Non-Polynomial Scalar Interactions | M.E. Irizarry-Gelpí

Thu 20 April 2017
Physics
#scalars

A common interaction in quantum field theory is a non-linear term in the lagrangian of the form

$$ g_{n} \vert \phi \vert^{2} A^{n}, $$

with $n$ being an integer greater or equal to one. According to dimensional analysis, in $D$ spacetime dimensions the coupling $g_{n}$ has a mass dimension given by

$$ \Delta = \frac{4 +2n - nD}{2}. $$

The cases when $\Delta = 0$ are very important:

$$ (D = 3, \, n = 4), \qquad (D = 4, \, n = 2), \qquad (D = 6, \, n = 1). $$

For many important reasons, any serious quantum field theory should live in a spacetime with integer number of dimensions, and only have positive integer powers of fields in the lagrangian. However, we can relax a bit and ask two different questions:

What values of $n$ give $\Delta = 0$ for given value of $D$?
What values of $D$ give $\Delta = 0$ for given value of $n$?

If $\Delta = 0$, then you have

$$ n = \frac{4}{D - 2}. $$

Obviously, $D = 2$ is problematic. The first non-trivial case is $D = 5$, giving $n = 4/3$. Beyond six spacetime dimensions, you get:

$$ (D, n) = (7, 4/5), \quad (8, 2/3), \quad (9, 4/7), \quad (10, 1/2), \quad (11, 4/9), \quad ... $$

On the other hand, if $\Delta = 0$, then you also have

$$ D = 2 + \frac{4}{n}. $$

Again $D = 2$ is problematic. The first non-trivial case is $n = 3$, giving $D = 10/3$. Note that

$$ 3 < \frac{10}{3} < 4. $$

There are spaces with fractional number of dimensions: fractals. So I am curious if one could define a field theory living in a space with a rational number of dimensions.

Beyond the sixth-order interaction, you get:

$$ (D, n) = (14/5, 5), \quad (8/3, 6), \quad (18/7, 7), \quad (5/2, 8), \quad (22/9, 9), \quad ...$$

A coupling with vanishing mass dimension is a good candidate for a conformal field theory.