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Add Chan-Paton factors and SM-like scenario building

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-09-04 18:58:06 +02:00
parent 753172b4dc
commit 6322bef283
5 changed files with 433 additions and 4 deletions
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sec/part1/introduction.tex
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@@ -6,6 +6,7 @@ As a first test of validity, the string theory should properly extend the known
In particular its description in terms of fundamental strings should be able to include a gauge algebra isomorphic to that of
\begin{equation}
\SU{3}_{\rC} \otimes \SU{2}_{\rL} \otimes \U{1}_{\rY}
\label{eq:intro:smgroup}
\end{equation}
in order to reproduce known results.
For instance, string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm{} as a subset.
@@ -217,14 +218,14 @@ In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$
\begin{figure}[tbp]
\centering
\begin{subfigure}[c]{0.45\linewidth}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{\linewidth}
\import{img}{complex_plane.pdf_tex}
\caption{Radial ordering.}
\end{subfigure}
\hfill
\begin{subfigure}[c]{0.45\linewidth}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{\linewidth}
\import{img}{radial_ordering.pdf_tex}
@@ -949,7 +950,7 @@ The diamond in this case is
& & & 1 & & &
},
\end{equation}
where we used $h^{r,s} = h^{d-r, d-s}$ to stress the fact that the only independent Hodge numbers are $h^{1,1}$ and $h^{2,1}$ for $m = 3$.
where we used $h^{r,s} = h^{m-r, m-s}$ to stress the fact that the only independent Hodge numbers are $h^{1,1}$ and $h^{2,1}$ for $m = 3$.
These results will also be the starting point of~\Cref{part:deeplearning} in which the ability to predict the values of the Hodge numbers using \emph{artificial intelligence} is tested.
@@ -1159,7 +1160,7 @@ In fact the original Neumann boundary condition~\eqref{eq:tduality:bc} becomes a
0.
\end{split}
\end{equation}
The coordinate of the endpoint in the compact direction is therefore fixed and constrained on a hypersurface called \emph{Dp-brane}, where $p$ stands for the dimension of the surface (in this case $p = D - 1$):
The coordinate of the endpoint in the compact direction is therefore fixed and constrained on a hypersurface called \emph{Dp-brane}, where $p+1$ is the dimension of the surface (in this case $p = D - 2$):
\begin{equation}
\begin{split}
Y^{D-1}( \tau, \pi ) - Y^{D-1}( \tau, 0 )
@@ -1189,4 +1190,136 @@ They also present physical properties such as tension and charge~\cite{DiVecchia
However these aspects will not be discussed here as the following analysis will mainly focus on geometrical aspects of D-branes in spacetime.
\subsubsection{Gauge Groups from D-branes}
As previously stated, in order to recover $4$-dimensional physics we need to compactify the $6$ extra-dimensions of the superstring.
There are in general multiple ways to do such operation consistently~\cite{Brown:1988:NeutralizationCosmologicalConstant,Bousso:2000:QuantizationFourformFluxes,Susskind:2003:AnthropicLandscapeString,tHooft:2009:DimensionalReductionQuantum,Kachru:2003:SitterVacuaString}.
Reproducing the \sm or beyond \sm spectra are however strong constraints on the possible compactification procedures~\cite{Cleaver:2007:SearchMinimalSupersymmetric,Lust:2009:LHCStringHunter}.
Many of the physical requests usually involve the introduction of D-branes and the study of open strings in order to be able to define chiral fermions and realist gauge groups.
As seen in the previous section, D-branes introduce preferred directions of motion by restricting the hypersurface on which the open string endpoints live.
Specifically a Dp-branes breaks the original \SO{1, D-1} symmetry to $\SO{1, p} \otimes \SO{D - 1 - p}$.\footnotemark{}
\footnotetext{%
Notice that usually $D = 10$, but we keep a generic indication of the spacetime dimensions when possible.
}
The massless spectrum of the theory on the D-brane is easily computed in lightcone gauge~\cite{Goddard:1973:QuantumDynamicsMassless,Polchinski:1998:StringTheoryIntroduction,Green:1988:SuperstringTheoryIntroduction,Angelantonj:2002:OpenStrings}.
Using the residual symmetries of the two-dimensional diffeomorphism (i.e.\ armonic functions of $\tau$ and $\sigma$) we can set
\begin{equation}
X^+( \tau, \sigma ) = x_0^+ + 2 \ap\, p^+\, \tau,
\end{equation}
where $X^{\pm} = \frac{1}{\sqrt{2}} (X^0 \pm X^{D-1})$.
The vanishing of the stress-energy tensor fixes the oscillators in $X^-$ in terms of the physical transverse modes.
The mass shell condition for open strings then becomes:\footnotemark{}
\footnotetext{%
The constant $a$ in~\eqref{eq:dbranes:closedspectrum} takes here the value $-1$ from the imposition of the canonical commutation relations and a $\zeta$-regularisation.
}
\begin{equation}
M^2 = \frac{1}{\ap} \left( N - 1 \right).
\end{equation}
Consider for a moment bosonic string theory and define the usual vacuum as
\begin{equation}
\alpha_n^i \regvacuum = 0,
\qquad
n \ge 0,
\qquad
i = 1, 2, \dots, D - 2,
\end{equation}
we find that at the massless level we have a single \U{1} gauge field in the representation of the Little Group \SO{D-2}:
\begin{equation}
\cA^i
\qquad
\rightarrow
\qquad
\alpha_{-1}^i \regvacuum.
\end{equation}
The introduction of a Dp-brane however breaks the Lorentz invariance down to $\SO{1, p} \otimes \SO{D - 1 - p}$.
Thus the gauge field in the original theory is split into
\begin{equation}
\begin{split}
\cA^A
\qquad
& \rightarrow
\qquad
\alpha_{-1}^A \regvacuum,
\qquad
A = 1, \dots, p - 2,
\\
\cA^a
\qquad
& \rightarrow
\qquad
\alpha_{-1}^a \regvacuum,
\qquad
a = 1, 2, \dots, D - 1 -p.
\end{split}
\end{equation}
In the last expression $\cA^A$ forms a representation of the Little Group \SO{p-2} and as such it is a vector gauge field in $p$ dimensions.
The field $\cA^a$ form a vector representation of the group \SO{D-1-p} and from the point of view of the Lorentz group they are $D - 1 - p$ scalars in the light spectrum.
\begin{figure}[tbp]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{0.8\linewidth}
\import{img}{chanpaton.pdf_tex}
\caption{Chan-Paton factors labelling strings.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{0.7\linewidth}
\import{img}{quark.pdf_tex}
\caption{Naive model of a left handed massive quark.}
\end{subfigure}
\caption{Strings attached to different D-branes.}
\label{fig:dbranes:chanpaton}
\end{figure}
It is also possible to add non dynamical degrees of freedom to the open string endpoints.
They are known as \emph{Chan-Paton factors}~\cite{Paton:1969:GeneralizedVenezianoModel}.
They have no dynamics and do not spoil Poincaré or conformal invariance in the action of the string.
Each state can then be labelled by $i$ and $j$ running from $1$ to $N$.
Matrices $\tensor{\lambda}{^a_{ij}}$ thus form a basis for expanding wave functions and states:
\begin{equation}
\ket{n;\, a} = \sum\limits_{i,\, j = 1}^N \ket{n;\, i, j}\, \lambda^a_{ij}.
\end{equation}
In general Chan-Paton factors label the D-brane on which the endpoint of the string lives as in the left of~\Cref{fig:dbranes:chanpaton}.
Notice that strings stretching across different D-branes present an additional term in the mass shell condition proportional to the distance between the hypersurfaces: fields built using strings with Chan-Paton factors $\lambda^a_{ij}$ for which $i \neq j$ will therefore be massive.
However when $N$ D-branes coincide in space and form a stack their mass vanishes again: it then possible to organise the $N^2$ resulting massless fields in a representation of the gauge group \U{N}, thus promoting the symmetry $\bigotimes\limits_{a = 1}^N \rU_a( 1 )$ of $N$ separate D-branes.
It is also possible to show that in the field theory limit the resulting gauge theory is a Yang-Mills gauge theory.
Eventually the massless spectrum of $N$ coincident $Dp-branes$ is formed by \U{N} gauge bosons in the adjoint representation, $N^2 \times (D - 1 - p)$ scalars and $N^2$ sets of $(p+1)$-dimensional fermions~\cite{Uranga:2005:TASILecturesString}.
These are the basic building blocks for a consistent string phenomenology involving both gauge bosons and matter.
\subsubsection{Standard Model Scenarios}
Being able to describe gauge bosons and fermions is not enough.
Physics as we test it in experiments poses stringent constraints on what kind of string models we can use.
For instance there is no way to describe chirality by simply using parallel D-branes and strings stretching among them, while requiring the existence of fermions transforming in different representations of the gauge group is necessary to reproduce \sm results~\cite{Aldazabal:2000:DBranesSingularitiesBottomUp}.
For instance, in the low energy limit it is possible to build a gauge theory of the strong force using a stack of $3$ coincident D-branes and an electroweak sector using $2$ D-branes.
These stacks would separately lead to a $\U{3} \times \U{2}$ gauge theory.
It would however be theory of pure force, without matter content.
Moreover we should also worry about the extra \U{1} groups appearing: these need careful consideration but go beyond the necessary analysis for what follows.
Matter fields are notoriously fermions transforming in the bi-fundamental representation $(\vb{N}, \vb{M})$ of the \sm gauge group~\eqref{eq:intro:smgroup}.
For example left handed quarks in the \sm transform under the $(\vb{3}, \vb{2})$ representation of the group $\SU{3}_C \otimes \SU{2}_L$.
This is realised in string theory by a string stretched across two stacks of $3$ and $2$ D-branes as in the right of~\Cref{fig:dbranes:chanpaton}.
The fermion would then be characterised by the charge under the gauge bosons living on the D-branes.
The corresponding anti-particle would then simply be a string oriented in the opposite direction.
Things get complicated when introducing also left handed leptons transforming in the $(\vb{1}, \vb{2})$ representation: they cannot have endpoints on the same stack of D-branes as quarks since they do not have colour charge.
We therefore need to introduce more D-branes to account for all the possible combinations.
An additional issue comes from the requirement of chirality.
Strings stretched across D-branes are naturally massive but, in the field theory limit, a mass term would mix different chiralities.
We thus need to include a symmetry preserving mechanism for generating the mass of fermions.
In string theory there are ways to deal with the requirement~\cite{Uranga:2003:ChiralFourdimensionalString,Uranga:2005:TASILecturesString,Zwiebach::FirstCourseString,Aldazabal:2000:DBranesSingularitiesBottomUp}.
These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles~\cite{Finotello:2019:ClassicalSolutionBosonic}.
We focus in particular on the latter.
Specifically we focus on intersecting D6-branes filling the $4$-dimensional spacetime and whose additional $3$ dimensions are embedded in a \cy 3-fold (e.g.\ as lines in a factorised torus $T^6 = T^2 \times T^2 \times T^2$).
% vim ft=tex