NIC, DESY Zeuthen · 2016. 7. 29. · The goal Combined ts Ratios Summary & Outlook Extraction of...

The goal Combined fits Ratios Summary & Outlook

Extraction of the bare form factors forsemi-leptonic Bs decays

Mateusz Koren

NIC, DESY Zeuthen

In collaboration with F. Bahr, D. Banerjee, H. Simma, and R. Sommer

LPHAACollaboration

Lattice 2016, Southampton, 29-07-2016

Mateusz Koren Extraction of the bare form factors for Bs → K`ν decays


Table of contents

1 The goal

2 Combined fits

3 Ratios

4 Summary & Outlook



The goal

Extract the bare ground-state matrix elements for staticHQET Bs → K`ν decay at ≈2% precision.

hstat,bare‖ = (2mBs )

−1/2〈K(pK)|V 0(0)|Bs(0)〉 = ϕ(0,0)0

√2E

(0)K ,

pkKhstat,bare⊥ = (2mBs )

−1/2〈K(pK)|V k(0)|Bs(0)〉 = ϕ(0,0)k

√2E

(0)K

Correlation functions (three smearings for Bs, one for Kaon):

CK(tK) ∼=

NK

∑m

(κ(m))2e−E(m)K tK ,

CBs

ij (tBs )∼=

NBs

∑n

β(n)i β

(n)j e−E

(n)Bs

tBs ,

CBs→Kµ,i (tK, tBs )

∼=

NK,NBs

∑m,n

κ(m)ϕ(m,n)µ β

(n)i e−E

(m)K tK−E

(n)Bs

tBs


ub

s

Vµ

tBstK

u

s

tK

b

s

tBs


The goal

Extract the bare ground-state matrix elements for staticHQET Bs → K`ν decay at ≈2% precision.

hstat,bare‖ = (2mBs )

−1/2〈K(pK)|V 0(0)|Bs(0)〉 = ϕ(0,0)0

√2E

(0)K ,

pkKhstat,bare⊥ = (2mBs )

−1/2〈K(pK)|V k(0)|Bs(0)〉 = ϕ(0,0)k

√2E

(0)K

Correlation functions (three smearings for Bs, one for Kaon):

CK(tK) ∼=NK∑m

(κ(m))2e−E(m)K tK ,

CBs

ij (tBs )∼=

NBs∑n

β(n)i β

(n)j e−E

(n)Bs

tBs ,


∼=NK,NBs∑

m,n

κ(m)ϕ(m,n)µ β

(n)i e−E

(m)K tK−E

(n)Bs

tBs


ub

s

Vµ

tBstK

u

s

tK

b

s

tBs


The goal cntd.

CLS Nf = 2 lattices:

Three ensembles for the continuum limit @ fixed q2

All the plots shown here are for the finest lattice N6,483 × 96, a = 0.0483(4) fm, mπ = 340 MeV

apkK = 2π

L δk1 on N6, twisted b.c. for equal pK on other

ensembles

We use stochastic sources with full-time dilution formeasurements ⇒ all tK, tBs accessible.

Especially the Bs-sector (treated in HQET) problematic dueto large contamination by excited states + signal-to-noiseproblem.



The goal cntd.

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

EK [i

n G

eV]

t [fm]

EK from fiteff mass

5.26

5.28

5.3

5.32

5.34

5.36

5.38

5.4

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

EB

s [in

GeV

]

t [fm]

EBs from 2-exp fit

eff mass HYP1

Effective energies of CK and CBs , ensemble N6. Both plots are scaled to

show the same y-axis span. Red bands show the results of two-exp fits.



Table of contents

1 The goal

2 Combined fits

3 Ratios

4 Summary & Outlook



Combined fits

Strategy: do a combined non-linear fit to both two-point andthree-point functions, to get energies, amplitudes and theform factors.

In practice NBs = 3 and NK = 1 or 2 needed for stable fitresults.

Many parameters (≈20 parameters to O(103) data points)⇒ good starting values:

1 Find E(m)K and κ(m) from CK,

2 From CBs

ij , find E(n)Bs

with GEVP and feed them to fit for β(n)i ,

3 Find a region in CBs→Kµ,i free of wrap-around states coming

from finite T ,4 Do a linear fit for ϕ

(m,n)µ .

Feed these as starting values to the (uncorrelated) combinedfit. Analyze stability w.r.t. fit ranges.



Fit ranges, finite T effects

ub

s

Vµ

0 T

u u

bs s

Vµ

0 T

T

Tt

tK

B

B'

K'

B noise

wrapper

s

tBstK

Selection of data points for fitting in CBs→Kµ (tK, tBs ), schematic plot



Fit ranges, finite T effectsµ = 0 µ = 1

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40 45 50

t K/a

tB /as

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40 45 50

t K/a

tB /as

Selection of data points for fitting in CBs→Kµ (tK, tBs ), ensemble N6

tK3min ≈ 0.82 fm (for NK = 1), tB3

min ≈ 0.67 fm (NBs = 3)

Wrapper criterion limits the relative wrapper contribution to 1/3 ofthe statistical noise, noise criterion is SNR ≥ 10.

Wrapper contribution larger for ϕk than ϕ0.



Combined fit stability w.r.t. tB2min, t

K3min, t

B3min

(NK ,NBs) = (1, 3) (2, 3)

ϕ(0,0)0

1.13

1.14

1.15

1.16

1.17

1.18

1.19

0 10 20 30 40 50 60 70 80

tmin-s (tB2, tK2, tB3)

1.13

1.14

1.15

1.16

1.17

1.18

1.19

0 5 10 15 20 25 30


ϕ(0,0)1

0.59

0.6

0.61

0.62

0.63

0.64

0 10 20 30 40 50 60 70 80


0.59

0.6

0.61

0.62

0.63

0.64

0 5 10 15 20 25 30




Table of contents

1 The goal

2 Combined fits

3 Ratios

4 Summary & Outlook



Ratios

A ratio should exponentially converge to ϕ(0,0)µ . Many

different ratios can be defined (τ = tK + tBs):

RIµ,i (tK, tBs ) =

CBs→Kµ,i (tK, tBs )[CK(τ)CBs

ii (τ)]1/2

e(E effBs

(τ)−E effK (τ))

tBs−tK2

RIIµ,i (tK, tBs ) =

CBs→Kµ,i (tK, tBs )[

CK(tK)CBs

ii (tBs )]1/2

eE effBs

(τ)tBs

2 +E effK (τ)

tK2

RIIIµ,i (tK, tBs ) =


NKCK(tK)NBs

i CBs

ii (tBs )

Rfµ,i (tK, tBs ) =


ii (τ)]1/2

[CK(tBs )C

Bs

ii (tK)

CK(tK)CBs

ii (tBs )

]1/2

A priori, none of them is clearly superior.In practice, statistical errors & convergence vastly differ.

Rather late onset of plateaux – one possible solution: fitfunctional form, including excited state.But: convergence ∼ e−∆Emint . Can we do better?



Ratios



RIµ,i (tK, tBs ) =


ii (τ)]1/2

e(E effBs

(τ)−E effK (τ))

tBs−tK2

Rfµ,i (tK, tBs ) =


ii (τ)]1/2

[CK(tBs )C

Bs

ii (tK)

CK(tK)CBs

ii (tBs )

]1/2

Rather late onset of plateaux – one possible solution: fitfunctional form, including excited state.

But: convergence ∼ e−∆Emint . Can we do better?



Ratios



Particularly simple form for tK = tBs = t:

RI/fµ,i (t, t) =

CBs→Kµ,i (t, t)[

CK(τ)CBs

ii (τ)]1/2





Ratios

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

4 6 8 10 12 14 16 18 20

RI/

f µ=0,

i(t,t)

t/a

i=1i=2i=3

Combined fit 0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

4 6 8 10 12 14 16 18 20R

I/f µ=

1,i(t

,t)t/a

i=1i=2i=3

Combined fit

Ratio RI/fµ,i as function of t = tK = tBs . Large contribution from

excited states visible.



Ratios



Particularly simple form for tK = tBs = t:

RI/fµ,i (t, t) =

CBs→Kµ,i (t, t)[

CK(τ)CBs

ii (τ)]1/2





Summed ratios

Better suppresion of excited states may be provided bysumming the ratios:

MX ,effµ,i (τ) = −∂τaSX

µ,i (τ), where SXµ,i (τ) =

∑tBs

RXµ,i (τ−tBs , tBs)

Improved convergence ∼ τ∆Emine−∆Eminτ holds for RI

[Maiani et al. 1987, Capitani et al. 2010, Bulava et al. 2010,2011]

One can extract ϕ(0,0)µ from numerical derivative or linear fit

to Sµ(τ) = ϕµτ + C

Approach to the plateaux clearly improved



Summed ratios

1.1

1.12

1.14

1.16

1.18

1.2

1.22

1.24

6 8 10 12 14 16 18 20

ϕ 0 fr

om s

umm

ed r

atio

(τ,τstart)

Linear fitNum-derivative

Combined fit

0.4

0.45

0.5

0.55

0.6

0.65

6 8 10 12 14 16 18ϕ 1

from

sum

med

rat

io(τ,τstart)

Linear fitNum-derivative

Combined fit

Comparison of ϕ(0,0)µ extracted from summed ratio MI

µ,3(τ) and the

combined fit. The fit is done in a fit range τstart to 32a.



Summed ratios

Better suppresion of excited states may be provided bysumming the ratios:

MX ,effµ,i (τ) = −∂τaSX

µ,i (τ), where SXµ,i (τ) =

∑tBs

RXµ,i (τ−tBs , tBs)

Improved convergence ∼ τ∆Emine−∆Eminτ holds for RI

[Maiani et al. 1987, Capitani et al. 2010, Bulava et al. 2010,2011]

One can extract ϕ(0,0)µ from numerical derivative or linear fit

to Sµ(τ) = ϕµτ + C

Approach to the plateaux clearly improved



Table of contents

1 The goal

2 Combined fits

3 Ratios

4 Summary & Outlook



Summary & Outlook

Full-time dilution of stochastic propagators⇒ Access to all values of tK and tBs

⇒ Good control over systematics from excited states andfinite T

Combined fits: stable results with NBs = 3; starting valuesfrom 2pt functions and linear fits to 3pt functions

Ratios: good statistical signal for selected ratios, but lateonset of plateaux

Summed ratios: improved plateaux convergence, in agreementwith theory

Outlook:

1/m insertions, non-perturbative HQET parameters, morevalues of q2

Chiral extrapolation ⇒ B→ π


NIC, DESY Zeuthen · 2016. 7. 29. · The goal Combined ts Ratios Summary & Outlook Extraction of...

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