1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Pierre Ramond-Journeys Beyond the Standard Model

  1. Jun 21, 2016 #1
    1. The problem statement, all variables and given/known data
    I want to diagonalize the quadratic form
    $$ m_0((m_u+m_d)\pi^3\pi^3+\frac{2}{\sqrt{3}}(m_u-m_d)\pi^3\pi^8+\frac{1}{3}(m_u+m_d+4m_s)\pi^8\pi^8)$$
    which can be found under equation 5.47, in order to get the mass of the η and ##\pi^0## pions. This quadratic form is produced by the term that breaks the Chiral Symmetry ##SU_L(3)\times SU_R(3) ##

    2. Relevant equations
    I want to know how to produce the result 5.48 as i am unable till now to do the calculation.

    3. The attempt at a solution
    I tried to diagonalize the expression using the standard procedure of orthogonal diagonalization. After doing all the calculations i get for the ##\pi^0##: $$m^2_{\pi^0} = m_0\bigg(m_u+m_d-\frac{(m_u-m_d)^2}{2\sqrt{m^2_u+m^2_d+m^2_s-m_um_d-m_um_s-m_dm_s}+2m_s-m_u-m_d}\bigg)$$

    which is incorrect. Can anyone help me to figure out what i am doing wrong.
    Thank you very much!!
    Last edited: Jun 21, 2016
  2. jcsd
  3. Jun 21, 2016 #2
    Actually i found it. Taking the taylor expansion for the square root at ##m_s>>m_{u,d}## produces the right answer!! Thank you very much!!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted