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Hyperon and charmed baryon masses from twisted mass Lattice QCD(N f = 2 + 1 + 1, N f = 2 plus clover)

Christos KallidonisComputation-based Science and Technology Research Center

The Cyprus Institute

C. Alexandrou et al. arXiv:1406.4310

withC. Alexandrou, V. Drach, K, Hadjiyiannakou, K. Jansen, G. Koutsou

Rheinische Friedrich-Wilhelms-Universitat Bonn

Bonn, Germany

1 April 2015

C. Kallidonis (CyI) Baryon Spectrum Bonn University 1 / 27

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Outline

1 Introduction - Motivation

2 Lattice evaluationWilson twisted mass actionSimulation detailsScale settingInterpolating fields - Effective mass

3 Tuning of the strange and charm quark mass

4 ResultsChiral and continuum extrapolation for N f = 2 + 1 + 1Isospin symmetry breaking

5 Comparison

6 Conclusions


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Introduction - Motivation

Why we want to calculate baryon masses?

easy to calculatefirst quantities one calculates before proceeding with more complex observables

large signal to noise ratio

reliable way to study lattice effects

significant for on-going experiments

observation of doubly-charmed Ξ baryons (SELEX, hep-ex/0208014, hep-ex/0209075,hep-ex/0406033) - interest in charmed baryon spectroscopy (G. Bali et al. arXiv:1503.08440, M.

Padmanath et al. arXiv:1502.01845)

are the experimentally known masses reproduced?safe and reliable predictions for the rest


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Lattice evaluationWilson twisted mass action for N f = 2 + 1 + 1

doublet of light quarks: ψ =

u

d

R. Frezzotti et al. arXiv:hep-lat/0306014

Transformation of quark fields:ψ(x) = 1√

2

11 + iτ 3γ 5

χ(x)

ψ(x) = χ(x) 1√ 2

11 + iτ 3γ 5

mass term

ψmψ → χiγ 5τ 3mχ

S (l)F = a4

x

χ(x)

1

2γ µ(∇µ + ∇∗µ)−

ar

2 ∇µ∇

∗µ + m0,l + iγ 5τ 3µ

χ(x)

heavy quarks: χh =

s

c

In the sea we use the action: R. Frezzotti et al. arXiv:hep-lat/0311008

S (h)F = a4

x

χh(x)

1

2γ µ(∇µ + ∇∗µ)−

ar

2 ∇µ∇

∗µ + m0,h + iµσγ 5τ 1 + τ 3µδ

χh(x)

presence of τ

1

introduces mixing of the strange and charm flavorsvalence sector: use Osterwalder-Seiler valence heavy quarks χ(s) = (s+, s−) , χ(c) = (c+, c−)

re-tuning of the strange and charm quark masses required

Wilson TM at maximal twist

cut-off effects are automatically O(a) improved

no operator improvement is needed (important for nucleon structure)


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Lattice evaluationWilson twisted mass action for N f = 2 plus clover

S (l)F = a4

x

χ(x)

1

2γ µ(∇µ + ∇∗µ)−

ar

2 ∇µ∇

∗µ + m0,l + iγ 5τ 3µ +

i

4C SW σ

µν F µν (U )

χ(x)

Clover term

stable simulations

control O(a2) effects

O(a) improvement remains!

C SW = 1.57551

B. Sheikholeslami et al. Nucl.Phys. B259 (1985), S. Aoki et al. hep-lat/0508031


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Lattice evaluationSimulation details

Total of 10 N f = 2 + 1 + 1 gauge ensembles produced by ETMC

N f = 2 plus clover ensemble at the physical pion mass

R. Baron et al. (ETMC) arXiV:1004.5284, A. Abdel-Rehim et al. arXiv:1311.4522

β = 1.90, a = 0.0936(13) fm

323 × 64, L = 3.0 fm

aµ 0.0030 0.0040 0.0050

No. of Confs 200 200 200

mπ (GeV) 0.2607 0.2975 0.3323

mπL 3.97 4.53 5.05

β = 1.95, a = 0.0823(10) fm

323 × 64, L = 2.6 fm

aµ 0.0025 0.0035 0.0055 0.0075

No. of Confs 200 200 200 200

mπ (GeV) 0.2558 0.3018 0.3716 0.4316

mπL 3.42 4.03 4.97 5.77

β = 2.10, a = 0.0646(7) fm

483 × 96, L = 3.1 fm

aµ 0.0015 0.002 0.003

No. of Confs 196 184 200

mπ (GeV) 0.2128 0.2455 0.2984

mπL 3.35 3.86 4.69

β = 2.10, a = 0.0941(12) fm

483 × 96, L = 4.5 fm

aµ 0.0009No. of Confs 524

mπ (GeV) 0.1303

mπL 2.99

two lattice volumes

pion masses from 210-430 MeV → chiral extrapolations

three values of the lattice spacing → investigation of finite lattice effectsC. Kallidonis (CyI) Baryon Spectrum Bonn University 6 / 27

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Lattice evaluationScale setting

for baryon masses → physical nucleon mass

dedicated high statistics analysis on 17 N f = 2 + 1 + 1 ensembles

use HBχPT leading one-loop order result mN = m(0)N − 4c1m2

π − 3g2A16πf 2π

m3π

fit simultaneously for N f = 2 + 1 + 1 and N f = 2 plus clover for all β values

systematic error due to the chiral extrapolation → use O( p4) HBχPT with explicit ∆-degrees of freedom

β a (fm)

1.90 0.0936(13)(35)

1.95 0.0823(10)(35)

2.10 0.0646(7)(25)

2.10 0.0941(12)(2)

fitting for each β separately yields consistent values - negligible cut-off effects for the nucleon case

light σ-term for nucleon σπN = 64.9(1.5)(13.2) MeVC. Kallidonis (CyI) Baryon Spectrum Bonn University 7 / 27

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Lattice evaluationEffective mass

Effective masses are obtained from two-point correlation

functions

C ±B (t, p = 0) =xsink

1

4Tr (1± γ 0) J B (xsink) J B (xsource)

, t = tsink − tsource

Gaussian smearing at source and sink, APE smearing at spatial links

source position chosen randomly

amBeff (t) = log

C B(t)

C B(t + 1)


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Lattice evaluationInterpolating fields

constructed such that they have the quantum numbers of the baryon in interest

4 quark flavors

baryons (qqq)

SU(3) subgroups

of SU(4)

Examples

p (uud) J = abcuT a Cγ 5db

uc

Σ0 (uds) J = 1√ 2abc

uT a Cγ 5sb

dc +

dT a Cγ 5sb

uc

Ξ+c (usc) J = abc

uT a Cγ 5sb

cc

Ξ0 (uss) J µ = abcsT a Cγ µub

sc

Σ++c (uuc) J µ = 1√

3abc

uT a Cγ µub

cc + 2

cT a Cγ µub

uc

Ω0c (ssc) J µ = abcsT a Cγ µcb

sc

20plet of spin-1/2 baryons20 = 8 ⊕ 6⊕ 3⊕ 3

20plet of spin-3/2 baryons20 = 10⊕ 6⊕ 3⊕ 1


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Lattice evaluationInterpolating fields

incorporation of spin-3/2 and spin-1/2 projectors

P µν 3/2 = δ µν −

1

3γ µγ ν , J

µB3/2

= P µν 3/2J νB

P µν 1/2 = δ µν − P

µν 3/2 , J

µB1/2

= P µν 1/2J νB

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 4 8 12 16 20

a m e ff

Σ ∗ +

t / a

3 / 2 projection

1 / 2 projection

No projection

0.4

0.5

0.60.7

0.8

0.9

1

1.1

1.2

2 4 6 8 10 12 14 16

a m e ff

t /a

J Ξ∗− 3 / 2 projection

J Ξ∗− 1 / 2 projection

J Ξ∗− No projection

J Ξ−


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Tuning of the strange and charm quark mass (N f = 2 + 1 + 1)

use Ω− for strange quark and Λ+c

for charm quark

fix renormalized strange and charm masses using non-perturbatively determined renormalization

constants (N. Carrasco et al. arXiv:1403.4504

) in the M S scheme at 2 GeV

Strange quark mass tuning

use a set of strange quark masses to interpolate the mass of Ω− to a given value of

mRs and extrapolate to the continuum and physical pion mass using

mΩ = m0Ω − 4c

(1)Ω m2

π + da2

match with physical mass of Ω−

M S : mRs (2 GeV) = 92.4(6)(2.0) MeV


( )

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Tuning of the strange and charm quark mass (N f = 2 + 1 + 1)

Charm quark mass tuning

follow the same procedure using Λ+c and fit using

mΛc = m0Λc + c1m2π + c2m3π + da2

M S : mRc (2 GeV) = 1173.0(2.4)(17.0) MeV


T i f h d h k (N 2 l l )

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Tuning of the strange and charm quark mass (N f = 2 plus clover)

use Ω− for strange quark and Λ+c

for charm quark

use a set of strange and charm quark masses and interpolate to the physical Ω− and Λ+c mass

interpolate all the rest hyperons and charmed baryons to the tuned valuesof aµs and aµc


T i f th t d h k (N 2 l l )

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Tuning of the strange and charm quark mass (N f = 2 plus clover)Interpolation

Hyperons - Charmed baryons

mass

Min t

!1 !2 ! t !3 !

Charmed baryons with strange quarks

mass !c = !c,1

Ms,1

!s,1 !s,2 !s,t !s,3 !s

mass !c = !c,2

Ms,2

!s,1 !s,2 !s,t !s,3 !s

...

mass

Mint

!c,1 !c,2 !c,t !c,3 !c


Res lts I: Chiral and contin m extrapolation for N 2 + 1 + 1

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Results I: Chiral and continuum extrapolation for N f = 2 + 1 + 1

fit in the whole pion mass range 210-430 MeV

include all β ’s

allow for cut-off effects by including a term ∝ a

2

Hyperons

use leading one-loop order continuum HBχPT

systematic error due to the chiral extrapolation → use O( p4) HBχPT


Results I: Chiral and continuum extrapolation for N 2 + 1 + 1

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Results I: Chiral and continuum extrapolation for N f = 2 + 1 + 1

Charmed baryons

use Ansatz mB = m(0)B + c1m2

π + c2m3π + da2

systematic error due to the chiral extrapolation → set c2 = 0 and restrict mπ < 300 MeV

systematic error due to the tuning for all baryons

finite-a corrections ∼ 1%− 9% - cut-off effects are small

reproduction of experimentally known baryon masses → Predictions


Results I: Chiral and continuum extrapolation for Nf = 2 + 1 + 1

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Results I: Chiral and continuum extrapolation for N f = 2 + 1 + 1Cut-off effects

Baryon d (GeV3) % correction

β = 1.90 β = 1.95 β = 2.10

Ξcc 1.08(7) 6.3 5.0 3.1Ξ∗cc 1.01(10) 5.9 4.6 2.9

Ωcc 1.20(5) 6.9 5.4 3.4

Ω∗cc 1.10(7) 6.2 4.9 3.0

Ωccc 1.15(5) 5.1 4.1 2.6


Results II: Isospin symmetry breaking

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Wilson twisted mass action breaks isospin symmetry explicitly to O(a2)

it is expected to be zero in the continuum limit

manifests itself as mass splitting between baryons belonging to the same isospin multiplets due tolattice artifacts

u ←→ d is a symmetry, e.g. ∆++(uuu), ∆−(ddd) and ∆+(uud), ∆0(ddu) are degenerate



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∆ baryons

isospin splitting effects are consistent with zero for all lattice spacings and pion masses



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Hyperons

small mass splittings for the spin-1/2 hyperons - decreased as a−→ 0

splitting is smaller for the N f = 2 plus clover ensemble

isospin splitting consistent with zero for spin-3/2 hyperons



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p y y g

Charmed baryons

very small effects for spin-1/2 charmed baryons

no isospin symmetry breaking for spin-3/2 charmed baryons


Comparison

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pLattice results from other schemes

BMW: N f = 2 + 1 clover fermions S. Durr et al. arXiV:0906.3599

PACS-CS: N f = 2 + 1 O(a) improved clover fermions A. Aoki et al. arXiV:0807.1661

LHPC: domain wall valence quarks on a staggered fermions sea (hybrid) A. Walker-Loud et al. arXiV:0806.4549

MILC: N f = 2 + 1 + 1 Kogut-Susskind fermion action C.W. Bernard et al. hep/lat 0104002

QCDSF-UKQCD: N f = 2 Wilson fermions G. Bali et al. arXiV:1206.7034


Comparison

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Experiment

Octet - Decuplet spectrum

S. Durr et al. arXiV:0906.3599, A. Aoki et al. arXiV:0807.1661, W. Bietenholz et al. arXiV:1102.5300, Particle Data Group


Comparison

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Experiment

Charm baryons, spin-1/2 spectrum

R. A. Briceno et al. arXiV:1207.3536, H. Na et al. arXiV:0812.1235, H. Na et al. arXiV:0710.1422, L. Liu et al.

arXiV:0909.3294, G. Bali et al. arXiv:1503.08440, Particle Data Group


Comparison

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Experiment

Charm baryons, spin-3/2 spectrum

R. A. Briceno et al. arXiV:1207.3536, H. Na et al. arXiV:0812.1235, H. Na et al. arXiV:0710.1422, G. Bali et al.

arXiv:1503.08440, Particle Data Group


Ongoing - Future work

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finalize work on baryon spectrum for the N f = 2 plus clover ensemble

proceed with calculation of other observables (gA,...)

new implementation in twisted mass CG inverter to accelerate inversions using deflation leads tolarge speed-up! (might become even larger...) - Arnoldi algorithm and ARPACK package

more gauge ensembles from ETMC at the physical pion mass / with N f = 2 plus clover action (?)


Conclusions

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twisted mass formulation with N f = 2 + 1 + 1 flavors provides a good framework to study baryonspectrum

promising results from N f = 2 plus clover ensemble at the physical pion mass physical nucleon mass appropriate to fix lattice spacing when studying baryon masses

isospin symmetry breaking effects are small and vanish as the continuum limit is approached

cut-off effects are small and under control

good agreement with other lattice calculations and with experiment - reliable predictions of theΞ∗cc, Ωcc, Ω∗cc and Ωccc masses

Thank you

The Project Cy-Tera (NEA YΠO∆OMH/ΣTPATH/0308/31) is co-financed by the European Regional Development Fund and theRepublic of Cyprus through the Research Promotion Foundation


Lattice evaluationEff i

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Effective mass

mB

eff (t) = log C B(t)

C B(t + 1) = mB + log 1 +

∞i=1 cie−∆it

1 +∞i=1 cie−∆i(t+1)

−→t→∞

mB , ∆i = mi −mB

mBeff (t) ≈ me

B + log

1 + c1e−∆1t

1 + c1e−∆1(t+1)

criterion for plateau selection

mc

B −meB

12 (mcB + meB) ≤

1

2σ

m

c

B


Backup slidesN l t

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Nucleon σ-term

20 30 40 50 60 70 80 90 100σπN (MeV)

ETMC N f = 2 + 1 + 1 (this work)

C. Alexandrou et al. (ETMC) arXiv:0910.2419

G. Bali et al. (QCDSF) arXiv:1111.1600

L. Alvarez-Ruso et al. arXiv:1304.0483X.-L. Ren et al. arXiv:1404.4799

M.F.M. Lutz et al. arXiv:1401.7805

S. Durr et al. (BMW) arXiv:1109.4265

R. Horsley et al. (QCDSF-UKQCD) arXiv:1110.4971


Backup slidesHyperon σ terms

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Hyperon σ-terms

20 40 60 80

Λ

20 40 60 80

σπB (MeV)

Σ

0 10 20 30

Ξ

ETMC N f = 2 + 1 + 1 (this work)

C. Alexandrou et al. (ETMC) [1]

X.-L. Ren et al. [2]M.F.M. Lutz et al. [3]

S. Durr et al. (BMW) [4]

R. Horsley et al. (QCDSF-UKQCD) [5]

[1] C. Alexandrou et al. (ETMC) arXiv:0910.2419

[2] X.-L. Ren et al. arXiv:1404.4799

[3] M.F.M. Lutz et al. arXiv:1401.7805

[4] S. Durr et al. (BMW) arXiv:1109.4265[5] R. Horsley et al. (QCDSF-UKQCD) arXiv:1110.4971


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