Amplitudes at SCGP

The Simons Center for Geometry and Physics in Stony Brook is going to be running the program Physics and Mathematics of Scattering Amplitudes during the Fall 2013 semester. It started last week with talks by Henrietta Elvang and Johannes Henn. Sadly, I am going to miss the talks in person. But thankfully, there are going to be videos of most of the talks. The talks from last week can be found here and here.

Defense Success!

Yesterday, I successfully defended my dissertation in Stony Brook. It was a great moment for me. I thank my committee for helping me understand my own work. You can read my dissertation here, and my slides here. Maybe I will talk more about my work soon.

Three-body interaction for particle action

Consider a scalar quantum field theory with the following interaction vertices:\[g_{F} |\Phi|^{2} A + \frac{h_{F}}{6}A^{3}\]where \(g_{F}\) and \(h_{F}\) are two coupling strengths for cubic interactions. In the particle theory we can consider a three-body system with a three-body interaction (i.e. not pair-wise) that has the form\[S_{3}\left[ q_{14}, \, q_{25}, \, q_{36} \right] \equiv g_{P}^{3} h_{F} \int\limits_{0}^{T_{14}} \mathrm{d}\tau \int\limits_{0}^{T_{25}} \mathrm{d}\sigma \int\limits_{0}^{T_{36}} \mathrm{d}\rho \int \mathrm{d}Y \, G_{A}\left[ q_{14}(\tau)| Y \right] G_{A}\left[ q_{25}(\sigma)| Y \right] G_{A}\left[ q_{36}(\rho)| Y \right] \]In the eikonal approximation we will get a three-body Gram invariant from this term in the action.

Gram invariants in momentum space

Given two vectors \(k_{1}\) and \(k_{2}\) in a $D$-dimensional euclidean space we can consider the parallelogram formed with them. The area of this parallelogram is given by the square root of the determinant of the Gram matrix \(G_{12}\):\[G_{12} = \begin{pmatrix} k_{1}^{2} & k_{1} \cdot k_{2} \\ k_{1} \cdot k_{2} & k_{2}^{2} \end{pmatrix} \quad \Longrightarrow \quad \det{(G_{12})} = k_{1}^{2} k_{2}^{2} - (k_{1} \cdot k_{2})^{2}\]We will denote the magnitude of each vector by \(|k_{i}| \equiv m_{i}\). Then it follows that\[\det{(G_{12})} = \left[m_{1}m_{2} - k_{1} \cdot k_{2} \right]\left[m_{1}m_{2} + k_{1} \cdot k_{2} \right]\]Introducing the invariant \(s_{12} \equiv (k_{1} + k_{2})^{2}\) we can write\[\det{(G_{12})} = \frac{1}{4} \left[ s_{12} - (m_{1} - m_{2})^{2} \right] \left[ (m_{1} + m_{2})^{2} - s_{12} \right]\]Hence, the area of the parallelogram generated by \(k_{1}\) and \(k_{2}\) is\[A_{12} = \frac{1}{2} \sqrt{\left[ s_{12} - (m_{1} - m_{2})^{2} \right] \left[ (m_{1} + m_{2})^{2} - s_{12} \right]}\]Note that this area is real in the domain\[(m_{1} - m_{2})^{2} \leq s_{12} \leq (m_{1} + m_{2})^{2}\]The notation that we have used is very suggestive. If the two vectors were momentum vectors, then \(m_{i}\) would correspond to masses and \(s_{12}\) would correspond to a Mandelstam invariant. Then \( (m_{1}+m_{2})^{2} \) is a mass threshold and \( (m_{1} - m_{2})^{2} \) a mass pseudothreshold.

We can study the case with three vectors. Now we have three masses and hence three three-body mass pseudothresholds:\[(m_{1} - m_{2} + m_{3})^{2} \qquad (m_{1} + m_{2} - m_{3})^{2} \qquad (m_{1} - m_{2} - m_{3})^{2}\]We also have three (two-body) Mandelstam invariants:\[s_{12} \equiv (k_{1} + k_{2})^{2} \qquad s_{23} \equiv (k_{2} + k_{3})^{2} \qquad s_{31} \equiv (k_{3} + k_{1})^{2}\]It might be convenient to introduce a three-body Mandelstam invariant:\[t_{123} \equiv (k_{1} + k_{2} + k_{3})^{2} = s_{12} + s_{23} + s_{31} - m_{1}^{2} - m_{2}^{2} - m_{3}^{2}\]Note that\[4 (m_{1}^{2} + m_{2}^{2} + m_{3}^{2}) = (m_{1} + m_{2} + m_{3})^{2} + (m_{1} - m_{2} + m_{3})^{2} +(m_{1} + m_{2} - m_{3})^{2} + (m_{1} - m_{2} - m_{3})^{2}\]Now the three vectors will form a parallelepiped whose volume is given by the square root of the Gram determinant:\[V_{123} \equiv \sqrt{\det{(G_{123})}}\]What is driving me nuts is how to write this determinant in a nice form that involves the above invariants. Naively we have\[\det{(G_{123})} = m_{1}^{2}m_{2}^{2}m_{3}^{2} - m_{1}^{2} (k_{2} \cdot k_{3})^{2} - m_{2}^{2}(k_{1} \cdot k_{3})^{2} - m_{3}^{2} (k_{1} \cdot k_{2})^{2} + 2 (k_{1} \cdot k_{2})(k_{2} \cdot k_{3})(k_{3} \cdot k_{1})\]The secret might be with traces.

Multivariate weight functions

These two look promising. One for bivariate Hermite polynomials:\[ w_{H}(x, \, y|z) = \frac{1}{\sqrt{1 - z^{2}}} \exp{\left( \frac{2 x y z -x^{2} - y^{2}}{1 - z^{2}} \right)} \qquad -1 < z < 1\]and one for bivariate Gegenbauer polynomials:\[w_{G}(x, \, y|z) = \frac{1}{\sqrt{1 - z^{2}}} \left( \frac{1 + 2xyz - x^{2} - y^{2} - z^{2}}{1 - z^{2}} \right)^{\alpha} \qquad -1 < z < 1\]

Four-point Kinematics: Momentum Basis

We have four external momentum vectors:\[k_{1} \qquad k_{2} \qquad k_{3} \qquad k_{4}\]In the $s$-channel we take $k_{1}$ and $k_{2}$ to be incoming. The external momenta are constrained by the conservation condition:\[k_{1} + k_{2} = k_{3} + k_{4}\]and the on-shell conditions:\[k_{1}^{2} = -m_{1}^{2} \qquad k_{2}^{2} = -m_{2}^{2} \qquad k_{3}^{2} = -m_{3}^{2} \qquad k_{4}^{2} = -m_{4}^{2}\]The Mandelstam invariants are defined as\[s \equiv - (k_{1} + k_{2})^{2} \qquad t \equiv - (k_{1} - k_{4})^{2} \qquad u \equiv - (k_{1} - k_{3})^{2}\]Note that these three invariants are not independent,\[s + t + u = m_{1}^{2} + m_{2}^{2} + m_{3}^{2} + m_{4}^{2}\]Using momentum conservation, we can also write\[s = -(k_{3} + k_{4})^{2} \qquad t = -(k_{3} - k_{2})^{2} \qquad u = -(k_{4} - k_{2})^{2}\]We now write all other inner products in terms of the massess and the three Mandelstam invariants:\[k_{1} \cdot k_{2} = \frac{1}{2} \left[ m_{1}^{2} + m_{2}^{2} - s \right] \qquad k_{1} \cdot k_{3} = \frac{1}{2} \left[ u - m_{1}^{2} - m_{3}^{2} \right] \qquad k_{1} \cdot k_{4} = \frac{1}{2} \left[ t - m_{1}^{2} - m_{4}^{2} \right]\]\[k_{2} \cdot k_{3} = \frac{1}{2} \left[ t - m_{2}^{2} - m_{3}^{2} \right] \qquad k_{2} \cdot k_{4} = \frac{1}{2} \left[ u - m_{2}^{2} - m_{4}^{2} \right]  \qquad k_{3} \cdot k_{4} = \frac{1}{2} \left[m_{3}^{2} + m_{4}^{2} - s \right]\]

Massive Scalar Propagator in Euclidean Space

In Euclidean space, the propagator for a massive scalar field with mass \(M > 0\) is\[G(x|y) \equiv \left( \frac{\hbar}{\mu} \right) \int\limits_{0}^{\infty} \mathrm{d}\xi \left( \frac{\mu}{\hbar \xi} \right)^{D/2} \exp{\left( -\frac{\mu (x - y)^{2}}{2 \hbar \xi} - \frac{M^{2} \xi}{2 \hbar \mu} \right)} \]where $D$ is the dimension of space and $\mu$ is a constant with units of mass. We have used a Schwinger parameter $\xi$ to write the propagator. This Schwinger parameter has units of length. The integral over $\xi$ can be performed explicitly to give\[G(x|y) = 2 \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \left( \frac{\hbar}{\mu |x - y|} \right)^{\Delta} \left( \frac{M}{\mu} \right)^{\Delta} K_{\Delta} \left( \frac{M |x - y|}{\hbar} \right) \qquad \Delta \equiv \frac{D - 2}{2}\]where $K_{\Delta}$ is a modified Bessel function.

The semiclassical propagator is defined by taking $\hbar \rightarrow 0$. This corresponds to either using the asymptotic expansion for the Bessel function, or performing the integral over $\xi$ with saddle-point approximations. The result is\[G(x|y) \approx \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \left( \frac{M}{\mu} \right)^{(D - 3) / 2} \left( \frac{\hbar^{2}}{\mu^{2} (x - y)^{2}} \right)^{(D - 1)/2} \exp{\left(- \frac{M |x - y|}{\hbar} \right)}\]We write this as an infinite series:\[G(x|y) \approx \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{\Gamma(n + 1)} \left( \frac{M}{\mu} \right)^{(2\Delta + 2n - 1)/2} \left( \frac{\hbar^{2}}{\mu^{2} (x - y)^{2}} \right)^{(2 \Delta - n + 1)/2}\]

Numerology with scalar fields

Today is a good day to do some numerology, being March 14th and Einstein's birthday. Consider a quantum field theory in \(D\) spacetime dimensions with scalars \(\varphi\) and an interaction of the form \(g_{n}\varphi^{n}\). We are going to measure everything in units of mass and units of \( \hbar \). Taking the speed of light to be dimensionless leads to length, time, momentum and energy written in terms of mass and \( \hbar \). For example, the action has units of \( \hbar \). The scalar field has units\[ \left[ \varphi \right] =  \left( \frac{3 - D}{2} \right) \left[ \hbar \right] + \left( \frac{D - 2}{2} \right) \left[ M \right] \]The dimension of the coupling satisfies\[ \left[ g_{n} \right] = \left[ \hbar \right] - n \left[ \varphi \right] - D \left[ L \right] \]Using the dimension for \(\varphi\) and the fact that \(\left[ L \right] = \left[ \hbar \right] - \left[ M \right] \) we get\[\left[ g_{n} \right] = \left( \frac{(n - 2)D + 2 - 3n}{2} \right) \left[ \hbar \right] + \left( \frac{(2-n)D + 2n}{2} \right) \left[ M \right]\]Here comes the numerology. For \(n = 3\) one gets\[\left[ g_{3} \right] = \left( \frac{D - 7}{2} \right) \left[ \hbar \right] + \left( \frac{6 - D}{2} \right) \left[ M \right]\]For \(n = 4\) one gets\[\left[ g_{4} \right] = \left( D - 5 \right) \left[ \hbar \right] + \left( 4 - D \right) \left[ M \right]\]And finally, for \(n = 6\) one get\[\left[ g_{6} \right] = 2\left( D - 4 \right) \left[ \hbar \right] + 2\left( 3 - D \right) \left[ M \right]\]In all three cases the mass dimension of the coupling vanishes in a dimension lower than the \(\hbar\) dimension. For \(n = 3\) we have vanishing mass dimension in \(D = 6\) and vanishing \(\hbar\) dimension in \(D = 7\). For \(n = 4\) we have vanishing mass dimension in \(D = 4\) and vanishing \(\hbar\) dimension in \(D = 5\). And finally for \(n = 6\) we have vanishing mass dimension in \(D = 3\) and vanishing \(\hbar\) dimension in \(D = 4\). These three cases coincide with the three canonical cases of \(adS_{D+1} / CFT_{D}\). Indeed, \(\mathcal{N} = 4\) super Yang-Mills in four spacetime dimensions contains scalar fields that interact with a quartic vertex and ABJM theory in three spacetime dimensions has scalar fields that interact via a sextic vertex. I guess this means that the theory related to the \(\mathcal{N} = (2, \, 0)\) theory in six spacetime dimensions has scalars interacting via a cubic vertex.

Actually, the coefficient of \(\left[M\right]\) in \(D\) dimensions equals the negative of the coefficient of \(\left[ \hbar \right]\) in \(D+1\) dimensions for any value of \(n\). However, only for the three cases above we have integer solutions.

If one performs the same analysis on the coupling constant in Yang-Mills theory, the dimension comes out like\[\left[ g_{YM} \right] = \left( \frac{D - 3}{2} \right)\left[\hbar \right] + \left( \frac{4 - D}{2} \right) \left[ M \right]\]However, the coupling appears in the field theory action in the combination \(g_{YM} / \hbar\) and this leads to\[\left[ g_{YM} \right] - \left[\hbar \right] = \left( \frac{D - 5}{2} \right)\left[\hbar \right] + \left( \frac{4 - D}{2} \right) \left[ M \right]\]

Coulomb electrostatics in arbitrary spatial dimensions

In three spatial dimensions, the electrostatic potential \(\phi(\mathbf{x})\) satisfies Poisson's equation:\[\left(\frac{\partial}{\partial \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \right) \phi(\mathbf{x}) = - \frac{\rho(\mathbf{x})}{\varepsilon_{0}}\]The solution to this partial differential equation can be written down in terms of the Green function for Laplace's operator:\[\phi(\mathbf{x}) = \phi_{0}(\mathbf{x}) + \int dy \left[ G(\mathbf{x}, \, \mathbf{y}) \rho(\mathbf{y})\right] \]where \(\phi_{0}\) is a harmonic function and \(G\) satisfies\[\left(\frac{\partial}{\partial \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \right) G(\mathbf{x}, \, \mathbf{y}) = -\frac{1}{\varepsilon_{0}}\delta(\mathbf{x} - \mathbf{y})\]In order to find the Green function, we write it as the Fourier transform of another function:\[G(\mathbf{x}, \, \mathbf{y}) = \int \int dbdc \left[ g(\mathbf{b}, \, \mathbf{c}) \exp{\left( -i \mathbf{x} \cdot \mathbf{b} +i\mathbf{y} \cdot \mathbf{c}\right)} \right]\]This leads to\[g(\mathbf{b}, \, \mathbf{c}) = \frac{1}{\varepsilon_{0}} \frac{\delta(\mathbf{b} - \mathbf{c})}{\mathbf{b}^{2}}= \frac{1}{2\varepsilon_{0}} \delta(\mathbf{b} - \mathbf{c}) \int_{0}^{\infty}d\tau \, \exp{\left(- \frac{\tau}{2}\mathbf{b}^{2} \right)} \]Then, the Green function yields\[G(\mathbf{x}, \, \mathbf{y}) = \frac{1}{2\varepsilon_{0}} \int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{3} \exp{\left(-\frac{1}{2\tau}(\mathbf{x} - \mathbf{y})^{2} \right)}\]This expression can be generalize to \(D\) spatial dimensions: \[G(\mathbf{x}, \, \mathbf{y}) = \frac{1}{2 \varepsilon_{D}}\int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{D} \exp{\left(- \frac{1}{2\tau} (\mathbf{x} - \mathbf{y})^{2} \right)}\]where we have introduced \(\varepsilon_{D}\) as the \(D\)-dimensional analog of \(\varepsilon_{0}\). The Coulomb potential corresponds to a localized charge density:\[\rho(\mathbf{x}) = e \delta(\mathbf{x} - \mathbf{x}_{e})\]In \(D\) spatial dimensions, the Coulomb potential is\[\phi_{C}(\mathbf{x}) = \frac{e}{2 \varepsilon_{D}}\int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{D} \exp{\left(- \frac{1}{2\tau} (\mathbf{x} - \mathbf{x}_{e})^{2} \right)} = \frac{e}{2 \varepsilon_{D}} \left( \frac{2}{(\mathbf{x} - \mathbf{x}_{e})^{2}} \right)^{\nu}\Gamma(\nu) \qquad \nu \equiv \frac{D - 2}{2}\]This expression reduces to the familiar form when \(D = 3\).

In three spatial dimensions, the Green function can be written as a Legendre series. For arbitrary spatial dimension \(D\) the Legendre series generalizes to a Gegenbauer series. With \(|\mathbf{x}| > |\mathbf{y}|\) this yields\[G(\mathbf{x}, \, \mathbf{y}) = \frac{\Gamma(\nu)}{2\varepsilon_{D}}\left(\frac{2}{|\mathbf{x}| |\mathbf{y}|}\right)^{\nu}\sum_{n = 0}^{\infty}\eta^{n+\nu}C_{n}^{\hphantom{n}\nu}(\xi)\]where\[\eta \equiv \frac{|\mathbf{y}|}{|\mathbf{x}|} \qquad \xi \equiv \frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}| |\mathbf{y}|}\]

Multivariate polynomials

While doing research, I came across some interesting multivariate versions of familiar orthogonal polynomials like Legendre or Chebyshev. In the end, they appear to not be very useful for my particular research problem. Still, I want to write a bit about what I found because they are kind of cute.