Determination of mc from Nf = 2 + 1 QCD with Wilsonfermions

Sjoerd Bouma

University of Regensburg

RQCD collaborators: Gunnar Bali, Sara Collins, Wolfgang Söldner

6 August 2020

QCD

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Overview

1 Introduction

2 AnalysisPCAC massesChiral-continuum extrapolationEnsembles

3 Preliminary results

4 Conclusion and outlook

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Motivation

• Quark masses are fundamental parametersof the standard model

• Input for many phenomenologicalpredictions, including for BSM physics.

• Not directly measurable (confinement) -values depend on renormalization scheme.

• Charm observables difficult to simulate onthe lattice - in between relativistic andnon-relativistic regimes; amc large formany lattice spacings currently used.

• We use 5 lattice spacings down to 0.04 fmto control discretization effects

1.25 1.30 1.35 1.40

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Motivation

• Using the lattice PCAC relation:

amij =∂0CA0P + cAa∂

20CPP

2CPP, (1)

and the Vector-Ward Identity (VWI) quark masses:

amq,ij =

(1

4κi+

1

4κj− 1

2κcrit

), (2)

we can determine the renormalization-group independent (RGI) mass:

mRGIij = ZMmij[1 + (bA − bP)amq,ij + (b̃A − b̃P)aTr[Mq]

]+O(a2), (3)

where, following Divitiis et al. 2019, Bulava et al. 2015, Campos et al.2018:

ZM =M

m(µhad)

ZA(g20 )

ZP(g 20 , aµhad),

Tr[Mq] ≈ 2m` + ms ,

COO′(t) = 〈O(t)O ′(0)〉,

∂0f (t) =f (t + a)− f (t − a)

2a.

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Motivation

• Renormalization-group independent (RGI) mass:

mRGIij = ZMmij[1 + (bA − bP)amq,ij + (b̃A − b̃P)aTr[Mq]

]+O(a2), (3)

• b̃A − b̃P is poorly constrained, but compatible with 0, and Tr[Mq]� mq,ijfor the charm quark → (for now) we take b̃A − b̃P = 0.

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PCAC masses

0 20 40 60 80 100 120t/a

0.2548

0.2550

0.2552

0.2554

0.2556

0.2558

0.2560am

PCAC

D200, = 3.55, ts = 37fit ( 2 = 751 / 760)fit limits:[64, 101]

0 5 10 15 20 25 30(t ts)/a

0.3494

0.3496

0.3498

0.3500

0.3502

0.3504

0.3506

amPC

AC

B452, = 3.46fit ( 2 = 14 / 22)fit limits:[18, 32]

• PCAC masses determined by fitting to a constant in a ’plateau’ region.• Ansatz for boundary effects and contact terms:

amPCAC(t) ≈ amPCAC + c1e−b1t + c2e−b2(Tbd−t). (4)

• Plateau defined as the region where

4 ·(c1 · exp−b1t +c2e−b2(Tbd−t)

)≤ ∆statamPCAC(t). (5)

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Error extrapolation

• Have to deal with autocorrelationsin lattice and Monte-Carlo time

• Strategy: estimate errors througha binned jackknife procedure

• Bin in Monte-Carlo time andobtain jackknife error on mPCACat each bin size S .

2.5 5.0 7.5 10.0 12.5 15.0 17.5bin size S

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2 am(S

)/2 am

(1)

3-param. fit ( 2 = 4.4 / 13)2 int = 1.623Variance (normalized)

• Then extrapolate to infinite bin size using the formula:

σ2[S ]

σ2[1]≈ 2τint

(1− cA

S+

dASe−S/τint

). (6)

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Chiral-continuum extrapolation

• Once we have obtained the PCAC masses for all ensembles, want toextrapolate to the chiral and continuum limits.

• Ansatz:

fc(a2/(8t∗0 ),mπ,mK ) = fc(0,mπ,mK )

×(

1 +a2

t∗0(p1 + p2M

2+ p3δM

2) +a3

(t∗0 )3/2

p4

), (7)

where

mP =√

8t0mP ,

fc(0,mπ,mK ) = (c0 + c1M2

+ c2M2δM2),

M2

=2m2K + m

2π

3,

δM2 = 2(m2K − m2π).• t0 is the Wilson flow parameter determined per ensemble• t∗0 defined by 12t∗0m2π = 1.110 at ms = m`.• Ensembles generated at fixed bare couplings g0 → extrapolation in t0

times the masses ensures O(a) improvement.

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Chiral-continuum extrapolation

• We fit the results from each ensemble to the fit function in (7)• As σ(mπ,K ) & σ(mc), we use a generalized chi-squared fit allowing also

mπ,mK , t∗0 to vary from their expectation values.

• We also take into account the (infinite bin size extrapolated) correlationsbetween the variables.

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Ensembles

• CLS-generated ensemblesNf = 2 + 1 Wilson-Clover O(a)improved fermions.

• 5 lattice spacings∼ 0.085 to 0.04 fm• Pion masses from 420 MeV down

to the physical point• Three different chiral trajectories:

• ms = mphyss• ms = m`• Tr[Mq ] = 2m` +ms = constant

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

8t0(2M

2 K−

M2 π)∝

ms

8t0M2π ∝ ml

ms = m̃physs

ms = mℓ2ml +ms = const

β = 3.40β = 3.46β = 3.55β = 3.70β = 3.85

physical point

• ⇒ we can adjust for any ’mistuning’ in the fit.• For each ensemble, simulated 2 heavy quark masses around mphysc and

interpolate the PCAC mass to√

8t0mDs :=√

8tphys0 mphysDs

.

Coordinated Lattice Simulations (CLS): Berlin, CERN, Mainz, UA Madrid, MilanoBicocca, Münster, Odense, Regensburg, Rome I and II, Wuppertal, DESY-Zeuthen,Kraków.

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Preliminary results

0.00 0.01 0.02 0.03 0.04 0.05a2/(8t *0 )

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

8t0m

c

Heavy-HeavyHeavy-LightHeavy-StrangeFLAG

0.00 0.01 0.02 0.03 0.04 0.05a2/(8t *0 )

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

8t0m

c

Heavy-HeavyHeavy-LightHeavy-StrangeFLAG

• Preliminary results• Left: c2 = 0, right: c2 6= 0.• Uncertainties due to fit parametrisation still need to be explored.• ∼ 1% errors due to scale determination (in ZM) and tphys0 need to be

added to any final values.

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Preliminary Results

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.88t0m2

3.10

3.15

3.20

3.25

3.30

3.35

8t0m

c

Heavy-HeavyHeavy-LightHeavy-StrangeFLAG

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.88t0m2

3.075

3.100

3.125

3.150

3.175

3.200

3.225

8t0m

c

Heavy-HeavyHeavy-LightHeavy-StrangeFLAG

Left: c2 = 0, right: c2 6= 0.

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Conclusion and outlook

• What’s next?• Combined fit for all three flavour combinations (heavy-heavy, heavy-light,

heavy-strange)

• Check for consistency with higher-order discretizations of the derivative inthe PCAC relation.

• Perform the chiral-continuum extrapolation for the ratio mc/ms , where weare not affected by the uncertainties on tphys0 and ZM

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References

Bulava, John et al. (2015). In: Nucl. Phys. B 896, pp. 555–568. doi:10.1016/j.nuclphysb.2015.05.003. arXiv: 1502.04999 [hep-lat].

Campos, Isabel et al. (2018). In: Eur. Phys. J. C 78.5, p. 387. doi:10.1140/epjc/s10052-018-5870-5. arXiv: 1802.05243 [hep-lat].

Divitiis, Giulia Maria de et al. (2019). In: Eur. Phys. J. C 79.9, p. 797. doi:10.1140/epjc/s10052-019-7287-1. arXiv: 1906.03445 [hep-lat].

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https://doi.org/10.1016/j.nuclphysb.2015.05.003https://arxiv.org/abs/1502.04999https://doi.org/10.1140/epjc/s10052-018-5870-5https://arxiv.org/abs/1802.05243https://doi.org/10.1140/epjc/s10052-019-7287-1https://arxiv.org/abs/1906.03445

IntroductionAnalysisPCAC massesChiral-continuum extrapolationEnsembles

Preliminary resultsConclusion and outlookReferences