gamma basis of su3spinor.h inconsistent with the one used in the Dirac operator

[kostrzewa]

I realised yesterday that the gamma basis which is used in su3spinor.h of tmLQCD is not consistent with the one used in the Dirac operator. The difference amounts to a sign difference to all elementary gamma matrices (gamma_mu -> -gamma_mu \forall mu \in [0,1,2,3]) so all it will do is

• change signs of all correlators with an odd number of vector or time gamma matrices
• exchange |l=1, m=1> and |l=1, m=-1> basis states

[kostrzewa]

The inconsistency stems, I believe, from notes dating back some time which had the (1+\gamma_\mu) projector associated with positive hops and the (1-\gamma_\mu) projector associated with negative hops. Writing down (on paper) the Dirac operator like that of course means that one ends up extracting the wrong sign for the gamma matrices from the implementation in the code and concluding that what is written in su3spinor.h is actually correct...

[kostrzewa]

I believe that up to signs, the difference is essentially inconsequential up to the point where one begins to employ parity projectors.

[kostrzewa]

On the other hand, @maowerner if the signs of the vectorial Dirac matrices in the contractions are all the opposites of what they should be, does this also affect the isospin projection at non-zero momentum?

[maowerner]

If they are consistently switched it does not affact the angular momentum projection. The sign propagates through and the correlator simply becomes negative. To project isospin only the 2-particle operators must be projected, the rho is just a single particle and an isospin eigenstate by construction. If we use pions with different operators it might amound to a minus sign between the 2pt and 4pt operator, but as both pions are gamma_5 so far, the -1 drops out and it can not affect the current rho data

When we consider gamma_0 gamma_5 for the pions, we might need to be careful

[kostrzewa]

the -1 drops out and it can not affect the current rho data

what about the < \pi \pi \rho > correlation function?

[kostrzewa]

ps: aren't you on holidays? I wasn't really expecting a reply, just wanted to jot this down here to not forget it...

[maowerner]

It contains 2 pions. That's why all factors enter squared. This changes when we have something like
\langle (\gamma_5 * \gamma_0 \gamma_5) * (\gammi_i)^\dagger \rangle

[kostrzewa]

What I meant is that:

< \gamma_5 \gamma_5 (\gamma_i)^\dagger >

changes sign when \gamma_i -> -\gamma_i while < \gamma_i (\gamma_i)^\dagger > does not. Similarly,

< \gamma_5 \gamma_0 \gamma_5 (\gamma_i)^\dagger >

does not change sign (because both gamma_0 and gamma_i have the "wrong" sign), while other contributions do change sign.
As long as this doesn't mess up things down the line, it's probably fine to stick with the wrong signs in the gamma basis for now.