CKM factor in K -> pi nu nu amplitude

[girishky]

Hi, Flavio basis convention (https://wcxf.github.io/assets/pdf/WET-3.flavio.pdf) for s -> d nunu transition has CKM factor (Vts* Vtd) in operators' definition. But in flavio implementation, amplitude for K -> pi nu nu has xi('t', 'sd') which is basically (Vts Vtd*) https://github.com/flav-io/flavio/blob/3d11de09d25507f093cce1ad6e3d970a379baf88/flavio/physics/kdecays/kpinunu.py#L11

Shouldn't it be xi('t', 'ds') in the above to be consistent with flavio operator definition?

[DavidMStraub]

In the observables, only the absolute value or the square of the imaginary part of the amplitude enters. so it shouldn't matter whether it's the d->s or the s->d amplitude. Do you agree?

[girishky]

Hi David, Thanks for reply. I agree with what you said. But my point is that in K -> pi nunu amplitude function, X is basically WC for s -> d transition https://github.com/flav-io/flavio/blob/3d11de09d25507f093cce1ad6e3d970a379baf88/flavio/physics/kdecays/kpinunu.py#L20

but the CKM factor that is multiplied by WC here https://github.com/flav-io/flavio/blob/3d11de09d25507f093cce1ad6e3d970a379baf88/flavio/physics/kdecays/kpinunu.py#L24 is related to conjugate (d->s) transition.

For absolute value, I agree this won't matter. But imaginary part of amplitude (even square of it) would be different if X also contain complex phase due to NP. So shouldn't amplitude be amp = xi_t.conj() * X or amp = xi_t * X.conj() ?

[hoodyn]

I agree with @girishky. Simply taking imaginary part for different neutrino species is inappropriate. It is the same kind of error that has been reported in #138 for K0->ll'.

[DavidMStraub]

But that's a different issue...

[girishky]

Hi @hoodyn , even in cases where nu flavor is conserved, so that NP contribution can only rescales X function, the current flavio implementation has issue (imo). The operator $(\bar s d )_{V-A} (\bar nu nu)_{V-A}$ comes with VtdVts* in the normalization but its WC (i.e. X) is being multiplied with VtsVtd*; but this ckm factor appears in normalization of conjugate operator $(\bar d s)_{V-A} (\bar nu nu)_{V-A}$. The results of this is that even a CP conserving NP will affect KL decay mode which it shouldn't. people can crosscheck the above observation.

best,

[peterstangl]

In the observables, only the absolute value or the square of the imaginary part of the amplitude enters. so it shouldn't matter whether it's the d->s or the s->d amplitude. Do you agree?

I agree with this as long as in https://github.com/flav-io/flavio/blob/3d11de09d25507f093cce1ad6e3d970a379baf88/flavio/physics/kdecays/kpinunu.py#L24

the X is strictly real. However, I agree with @girishky that if X also has an imaginary part, then it does matter whether xi('t', 'sd') or xi('t', 'ds') is used.

I suggest to simply replace https://github.com/flav-io/flavio/blob/3d11de09d25507f093cce1ad6e3d970a379baf88/flavio/physics/kdecays/kpinunu.py#L11-L12 by

 xi_t = flavio.physics.ckm.xi('t', 'ds')(par) 
 xi_c = flavio.physics.ckm.xi('c', 'ds')(par) 

@DavidMStraub @girishky do you both agree with this solution?

[girishky]

@peterstangl Yes. This is better. Thanks.

best,

[peterstangl]

This issue has been fixed in PR #189.