# Orthogonal and linear

1. Sep 4, 2007

### touqra

I was reading the literature, and come across this. But I couldn't decipher what it means by linear combination and orthogonal combination.

It is sufficient to consider two axionic fields, A and B, with a potential:

$$V = \lambda_{1}^4 [1 - cos(\frac{\theta}{f_1} + \frac{\rho}{g_1})] + \lambda_{2}^4 [1 - cos(\frac{\theta}{f_2} + \frac{\rho}{g_2})]$$

It is easy to see that, when the condition

$$\frac{f_1}{g_1} = \frac{f_2}{g_2}$$

is met, the same linear combination of the two axions appears in both terms. Hence, the orthogonal combination is a flat direction of V.

Last edited: Sep 4, 2007