Talk given at the University of Bonn, Germany. We present results on the masses of all forty light, strange and charm baryons from Lattice QCD simulations. We elaborate on the various methods and techniques followed and examine systematic uncertainties related to isospin breaking effects and finite lattice spacing.

1. Hyperon and charmed baryon masses from twisted mass Lattice QCD
(Nf = 2 + 1 + 1, Nf = 2 plus clover)
Christos Kallidonis
Computation-based Science and Technology Research Center
The Cyprus Institute
C. Alexandrou et al. arXiv:1406.4310
with
C. Alexandrou, V. Drach, K, Hadjiyiannakou, K. Jansen, G. Koutsou
Rheinische Friedrich-Wilhelms-Universit¨at Bonn
Bonn, Germany
1 April 2015
C. Kallidonis (CyI) Baryon Spectrum Bonn University 1 / 27

2. Outline
1 Introduction - Motivation
2 Lattice evaluation
Wilson twisted mass action
Simulation details
Scale setting
Interpolating fields - Effective mass
3 Tuning of the strange and charm quark mass
4 Results
Chiral and continuum extrapolation for Nf = 2 + 1 + 1
Isospin symmetry breaking
5 Comparison
6 Conclusions
C. Kallidonis (CyI) Baryon Spectrum Bonn University 2 / 27

3. Introduction - Motivation
Why we want to calculate baryon masses?
easy to calculate
first 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
C. Kallidonis (CyI) Baryon Spectrum Bonn University 3 / 27

4. Lattice evaluation
Wilson twisted mass action for Nf = 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 + iτ3γ5 χ(x)
ψ(x) = χ(x) 1√
2
11 + iτ3γ5



mass term
ψmψ → χiγ5τ3mχ
S
(l)
F = a4
x
χ(x)
1
2
γµ( µ + ∗
µ) −
ar
2
µ
∗
µ + m0,l + iγ5τ3
µ χ(x)
C. Kallidonis (CyI) Baryon Spectrum Bonn University 4 / 27

5. Lattice evaluation
Wilson twisted mass action for Nf = 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 + iτ3γ5 χ(x)
ψ(x) = χ(x) 1√
2
11 + iτ3γ5



mass term
ψmψ → χiγ5τ3mχ
S
(l)
F = a4
x
χ(x)
1
2
γµ( µ + ∗
µ) −
ar
2
µ
∗
µ + m0,l + iγ5τ3
µ 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
2
γµ( µ + ∗
µ) −
ar
2
µ
∗
µ + m0,h + iµσγ5τ1
+ τ3
µδ χh(x)
presence of τ1 introduces mixing of the strange and charm flavors
valence 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)
C. Kallidonis (CyI) Baryon Spectrum Bonn University 4 / 27

6. Lattice evaluation
Wilson twisted mass action for Nf = 2 plus clover
S
(l)
F = a4
x
χ(x)
1
2
γµ( µ + ∗
µ) −
ar
2
µ
∗
µ + m0,l + iγ5τ3
µ +
i
4
CSW σµν
Fµν
(U) χ(x)
Clover term
stable simulations
control O(a2) effects
O(a) improvement remains!
CSW = 1.57551
B. Sheikholeslami et al. Nucl.Phys. B259 (1985), S. Aoki et al. hep-lat/0508031
C. Kallidonis (CyI) Baryon Spectrum Bonn University 5 / 27

7. Lattice evaluation
Simulation details
Total of 10 Nf = 2 + 1 + 1 gauge ensembles produced by ETMC
Nf = 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 0.0009
No. 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 effects
C. Kallidonis (CyI) Baryon Spectrum Bonn University 6 / 27

8. Lattice evaluation
Scale setting
for baryon masses → physical nucleon mass
dedicated high statistics analysis on 17 Nf = 2 + 1 + 1 ensembles
use HBχPT leading one-loop order result mN = m
(0)
N − 4c1m2
π −
3g2
A
16πf2
π
m3
π
fit simultaneously for Nf = 2 + 1 + 1 and Nf = 2 plus clover for all β values
systematic error due to the chiral extrapolation → use O(p4) HBχPT with explicit ∆-degrees of
freedom
0.8
0.9
1
1.1
1.2
1.3
1.4
0 0.05 0.1 0.15 0.2 0.25
mN(GeV)
mπ
2 (GeV2)
β=1.90, L/a=32, L=3.0fm
β=1.90, L/a=24, L=2.2fm
β=1.90, L/a=20, L=1.9fm
β=1.95, L/a=32, L=2.6fm
β=1.95, L/a=24, L=2.0fm
β=2.10, L/a=48, L=3.1fm
β=2.10, L/a=32, L=2.1fm
β=2.10, CSW=1.57551, L/a=48, L=4.5fm
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) MeV
C. Kallidonis (CyI) Baryon Spectrum Bonn University 7 / 27

9. Lattice evaluation
Effective mass
Effective masses are obtained from two-point correlation
functions
C±
B (t, p = 0) =
xsink
1
4
Tr (1 ± γ0) JB (xsink) ¯JB (xsource) , t = tsink − tsource
Gaussian smearing at source and sink, APE smearing at spatial links
source position chosen randomly
C. Kallidonis (CyI) Baryon Spectrum Bonn University 8 / 27

10. Lattice evaluation
Effective mass
Effective masses are obtained from two-point correlation
functions
C±
B (t, p = 0) =
xsink
1
4
Tr (1 ± γ0) JB (xsink) ¯JB (xsource) , t = tsink − tsource
Gaussian smearing at source and sink, APE smearing at spatial links
source position chosen randomly
amB
eff(t) = log
CB(t)
CB(t + 1)
0
0.4
0.8
1.2
1.6
2 4 6 8 10 12 14 16 18
ameff
t/a
Σc
*++
Λc
+
Ω
Ξ0
N
C. Kallidonis (CyI) Baryon Spectrum Bonn University 8 / 27

11. Lattice evaluation
Interpolating 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 = abc uT
a Cγ5db uc
Σ0 (uds) J = 1√
2 abc uT
a Cγ5sb dc + dT
a Cγ5sb uc
Ξ+
c (usc) J = abc uT
a Cγ5sb cc
Ξ 0 (uss) Jµ = abc sT
a Cγµub sc
Σ ++
c (uuc) Jµ = 1√
3 abc uT
a Cγµub cc + 2 cT
a Cγµub uc
Ω 0
c (ssc) Jµ = abc sT
a Cγµcb sc
20plet of spin-1/2 baryons
20 = 8 ⊕ 6 ⊕ 3 ⊕ 3
20plet of spin-3/2 baryons
20 = 10 ⊕ 6 ⊕ 3 ⊕ 1
C. Kallidonis (CyI) Baryon Spectrum Bonn University 9 / 27

12. Lattice evaluation
Interpolating fields
incorporation of spin-3/2 and spin-1/2 projectors
Pµν
3/2 = δµν
−
1
3
γµγν , J µ
B3/2
= Pµν
3/2JνB
Pµν
1/2 = δµν
− Pµν
3/2 , J µ
B1/2
= Pµν
1/2JνB
C. Kallidonis (CyI) Baryon Spectrum Bonn University 10 / 27

13. Lattice evaluation
Interpolating fields
incorporation of spin-3/2 and spin-1/2 projectors
Pµν
3/2 = δµν
−
1
3
γµγν , J µ
B3/2
= Pµν
3/2JνB
Pµν
1/2 = δµν
− Pµν
3/2 , J µ
B1/2
= Pµν
1/2JνB0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 4 8 12 16 20
ame↵⌃⇤++
c
t/a
3/2 projection
1/2 projection
No projection
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 4 8 12 16 20
ame↵⌃⇤+
t/a
3/2 projection
1/2 projection
No projection
C. Kallidonis (CyI) Baryon Spectrum Bonn University 10 / 27

14. Lattice evaluation
Interpolating fields
incorporation of spin-3/2 and spin-1/2 projectors
Pµν
3/2 = δµν
−
1
3
γµγν , J µ
B3/2
= Pµν
3/2JνB
Pµν
1/2 = δµν
− Pµν
3/2 , J µ
B1/2
= Pµν
1/2JνB0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 4 8 12 16 20
ame↵⌃⇤++
c
t/a
3/2 projection
1/2 projection
No projection
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 4 8 12 16 20
ame↵⌃⇤+
t/a
3/2 projection
1/2 projection
No projection
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
2 4 6 8 10 12 14 16
ame↵
t/a
J⌅⇤0 3/2 projection
J⌅⇤0 1/2 projection
J⌅⇤0 No projection
J⌅0
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
2 4 6 8 10 12 14 16
ame↵
t/a
J⌅⇤ 3/2 projection
J⌅⇤ 1/2 projection
J⌅⇤ No projection
J⌅
C. Kallidonis (CyI) Baryon Spectrum Bonn University 10 / 27

15. Tuning of the strange and charm quark mass (Nf = 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 MS scheme at 2 GeV
C. Kallidonis (CyI) Baryon Spectrum Bonn University 11 / 27

16. Tuning of the strange and charm quark mass (Nf = 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 MS 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
mR
s and extrapolate to the continuum and physical pion mass using
mΩ = m0
Ω − 4c
(1)
Ω m2
π + da2
match with physical mass of Ω−
mΩ
phys
1.6
1.65
1.7
1.75
1.8
85 90 95 100 105 110 115
mΩ-(GeV)
ms
R (MeV)
ms
R = 92.4(6) MeV
1.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85
0 0.05 0.1 0.15 0.2 0.25
mΩ-(GeV)
mπ
2 (GeV2)
β=1.90, L/a=32
β=1.95, L/a=32
β=2.10, L/a=48
Continuum limit
MS : mR
s (2 GeV) = 92.4(6)(2.0) MeV
C. Kallidonis (CyI) Baryon Spectrum Bonn University 11 / 27

17. Tuning of the strange and charm quark mass (Nf = 2 + 1 + 1)
Charm quark mass tuning
follow the same procedure using Λ+
c and fit using
mΛc = m0
Λc
+ c1m2
π + c2m3
π + da2
mΛc
phys
2.26
2.28
2.3
2.32
1150 1160 1170 1180 1190 1200
mΛc
+(GeV)
mc
R (MeV)
mc
R = 1173.0(2.4) MeV
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
0 0.05 0.1 0.15 0.2 0.25
mΛc
+(GeV)
mπ
2 (GeV2)
β=1.90, L/a=32
β=1.95, L/a=32
β=2.10, L/a=48
Continuum limit
MS : mR
c (2 GeV) = 1173.0(2.4)(17.0) MeV
C. Kallidonis (CyI) Baryon Spectrum Bonn University 12 / 27

18. Tuning of the strange and charm quark mass (Nf = 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
mphys
Ω
1.6
1.65
1.7
1.75
0.023 0.024 0.025 0.026 0.027 0.028
mΩ(GeV)
aμs
aμs
phys = 0.0264(3)
mphys
Λc
+
2.15
2.2
2.25
2.3
2.35
2.4
0.3 0.31 0.32 0.33 0.34 0.35 0.36
mΛc
+
(GeV)
aμc
aμc
phys = 0.3346(15)
C. Kallidonis (CyI) Baryon Spectrum Bonn University 13 / 27

19. Tuning of the strange and charm quark mass (Nf = 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
mphys
Ω
1.6
1.65
1.7
1.75
0.023 0.024 0.025 0.026 0.027 0.028
mΩ(GeV)
aμs
aμs
phys = 0.0264(3)
mphys
Λc
+
2.15
2.2
2.25
2.3
2.35
2.4
0.3 0.31 0.32 0.33 0.34 0.35 0.36
mΛc
+
(GeV)
aμc
aμc
phys = 0.3346(15)
interpolate all the rest hyperons and charmed baryons to the tuned values
of aµs and aµc
C. Kallidonis (CyI) Baryon Spectrum Bonn University 13 / 27

20. Tuning of the strange and charm quark mass (Nf = 2 plus clover)
Interpolation
Hyperons - Charmed baryons
mass
Mint
μ1 μ2 μt μ3 μ
Charmed baryons with strange quarks
mass μc = μc,1
Ms,1
μs,1 μs,2 μs,t μs,3 μs
mass μc = μc,2
Ms,2
μs,1 μs,2 μs,t μs,3 μs
...
mass
Mint
μc,1 μc,2 μc,t μc,3 μc
C. Kallidonis (CyI) Baryon Spectrum Bonn University 14 / 27

21. Results I: Chiral and continuum extrapolation for Nf = 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 ∝ a2
C. Kallidonis (CyI) Baryon Spectrum Bonn University 15 / 27

22. Results I: Chiral and continuum extrapolation for Nf = 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 ∝ a2
Hyperons
use leading one-loop order continuum HBχPT
systematic error due to the chiral extrapolation → use O(p4) HBχPT
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
0 0.05 0.1 0.15 0.2 0.25
mΣ0(GeV)
mπ
2 (GeV2)
NLO HBχPT
LO HBχPT
β=2.10, L/a=48
β=1.95, L/a=32
β=1.90, L/a=32
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
1.8
0 0.05 0.1 0.15 0.2 0.25
mΞ*(GeV)
mπ
2 (GeV2)
NLO HBχPT
LO HBχPT
β=2.10, L/a=48
β=1.95, L/a=32
β=1.90, L/a=32
C. Kallidonis (CyI) Baryon Spectrum Bonn University 15 / 27

23. Results I: Chiral and continuum extrapolation for Nf = 2 + 1 + 1
Charmed baryons
use Ansatz mB = m
(0)
B + c1m2
π + c2m3
π + da2
systematic error due to the chiral extrapolation → set c2 = 0 and restrict mπ < 300 MeV
2.35
2.4
2.45
2.5
2.55
2.6
0 0.05 0.1 0.15 0.2 0.25
mΞc
0(GeV)
mπ
2 (GeV2)
mπ < 0.300GeV
mπ < 0.432GeV
β=2.10, L/a=48
β=1.95, L/a=32
β=1.90, L/a=32
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
0 0.05 0.1 0.15 0.2 0.25
mΣc
*(GeV)
mπ
2 (GeV2)
mπ < 0.300GeV
mπ < 0.432GeV
β=2.10, L/a=48
β=1.95, L/a=32
β=1.90, L/a=32
C. Kallidonis (CyI) Baryon Spectrum Bonn University 16 / 27

24. Results I: Chiral and continuum extrapolation for Nf = 2 + 1 + 1
Charmed baryons
use Ansatz mB = m
(0)
B + c1m2
π + c2m3
π + da2
systematic error due to the chiral extrapolation → set c2 = 0 and restrict mπ < 300 MeV
2.35
2.4
2.45
2.5
2.55
2.6
0 0.05 0.1 0.15 0.2 0.25
mΞc
0(GeV)
mπ
2 (GeV2)
mπ < 0.300GeV
mπ < 0.432GeV
β=2.10, L/a=48
β=1.95, L/a=32
β=1.90, L/a=32
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
0 0.05 0.1 0.15 0.2 0.25
mΣc
*(GeV)
mπ
2 (GeV2)
mπ < 0.300GeV
mπ < 0.432GeV
β=2.10, L/a=48
β=1.95, L/a=32
β=1.90, L/a=32
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
C. Kallidonis (CyI) Baryon Spectrum Bonn University 16 / 27

25. Results I: Chiral and continuum extrapolation for Nf = 2 + 1 + 1
Cut-off effects
Baryon d (GeV3)
% correction
β = 1.90 β = 1.95 β = 2.10
Ξcc 1.08(7) 6.3 5.0 3.1
Ξ∗
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
1.6
1.65
1.7
1.75
1.8
1.85
1.9
0 0.05 0.1 0.15 0.2 0.25
mΩ(GeV)
a2 (1/GeV2)
mΩ = 1.672(7) + 0.466(4) a2
4.7
4.8
4.9
5
5.1
0 0.05 0.1 0.15 0.2 0.25
mΩccc
(GeV)
a2 (1/GeV2)
mΩccc
= 4.734(9) + 1.154(10) a2
C. Kallidonis (CyI) Baryon Spectrum Bonn University 17 / 27

26. Results II: Isospin symmetry breaking
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 to
lattice artifacts
u ←→ d is a symmetry, e.g. ∆++(uuu), ∆−(ddd) and ∆+(uud), ∆0(ddu) are degenerate
C. Kallidonis (CyI) Baryon Spectrum Bonn University 18 / 27

27. Results II: Isospin symmetry breaking
∆ baryons
-0.08
-0.04
0
0.04
0.08
0.12
0 0.002 0.004 0.006 0.008 0.01
Δm(GeV)
a2 (fm2)
Δ++,- - Δ+,0
isospin splitting effects are consistent with zero for all lattice spacings and pion masses
C. Kallidonis (CyI) Baryon Spectrum Bonn University 19 / 27

28. Results II: Isospin symmetry breaking
Hyperons
-0.08
-0.04
0
0.04
0.08
0.12
0 0.002 0.004 0.006 0.008 0.01
Δm(GeV)
a2 (fm2)
Ξ0 - Ξ-
-0.08
-0.04
0
0.04
0.08
0.12
0 0.002 0.004 0.006 0.008 0.01
Δm(GeV)
a2 (fm2)
Ξ*0 - Ξ*-
small mass splittings for the spin-1/2 hyperons - decreased as a −→ 0
splitting is smaller for the Nf = 2 plus clover ensemble
isospin splitting consistent with zero for spin-3/2 hyperons
C. Kallidonis (CyI) Baryon Spectrum Bonn University 20 / 27

29. Results II: Isospin symmetry breaking
Charmed baryons
-0.08
-0.04
0
0.04
0.08
0.12
0 0.002 0.004 0.006 0.008 0.01
Δm(GeV)
a2 (fm2)
Ξcc
++ - Ξcc
+
-0.08
-0.04
0
0.04
0.08
0.12
0 0.002 0.004 0.006 0.008 0.01
Δm(GeV)
a2 (fm2)
Ξcc
*++ - Ξcc
*+
very small effects for spin-1/2 charmed baryons
no isospin symmetry breaking for spin-3/2 charmed baryons
C. Kallidonis (CyI) Baryon Spectrum Bonn University 21 / 27

30. Comparison
Lattice results from other schemes
0.6
0.8
1
1.2
1.4
1.6
0 0.05 0.1 0.15 0.2 0.25
mN(GeV)
mπ
2 (GeV2)
ETMC Nf=2+1+1
ETMC Nf=2 with CSW
BMW
LHPC
QCDSF-UKQCD
MILC
PACS-CS
0.9
1
1.1
1.2
1.3
1.4
1.5
0 0.05 0.1 0.15 0.2 0.25
mΛ(GeV)
mπ
2 (GeV2)
ETMC
ETMC Nf=2 with CSW
BMW
PACS-CS
LHPC
BMW: Nf = 2 + 1 clover fermions S. Durr et al. arXiV:0906.3599
PACS-CS: Nf = 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: Nf = 2 + 1 + 1 Kogut-Susskind fermion action C.W. Bernard et al. hep/lat 0104002
QCDSF-UKQCD: Nf = 2 Wilson fermions G. Bali et al. arXiV:1206.7034
C. Kallidonis (CyI) Baryon Spectrum Bonn University 22 / 27

31. Comparison
Experiment
Octet - Decuplet spectrum
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
N Λ Σ Ξ Δ Σ* Ξ* Ω
M(GeV)
ETMC Nf=2+1+1
ETMC Nf=2 with CSW
BMW Nf=2+1
PACS-CS Nf=2+1
QCDSF-UKQCD Nf=2+1
S. Durr et al. arXiV:0906.3599, A. Aoki et al. arXiV:0807.1661, W. Bietenholz et al. arXiV:1102.5300, Particle Data Group
C. Kallidonis (CyI) Baryon Spectrum Bonn University 23 / 27

32. Comparison
Experiment
Charm baryons, spin-1/2 spectrum
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
Λc Σc Ξc Ξ'
c
Ωc Ξcc Ωcc
M(GeV)
ETMC Nf=2+1+1
ETMC Nf=2 with CSW
PACS-CS Nf=2+1
Na et al. Nf=2+1
Briceno et al. Nf=2+1+1
Liu et al. Nf=2+1
G. Bali et al. Nf=2+1
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
C. Kallidonis (CyI) Baryon Spectrum Bonn University 24 / 27

33. Comparison
Experiment
Charm baryons, spin-3/2 spectrum
2.5
3
3.5
4
4.5
5
Σc
* Ξc
* Ωc
* Ξcc
* Ωcc
* Ωccc
M(GeV)
ETMC Nf=2+1+1
ETMC Nf=2 with CSW
PACS-CS Nf=2+1
Na et al. Nf=2+1
Briceno et al. Nf=2+1+1
G. Bali et al. Nf=2+1
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
C. Kallidonis (CyI) Baryon Spectrum Bonn University 25 / 27

34. Ongoing - Future work
finalize work on baryon spectrum for the Nf = 2 plus clover ensemble
proceed with calculation of other observables (gA,...)
new implementation in twisted mass CG inverter to accelerate inversions using deflation leads to
large speed-up! (might become even larger...) - Arnoldi algorithm and ARPACK package
more gauge ensembles from ETMC at the physical pion mass / with Nf = 2 plus clover action (?)
C. Kallidonis (CyI) Baryon Spectrum Bonn University 26 / 27

35. Conclusions
twisted mass formulation with Nf = 2 + 1 + 1 flavors provides a good framework to study baryon
spectrum
promising results from Nf = 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 the
Ξ∗
cc, Ωcc, Ω∗
cc and Ωccc masses
C. Kallidonis (CyI) Baryon Spectrum Bonn University 27 / 27

36. Conclusions
twisted mass formulation with Nf = 2 + 1 + 1 flavors provides a good framework to study baryon
spectrum
promising results from Nf = 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 the
Ξ∗
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 the
Republic of Cyprus through the Research Promotion Foundation
C. Kallidonis (CyI) Baryon Spectrum Bonn University 27 / 27

37. Lattice evaluation
Effective mass
mB
eff(t) = log
CB(t)
CB(t + 1)
= mB + log
1 + ∞
i=1 cie−∆it
1 + ∞
i=1 cie−∆i(t+1)
−→
t→∞
mB , ∆i = mi − mB
mB
eff(t) ≈ me
B + log
1 + c1e−∆1t
1 + c1e−∆1(t+1)
criterion for plateau selection
mc
B − me
B
1
2(mc
B + me
B)
≤
1
2
σmc
B
0.3
0.4
0.5
0.6
0.7
0 4 8 12 16 20
ameff
t/a
Ξ0
Exponential fit
Constant fit
C. Kallidonis (CyI) Baryon Spectrum Bonn University 1 / 3

38. Backup slides
Nucleon σ-term
20 30 40 50 60 70 80 90 100
σπN (MeV)
ETMC Nf = 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:1304.0483
X.-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
C. Kallidonis (CyI) Baryon Spectrum Bonn University 2 / 3

39. Backup slides
Hyperon σ-terms
20 40 60 80
Λ
20 40 60 80
σπB (MeV)
Σ
0 10 20 30
Ξ
ETMC Nf = 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
C. Kallidonis (CyI) Baryon Spectrum Bonn University 3 / 3