# Quick Question (EW gaugino mass relation)

1. Jun 7, 2010

### samr

Can't for the life of me remember the characteristic EW gaugino mass relation for CMSSM, nor can I track it down.

Anyone know it off the top of their head or link me a paper?

(to clarify, I mean that mSUGRA is commonly defined by the gaugino mass relation 1:2:6 for $$M_1:M_2:M_3$$)

(maybe more usefully that's $$M_\textrm{bino}:M_\textrm{wino}:M_\textrm{gluino}$$)

2. Jun 7, 2010

### BenTheMan

What's the difference between msugra and cmssm, in terms of the boundary conditions? I thought they were interchangable: both have universal gaugino masses at the GUT scale, along with a single A term and universal scalars.

3. Jun 7, 2010

### samr

To be honest it doesn't seem to matter since most papers use them interchangeably. However, they are mostly either oversimplifying (since both are considered gravity mediated) or (possibly) mistaken.

The CMSSM is parametrized in terms of 4 variables and a sign at the GUT scale.
That is $$m_0$$ the sfermion (scalar) mass, $$m_\frac{1}{2}$$ the gaugino mass, $$A_0$$ the trilinear coupling and $$\tan(\beta)$$ the ratio of the vacuum expectation values of the Higgs fields, and the sign of $$\mu$$ the Higgs mixing parameter..

mSUGRA however is a more constrained model, it implements a boundary condition relating (among other things, but only directly) $$m_0,A_0,B_0$$ where $$B_0$$ is the bilinear coupling. This boundary condition is:
$$B_0=A_0-m_0$$
While the implication of this isn't obvious if you think of it interms of the pseudo scalar higgs mass $$M_3 = \mu B_0$$ this is the same as setting $$M_3^2=\mu(A_0-m_0)$$ at the GUT scale.
This reduces mSUGRA to 3 parameters and a sign ($$m_0,m_\frac{1}{2},A_0,\textrm{sign}(\mu)$$) and now $$\tan(\beta)$$ is an output.

edit: to clarify the $$M_3$$ here is NOT the same as the $$M_3$$ above. In the first post it is the gluino mass, here it is the pseudoscalar higgs mass (with the $$H_1H_2$$ term in the lagrangian). A slight ambiguity in my notation - sorry.

Last edited: Jun 7, 2010
4. Jun 7, 2010

### samr

Without having worked through it , I can't see why it would be any different from mSUGRA (the gaugino mass relation), the RGEs aren't horribly affected by the mSUGRA boundary condition so it should be about the same. Oh well.

5. Jun 7, 2010

### BenTheMan

I seem to recall a paper by Nilles and someone called The gaugino code''. You might check that one out. Why don't you try downloading softSUSY and just do the running and see?

6. Jun 8, 2010

### samr

Already have done that and it comes out roughly the same as mSUGRA (slight deviations in the last term); and yes I have the Gaugino code about - but in any case they should be the same as I think I said earlier. So it was a bit of a pointless question :D

(And they'd need to be 1:2:6 to be consistent any way)

7. Jun 8, 2010

### BenTheMan

Yeah, the slight deviations come from higher order loopy effects that softSUSY calculates, I bet. Most of the RGE flow in both cases comes from the coupling constants, as you probably have surmised :)