\,
|
|
-
|
|
\qty( (2) \rightarrow (3) )
|
|
\right\rbrace,
|
|
\label{eq:sQED_cubic_final}
|
|
\end{split}
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\begin{split}
|
|
\ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)})
|
|
& =
|
|
\frac{\norm{\vec{k}_{\qty(2)}}^2 + r_{\qty(2)}}{2\, k_{\qty(2)\, +}} \cI_{\qty{3}}^{\qty[0]}
|
|
+
|
|
i
|
|
\frac{k_{\qty(2)\, +}}{2\, k_{\qty(1)\, +}}
|
|
\cI_{\qty{3}}^{\qty[-1]}
|
|
\\
|
|
& +
|
|
\frac{1}{2} \frac{k_{\qty(2)\, +}}{\Delta^2}
|
|
\qty(%
|
|
\frac{l_{\qty(1)}}{k_{\qty(1)\, +}}
|
|
-
|
|
\frac{l_{\qty(2)}}{k_{\qty(2)\, +}}
|
|
)^2
|
|
\cI_{\qty{3}}^{\qty[-2]}.
|
|
\end{split}
|
|
\end{equation}
|
|
In the previous expressions we also defined for future use:
|
|
\begin{eqnarray}
|
|
\cI_{(1) \dots (N)}^{\qty[\nu]}
|
|
=
|
|
\cI_{\qty{N}}^{\qty[\nu]}
|
|
& = &
|
|
\infinfint{u}\,
|
|
\abs{\Delta u}\, u^{\nu}\,
|
|
\finiteprod{i}{1}{N}
|
|
\tphi_{\kmkrN{i}}
|
|
\\
|
|
\cJ_{\qty{N}}^{\qty[\nu]}
|
|
& = &
|
|
\infinfint{u}\,
|
|
\abs{\Delta}\, \abs{u}^{1 + \nu}
|
|
\finiteprod{i}{1}{N} \tphi_{\kmkrN{i}}.
|
|
\end{eqnarray}
|
|
For the sake of brevity from now on we use
|
|
\begin{eqnarray}
|
|
\tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}},
|
|
\\
|
|
\tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}}
|
|
\end{eqnarray}
|
|
when not causing confusion.
|
|
|
|
|
|
\subsubsection{Quartic Interactions and Divergences}
|
|
|
|
The issue with the divergent vertex is even more visible when considering the quartic terms:
|
|
\begin{equation}
|
|
\rS_{\text{s}\qed}^{(\text{quartic})}\qty[ \phi,\, a ]
|
|
=
|
|
\int\limits_{\Omega} \dd[D]{x}\,
|
|
\sqrt{- \det g}\,
|
|
\qty(%
|
|
e^2\, g^{\mu\nu}\, a_{\mu} a_{\nu}\, \abs{\phi}^2
|
|
-
|
|
\frac{g_4}{4}\abs{\phi}^4
|
|
),
|
|
\end{equation}
|
|
which can be expressed using the modes as:
|
|
\begin{equation}
|
|
\begin{split}
|
|
\rS_{\text{s}\qed}^{(\text{quartic})}\qty[ \phi,\, a ]
|
|
& =
|
|
\finiteprod{i}{1}{4}
|
|
\qty[%
|
|
\int \dd[D-3]{\vec{k}_{\qty(i)}}
|
|
\dd{k_{\qty(i)\, +}}
|
|
\dd{r_{\qty(i)}}
|
|
\sum_{l_{\qty(i)}}
|
|
]\,
|
|
\qty(2\pi)^{D-1}
|
|
\\
|
|
& \times
|
|
\delta\qty( \finitesum{i}{1}{4} \vec{k}_{\qty(i)} )\,
|
|
\delta\qty( \finitesum{i}{1}{4} k_{\qty(i)\, +} )\,
|
|
\delta_{l_{\qty(1)} + l_{\qty(2)} + l_{\qty(3)} + l_{\qty(4)},\, 0}
|
|
\\
|
|
& \times
|
|
\left\lbrace
|
|
e^2\,
|
|
\qty(\cA_{\mkmkrN{3}})^* \cA_{\kmkrN{4}}
|
|
\right\rbrace
|
|
\\
|
|
& \times
|
|
\left[
|
|
\qty(\cE_{\kmkrN{1}} \circ \cE_{\kmkrN{2}})\,
|
|
\cI_{\qty{4}}^{\qty[0]}
|
|
\right.
|
|
\\
|
|
& -
|
|
\frac{i}{2}
|
|
\cE_{\kmkrN{1}\, \underline{v}}\, \cE_{\kmkrN{2}\, \underline{v}}
|
|
\qty(%
|
|
\frac{1 }{k_{\qty(2)\, +}}
|
|
+
|
|
\frac{1}{k_{\qty(1)\, +}}
|
|
)\,
|
|
\cI_{\qty{4}}^{\qty[-1]}
|
|
\\
|
|
& +
|
|
\left.
|
|
\frac{1}{2}\,
|
|
\frac{\cE_{\kmkrN{1}\, \underline{v}} \cE_{\kmkrN{2}\, \underline{v}} }{\Delta^2}
|
|
\qty(%
|
|
\frac{l_{\qty(1)}}{k_{\qty(1)\, +}}
|
|
-
|
|
\frac{l_{\qty(2)}}{k_{\qty(2)\, +}}
|
|
)^2\,
|
|
\cI_{\qty{4}}^{\qty[-2]}
|
|
\right]
|
|
\\
|
|
& -
|
|
\left.
|
|
\frac{g_4}{4}\,
|
|
\ccA\qty(\qty{k_+,\, l,\, \vec{k},\, r})\,
|
|
\cI_{\qty{4}}^{\qty[0]}
|
|
\right\rbrace,
|
|
\end{split}
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\begin{split}
|
|
\ccA\qty(\qty{k_+,\, l,\, \vec{k},\, r})
|
|
& =
|
|
\qty(\cA_{\mkmkrN{1}})^*\,
|
|
\qty(\cA_{\mkmkrN{2}})^*\,
|
|
\\
|
|
& \times
|
|
\cA_{\kmkrN{3}}\,
|
|
\cA_{\kmkrN{4}}.
|
|
\end{split}
|
|
\end{equation}
|
|
When setting $l_{\qty(*)} = 0$ all the surviving terms are divergent.
|
|
The explicit behaviour is $\cI_{\qty{4}}^{\qty[0]} \sim \int \dd{u}\, \abs{u}^{1 -4 \times \frac{1}{2}}$ and $\cI_{\qty{4}}^{\qty[-1]} \sim \int \dd{u}\, u^{-1}\, \abs{u}^{1 - 4 \times \frac{1}{2}}$ since $\eval{\tphi}_{l = 0} \sim \abs{u}^{-\frac{1}{2}}$.
|
|
Higher order terms in the effective field theory have even worse behaviour.
|
|
This makes the theory ill defined and the string theory which should give this effective theory ill defined too.
|
|
|
|
|
|
\subsubsection{Failure of Obvious Divergence Regularizations}
|
|
\label{sec:saving}
|
|
|
|
From the discussion in the previous section the origin of the divergences is the sector $l = 0$.
|
|
When $l = 0$ the highest order singularity of the Fourier transformed d'Alembertian equation vanishes.
|
|
Explicitly we have:
|
|
\begin{equation}
|
|
A\, \ipd{u} \tphi_{\kmkr}
|
|
+
|
|
B(u)\, \tphi_{\kmkr}
|
|
=
|
|
A\, e^{-\int^u \frac{B(u)}{A} du}\,
|
|
\ipd{u} \qty[ e^{+\int^u \frac{B(u)}{A} du} \tphi_{\kmkr} ]
|
|
=
|
|
0,
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
A = \qty(-2\, i\, k_+),
|
|
\qquad
|
|
B(u)
|
|
=
|
|
-\qty(\vec{k}^2 + r)
|
|
-
|
|
i\, k_+\, \frac{1}{u}
|
|
-
|
|
\frac{l^2}{\Delta^2}\, \frac{1}{u^2}.
|
|
\end{equation}
|
|
This implies the absence of the oscillating factor $e^{i \frac{\cA}{u} }$ when $l$ vanishes.
|
|
It follows that any deformation which prevents the coefficient of the highest order singularity from vanishing will do the trick.
|
|
|
|
The first and easiest possibility is to add a Wilson line along $z$, i.e.\ $a = \theta \dd{z}$.
|
|
This shifts $l \rightarrow l - e\, \theta$ and regularises the scalar \qed.
|
|
Unfortunately this does not work in the string theory where Wilson lines on D25-branes are not felt by the neutral strings starting and ending on the same D-brane.
|
|
In fact not all interactions involve commutators of the Chan-Paton factors which vanish for neutral strings.
|
|
For instance the interaction of two tachyons with the first massive state involves an anti-commutator as we discuss later.
|
|
The anti-commutators are present also in amplitudes of supersymmetric strings with massive states and therefore the issue is not solved by supersymmetry.
|
|
|
|
A second possibility is to include higher derivative couplings to curvature as natural in the string theory.
|
|
If we regularise the metric in a minimal way as shown at the end of~\Cref{sec:geometric_preliminaries_nbo}, only $\tensor{Ric}{_u_u}$ does not vanish.
|
|
We can introduce:
|
|
\begin{equation}
|
|
\begin{split}
|
|
&
|
|
S_{\mathrm{HE}}^{(\text{higher R})}\qty[ \phi,\, g ]
|
|
\\
|
|
= &
|
|
\int\limits_{\Omega} \dd[D]{x}\,
|
|
\sqrt{- \det g}\,
|
|
\qty(%
|
|
\finitesum{k}{1}{+\infty}
|
|
\qty(\ap)^{2 k-1}\,
|
|
\finiteprod{j}{1}{k}\,
|
|
g^{\mu_j \nu_j}\, g^{\rho_j\sigma_j}\,
|
|
\tensor{Ric}{_{\mu_j}_{\rho_j}}\,
|
|
\qty(%
|
|
\finitesum{s}{0}{2k}
|
|
c_{k s}\, \ipd{\nu_j}^{2k - s}\phi^*\, \ipd{\sigma_j}^s \phi
|
|
)
|
|
)
|
|
\\
|
|
= &
|
|
\int\limits_{\Omega} \dd[D]{x}\,
|
|
\sqrt{- \det g}\,
|
|
\qty(%
|
|
\ap\,
|
|
g^{\mu\nu}\, g^{\rho\sigma}\,
|
|
\tensor{Ric}{_{\mu}_{\rho}}\,
|
|
\qty(%
|
|
c_{12}
|
|
\phi^*\, \ipd{\nu\sigma}^2 \phi
|
|
+
|
|
c_{11}
|
|
\ipd{\nu} \phi^*\, \ipd{\sigma} \phi
|
|
+
|
|
c_{10}
|
|
\ipd{\nu\sigma}^2 \phi^*\, \phi
|
|
)
|
|
),
|
|
\end{split}
|
|
\end{equation}
|
|
where $\ap$ has been introduced after dimensional analysis and in order to have all adimensional $c$ factors.
|
|
Since only $\tensor{Ric}{_u_u}$ is non vanishing and it depends only on $u$,
|
|
the regularised d'Alembertian eigenmode problem now reads:
|
|
\begin{equation}
|
|
\begin{split}
|
|
-
|
|
2 \ipd{u} \ipd{v} \phi_r
|
|
& -
|
|
\frac{u}{u^2 + \epsilon^2} \ipd{v} \phi_r
|
|
+
|
|
\frac{1}{\Delta^2 (u^2+ \epsilon^2)} \ipd{z}^2 \phi_r
|
|
\\
|
|
& +
|
|
\finitesum{k}{1}{+\infty} \qty(\ap)^{2k-1}\,
|
|
C_k\,
|
|
\tensor{Ric}{_u_u}^k\,
|
|
\ipd{v}^{2k} \phi
|
|
+
|
|
\ipd{i}^2 \phi_r
|
|
-
|
|
r\, \phi_r
|
|
=
|
|
0,
|
|
\end{split}
|
|
\end{equation}
|
|
with $C_k = \finitesum{s}{0}{2k} (-1)^s\, c_{k s}$.
|
|
We can perform the usual Fourier transform and the function $B(u)$ becomes
|
|
\begin{equation}
|
|
\begin{split}
|
|
B(u)
|
|
& =
|
|
-
|
|
(\vec{k}^2 + r)
|
|
-
|
|
i\, k_+\, \frac{u}{u^2 + \epsilon^2}
|
|
-
|
|
\frac{l^2}{\Delta^2}\, \frac{1}{u^2+\epsilon^2}
|
|
\\
|
|
& +
|
|
\finitesum{k}{1}{+\infty}
|
|
\qty(\ap)^{2k-1}\,
|
|
C_k\,
|
|
\qty(\frac{\epsilon^2}{(u^2 + \epsilon^2)^2})^k
|
|
(-i k_+)^{2k}.
|
|
\end{split}
|
|
\end{equation}
|
|
When $u = 0$ we have:
|
|
\begin{equation}
|
|
B(0)
|
|
\sim
|
|
- \frac{l^2}{\Delta^2}\, \frac{1}{\epsilon^2}
|
|
+
|
|
\finitesum{k}{1}{+\infty}
|
|
\qty(\ap)^{2k-1}\,
|
|
C_k\,
|
|
\frac{(-i k_+)^{2k}}{\epsilon^{2 k}}.
|
|
\end{equation}
|
|
Though the correction seems to lead to a cure for the divergence, ff we consider $\ap$ and $\epsilon^2$ uncorrelated we lose predictability.
|
|
However if $\ap \sim \epsilon^2$ as natural in string theory we do not solve the problem since
|
|
\begin{equation}
|
|
B(0)
|
|
\stackrel{\ap \sim \epsilon^2}{\sim}
|
|
- \frac{l^2}{\Delta^2}\, \frac{1}{\epsilon^2}
|
|
+
|
|
\finitesum{k}{1}{+\infty}
|
|
C_k\,
|
|
(-i k_+)^{2k}
|
|
\epsilon^{2k - 2}
|
|
\end{equation}
|
|
and the curvature terms are no longer singular.
|
|
|
|
|
|
\subsubsection{A Hope from Twisted State Background}
|
|
|
|
The issue with the divergences is associated with the dipole string and its charge neutral states since the charged ones can be cured rather trivially by a Wilson line.
|
|
|
|
On the other hand we know that the usual time-like orbifolds are well defined because of a presence of a $B_{\mu\nu}$ background and this field is sourced by strings.
|
|
We may switch on such a background in the open string.
|
|
For open strings $F$ is equivalent to such $B$ field so we can consider what happens to an open string in an electromagnetic background.
|
|
|
|
The choice of such a background is limited first of all by the request that it must be an exact string solution, i.e.\ it needs to obey the \eom derived from the Dirac--Born--Infeld action.
|
|
If a closed string winds the compact direction $z$ then it is coupled to $B_{z u}$, $B_{z v}$ and $B_{z i}$ but if we choose
|
|
\begin{equation}
|
|
\frac{1}{2\pi\ap} B(u)
|
|
=
|
|
f(u) \dd{u} \wedge \dd{z}.
|
|
\label{eq:F_bck}
|
|
\end{equation}
|
|
then
|
|
\begin{equation}
|
|
\det( g + 2 \pi \ap f(u) ) = \det(g).
|
|
\end{equation}
|
|
It is therefore a solution of the open string \eom for any $f\qty(u,\, v,z,x^i)$.
|
|
As the two-tachyons---two-photons amplitude suggests, suppose that the action for a real neutral scalar $\phi$ is given by:
|
|
\begin{equation}
|
|
\begin{split}
|
|
S_{\text{scalar}}^{(\text{kinetic})}\qty[ \phi ]
|
|
& =
|
|
\int\limits_{\Omega} \dd[D]{x}\,
|
|
\sqrt{- \det g}\,
|
|
\frac{1}{2}
|
|
\qty(
|
|
- g^{\alpha\beta}
|
|
\ipd{\alpha} \phi\, \ipd{\beta} \phi
|
|
- M^2 \phi^2
|
|
+ c_1
|
|
\qty(\ap)^2\, \ipd{\mu} \phi\, \ipd{\nu} \phi
|
|
\tensor{f}{^{\mu}^\kappa} \tensor{f}{^{\nu}_\kappa}
|
|
)
|
|
\\
|
|
& =
|
|
\int \dd[D-3]{\vec{x}}\,
|
|
\int \dd{u}\,
|
|
\int \dd{v}\,
|
|
\finiteint{z}{0}{2\pi}
|
|
\abs{\Delta u}\,
|
|
\frac{1}{2}\,
|
|
\Biggl(
|
|
2\, \ipd{u} \phi\,\ipd{v} \phi\,
|
|
\\
|
|
& -
|
|
\frac{1}{\qty(\Delta u )^2} \qty(\ipd{z} \phi)^2
|
|
-
|
|
\eta^{ij} \ipd{i}\phi\, \ipd{j} \phi
|
|
-
|
|
M^2 \phi^2
|
|
+
|
|
c_1 \qty(\ap)^2 \frac{1}{\qty(\Delta u)^2} \qty(\ipd{v} \phi)^2 f^2(u)
|
|
\Biggr)
|
|
,
|
|
\end{split}
|
|
\end{equation}
|
|
Performing the same steps as before we get
|
|
\begin{equation}
|
|
B(u)
|
|
=
|
|
- (\vec{k}^2 + r)
|
|
-
|
|
i\, k_+\, \frac{1}{u}
|
|
+
|
|
\frac{\qty(c_1 \qty(\ap)^2 f(u)^2 k_+^2 - l^2)}{\Delta^2\, u^2},
|
|
\end{equation}
|
|
so even for a constant $f(u) = f_0$ we get a solution which solves the issues.
|
|
Notice however that the ``trivial'' solution $f = f_0 \dd{u} \wedge \dd{z}$ is not trivial in Minkowski coordinates where it reads $f = \frac{f_0}{x^-} \dd{x^-} \wedge \dd{x^2}$.
|
|
Though appealing, the study of the string in the presence of this non trivial background needs a deeper analysis and it surely is a direction to cover in the future.
|
|
|
|
|
|
\subsection{NBO Eigenfunction from the Covering Space}
|
|
|
|
We recover the eigenfunctions from the covering Minkowski space in order to elucidate the connection between the polarizations in \nbo and in Minkowski.
|
|
Moreover we generalise the result to a symmetric two index tensor which is the polarization of the first massive state to compute the two-tachyons--one-massive-state amplitude in the next section and to show that it diverges.
|
|
|
|
|
|
\subsubsection{Spin 0 Wave Function from Minkowski space}
|
|
|
|
We start with the usual plane wave in flat space and we express it in the new coordinates (we do not write the dependence on $\vec{x}$ since it is trivial):
|
|
\begin{equation}
|
|
\begin{split}
|
|
\psi_{k_+\, k_-\, k_2}\qty(x^+,\, x^-,\, x^2)
|
|
& =
|
|
e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )}
|
|
\\
|
|
& =
|
|
e^{%
|
|
i\, \qty[%
|
|
k_+ v
|
|
+
|
|
\frac{2\, k_+ k_- - k_2^2}{2 k_+} u
|
|
+
|
|
\frac{1}{2} \Delta^2 k_+ u
|
|
\qty( z + \frac{k_2}{\Delta k_+} )^2
|
|
]
|
|
}
|
|
\\
|
|
& =
|
|
\psi_{k_+\, k_-\, k_2}\qty(u,\, v,\, z).
|
|
\end{split}
|
|
\end{equation}
|
|
The corresponding wave function on the \nbo is obtained by the periodicity of $z$.
|
|
This can be done in two ways either in $\qty(x^{\mu})$ coordinates or in $\qty(x^{\alpha}) = \qty(u\, v\, z)$.
|
|
From the first we study how the map to the orbifold gives the function a dependence on the equivalence class of momenta.
|
|
Implementing the projection on periodic $z$ functions we get:
|
|
\begin{equation}
|
|
\begin{split}
|
|
\Psi_{\qty[k_+\, k_-\, k_2]}\qty(\qty[x^+,\, x^-,\, x^2])
|
|
& =
|
|
\infinfsum{n}
|
|
\psi_{k_+\, k_-\, k_2}\qty( \cK^n\qty(x^+,\, x^-,\, x^2) )
|
|
\\
|
|
& =
|
|
\infinfsum{n}
|
|
\psi_{\cK^{-n}\qty( k_+\, k_-\, k_2 )}\qty( x^+,\, x^-,\, x^2 ),
|
|
\end{split}
|
|
\end{equation}
|
|
where we write $\qty[k_+\, k_-\, k_2]$ since the function depends on the equivalence class of $\qty(k_+\, k_-\, k_2)$ only.
|
|
The equivalence relation is given by
|
|
\begin{equation}
|
|
k =
|
|
\mqty(
|
|
k_+\\ k_-\\ k_2
|
|
)
|
|
\equiv
|
|
\cK^{-n} k
|
|
=
|
|
\mqty(
|
|
k_+
|
|
\\
|
|
k_- + n \qty(2 \pi \Delta) k_2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 k_+
|
|
\\
|
|
k_2 + n \qty(2 \pi \Delta) k_+
|
|
).
|
|
\end{equation}
|
|
It allows us to choose a representative with
|
|
\begin{equation}
|
|
\begin{cases}
|
|
0 \le \frac{k_2}{\Delta \abs{k_+}} < 2 \pi,
|
|
& \qquad
|
|
k_+ \neq 0
|
|
\\
|
|
0 \le \frac{k_-}{\Delta \abs{k_2}} < 2 \pi,
|
|
& \qquad
|
|
k_+ = 0, \quad k_2 \neq 0
|
|
\end{cases}.
|
|
\end{equation}
|
|
|
|
If we perform the computation in $\qty(u,\, v,\, z)$ coordinates we get:
|
|
\begin{equation}
|
|
\begin{split}
|
|
\Psi_{\qty[k_+\, k_-\, k_2]}\qty(u,\, v,\, z)
|
|
& =
|
|
\infinfsum{n}
|
|
\psi_{k_+\, k_-\, k_2}\qty(u,\, v,\, z + 2 \pi n)
|
|
\\
|
|
& =
|
|
\infinfsum{n}
|
|
e^{%
|
|
i\, \qty[%
|
|
k_+ v
|
|
+
|
|
\frac{r}{2 k_+} u
|
|
+
|
|
\frac{1}{2} \qty(2 \pi \Delta)^2 k_+ u
|
|
\qty[ n + \frac{1}{2\pi} \qty( z + \frac{k_2}{\Delta k_+} ) ]^2
|
|
]
|
|
},
|
|
\end{split}
|
|
\end{equation}
|
|
with $r = 2\, k_+ k_- - k_2^2$ and $\Im(k_+ u) > 0$, i.e.\ $k_+ u = \abs{k_+ u} e^{i \epsilon}$ and $\pi > \epsilon > 0$.
|
|
There is no separate dependence on $z$ and on $\frac{k_2}{\Delta k_+}$: we could fix the range $0 \le z + \frac{k_2}{\Delta k_+} < 2\pi$.
|
|
However this symmetry is broken when considering the photon eigenfunction.
|
|
|
|
We can now use the Poisson resummation
|
|
\begin{equation}
|
|
\infinfsum{n}
|
|
e^{i\, a\, (n + b)^2}
|
|
=
|
|
\int \dd{s}\,
|
|
\delta_P(s) e^{i\, a\, (s + b)^2}
|
|
=
|
|
\qty(2\pi)^2
|
|
\frac{e^{-i\, \qty( \frac{\pi}{4} + \frac{1}{2} arg(a) ) }}{2 \sqrt{\pi \abs{a}}}
|
|
\infinfsum{m} e^{-i\, \frac{\pi^2 m^2}{a} + i\, 2 \pi b m},
|
|
\end{equation}
|
|
to finally get:\footnotemark{}
|
|
\footnotetext{%
|
|
In the expression we insert the variables $\vec{k}$ and $\vec{x}$ for completeness.
|
|
We also set $r = 2\, k_+ k_- -k_2^2 - \vec{k}^2$.
|
|
}
|
|
\begin{equation}
|
|
\begin{split}
|
|
\Psi_{[k_+\, k_-\, k_2\, \vec{k}]}\qty(u,\, v,\, z,\, \vec{x})
|
|
& =
|
|
\sqrt{2\pi}~
|
|
\frac{2 e^{-i \frac{\pi}{4}}}{\Delta}
|
|
\\
|
|
& \times
|
|
\infinfsum{l}
|
|
\qty[
|
|
\frac{1}{\sqrt{\abs{k_+ u}}}
|
|
e^{%
|
|
i\, \qty[%
|
|
k_+ v
|
|
+
|
|
l z
|
|
-
|
|
\frac{l^2}{2 \Delta^2 k_+}\, \frac{1}{u}
|
|
+
|
|
\frac{r + \vec{k}^2}{2 k_+} u
|
|
+
|
|
\vec{k} \cdot \vec{x}
|
|
]
|
|
}
|
|
]
|
|
e^{i\, l\, \frac{k_2}{\Delta k_+}}
|
|
\\
|
|
& =
|
|
\cN\,
|
|
\infinfsum{l}
|
|
\phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x})
|
|
e^{i\, l\, \frac{k_2}{\Delta k_+}},
|
|
\end{split}
|
|
\label{eq:Psi_phi}
|
|
\end{equation}
|
|
when $k_+ \neq 0$ and where
|
|
\begin{equation}
|
|
\cN
|
|
=
|
|
\sqrt{\frac{\qty(2\pi)^D}{\pi \Delta}}
|
|
\frac{e^{-i \frac{\pi}{4}}}{\pi}.
|
|
\end{equation}
|
|
The fact that $\Psi$ depends only on the equivalence class $\qty[k_+\, k_-\, k_2\, k]$ allows us to restrict $0 \le \frac{k_2}{\Delta\, \abs{k_+}} < 2 \pi$ so that we can invert the previous expression and get:
|
|
\begin{equation}
|
|
\begin{split}
|
|
\phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x})
|
|
& =
|
|
\frac{1}{\cN}\,
|
|
\frac{1}{2 \pi \Delta \abs{k_+}}
|
|
\finiteint{k_2}{0}{2 \pi \Delta \abs{k_+}}
|
|
e^{-i\, l\, \frac{k_2}{\Delta k_+}}\,
|
|
\Psi_{\qty[k_+\, k_-\, k_2\, k]}\qty(u,\, v,\, z,\, \vec{x}).
|
|
\end{split}
|
|
\end{equation}
|
|
|
|
|
|
\subsubsection{Spin 1 Wave Function from Minkowski space}
|
|
|
|
We go through the steps in the previous case for an electromagnetic wave.
|
|
We concentrate on $x^+$, $x^-$ and $x^2$ coordinates and reinstate $\vec{x}$ at the end.
|
|
We start with the usual plane wave in flat space $\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}$ and we express it in both Minkowski and orbifold coordinates.
|
|
We use the notation $\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}$ to stress that it is the eigenfunction and not the field which is obtained as
|
|
\begin{equation}
|
|
A_{\mu}(x)\, \dd{x}^{\mu}
|
|
=
|
|
\int \dd[3]{k}\,
|
|
\sum_{\qty{\epsilon_+,\, \epsilon_-,\, \epsilon_2}}
|
|
\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2},
|
|
\end{equation}
|
|
where the sum is performed over $\epsilon_+$, $\epsilon_-$, $\epsilon_2$ independent and compatible with $k$.
|
|
The explicit expression for the eigenfunction with constant $\epsilon_+$, $\epsilon_-$ and $\epsilon_2$ is:\footnotemark{}
|
|
\footnotetext{%
|
|
We introduce the normalization factor $\cN$ in order to have a less cluttered relation between $\epsilon$ and $\cE$.
|
|
}
|
|
\begin{equation}
|
|
\begin{split}
|
|
\cN\,
|
|
\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}\qty(x^+,\, x^-,\, x^2)
|
|
& =
|
|
\qty(\epsilon_+ \dd{x^+} + \epsilon_- \dd{x^-} + \epsilon_2 \dd{x^2})\,
|
|
e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )}
|
|
\\
|
|
& =
|
|
\qty( \epsilon_u\, \dd{u} + \epsilon_z\, \dd{z} + \epsilon_v\, \dd{v})\,
|
|
\\
|
|
& \times
|
|
e^{%
|
|
i\, \qty[%
|
|
k_+ v
|
|
+
|
|
\frac{2\, k_+ k_- - k_2^2}{2 k_+} u
|
|
+
|
|
\frac{1}{2} \Delta^2 k_+ u \qty( z + \frac{k_2}{\Delta k_+})^2
|
|
]
|
|
}
|
|
\\
|
|
& =
|
|
\cN
|
|
\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}\qty(u,\, v,\, z),
|
|
\end{split}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\begin{split}
|
|
\epsilon_v & = \epsilon_+,
|
|
\\
|
|
\epsilon_u(z)
|
|
& =
|
|
\epsilon_- + (\Delta z)\, \epsilon_2 + (\frac{1}{2} \Delta^2 z^2)\, \epsilon_+,
|
|
\\
|
|
\epsilon_z(u,\, z)
|
|
& =
|
|
\qty(\Delta u)\,
|
|
\qty(\epsilon_2 + \Delta z\, \epsilon_+ ).
|
|
\end{split}
|
|
\end{equation}
|
|
Notice that we are not imposing any gauge condition.
|
|
Moreover if $(\epsilon_+,\, \epsilon_-,\, \epsilon_2)$ are constant then $(\epsilon_u,\, \epsilon_v,\, \epsilon_z)$ are generic functions.
|
|
It is worth stressing that they are not the polarizations in the orbifold which are in any case constant: the fact that they depend on the coordinates is simply the statement that not all eigenfunctions of the vector d'Alembertian are equal.
|
|
|
|
Building the corresponding function on the orbifold amounts to summing the images created by the orbifold group:
|
|
\begin{equation}
|
|
\begin{split}
|
|
\cN
|
|
\Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x])
|
|
=
|
|
\infinfsum{n}
|
|
\vec{\epsilon} \cdot \qty( \cK^{-n} \dd{x})~
|
|
\psi_{k}\qty( \cK^{-n} x)
|
|
=
|
|
\infinfsum{n}
|
|
\cK^n
|
|
\vec{\epsilon} \cdot \dd{x}~
|
|
\psi_{\cK^n k}\qty(x).
|
|
\end{split}
|
|
\end{equation}
|
|
Under the action of the Killing vector $\epsilon$ transforms exactly as the $k$ since it is induced by $\epsilon \cdot \cK^n \dd{x} = \cK^{-n} \epsilon \cdot \dd{x}$, that is:
|
|
\begin{equation}
|
|
\epsilon
|
|
=
|
|
\mqty(%
|
|
\epsilon_+ \\ \epsilon_2 \\ \epsilon_-
|
|
)
|
|
\equiv
|
|
\cK^{-n} \epsilon
|
|
=
|
|
\mqty(%
|
|
\epsilon_+
|
|
\\
|
|
\epsilon_2 + n\, \qty(2 \pi \Delta)\, \epsilon_+
|
|
\\
|
|
\epsilon_- + n\, \qty(2 \pi \Delta)\, \epsilon_2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 \epsilon_+
|
|
).
|
|
\end{equation}
|
|
However the pair $\qty(\vec{k},\, \vec{\epsilon})$ transforms with the same $n$ since both are ``dual'' to $x$, i.e.\ their transformation rules are dictated by $x$.
|
|
There is therefore only one equivalence class $\qty[\vec{k},\, \vec{\epsilon}]$ and not two separate classes $\qty[\vec{k}]$, $\qty[\vec{\epsilon}]$.
|
|
In other words, a representative of the combined equivalence class is the one with $0 \le k_2 < 2 \pi \Delta \abs{k_+}$ when $k_+ \neq 0$.
|
|
|
|
%%% TODO %%%
|
|
|
|
We now proceed to find the eigenfunctions on the orbifold
|
|
in orbifold coordinates.
|
|
We notice that $\dd{u}, \dd{v}$ and $\dd{z}$ are
|
|
invariant and therefore their coefficients in $a$ are as well.
|
|
So we write
|
|
\begin{align}
|
|
\cN
|
|
\Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x])
|
|
=&
|
|
\sum_n \epsilon \cdot ( \cK^{ n } \dd x)\,
|
|
\psi_{k}( \cK^{ n } x)
|
|
\nonumber\\
|
|
=&
|
|
\dd{v}\,
|
|
\left[ \epsilon_+ \sum_n \psi_{ k}( \cK^{ n } x) \right]
|
|
+
|
|
\dd{z}\, (\Delta u)
|
|
\left[
|
|
\epsilon_2 \sum_n \psi_{ k}( \cK^{ n } x)
|
|
+
|
|
\epsilon_+ \Delta \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x)
|
|
\right]
|
|
\nonumber\\
|
|
&+
|
|
\dd{u}
|
|
\left[
|
|
\epsilon_- \sum_n \psi_{ k}(\cK^{ n }x)
|
|
+
|
|
\epsilon_2 \Delta \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x)
|
|
+
|
|
\frac{1}{2}
|
|
\epsilon_+ \Delta^2
|
|
\sum_n (z + 2\pi n)^2 \psi_{ k}(\cK^{ n }x)
|
|
\right]
|
|
.
|
|
\end{align}
|
|
From direct computation we get\footnote{Notice that these expressions may be written using Hermite polynomials.}
|
|
\begin{align}
|
|
&
|
|
\sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x)
|
|
=
|
|
\qty(
|
|
\frac{1}{i \Delta\, u}
|
|
\frac{\partial}{\partial k_2}
|
|
-
|
|
\frac{k_2}{\Delta k_+}
|
|
) \Psi_{[k]}\qty(\qty[x])
|
|
\nonumber\\
|
|
&
|
|
\sum_n (z + 2\pi n)^2 \psi_{ k}(\cK^{ n }x)
|
|
=
|
|
\qty(
|
|
\frac{1}{i \Delta\, u}
|
|
\frac{\partial}{\partial k_2}
|
|
-
|
|
\frac{k_2}{\Delta k_+}
|
|
)^2 \Psi_{[k]}\qty(\qty[x])
|
|
.
|
|
\label{eq:sum_z_psi}
|
|
\end{align}
|
|
Then it follows that
|
|
\begin{align}
|
|
\cN
|
|
\Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x])
|
|
=
|
|
&
|
|
\dd{v}\,
|
|
\left[ \epsilon_+\, \Psi_{[k]}\qty(\qty[x])
|
|
\vphantom{\frac{\partial}{\partial k_2}}
|
|
\right]
|
|
\nonumber\\
|
|
&
|
|
+
|
|
\dd{z}\, (\Delta u)
|
|
\left[
|
|
\frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+}
|
|
\Psi_{[k]}\qty(\qty[x])
|
|
+
|
|
\epsilon_+ \, \frac{-i}{u} \frac{\partial}{\partial k_2} \Psi_{[k]}\qty(\qty[x])
|
|
\right]
|
|
\nonumber\\
|
|
&+
|
|
\dd{u}
|
|
\Biggl[
|
|
\qty(
|
|
\epsilon_- - \epsilon_2 \frac{k_2}{k_+}
|
|
+ \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2
|
|
)
|
|
\, \Psi_{[k]}\qty(\qty[x])
|
|
+
|
|
\frac{i}{2 u} \frac{\epsilon_+}{k_+ } \, \Psi_{[k]}\qty(\qty[x])
|
|
\nonumber\\
|
|
&\phantom{d u [}
|
|
+
|
|
\frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+}
|
|
\frac{-i}{ u} \frac{\partial}{\partial k_2} \Psi_{[k]}\qty(\qty[x])
|
|
+
|
|
\frac{1}{2}
|
|
\epsilon_+
|
|
\frac{-1}{ u^2} \frac{\partial^2}{\partial k_2{}^2} \Psi_{[k]}\qty(\qty[x])
|
|
\Biggr]
|
|
,
|
|
\label{eq:a_uvz_from_covering}
|
|
\end{align}
|
|
where many coefficients of $\Psi$ or its derivatives contain
|
|
$k_2$.
|
|
They cannot be expressed using the quantum
|
|
numbers of the orbifold $\kmkr$ but are invariant on the orbifold and
|
|
therefore are new orbifold quantities which we can interpret as
|
|
orbifold polarizations.
|
|
Using \eqref{eq:Psi_phi} we can finally write
|
|
\begin{align}
|
|
% \cN
|
|
\Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x])
|
|
=
|
|
% \cN
|
|
\sum_l
|
|
&
|
|
% \frac{1}{\sqrt{|k_+|}}
|
|
\phi_{\kmkr}(u,v,z,\vec{x})
|
|
e^{ i l \frac{k_2}{ \Delta k_+} }
|
|
\Bigg\{
|
|
\dd{v} \Bigl[ \epsilon_+ \Bigr]
|
|
\nonumber\\
|
|
&+
|
|
\dd{z}\, (\Delta u)
|
|
\Biggl[
|
|
\frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+}
|
|
+
|
|
\epsilon_+ \frac{1}{\Delta u} \frac{l}{k_+}
|
|
\Biggr ]
|
|
\nonumber\\
|
|
&+
|
|
\dd{u}
|
|
\Biggl[
|
|
% \epsilon_-
|
|
\qty(
|
|
\epsilon_- - \epsilon_2 \frac{k_2}{k_+}
|
|
+ \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2
|
|
)
|
|
+
|
|
\frac{i}{2 u} \frac{\epsilon_+}{k_+ }
|
|
\nonumber\\
|
|
&
|
|
\phantom{ \dd{u} [ }
|
|
+
|
|
\frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+}
|
|
\frac{1}{u} \frac{l}{\Delta k_+}
|
|
+
|
|
\epsilon_+ \frac{1}{2 u^2}
|
|
\qty(\frac{l}{ \Delta k_+} )^2
|
|
\Biggr]
|
|
\Bigg\}
|
|
.
|
|
\label{eq:spin1_from_covering}
|
|
\end{align}
|
|
|
|
If we compare with \eqref{eq:Orbifold_spin1_pol} we find
|
|
\begin{align}
|
|
\cE_{\kmkr\, \underline{v}} &= \epsilon_+
|
|
\nonumber\\
|
|
\cE_{\kmkr\, \underline{z}} &= \mathrm{sign}(u)
|
|
\frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+}
|
|
\nonumber\\
|
|
\cE_{\kmkr\, \underline{u}} &=
|
|
\epsilon_- - \epsilon_2 \frac{k_2}{k_+}
|
|
+ \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2
|
|
,
|
|
\label{eq:eps_calE}
|
|
\end{align}
|
|
which implies that the true polarizations $(\epsilon_+, \epsilon_-, \epsilon_2)$
|
|
and
|
|
$\cE_{\kmkr\, \underline{*}}$ are constant as it turns out from direct computation.
|
|
|
|
A different way of reading the previous result is that the
|
|
polarizations on the orbifold are the coefficients of the highest power
|
|
of $u$.
|
|
|
|
We can also invert the previous relations to get
|
|
\begin{align}
|
|
\epsilon_+
|
|
&=
|
|
\cE_{\kmkr\, \underline{v}}
|
|
\nonumber\\
|
|
\epsilon_2
|
|
&=
|
|
\cE_{\kmkr\, \underline{z}} \mathrm{sign}(u) + \frac{k_2}{k_+} \cE_{\kmkr\, \underline{v}}
|
|
\nonumber\\
|
|
\epsilon_- &=
|
|
\cE_{\kmkr\, \underline{u}}
|
|
+ \frac{k_2}{k_+} \cE_{\kmkr\, \underline{z}} \mathrm{sign}(u)
|
|
+ \frac{1}{2} \qty( \frac{k_2}{k_+} )^2 \cE_{\kmkr\, \underline{v}}
|
|
,
|
|
\label{eq:calE_eps}
|
|
\end{align}
|
|
and use them in Lorenz gauge $k \cdot \epsilon=0$ in order to get the
|
|
expression of Lorenz gauge with orbifold polarizations.
|
|
If the definition of orbifold polarizations is right the result cannot
|
|
depend on $k_2$ since $k_2$ is not a quantum number of orbifold
|
|
eigenfunctions.
|
|
Taking in account $k_- = \frac{\vec{k}^2+ k_2^2 + r}{2 k_+}$ in $k \cdot
|
|
\epsilon=0$
|
|
we get
|
|
exactly the expression for the Lorenz gauge for orbifold polarizations
|
|
\eqref{eq:Lorenz_gauge}.
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\subsection{Tensor Wave Function from Minkowski space}
|
|
Once again, we can use the analysis of the previous section in the case of a
|
|
second order symmetric tensor wave function.
|
|
Again we suppress the dependence on $\vec{x}$ and $\vec{k}$ with a caveat: the Minkowskian polarizations $S_{+\, i}$, $S_{-\, i}$ and $S_{2\, i}$ do transform non trivially, therefore we give the full expressions in Appendix~\ref{app:NO_tensor_wave} even if these components contribute in a somewhat trivial way since they behave effectively as a vector of the orbifold.
|
|
|
|
We start with the usual wave in flat space and we express either in
|
|
the Minkowskian coordinates
|
|
\begin{alignat}{4}
|
|
\cN
|
|
\psi^{[2]}_{k\, S}(x^+,x^-,x^2)
|
|
&= S_{\mu \nu}\, \psi_k(x)\, \dd x^{\mu}\, \dd x^{\nu}
|
|
\nonumber\\
|
|
&=
|
|
% (\epsilon_+ d x^+ + \epsilon_- dx^- + \epsilon_2 d x^2)
|
|
% \otimes
|
|
% (\etp d x^+ + \etm dx^- + \etd d x^2)
|
|
% \qty(
|
|
(
|
|
S_{+\, +}\, \dd x^+\, \dd x^+
|
|
&
|
|
+
|
|
2 S_{+\, 2}\, \dd x^+\, \dd x^2
|
|
&
|
|
+
|
|
2 S_{+\, -}\, \dd x^+\, \dd x^-
|
|
\nonumber\\
|
|
&
|
|
&
|
|
+
|
|
2 S_{2\, 2}\, \dd x^2\, \dd x^2
|
|
&
|
|
+
|
|
2 S_{2\, -}\, \dd x^2\, \dd x^-
|
|
\nonumber\\
|
|
&
|
|
&
|
|
&
|
|
+
|
|
2 S_{-\, -}\, \dd x^-\, \dd x^-
|
|
)
|
|
e^{i\qty( k_+ x^+ + k_- x^- + k_2 x^2 )}
|
|
\nonumber\\
|
|
,
|
|
\end{alignat}
|
|
or in orbifold coordinates
|
|
\begin{align}
|
|
\cN
|
|
\psi^{[2]}_{k\, S}(x)
|
|
=& S_{\alpha \beta}\, \psi_k(x)\, \dd x^\alpha\, \dd x^\beta
|
|
\nonumber\\
|
|
=&
|
|
\Bigl\{
|
|
(\dd{v})^2\,
|
|
[S_{+\, +}]
|
|
\nonumber\\
|
|
&
|
|
+
|
|
\dd{v}\, \dd{z}\,\Delta u
|
|
[ 2 S_{+\, 2}
|
|
+ S_{+\, +} \Delta z ]
|
|
\nonumber\\
|
|
&
|
|
+
|
|
\dd{v}\, \dd{u}\,
|
|
[ 2 S_{+\, -}
|
|
+ 2 S_{+\, 2} \Delta z
|
|
+ S_{+\, +} \Delta^2 z^2 ]
|
|
\nonumber\\
|
|
&
|
|
+
|
|
\dd{z}^2\,\Delta^2 u^2\,
|
|
[ S_{2\, 2}
|
|
+ 2 S_{+\, 2} \Delta z
|
|
+ S_{+\, +} \Delta^2 z^2 ]
|
|
% \nonumber\\
|
|
% &
|
|
% +
|
|
% d z\, d v\,
|
|
% [ S_{2\, 2} \Delta^2 u^2 + S_{+\, 2} \Delta^3 u^2 z
|
|
% + \frac{1}{4} S_{+\, +} \Delta^4 u^2 z^2 ]
|
|
\nonumber\\
|
|
&
|
|
+
|
|
\dd{z}\, \dd{v}\, \Delta u\,
|
|
[ 2 S_{-\, 2}
|
|
+ 2 (S_{2\, 2} + S_{+\, -} ) \Delta z
|
|
+ 3 S_{+\, 2} \Delta^2 z^2
|
|
+ S_{+\, +} \Delta^3 z^3
|
|
]
|
|
\nonumber\\
|
|
&
|
|
+
|
|
\dd{u}^2\,
|
|
[
|
|
S_{-\, -}
|
|
+ 2 S_{-\, 2} \Delta z
|
|
+ (S_{2\, 2} + S_{+\, -}) \Delta^2 z^2
|
|
+ S_{+\, 2} \Delta^3 z^3
|
|
+ \frac{1}{4} S_{+\, +} \Delta^4 z^4
|
|
]
|
|
\Bigr\}
|
|
\nonumber\\
|
|
&\times
|
|
e^{i\left[ k_+ v
|
|
+
|
|
\frac{2 k_+ k_- - k_2^2}{2 k_+} u
|
|
+ \frac{1}{2} \Delta^2 k_+ u \qty( z+ \frac{k_2}{\Delta k_+})^2
|
|
\right]}
|
|
.
|
|
\end{align}
|
|
Now we define the tensor on the orbifold as a sum over all images as
|
|
\begin{align}
|
|
\cN
|
|
\Psi^{[2]}_{[k\, S]}\qty(\qty[x])
|
|
&=
|
|
\sum_n ( \cK^{ n }d x) \cdot S \cdot ( \cK^{ n } \dd x)~
|
|
\psi_{k}( \cK^{ n } x)
|
|
\nonumber\\
|
|
&=
|
|
\sum_n \dd x \cdot ( \cK^{ - n }S ) \cdot \dd x~
|
|
\psi_{ \cK^{ -n } k}( x)
|
|
.
|
|
\end{align}
|
|
In the last line we have defined the induced action of the Killing vector on
|
|
$(k, S)$ which can be explicitely written as
|
|
\begin{align}
|
|
\cK^{-n}
|
|
\qty(
|
|
\begin{array}{c}
|
|
S_{ + + }\\
|
|
S_{ + 2 }\\
|
|
S_{ + - }\\
|
|
S_{ 2 2 }\\
|
|
S_{ 2 - }\\
|
|
S_{ - - }
|
|
\end{array}
|
|
)
|
|
=
|
|
%
|
|
\qty(
|
|
\begin{array}{c}
|
|
S_{ + + }\\
|
|
S_{ + 2 } + n \Delta S_{ + + }\\
|
|
S_{ + - } + n \Delta S_{ + 2 } + \frac{1}{2} n^2 \Delta^2 S_{ + + }\\
|
|
S_{ 2 2 } + 2 n \Delta S_{ + 2 } + n^2 \Delta^2 S_{ + + }\\
|
|
S_{ 2 - } + n \Delta (S_{ 2 2 } +S_{ + - })
|
|
+ \frac{3}{2} n^2 \Delta^2 S_{ + 2 } + \frac{1}{2} n^3 \Delta^3 S_{ + + }\\
|
|
S_{ - - } + 2 n \Delta S_{ - 2 }
|
|
+ n^2 \Delta^2 (S_{ 2 2 } + S_{ + - } ) + n^3 \Delta^3 S_{ + 2 }
|
|
+ \frac{1}{4} n^4 \Delta^4 S_{ + + }
|
|
\end{array}
|
|
)
|
|
.
|
|
\end{align}
|
|
|
|
In orbifold coordinates
|
|
to compute the tensor on the orbifold simply
|
|
amounts to sum over all the shifts
|
|
$z \rightarrow (z+2\pi n)$ and the use of the generalization of
|
|
\eqref{eq:sum_z_psi}, i.e. to substitute
|
|
$(\Delta\,z)^j \psi_k \rightarrow
|
|
\qty(
|
|
\frac{1}{i u} \frac{\partial}{\partial k_2}
|
|
- \frac{ k_2}{ \Delta k_+}
|
|
)^j
|
|
\Psi_{[k]}\qty(\qty[x])$.
|
|
When expressing all in the $\phi$ basis
|
|
this last step is equivalent to
|
|
$(\Delta\, z)^j \psi_k \rightarrow
|
|
\qty( \frac{l}{\Delta\, u\, k_+} )^j
|
|
+ \dots
|
|
$.
|
|
We identify the basic polaritazions on the orbifold by
|
|
considering the highest power in $u$ and get
|
|
\begin{align}
|
|
\cS_{u\,u}
|
|
&=
|
|
\frac{1}{4}{{K^4\,S_{+\,+}}}
|
|
+K^2\,S_{+\,-}
|
|
-K^3\,S_{+\,2}
|
|
+S_{-\,-}
|
|
-2\,K\,S_{-\,2}
|
|
+S_{2\,2}\,K^2
|
|
\nonumber\\
|
|
\cS_{u\,v}
|
|
&=
|
|
\frac{1}{2} {{K^2\,S_{+\,+}}}
|
|
+S_{+\,-}
|
|
-K\,S_{+\,2}
|
|
\nonumber\\
|
|
\cS_{u\,z}
|
|
&=
|
|
- \frac{1}{2} {{K^3\,S_{+\,+}}}
|
|
-K\,S_{+\,-}
|
|
+\frac{3}{2} {{K^2\,S_{+\,2}}}
|
|
+S_{-\,2}
|
|
-K\, S_{2\,2}
|
|
\nonumber\\
|
|
\cS_{v\,v}
|
|
&=
|
|
S_{+\,+}
|
|
\nonumber\\
|
|
\cS_{v\,z}
|
|
&=
|
|
S_{+\,2}-K\,S_{+\,+}
|
|
\nonumber\\
|
|
\cS_{z\,z}
|
|
&=
|
|
K^2\,S_{+\,+}-2\,K\,S_{+\,2}+S_{2\,2}
|
|
.
|
|
\end{align}
|
|
with $K= \frac{k_2}{ k_+}$.
|
|
The previous equations can be inverted into
|
|
\begin{align}
|
|
S_{-\,-}
|
|
&=
|
|
% {{K^2\,\qty(4\,\cS_{z\,z}+4\,\cS_{u\,v})+4\,K^3\,\cS_{v\,z}+K^4\,\cS_{v\,v}+8\,K\,\cS_{u\,z}+4\,\cS_{u\,u}}\over{4}}
|
|
K^2\,\qty(\cS_{z\,z}+\cS_{u\,v})
|
|
+K^3\,\cS_{v\,z}
|
|
+\frac{1}{4} K^4\,\cS_{v\,v}
|
|
+2\,K\,\cS_{u\,z}
|
|
+\cS_{u\,u}
|
|
\nonumber\\
|
|
S_{+\,-}
|
|
&=
|
|
%{{2\,K\,\cS_{v\,z}+K^2\,\cS_{v\,v}+2\,\cS_{u\,v}}\over{2}}
|
|
K\,\cS_{v\,z}
|
|
+\frac{1}{2} K^2\,\cS_{v\,v}
|
|
+\cS_{u\,v}
|
|
\nonumber\\
|
|
S_{-\,2}
|
|
&=
|
|
% {{K\,\qty(2\,\cS_{z\,z}+2\,\cS_{u\,v})+3\,K^2\,\cS_{v\,z}+K^3\,\cS_{v\,v}+2\,\cS_{u\,z}}\over{2}}
|
|
K\,\qty(\cS_{z\,z}+\cS_{u\,v})
|
|
+\frac{3}{2}\,K^2\,\cS_{v\,z}
|
|
+\frac{1}{2} K^3\,\cS_{v\,v}
|
|
+\cS_{u\,z}
|
|
\nonumber\\
|
|
S_{+\,+}
|
|
&=
|
|
\cS_{v\,v}
|
|
\nonumber\\
|
|
S_{+\,2}
|
|
&=
|
|
\cS_{v\,z}+K\,\cS_{v\,v}
|
|
\nonumber\\
|
|
S_{2\,2}
|
|
&=
|
|
\cS_{z\,z}+2\,K\,\cS_{v\,z}+K^2\,\cS_{v\,v}
|
|
.
|
|
\end{align}
|
|
Since we plan to use the previous quantities in the case of the first
|
|
massive string state we compute the relevant quantities: the trace
|
|
\begin{equation}
|
|
\tr(S)=\cS_{z\,z}-2\,\cS_{u\,v}
|
|
\end{equation}
|
|
and the transversality conditions
|
|
\begin{align}
|
|
%
|
|
% v
|
|
trans ~\cS_{v}
|
|
=&
|
|
(k\cdot S)_{+}
|
|
=
|
|
-\frac{\qty(r + \vec{k}^2)}{2\, k_+}\,
|
|
\cS_{v\,v}
|
|
-k_{+}\,\cS_{u\,v},
|
|
\nonumber\\
|
|
%
|
|
% z
|
|
trans ~\cS_{z}
|
|
=&
|
|
(k\cdot S)_{2}
|
|
-K (k\cdot S)_{+}
|
|
=
|
|
-\frac{\qty(r + \vec{k}^2)}{2\, k_+}\,
|
|
\cS_{v\,z}
|
|
-k_{+}\,\cS_{u\,z},
|
|
\nonumber\\
|
|
trans ~\cS_{u}
|
|
=&
|
|
(k\cdot S)_{-}
|
|
-K (k\cdot S)_{2}
|
|
%+ \frac{1}{2} K^2 (k\cdot S)_{+}
|
|
+ \frac{1}{2} K^2 (k\cdot S)_{+}
|
|
=
|
|
-\frac{\qty(r + \vec{k}^2)}{2\, k_+}\,
|
|
\cS_{u\,v}
|
|
-k_{+}\,\cS_{u\,u}
|
|
.
|
|
\end{align}
|
|
where we used $k_-= (r+\vec{k}^2+k_2^2)/(2 k_+)$.
|
|
These conditions correctly do no depend on $K$ since $k_2$ is
|
|
not an orbifold quantum number.
|
|
|
|
|
|
The final expression for the orbifold symmetric tensor is
|
|
\begin{align}
|
|
% \cN
|
|
\Psi^{[2]}_{[k,\, S]}\qty([x])
|
|
&
|
|
=
|
|
% \cN
|
|
\sum_l
|
|
\phi_{\kmkr}(u,v,z,\vec{x})
|
|
e^{ i l \frac{k_2}{ \Delta k_+} }
|
|
\nonumber\\
|
|
%
|
|
% v v
|
|
&
|
|
\Big\{
|
|
(\dd{v})^2\,
|
|
[\cS_{v v} ]
|
|
\nonumber\\
|
|
%
|
|
% v z
|
|
&
|
|
+
|
|
2
|
|
\Delta\, u\,
|
|
\dd{v}\, \dd{z}\,
|
|
% [ 2 S_{+\, 2}
|
|
% + S_{+\, +} \frac{l}{ \Delta\, u\,k_+} ]
|
|
\Bigl[
|
|
\cS_{v\,z}
|
|
+
|
|
\qty(
|
|
\frac{L \cS_{v\,v}}{\Delta}
|
|
)
|
|
\frac{1}{u}
|
|
\Bigr]
|
|
\nonumber\\
|
|
%
|
|
% v u
|
|
&
|
|
+
|
|
2
|
|
\dd{v}\, \dd{u}\,
|
|
% [ 2 S_{+\, -}
|
|
% + 2 S_{+\, 2} \frac{l}{\Delta\, u\, k_+}
|
|
% + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^2 ]
|
|
\Bigl[
|
|
\cS_{u\,v}
|
|
+
|
|
\qty(
|
|
\frac{L\,\cS_{v\,z}}{\Delta}+\frac{i\,\cS_{v\,v}}{2\,k_{+}}
|
|
)
|
|
\frac{1}{u}
|
|
+
|
|
\qty(
|
|
\frac{L^2\,\cS_{v\,v}}{2\,\Delta^2}
|
|
)
|
|
\frac{1}{u^2}
|
|
\Bigr]
|
|
\nonumber\\
|
|
%
|
|
% z z
|
|
&
|
|
+
|
|
(\Delta\, u)^2
|
|
\dd{z}^2\,
|
|
% [ S_{2\, 2}
|
|
% + 2 S_{+\, 2} \frac{l}{\Delta\, u\, k_+}
|
|
% + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^2 ]\,
|
|
\Bigl[
|
|
\cS_{z\,z}
|
|
+
|
|
\qty(
|
|
\frac{2\,L\,\cS_{v\,z}}{\Delta}+\frac{i\,\cS_{v\,v}}{k_{+}}
|
|
)
|
|
\frac{1}{u}
|
|
+
|
|
\qty(
|
|
\frac{L^2\,\cS_{v\,v}}{\Delta^2}
|
|
)
|
|
\frac{1}{u^2}
|
|
\Bigr]
|
|
\nonumber\\
|
|
%
|
|
% z v
|
|
&
|
|
+
|
|
2
|
|
\Delta u\,
|
|
\dd{z}\, \dd{u}\,
|
|
% [ 2 S_{-\, 2}
|
|
% + 2 ( S_{2\, 2} + S_{+\, -} ) \frac{l}{\Delta\, u\, k_+}
|
|
% + 3 S_{+\, 2} \qty(\frac{l}{\Delta\, u\,k_+})^2
|
|
% + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^3
|
|
% ]\,
|
|
\Bigl[
|
|
\cS_{u\,z}
|
|
+
|
|
\qty(
|
|
\frac{L\,\cS_{z\,z}}{\Delta}
|
|
+\frac{3\,i\,\cS_{v\,z}}{2\,k_{+}}+\frac{L\,\cS_{u\,v}}{\Delta}
|
|
)
|
|
\frac{1}{u}
|
|
+
|
|
\qty(
|
|
\frac{3\,L^2\,\cS_{v\,z}}{2\,\Delta^2}
|
|
+\frac{3\,i\,L\,\cS_{v\,v}}{2\,\Delta\,k_{+}}
|
|
)
|
|
\frac{1}{u^2}
|
|
\nonumber\\
|
|
&
|
|
\phantom{+\Delta u\,d z\, d v\,}
|
|
+
|
|
\qty(
|
|
\frac{L^3\,\cS_{v\,v}}{2\,\Delta^3}
|
|
)
|
|
\frac{1}{u^3}
|
|
\Bigr]
|
|
\nonumber\\
|
|
%
|
|
% u u
|
|
&
|
|
+
|
|
\dd{u}^2\,
|
|
% [
|
|
% S_{-\, -}
|
|
% + 2 S_{-\, 2} \frac{l}{\Delta\, u\,k_+}
|
|
% + (S_{2\, 2} + S_{+\, -}) \qty(\frac{l}{\Delta\, u\,k_+})^2
|
|
% + S_{+\, 2} \qty(\frac{l}{\Delta\, u\,k_+})^3
|
|
% + \frac{1}{4} S_{+\, +} \qty(\frac{l}{\Delta\, u\,k_+})^4
|
|
% ]
|
|
%
|
|
\Bigl[
|
|
\cS_{u\,u}
|
|
+
|
|
\qty(
|
|
\frac{i\,\cS_{z\,z}}{k_{+}}+\frac{2\,L\,\cS_{u\,z}}{\Delta}+\frac{i\,\cS_{u\,v}}{k_{+}}
|
|
)
|
|
\frac{1}{u}
|
|
+
|
|
\qty(
|
|
\frac{L^2\,\cS_{z\,z}}{\Delta^2}+\frac{3\,i\,L\,\cS_{v\,z}}{\Delta\,k_{+}}
|
|
-\frac{3\,\cS_{v\,v}}{4\,k_{+}^2}+\frac{L^2\,\cS_{u\,v}}{\Delta^2}
|
|
)
|
|
\frac{1}{u^2}
|
|
\nonumber\\
|
|
&
|
|
\phantom{ + d u^2\, }
|
|
+
|
|
\qty(
|
|
\frac{L^3\,\cS_{v\,z}}{\Delta^3}+\frac{3\,i\,L^2\,\cS_{v\,v}}{2\,\Delta^2\,k_{+}}
|
|
)
|
|
\frac{1}{u^3}
|
|
+
|
|
\qty(
|
|
\frac{L^4 \cS_{v\,v}}{4 \Delta^4}
|
|
)
|
|
\frac{1}{u^4}
|
|
\Bigr]
|
|
\Bigr\}
|
|
,
|
|
\end{align}
|
|
with $L=\frac{l}{k_+}$.
|
|
|
|
\subsection{Summary and Conclusions}
|
|
|
|
In the previous analysis it seems that string theory cannot do better than field theory when the latter does not exist, at least at the perturbative level where one deals with particles.
|
|
Moreover when spacetime becomes singular, the string massive modes are not anymore spectators.
|
|
Everything seems to suggest that issues with spacetime singularities are hidden into contact terms and interactions with massive states.
|
|
This would explain in an intuitive way why the eikonal approach to gravitational scattering works well: the eikonal is indeed concerned with three point massless interactions.
|
|
In fact it appears that the classical and quantum scattering on an electromagnetic wave~\cite{Jackiw:1992:ElectromagneticFieldsMassless} or gravitational wave~\cite{tHooft:1987:GravitonDominanceUltrahighenergy} in \bo and \nbo are well behaved.
|
|
From this point of view the ACV approach~\cite{Soldate:1987:PartialwaveUnitarityClosedstring,Amati:1987:SuperstringCollisionsPlanckian} may be more sensible, especially when considering massive external states~\cite{Black:2012:HighEnergyString}.
|
|
Finally it seems that all issues are related with the Laplacian associated with the space-like subspace with vanishing volume at the singularity.
|
|
If there is a discrete zero eigenvalue the theory develops divergences.
|
|
|
|
% vim: ft=tex
|