One year in Edinburgh, string amplitudes and the KLT relations

My second semester at the MSc in Mathematical Physics at the University of Edinburgh is almost done (2 weeks) and I'm getting ready for examinations. The assessment scheme is a pretty tough one: I've attended twelve courses since September -some of which ended up being much harder and workload intense than I thought (e.g. linear analysis, advanced QFT, QCD, Lie groups/algebras, standard model)- and 9 of my examinations will count between 80% and 100% towards my final mark; furthermore, I need an average of above 50% with no more than 3 marks below 50% in order to get the degree.

So yes, it's crazy. And I shouldn't be wasting time writing here I guess, but at least I've started by (kind of) blocking my access to facebook, which is a huge distraction.

It's a bit ironic, but throughout the year I became uninspired to follow the path of mathematical physics and even the academic one (completely the opposite to what I expected). The reasons are varied but I guess I can just put it simply as a feeling of not belonging. It's not that I suddenly found that I don't like mathematical physics, I obviously love it, but considering it a full time job in Academia is a somewhat different matter. I'm still considering the pursuit of a PhD in Quantum Information, though, but I think I'll narrow my options to the U.S/Canada and Mexico, if I decide otherwise, God knows (whatever that means) what I'll end up doing.

In other news, one of the courses I'm taking concerns giving two 25 minute talks related to the dissertation topic that one has chosen. My dissertation will be about the so-called colour-kinematics duality: to design and implement Mathematica code which creates the relevant (loop-level) diagrams, determines the Jacobi relations among them, and allows for these objects to be manipulated.

There is a whole other world -unknown to me until now- about this and the most surprising aspect of this colour-kinematics duality is that it enables to obtain a gravity theory scattering amplitudes very easily as (prop to) the square of a gauge theory amplitudes. This then connects with another set of relations between scattering amplitudes in string theory known as the KLT-relations, which so happen to reduce to this gravity = gauge theory squared relations in the low-energy (point-like) limit.

There are two very nice popular-science level stories about this:

• Gravity is Yang-Mills Squared at the blog 4 gravitons
• Betting on the Future of Quantum Gravity at Quanta Magazine

What I did for my first presentation was to talk about string amplitudes and KLT: what does these things mean and how to interpret what the KLT relations are saying. It was a bit of a detour because I won't really be working with string theory, however I wanted to find out at least the basics of where this thing come from. For anyone who knows string theory well, I didn't say anything new and most of the slides are contained merely in the introduction of Superstring Theory by Witten, Green and Schwarz.

For anyone really serious about the topic, this paper looks like a really good review:

Perturbative Gravity and Gauge Theory Relations -- A Review
Thomas Sondergaard
arXiv:1106.0033 [hep-th]

In my second talk I'll look in a bit of a detail (as time allows me to prepare) to colour-kinematics (BCJ duality, the Kleiss-Kuijf relations, the double copy and other such words) and probably I'll share something about it here.

These are my slides about string amplitudes and the KLT relations (handout mode):

SUSY en el LHC, Cuerdas sin SUSY y lo que todo físico debería saber sobre Cuerdas

Hace poco encontré estos vídeos sobre supersimetría y teoría de cuerdas, dos temas íntimamente relacionados:

Supersimetría: el LHC regresa a la búsqueda de nueva física
Este vídeo está dirigido al público en general. La anfitriona entrevista a varios físicos, entre ellos a John Ellis, quien habla acerca del Run II del LHC y en particular sobre la búsqueda de nueva física y supersimetría. La explicación a nivel divulgativo es la usual, pero las entrevistas son interesantes y se presenta también el lado más escéptico de la propuesta de SUSY con Alvaro De Rújula que bromea un poco sobre viajar al pasado.

Entre otras cosas, si no se encuentra SUSY en el Run II del LHC, John Ellis señala que tendrá que limpiar su oficina.

El escritorio de John Ellis @CERN

Teoría de cuerdas en un universo sin supersimetría
Luego del Run I del LHC, los medios y "algunos físicos" comenzaron a decir que SUSY estaba muerto. En realidad, y como se explica en el vídeo anterior, SUSY podría estar en cualquier escala de energía y podría incluso ser inalcanzable por el LHC. De cualquier modo esto mete en problemas a muchos teóricos y en particular a los que se dedican a la construcción de modelos realistas de partículas (tipo modelo estándar) a partir de teoría de cuerdas (fenomenología de cuerdas).

Algunos recursos para abordar este tema conceptualmente son:

What if the LHC doesn't see SUSY? en Physics SE
Why string theory implies supersymmetry en The Reference Frame

Saúl Ramos, investigador del IFUNAM, aborda el tema de Cuerdas sin SUSY (y explica exactamente a qué se refiere esto en el tiempo 27:00 gracias a una pregunta) precisamente motivado por el hecho de que no ha habido evidencia alguna de SUSY en el LHC, entre otros.

El squark stop es la superpartícula asociada al top quark y era una de las primeras que se esperaba en el LHC 

La charla no es divulgativa (seminario de Altas Energías en la UNAM) pero es accesible para quien esté interesado sin ser experto. El audio es relativamente malo pero vale la pena intentar seguirlo (como sea se agradece la iniciativa del camarógrafo) y la acción en específico sobre la construcción de N=0 comienza en el tiempo 21:00.


Finalmente, no recuerdo si durante la presentación se cita el artículo en que está basada la presentación, pero parece que es este trabajo:

Non-supersymmetric heterotic model building
Michael Blaszczyk, Stefan Groot Nibbelink, Orestis Loukas, Saúl Ramos-Sánchez

We investigate orbifold and smooth Calabi-Yau compactifications of the non-supersymmetric heterotic SO(16)×SO(16) string. We focus on such Calabi-Yau backgrounds in order to recycle commonly employed techniques, like index theorems and cohomology theory, to determine both the fermionic and bosonic 4D spectra. We argue that the N=0 theory never leads to tachyons on smooth Calabi-Yaus in the large volume approximation. As twisted tachyons may arise on certain singular orbifolds, we conjecture that such tachyonic states are lifted in the full blow-up. We perform model searches on selected orbifold geometries. In particular, we construct an explicit example of a Standard Model-like theory with three generations and a single Higgs field.

Lo que todo físico debería saber sobre teoría de cuerdas
Al fin una charla de Eddie Witten que todos podemos entender... todos los físicos... todos los que conocen, al menos de lejitos, QFT y obviamente relatividad general.

Las diapositivas de la charla están disponibles aquí a través del sitio del Strings 2015.

A groso modo, en la primera parte, Witten comienza con la que sería una formulación 1-dimensional de gravedad cuántica, en la que el tensor de Riemann es idénticamente nulo y así la curvatura de Ricci y la acción de Einstein-Hilbert, con D campos escalares y llega a una teoría de campo en un espaciotiempo D-dimensional curvo. Luego incluso, con este modelo unidimensional, continúa calculando el propagador de una partícula de un punto a otro e introduce interacciones. Finalmente considera el caso dos dimensional, explica por qué Cuerdas es una teoría de gravedad cuántica (no esperen conocer al gravitón, aunque sólo quedan unas patadillas para describirlo) y discute cómo es que en cuerdas desaparecen las divergencias ultravioleta.


Hay además disponible en YouTube una charla de 2013 del mismo Witten llamada Feynman diagrams in String Theory que parece ser esencialmente la misma, aunque quizá con un poco más de detalle (en la primera diapositiva menciona "I will aim to explain the minimum about string perturbation theory that every quantum physicist should know"). Finalmente en algo con relación únicamente a Witten: How does Edward Witten know so much math?.

http://abstrusegoose.com/332

Null geodesics and affine parameters

I've been away for a while preparing my IELTS test and I hope I'll be ok, whatever my fate is! So, concerning this entry, it came to mind this Physics SE answer I wrote some time ago, where the original poster commented he was only concerned with proper time. In turn I posted this link to another Physics SE answer which deals with what an affine parameter is.

So, the thing is pretty straightforward: the definition of proper time is just the arc length
\begin{equation}\tau=\int\sqrt{-ds^2}\end{equation} and for null geodesics, (by definition) $ds^2=0$, so that's why "proper time assigns the same value to all points (on a null) geodesic". For spacelike and timelike geodesics, $ds^2\neq0$ and $\dot{s}^2=\pm1$ when proper time is the parameter.

Now, recall the geodesic equations for a curve $x=x(\lambda)$ on an arbitrary parameter $\lambda$ with a tangent vector $U\equiv\frac{dx}{d\lambda}$,
\begin{equation}\left(\frac{dU}{d\lambda}\right)^\alpha=\frac{d^2x^\alpha}{d\lambda^2}+\frac{dx^\beta}{d\lambda}\frac{dx^\mu}{d\lambda}{\Gamma^\alpha}_{\mu\beta}=0\end{equation} usually written shorthand as
\begin{equation}(\nabla_{U}{U})^\alpha\equiv{U}^\beta\nabla_\beta{U}^\alpha=0\label{dagger}\end{equation}
So, let's now consider another arbitrary parameter ${\sigma=\sigma(\lambda)}$. Then,
\begin{align}\frac{d}{d\lambda}&=\frac{d\sigma}{d\lambda}\frac{d}{d\sigma}\nonumber\\[0.2in]
\frac{d^2}{d\lambda^2}&=\left(\frac{d\sigma}{d\lambda}\frac{d}{d\sigma}\right)\left(\frac{d\sigma}{d\lambda}\frac{d}{d\sigma}\right)\nonumber\\&=\frac{d\sigma}{d\lambda}\left[\frac{d}{d\sigma}\left(\frac{d\sigma}{d\lambda}\right)\frac{d}{d\sigma}+\frac{d\sigma}{d\lambda}\frac{d^2}{d\sigma^2}\right]\nonumber\\&=\frac{d\sigma}{d\lambda}\left[\frac{\left(\frac{d^2\sigma}{d\lambda^2}\right)}{\frac{d\sigma}{d\lambda}}\frac{d}{d\sigma}+\frac{d\sigma}{d\lambda}\frac{d^2}{d\sigma^2}\right]\end{align} and the geodesic equations on $\sigma$ are
\begin{equation}\frac{\left(\frac{d^2\sigma}{d\lambda^2}\right)}{\frac{d\sigma}{d\lambda}}\frac{dx^\alpha}{d\sigma}+\frac{d\sigma}{d\lambda}\frac{d^2x^\alpha}{d\sigma^2}+\frac{d\sigma}{d\lambda}\frac{dx^\beta}{d\sigma}\frac{dx^\mu}{d\sigma}{\Gamma^\alpha}_{\mu\beta}=0\end{equation} i.e., from (\ref{dagger}),
\begin{equation}U^\beta\nabla_\beta{U}^\alpha=-\frac{\left(\frac{d^2\sigma}{d\lambda^2}\right)}{\left(\frac{d\sigma}{d\lambda}\right)^2}U^\alpha\end{equation} so that from the start, in principle, one could've defined the geodesic equations as ${\nabla_UU\propto{U}}$; however, if
\begin{equation}\frac{d^2\sigma}{d\lambda^2}=0\,\,\Longrightarrow\,\,\sigma=a\lambda+b\label{ddagger}\end{equation} (\ref{dagger}) is recovered. These $\sigma$'s are called affine parameters, and this is why they have this simple form.

Now, (I think) this is where trouble comes. Some authors, e.g. Carroll, define an affine parameter as any parameter related to the proper time $\tau$ as $\sigma=a\tau+b$. The thing is that also, usually it is said that one should use an affine parameter for null geodesics (as in the previous Physics SE answer I mentioned before), but still with this last definition you end up with a constant parameter. So what is generally understood as an affine parameter is simply that with which (\ref{dagger}) is satisfied. For null geodesics, (\ref{dagger}) is trivially satisfied with $\tau$ (or any linear combination whatsoever) so that this parameter is indeed useless; however one is free to use any affine parameter as defined in (\ref{ddagger}) with non-constant $\lambda\in\mathbb{R}$ regardless of its physical relevance.

Now, the term $\nabla_{U}{U}=\frac{dU}{d\lambda}=\frac{d^2x}{d\lambda^2}$ is indeed the acceleration of an observer along the curve $x=x(\lambda)$, so that when using an affine parameter, the observer is not accelerating, while au contraire, with a non-affine parameter, the observer will be accelerated parallel to the direction of movement. This is the main relevance of this sort of parameter; in General Relativity it's the custom to deal with affine parameters where $\tau$, of course, is our favorite one for timelike geodesics, whereas with null geodesics people usually don't care what the parameter is as long as it works, at the end it's all about a goofy parameter ;-)