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*)

flavio/flavio/physics/kdecays/kpinunu.py

Line 11 in 3d11de0

xi_t = flavio.physics.ckm.xi('t', 'sd')(par)

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

flavio/flavio/physics/kdecays/kpinunu.py

Line 20 in 3d11de0

X = -( wc['CL_'+label] + wc['CR_'+label] ) * s2w

but the CKM factor that is multiplied by WC here

flavio/flavio/physics/kdecays/kpinunu.py

Line 24 in 3d11de0

amp = xi_t * X # top contribution

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

flavio/flavio/physics/kdecays/kpinunu.py

Line 24 in 3d11de0

amp = xi_t * X # top contribution

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

flavio/flavio/physics/kdecays/kpinunu.py

Lines 11 to 12 in 3d11de0

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

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.