tomomi@quark.phy.bnl.gov

1

RBC/UKQCD Collaborations

Neutral B meson mixingwith static heavy

and domain-wall light quarksTomomi Ishikawa (RBRC)

Lattice 20132013/7/28-8/03, Mainz, Germany

Collaborators:Yasumichi Aoki, Taku Izubuchi,

Christoph Lehner and Amarjit Soni

mailto:tomomi@quark.phy.bnl.govmailto:tomomi@quark.phy.bnl.gov

‣ Constraints to CKM triangle- Oscillation frequency

- Hadronic matrix elements

2

q = {d, s}

MBq = ⇥B0q|[b̄�µPLq][b̄�µPLq]|B0q ⇤ =

83m2Bqf

2BqBBq

constraints on Vtd, Vts

q

q̄

t

W

b̄

b

t̄

WB0 B̄0

q

q̄

W

t

b̄

b

W

t̄B0 B̄0

mixing and CKMB0 � B̄0

�mq = (known factors)⇥��V ⇤tqVtb

��2 1mBq

MBq

‣ Constraints to CKM triangle SU(3) breaking ratio

- The most attractive quantity in phenomena- Many of the uncertainties cancel in the ratio.- In the lattice calculation, the error is reduced due to correlation between denominator and numerator.

‣ Other important quantities B meson decay constants

Bag parameters

3

mixing and CKMB0 � B̄0

⇠

B0 � B̄0

fBd , fBs

����VtdVts

���� = ⇠

s�md�ms

mBsmBd

⇠ =mBdmBs

sMBsMBd

Bq =3

8

MBqm2Bqf

2Bq

‣ V. Gadiyak and O. Loktik, Lattice calculation of SU(3) flavor breaking ratios in mixing, Phys. Rev. D 72 (2005) 114504.

‣ O. Loktik and T. Izubuchi, Perturbative renormalization for static and domain-wall bilinears and four-fermion operators with improved gauge actions, Phys. Rev. D 75 (2007) 034504.

‣ C. Albertus, Y. Aoki, P. A. Boyle, N. H. Christ, T. T. Dumitrescu, J. M. Flynn, T. I, T. Izubuchi, O. Loktik, C. T. Sachrajda, A. Soni, R. S. Van de Water, J. Wennekers and O. Witzel, Neutral B-meson mixing from unquenched lattice QCD with domain-wall light quarks and static b-quarks, Phys. Rev. D 82 (2010) 014505.

‣ T. I, Y. Aoki, J. M. Flynn, T. Izubuchib, and O. Loktik, One-loop operator matching in the static heavy and domain-wall light quark system with O(a) improvement, JHEP 05 (2011) 040.

‣ Y. Aoki, T. I, T. Izubuchi, C. Lehner and A. Soni (on-going project).

4

RBC/UKQCD Static B Physics

B0 � B̄0

‣ Static approximation (leading order of HQET)- Easy to implement (Static quark propagator is almost free.)- Symmetries (HQ spin symmetry + chiral symmetry)

- Continuum limit exists even in the perturbative renormalization.- But, we always have the error coming from static approx.

- Always sitting as an anchor point for other HQ approach.

‣ Ratio quantities ( , ) in the static limit- Error coming from static approximation is reduced to:

5

reduced operator mixing

O(�QCD/mb) � 10%

�

O

�ms �md�QCD

⇥ �QCDmb

⇥⇤ 2%

Static limit

fBs/fBd

‣ Standard static action with link smearing

Reduced 1/a power divergence.

‣ Domain-wall light quark action 5 dimensional, controllable approximate chiral symmetry Unphysical operator mixing does not occur.

‣ Iwasaki gluon action

6

Lattice action setup

Sstat =�

⇧x,t

�h(⌥x, t)⇥�h(⌥x, t)� U†0 (⌥x, t� a)�h(⌥x, t� a)

⇤

- HYP1 [Hasenfratz and Knechtli, 2001] - HYP2 [Della Morte et al.(ALPHA), 2004]

Ls

‣ Operator matching

7

Matching between continuum QCD and continuum HQET at 2-loop RG running from to in continuum HQET PT matching between continuum HQET and lattice HQET at

Matching procedure

JHEP05(2011)040

2.3 Matching procedure

We adopt a two step matching procedure:

continuum QCD(CQCD)

continuum HQET

continuum HQET

(CHQET)

(CHQET)

lattice HQET(LHQET)

µ = mb

µ = a−1

RG-evolution

,

in which we first perform the matching between continuum QCD (CQCD) and continuum

HQET (CHQET) and subsequently match CHQET to lattice HQET (LHQET). Some

comments on this matching:

(1) The continuum QCD (CQCD) operators are renormalized in MS(NDR) at scale µbwhich is usually chosen to be the b quark mass mb. Fierz transformations in arbitrary

dimensions are specified in the NDR scheme introduced by Buras and Weisz [16]. The

introduction of evanescent operators gives vanishing finite terms at one-loop but is

needed to obtain the correct anomalous dimensions at two-loop.

(2) The CHQET operators are also renormalized in MS(NDR) at some scale µ. Matching

between the continuum theories is performed in perturbation theory by calculating

and comparing matrix elements of the operators between an initial state |i〉 and fi-nal state |f〉 for each theory. The calculation has been done for quark bilinears atone-loop [4] and two-loop [17] levels, and for the ∆B = 2 four-quark operator at

one-loop [18].

(3) The continuum matching between QCD and HQET is done at scale µ = mb to avoid

a large logarithm of µ/mb. We use renormalization group (RG) running in CHQET

to move to a lower scale at which the HQET matching between continuum and lattice

is done. We employ the two-loop anomalous dimension calculations in refs. [19, 20]

for the bilinear and in refs. [21–23] for the four-quark operator.

(4) Matching between CHQET and LHQET is performed at scale µ = a−1, where a

denotes the lattice spacing. The calculation is performed in one-loop perturbation

theory taking into account O(a) discretization errors on the lattice. For this we

introduce external momenta, e.g.,

〈f|O|i〉 = 〈h(p′)|O|q(p)〉 for bilinear operators,〈f|O|i〉 = 〈h(p′2), q(p2)|O|h(p′1), q(p1)〉 for four-quark operators, (2.11)

where O denotes the bilinear and four-quark operators. On-shell improvement is

used, in which we impose the equations of motion on the external quarks:

D0h = 0, (#D + mq) q = 0, (2.12)

– 4 –

Different renormalization Operator matching is needed.

µ = mb

µ = mb µ = a�1

µ = a�1

Static with link smearing + DWF O(a) disc error is taken into account.

[T.I, Aoki, Flynn, Izubuchi, Loktik (2011)]

‣ Gluon ensemble- Nf=2+1 dynamical DWF + Iwasaki gluon (RBC-UKQCD)

‣ Measurement parameters

- Gaussian smearing on fermion field (width ~ 0.45 fm)

8

Measurement

Table 1: 2 + 1 flavor dynamical DWF ensembles by RBC-UKQCD Collaborations [1]. mres is a residual mass inthe DWF. aml and amh represent the sea ud and sea s quark mass parameter, respectively.

label β L3 × T × Ls a−1 [GeV] a [fm] amres ml/mh mπ [MeV] mπaL24c1 2.13 243 × 64× 16 1.729(25) 0.114 0.003152(43) 0.005/0.04 327 4.5424c2 0.01/0.04 418 4.7932c1 2.25 323 × 64× 16 2.280(28) 0.0864 0.0006664(76) 0.004/0.03 289 4.0532c2 0.006/0.03 344 4.8332c3 0.008/0.03 393 5.52

1/a power divergence. This situation has been significantly improved, since ALPHA collaboration introduced linksmearing technique in the static action, which partly cured the S/N.

In this study, we calculate B meson decay constants and neutral B meson mixing matrix elements using thestatic approximation. The static approximation always has O(ΛQCD/mb) ∼ 10% uncertainty, since physical bquark mass is not infinite. To reduce this uncertainty in the HQET approach, higher order contributions in the1/mb expansion needs to be included. Taking into account these contributions requires nonperturbative matchingwith continuum using Schrëdinger functional with step scaling technique, which is seemingly hard task. Instead,we stay in static limit aiming at a use for interpolation to physical b quark mass combining with lighter quarkmass simulation. The static limit value would become a good reference point, if the calculation is made with highprecision. It also gives good comparison to the other approaches, such as Relativistic Heavy Quark (RHQ). Forratio quantities like ξ, the uncertainty coming from static approximation is down to around 2% level:

O

(

ms −mdΛQCD

×ΛQCDmb

)

∼ 2%, (5)

which means the static results could be competitive to other lattice approaches for heavy quark physics.

Simulation setup

! Lattice actions

We perform the lattice QCD simulation on HQET side. We employ the standard static quark action with gluon linksmearing for b quark. As the link smearing, we use HYP1 and HYP2 smearing. The link smearing is introducedaiming at a reduction of the power divergences. For the light quark (u, d and s) sector, we use domain-wall fermion(DWF) formalism. The DWF is 5 dimensional realization of chiral fermion, which holds controllable approximatechiral symmetry at large enough fifth-dimension size. The chiral symmetry is important to prevent unnecessaryoperator mixing. For the gluon part, we use Iwasaki action.

! Gluon ensemble and measurement setup

In our simulation, 2 + 1 flavor dynamical DWF gluon ensemble generated by RBC-UKQCD Collaborations [1] isused. The ensemble parameters are shown in Tab. 1. Two lattice spacings a ∼ 0.114 [fm] and 0.0864 [fm] are usedto take a continuum limit. We label the coarser and finer lattices as “24c” and “32c”, respectively. The physicalbox size is set to be modest, which is around 2.75 [fm]. The size of the fifth dimension is Ls = 16 making the chiralsymmetry breaking small enough. Degenerate u and d quark mass parameters are chosen so that the simulationcovers the pion mass range of 290-420 [MeV]. The smallest value of mπaL is about 4, which implies finite volumeeffect would be small at simulation points in this work.

3

Table 2: Measurement parameters. amq represents valence the quark mass parameter. ∆t denotes source-sinkpoint separation in three-point correlators.

label amq Measured MD traj # of data # of src ∆t24c1 0.005, 0.034, 0.040 900–8980 every 40 203 4 2024c2 0.010, 0.034, 0.040 1460–8540 every 40 178 232c1 0.004, 0.027, 0.030 520–6800 every 20 315 1 2432c2 0.006, 0.027, 0.030 1000–7220 every 20 312 132c2 0.008, 0.027, 0.030 520–5540 every 20 252 1

Measurement parameters are presented in Tab. 2. In the measurement, we use gauge-invariant Gaussiansmearing for source and sink operators. The Gaussian width is set to be around 0.45 fm.

Matching procedureIn this work, we adopt a two step matching: the first is a matching between QCD and HQET in continuum, thesecond is a matching between continuum and lattice in HQET. The matching is made by one-loop perturbation.We here summarize key points of the matching.

• The QCD operators in the continuum are renormalized in MS(NDR) scheme at µb = mb, b quark massscale. Fierz transformations in arbitrary dimensions are specified in the NDR scheme by Buras and Weiszintroducing evanescent operators.

• The HQET operators in the continuum are also renormalized in MS(NDR) scheme at some scale µ.

• The matching between QCD and HQET is performed at scale µ = mb to avoid a large logarithm of µ/mb.We then use renormalization group running in the HQET to go down to a lower energy scale.

• Matching HQET operators between continuum and lattice is perturbatively carried out at scale µ = a−1,where a denotes a lattice spacing.

• In the matching of HQET operators between continuum and lattice, O(a) discretization errors are taken intoaccount. We employ on-shell O(a) improvement program, in which we impose the equation of motion on theexternal heavy and light quark lines. In the improvement, we include both O(pa) and O(ma) contributions,where p and m depict light quark momentum and mass, respectively [2].

Chiral and continuum extrapolation

! NLO SU(2) HMχPT

Physical quantities at simulated light quark mass points are extrapolated to physical degenerate u and d quarkmass value. In this work, we use next-to-leading order SU(2) heavy-light meson chiral perturbation theory (NLOSU(2) HMχPT) described in Ref. [3]. In SU(2)χPT, s quark is integrated out of the theory; effects from s quarkare included in low-energy constants. The SU(2)χPT formula is obtained from SU(3)χPT assuming u and d quarkmasses are much smaller than s quark mass. The formula does not depend on s quark mass in explicit way. Theconvergence of the chiral fit is improved by using the SU(2)χPT as long as the u and d quark mass is sufficientlysmall [4]. In Ref. [4], it is argued that the RBC/UKQCD DWF ensemble does not show convergence of NLOSU(2)χPT above the pion mass of 420 MeV for the light hadron masses and decay constants. The ensembles weuse in this work stay below that border.

4

[Phys. Rev. D 83, 074508 (2011)]

‣ Operators- 2PT correlation functions

- 3PT correlation functions

9

Measurement

CL

(tf

, t, t0) =X

~x

hAS0 (~x0, tf )†OV V+AA(~x, t)AS0 (~x0, t0)†i,

CS

(tf

, t, t0) =X

~x

hAS0 (~x0, tf )†OSS+PP (~x, t)AS0 (~x0, t0)†i.

C

L̃S(t) =X

~x

hAL0 (~x, t)AS0 (~x0, 0)†i,

C

S̃S(t) =X

~x

hAS0 (~x, t)AS0 (~x0, 0)†i,

C

SS(t) = hAS0 (~x0, t)AS0 (~x0, 0)†i.

A

L0 (~x, t) : local

A

S0 (~x, t) : smeared both on heavy and light

AL0 (~x, t), OV V+AA(~x, t) : O(a) improved operators

‣ Matching (continuum QCD and lattice HQET)

‣ Correlator fitting

10

Data extraction

�! �latBq , MlatBq

0 5 10 15 20t

0.4

0.45

E0 =

-ln

C(t

+1, 0)/

C(t

, 0)

LS

SSSS

fit range

5

VV+AA

0 5 10 15 20

t

-5

0SS+PP

�2/d.o.f. = 0.4

�2/d.o.f. = 0.5

�2/d.o.f. = 0.5

32c, HYP1

(mh,ml,mq)

= (0.03, 0.004, 0.004)

C2PT(t) = A(e�E0t + e�E0(T�t)), C3PT(tf , t, t0) = A3PT

fBq = (matching factor)⇥�

latBqpmB

, MBq = (matching factor)⇥mBM latBq

‣ scaling and consistency check

- HYP1 shows larger scaling violation than HYP2.- HYP1 and HYP2 look consistent in the continuum.- In the ratio, scaling violation is quite small.

11

Chiral and continuum extrapolation

0 10 20m

l+m

res [MeV]

200

250

f Bd [M

eV

]

cont, HYP1

cont, HYP2

24c, HYP1

24c, HYP2

32c, HYP1

32c, HYP2

0 10 20m

l+m

res [MeV]

200

250

f Bd [M

eV

]

cont, HYP1

cont, HYP2

24c, HYP1

24c, HYP2

32c, HYP1

32c, HYP2

0 10 20m

l+m

res [MeV]

1

1.1

1.2

1.3

f Bs

/ f B

d

cont, HYP1

cont, HYP2

24c, HYP1

24c, HYP2

32c, HYP1

32c, HYP2

0 10 20m

l+m

res [MeV]

1

1.1

1.2

1.3

f Bs

/ f B

d

cont, HYP1

cont, HYP2

24c, HYP1

24c, HYP2

32c, HYP1

32c, HYP2

O(a) unimp O(a) imp O(a) unimp O(a) imp

NLO SU(2) HMChPT

a2

‣ Combined fits

12

Chiral and continuum extrapolation

Linear fits are also used to estimate an uncertainty from chiral fits.

0 10 20m

l+m

res [MeV]

190

200

210

220

230

240

250

260

270

280

290

f Bd [

Me

V]

24c, HYP1

24c, HYP2

32c, HYP1

32c, HYP2

cont, phys

χ2/d.o.f. = 1.5

p-val = 0.17

0 10 20m

l+m

res [MeV]

1

1.1

1.2

1.3

(MB

s/M

Bd)1

/224c, HYP1

24c, HYP2

32c, HYP1

32c, HYP2

cont, phys

χ2/d.o.f. = 0.1

p-val = 1.00

NLO SU(2) HMChPT

‣ Preliminary results

13

Results

Error budget on (total 5.3% error)

Reducing statistical andchiral/continuum extrapolation errors important.

HQET short noteTomomi Ishikawa July 25, 2013

Table 1: 2+1 flavor dynamical domain-wall fermion ensembles by RBC-UKQCD Collaborations.[?] Physicalquark masses are obtained using SU(2)χPT.

label β L3 × T × Ls a−1 [GeV] a [fm] aL [fm] amres ml/mh mπ [MeV] mπaL24c1 2.13 243 × 64× 16 1.729(25) 0.114 2.74 0.003152(43) 0.005/0.04 327 4.5424c2 0.01/0.04 418 4.7932c1 2.25 323 × 64× 16 2.280(28) 0.0864 2.76 0.0006664(76) 0.004/0.03 289 4.0532c2 0.006/0.03 344 4.8332c3 0.008/0.03 393 5.52

Table 2: Measurement parameters. NG and ω are source and sink Gaussian smearing parameters. ∆tsrc−sinkrepresents source-sink separation in three point functions.

label amq Measured MD traj # of data # of src24c1 0.005, 0.034, 0.040 900–8980 every 40 203 424c2 0.010, 0.034, 0.040 1460–8540 every 40 178 232c1 0.004, 0.027, 0.030 520–6800 every 20 315 132c2 0.006, 0.027, 0.030 1000–7220 every 20 312 132c2 0.008, 0.027, 0.030 520–5540 every 20 252 1

fBd [MeV] = 222(17), 222(31),

fBs [MeV] = 265(19), 265(37),

fBs/fBd = 1.192(43), 1.192(51),

MBd [(GeV)4] = 2.79(44), 2.79(56),MBs [(GeV)4] = 4.34(46), 4.34(69),√

MBs/MBd = 1.238(59), 1.238(66),B̂Bd = 1.15(13), 1.15(19),

B̂Bs = 1.22(10), 1.22(18),

BBs/BBd = 1.047(65), 1.047(80),

ξ = 1.218(58), 1.218(65).

1

incl 1/mbuncertainty

statistical3.3%

chiral-continuum

extrap2.9%

1/mb2.2%

phys quark mass 1.3%

perturbation1.0%

others1.1%

⇠

(Systematic errors are included in the error.)

not incl 1/mbuncertainty

‣ Comparison

14

Results

Decay constants have ~10% deviation from other works.

1/mb uncertainty is included in the error.

Ratio quantities do not have such a significant deviation.

160 180 200 220 240 260 200 220 240 260 280 300

fBd

[MeV] fBs

[MeV]

HPQCD ’12 (NRQCD)

FNAL/MILC ’11 (Fermilab)

ETM ’12 (Ratio method, Nf=2)

This work (Static)

ALPHA ’12 (HQET, Nf=2)

1 1.1 1.2 1.3 1 1.1 1.2 1.3

fBs

/ fBd

ξ

HPQCD ’12 ’09 (NRQCD)

FNAL/MILC ’11 ’12 (Fermilab)

ETM ’12 (Ratio method, Nf=2)

This work (Static)

RBC/UKQCD ’10 (Static)

1/mb uncertainty is not included in the error.

‣ Example (32c, lightest quark mass parameter) Translational invariance as a symmetry

- Many source points

Sloppy CG as an approximation

- deflation with 130 lowest eigenvectors- CG iter = 120 to achieve res = 3e-3

15

All-mode-averaging (AMA)

g 2 GNg = Nsrc = 64 (2⇥ 2⇥ 2⇥ 8)

divide lattice by 32⇥ 32⇥ 32⇥ 64 2⇥ 2⇥ 2⇥ 8

CG iter ~ 750 to achieve res = 1e-8CG iter ~ 4000 to achieve res = 1e-8 (no deflation)

[Blum, Izubuchi and Shintani (2012)]Plenary session 8/2 9:30AM Chulwoo Jung

16

All-mode-averaging (AMA)2PT, HYP2

[ Cost ] EXACT(315 conf) : AMA(25 conf) : AMA(40 conf) ~ 1 : 1 : 1.6

0 5 10 15 20t

0.35

0.4

0.45

E0 =

-ln

C(t

+1

, 0

)/C

(t,

0)

EXACT (1 src, 315 conf)

0 5 10 15 20t

AMA (64 src, 25 conf)

0 5 10 15 20t

CSS

(t, 0)

CSS

(t, 0)

CLS

(t, 0)

AMA (64 src, 40 conf)

17

All-mode-averaging (AMA)3PT, HYP2

[ Cost ] EXACT(315 conf) : AMA(25 conf) : AMA(40 conf) ~ 1 : 1 : 1.6

10

15

CL(t

f, t,

0)

EXACT (1 src, 315 conf) AMA (64 src, 25 conf) AMA (64 src, 40 conf)

0 5 10 15 20t

-10

-5

CS(t

f, t,

0)

0 5 10 15 20t

0 5 10 15 20t

‣ Impact on physical quantities

Gain (compared with deflated exact CG)- Decay constant : x2.6 (HYP1), x3.8 (HYP2)- Matrix element : x2.2 (HYP1), x3.1 (HYP2)

Approximation is only used for light sector, then the gain in our heavy-light system would be smaller than other light-light simulations.

18

All-mode-averaging (AMA)

245

250

255

260

EXACT (1src, 315conf)

AMA (64src, 25conf)

AMA (64src, 40conf)225

230

235

240

fBd

(HYP1)

fBd

(HYP2)

3.5

4

4.5

EXACT (1src, 315conf)

AMA (64src, 25conf)

AMA (64src, 40conf)2.5

3

3.5

4

MBd

(HYP1)

MBd

(HYP2)

‣ B meson decay constants and neutral B meson mixing matrix elements in the continuum limit are obtained using static approximation.

‣ Two kinds of link smearing in the static action are used (HYP1 and HYP2). They give consistent results in the continuum limit.

‣ Decay constants has ~10% deviation from other works, possibly due to 1/mb error.

‣ Ratio quantities does not have significant deviation from other work, because 1/mb error is largely suppressed.

‣ Reducing statistical and chiral extrapolation error is important as a next step.

‣ AMA can largely reduce statistical error.‣ Considering calculations at physical pion mass point.‣ Considering non-perturbative matching.

19

Summary and outlook