Strong contribution to octet baryon mass splittings

0.39 MB·English

by  P. E. Shanahan

ADP-12-35/T802
Strong contribution to octet baryon mass splittings
P.E.Shanahan,A.W.Thomas∗,R.D.Young
ARCCentreofExcellenceinParticlePhysicsattheTerascaleandCSSM,SchoolofChemistryandPhysics,
UniversityofAdelaide,AdelaideSA5005,Australia
2
1 Abstract
0
2 Wecalculatethemd−mucontributiontothemasssplittingsinbaryonicisospinmultipletsusing
p SU(3) chiral perturbation theory and lattice QCD. Fitting isospin-averaged perturbation theory
e functions to PACS-CS and QCDSF-UKQCD Collaboration lattice simulations of octet baryon
S
masses,andusingthephysicallightquarkmassratiomu/md asinput,allows Mn−Mp, MΣ− −MΣ+
0
and M − M to be evaluated from the full SU(3) theory. The resulting values for each mass
1 Ξ− Ξ0
splittingareconsistentwiththeexperimentalvaluesafterallowingforelectromagneticcorrections.

h] Inthecaseofthenucleon,wefind Mn−Mp = 2.9±0.4MeV,withthedominantuncertaintyarising
-t fromtheerrorinmu/md.
l
c
Keywords:
u
n Isospinbreaking,Baryonmasssplittings,Chargesymmetrybreaking,SU(3)chiralperturbation
[
theory

1
v
2
9
8 1. Introduction
1
. The physical mass splittings between members of baryonic isospin multiplets have been mea-
9
0 suredextremelyprecisely[1,2]:
2
1 M − M = 1.2933322(4)MeV, (1a)
: n p
v
i MΣ− − MΣ+ = 8.079(76)MeV, (1b)
X
M − M = 6.85(21)MeV. (1c)
r Ξ− Ξ0
a
However,thedecompositionofeachintoitstwocomponents,arisingfromelectromagneticeffects
and the d −u quark mass difference, is less well known. Clearly, once one contribution has been
welldeterminedtheothercanbeinferredfromthetotal.
In recent years, several research groups have presented lattice determinations of both the
QCD contribution to the baryon mass splittings, e.g., [3–6], and the electromagnetic contribu-
tion, e.g., [7–10]. This work uses SU(3) chiral perturbation theory to determine estimates of the
∗Correspondingauthor
PreprintsubmittedtoPhysicsLettersB September11,2012
B C(1) C(1)
Bl Bs
N 2α+2β+4σ 2σ
Λ α+2β+4σ α+2σ
Σ 5α+ 2β+4σ 1α+ 4β+2σ
3 3 3 3
Ξ 1α+ 4β+4σ 5α+ 2β+2σ
3 3 3 3
Table 1: Coefficients of terms linear in the non-strange quark mass, Bm → m2/2, and the strange quark mass,
l π
Bm →(m2 −m2/2),expressedintermsoftheleadingquark-massinsertionparametersα,βandσ.
s K π
strong contribution, by fitting currently available isospin-averaged lattice calculations [11, 12] of
theoctetbaryonmasses. Wedonotconsiderelectromagneticeffects.
Our evaluation of the strong contribution to M − M is of particular interest in the light of
n p
recent results which suggest that the accepted value for the electromagnetic contribution, namely
∆ = 0.76 ± 0.30 MeV [13], may be too small. Walker-Loud et al. (WLCM) claim to find
EM
an omission in the traditional analysis and present a larger value of 1.3 ± 0.6 MeV [14]. From
these estimates, one infers strong isospin breaking contributions of ∆ = 2.05 ± 0.30 MeV
md−mu
(traditional) and 2.6±0.6 MeV (WLCM) respectively. Clearly, independent theoretical estimates
ofthesizeofthestrongcontributionto M −M ,suchasreportedhere,areofconsiderablevalue.
n p
2. Method
2.1. Isospin-averagedfit
The fit to isospin-averaged PACS-CS lattice results [11] which we use for this work has been
reported on in previous papers; we refer to [15–17] for details. In brief, we use a standard heavy-
baryon chiral perturbation theory formulation with a finite-range regularization scheme (FRR),
discussedin[18–22]. Themassofabaryon Binthechiralexpansioniswrittenas
M = M(0) +δM(1) +δM(3/2) +..., (2)
B B B
where the leading term M(0) denotes the degenerate mass of the octet baryons in the SU(3) chiral
limit and is independent of the quark mass matrix M and baryon B. The notation M(n) denotes
q B
thecontributiontotheoctetbaryonmassatorderM(n).
q
Thecorrectionlinearinthequarkmassescanbeexpressedas
δM(1) = −C(1)Bm −C(1)Bm , (3)
B Bl l Bs s
withthecoefficientsgiveninTable1. TheleadingloopcorrectionswhichcontributetoδM(3/2) are
B
madeexplicitin[16]andincludebothoctetanddecupletbaryonintermediatestates.
We retain the octet-decuplet mass splitting δ in numerical evaluations, setting δ = 0.292 GeV
to the physical N − ∆ splitting. The baryon-baryon-meson coupling constants are taken from
phenomenology; D + F = g = 1.27, F = 2D and C = −2D, and f is set to f = 0.0871 GeV,
A 3
a chiral perturbation theory estimate for the pion decay constant in the SU(3) chiral limit [23].
2
The fit parameters are the octet baryon mass in the chiral limit M(0), the SU(3) chiral symmetry
breaking parameters α, β, σ, and the finite-range regulator mass Λ, and correspond to those in
Refs.[16,17].
The fit to the PACS-CS baryon octet data is shown in [17], and a comparison between the
experimentalvaluesandtheoctetbaryonmassesevaluatedatthephysicalpointisgiveninTable2.
B Mass(GeV) Experimental
N 0.959(24)(9) 0.939
Λ 1.129(15)(6) 1.116
Σ 1.188(11)(6) 1.193
Ξ 1.325(6)(2) 1.318
Table 2: Extracted masses for the octet baryons. The first uncertainty quoted is statistical, while the second allows
for variation of the form of the UV regulator and 10% deviation of f, F, C, and δ from their central values. The
experimentalbaryonmassesareshownforcomparison.
2.2. EvaluationofSU(2)masssplittings
AfeatureofSU(3)chiralperturbationtheoryisthatthesamecoefficientsappearinthebaryon
mass expansion both when including and excluding SU(2) breaking effects. These coefficients,
once evaluated by fitting to the isospin-averaged lattice results as done above, can thus be used
to provide information about the SU(2) breaking mass splittings. That is, we can use N = 2+1
f
flavor lattice simulations, which are currently available, and are computationally cheaper than the
SU(2)-broken N = 1+1+1,toderiveourresults.
f
Tocalculatethebaryonmasssplittings,wemodifytheSU(3)chiralperturbationtheoryexpan-
sionsused inthe previoussectionto allowfora non-zerolight quarkmasssplitting: m −m (cid:44) 0.
d u
Whilethisisastraightforwardextension,wenotethatitgeneratesaπ0ηmixingtermintheSU(3)
Lagrangian. Thefieldsmustthusbediagonalizedintothemassbasisviaafieldrotation.
Tobeexplicit,werecalltheusualdefinitionofthemesonfield:
Σ = exp(cid:32)2ifΦ(cid:33) = ξ2, Φ = √12  √12π0Kπ+−− √16η −√12πKπ0+0+ √16η −KK√2+0η , (4)
6
andmesonLagrangian:
f2
L = Tr(∂µΣ†∂ Σ)+λTr(M (Σ† +Σ)), (5)
eff µ q
8
whereM = diag(m ,m ,m )is,asabove,thequarkmassmatrix.
q u d s
3
ExpandingthisLagrangianinpowersofthemesonfield,themasstermcanbewrittenas
L =BTr(M Φ2) (6a)
kin q
=B(m +m )(π+π−)+ B(m +m )(K0K0) (6b)
u d s d
B
+ B(m +m )(K+K−)+ (m +m )((π0)2)
s u u u
2
B B
+ (m +m +4m )(η2)+ √ (m −m )(ηπ0).
d u s u d
6 3
Clearly,form (cid:44) m ,mixingoccursbetweentheπ0 andη.
u d
Toidentifythemesonmassesweremovethismixingandbringthekinetictermintothecanon-
icalformviaafieldrotation:
π0 → π0cos(cid:15) −ηsin(cid:15), (7a)
η → π0sin(cid:15) +ηcos(cid:15), (7b)
wherethemixingangle(cid:15) isgivenby
√
3(m −m )
tan2(cid:15) = d u . (8)
2m −(m +m )
s d u
Afterperformingthisrotation,theSU(3)mesonmassestaketheform:
m2 = B(m +m ) (9a)
π± u d
2B sin2(cid:15)
m2 = B(m +m )− (2m −(m +m )) (9b)
π0 u d 3 s u d cos2(cid:15)
m2 = B(m +m ) (9c)
K± s u
m2 = B(m +m ) (9d)
K0 s d
B 2B sin2(cid:15)
m2 = (4m +m +m )+ (2m −(m +m )) , (9e)
η 3 s u d 3 s u d cos2(cid:15)
wherem andm nowcontainsomedependenceonthemixingangle(cid:15).
π0 η
Thisextensiongeneratesaseparatemassexpansion,oftheformofEquation2,foreachmem-
berofthebaryonoctet. Thetermslinearinquarkmasscanbeexpressedas
δM(1) = −C(1)Bm −C(1)Bm −C(1)Bm , (10)
B Bu u Bd d Bs s
wherethecoefficientsC aregivenexplicitlyinTable3.
Bq
The loop contributions δM(3/2) have the same form as in the isospin-averaged case, with sep-
B
arate couplings and integrals for each of the mesons π±,π0,K±,K0,η. The π± and K± remain
pairwise mass-degenerate. Of course, because of the redefinition of the meson fields, the baryon-
baryon-mesoncouplingswillalso receivecontributionsdependingon(cid:15),andare nowcomplicated
functionsofquarkmassandthecouplingconstantsF,D,andC. ThesearegiveninTables4and5.
Asexpected,setting(cid:15) → 0returnstheusualisospin-averagedfunctions.
4
B C(1) C(1) C(1)
Bu Bd Bs
p 5α+ 2β+2σ 1α+ 4β+2σ 2σ
3 3 3 3
n 1α+ 4β+2σ 5α+ 2β+2σ 2σ
3 3 3 3
Σ+ 5α+ 2β+2σ 2σ 1α+ 4β+2σ
3 3 3 3
Σ− 2σ 5α+ 2β+2σ 1α+ 4β+2σ
3 3 3 3
Ξ0 1α+ 4β+2σ 2σ 5α+ 2β+2σ
3 3 3 3
Ξ− 2σ 1α+ 4β+2σ 5α+ 2β+2σ
3 3 3 3
Table 3: Values for the terms linear in the up, down and strange quark masses, expressed in terms of the SU(3)
breakingparametersα,βandσ.
χ C−2
Tφ
π0 π± K0 K± η
p 4cos2(cid:15) 8 2 1 4sin2(cid:15)
9 9 9 9 9
n 4cos2(cid:15) 8 1 2 4sin2(cid:15)
9 √ 9 9 9 √ 9
Σ+ 1(cos(cid:15) + 3sin(cid:15))2 1 2 8 1(− 3cos(cid:15) +sin(cid:15))2
9 √ 9 9 9 9 √
Σ− 1(−cos(cid:15) + 3sin(cid:15))2 1 8 2 1( 3cos(cid:15) +sin(cid:15))2
9 √ 9 9 9 9 √
Ξ0 1(cos(cid:15) + 3sin(cid:15))2 2 1 8 1(− 3cos(cid:15) +sin(cid:15))2
9 √ 9 9 9 9 √
Ξ− 1(−cos(cid:15) + 3sin(cid:15))2 2 8 1 1( 3cos(cid:15) +sin(cid:15))2
9 9 9 9 9
Table 4: Chiral SU(3) coefficients for the coupling of the octet baryons to decuplet (T) baryons through the pseu-
doscalaroctetmesonφ.
It is now straightforward to write expressions for the baryon mass splittings as a function of
quark mass only. All other free parameters, namely the SU(3) breaking parameters α, β and σ, as
wellastheregulatormassΛ,arespecifiedbytheisospin-averagedfitdescribedpreviously.
To evaluate the baryon mass splittings at the physical point, we input the physical light-quark
massratioR := mu. TheGell-Mann-OakesRennerrelationsuggeststhedefinition
md
B(m −m ) 1(1−R)
ω = d u := m2 , (11)
2 2(1+R) π(phys)
whichallowsustodefine
Bm = m2 /2−ω, (12a)
u π(phys)
Bm = m2 /2−ω, (12b)
d π(phys)
Bm = m2 −m2 /2. (12c)
s K(phys) π(phys)
Here, we take m = 137.3 MeV and m = 497.5 MeV to be the physical isospin-averaged
π(phys) K(phys)
mesonmasses[2].
Evaluating the mass splitting expressions at these ‘physical’ quark masses, with loop meson
masses calculated using Equation 9, then gives our estimate of the baryon mass differences at the
physicalpoint. Adiscussionoftheerroranalysisisgiveninsection3.1.
5
χ
Bφ
π0
√
p 1(2(D2 +3F2)+(D2 +6DF −3F2)cos(2(cid:15))− 3(D−3F)(D+F)sin(2(cid:15)))
6 √
n 1(2(D2 +3F2)+(D2 +6DF −3F2)cos(2(cid:15))+ 3(D−3F)(D+F)sin(2(cid:15)))
6 √
Σ+ F2 +F2cos(2(cid:15))+ 2Dsin(cid:15)(2 3Fcos(cid:15) +Dsin(cid:15))
3 √
Σ− F2 +F2cos(2(cid:15))+ 2Dsin(cid:15)(−2 3Fcos(cid:15) +Dsin(cid:15))
3 √
Ξ0 1(2(D2 +3F2)+(D2 −6DF −3F2)cos(2(cid:15))+ 3(D+3F)(D−F)sin(2(cid:15)))
6 √
Ξ− 1(2(D2 +3F2)+(D2 −6DF −3F2)cos(2(cid:15))− 3(D+3F)(D−F)sin(2(cid:15)))
6
η
√
p 1(2(D2 +3F2)−(D2 +6DF −3F2)cos(2(cid:15))+ 3(D−3F)(D+F)sin(2(cid:15)))
6 √
n 1(2(D2 +3F2)−(D2 +6DF −3F2)cos(2(cid:15))− 3(D−3F)(D+F)sin(2(cid:15)))
6 √
Σ+ 2(D2cos2(cid:15) −2 3DFcos(cid:15)sin(cid:15) +3F2sin2(cid:15))
3 √
Σ− 2(D2cos2(cid:15) +2 3DFcos(cid:15)sin(cid:15) +3F2sin2(cid:15))
3 √
Ξ0 1(2(D2 +3F2)+(−D2 +6DF +3F2)cos(2(cid:15))− 3(D+3F)(D−F)sin(2(cid:15)))
6 √
Ξ− 1(2(D2 +3F2)+(−D2 +6DF +3F2)cos(2(cid:15))+ 3(D+3F)(D−F)sin(2(cid:15)))
6
π± K0 K±
p (D+F)2 (D−F)2 2(D2 +3F2)
3
n (D+F)2 2(D2 +3F2) (D−F)2
3
Σ+ 2(D2 +3F2) (D−F)2 (D+F)2
3
Σ− 2(D2 +3F2) (D+F)2 (D−F)2
3
Ξ0 (D−F)2 2(D2 +3F2) (D+F)2
3
Ξ− (D−F)2 (D+F)2 2(D2 +3F2)
3
Table5: ChiralSU(3)coefficientsforthecouplingoftheoctetbaryonstooctet(B)baryonsthroughthepseudoscalar
octetmesonφ.
3. Results
The calculation outlined in the previous section, where the central value is obtained using the
statedphenomenologicalestimatesfor f, F,Candδ,gives
M − M = (ω/m2 )(20.3±1.2)MeV (13a)
n p π(phys)
MΣ− − MΣ+ = (ω/m2π(phys))(52.6±2.0)MeV (13b)
M − M = (ω/m2 )(32.3±1.6)MeV. (13c)
Ξ− Ξ0 π(phys)
The quoted uncertainties contain all statistical and systematic errors, discussed in the following
section,combinedinquadrature.
Wetakeasinputthecurrentbest-estimateforthephysicalup-downquarkmassratio[24],
m
R := u = 0.553±0.043, (14)
m
d
6
determined by a fit to meson decay rates. We note that this value is compatible with more recent
estimates of the ratio from 2+1 and 3 flavor QCD and QED [4, 25]. Including in quadrature the
uncertaintyduetothestatedrangeforR,wefind
M − M = 2.9±0.4MeV (15a)
n p
MΣ− − MΣ+ = 7.5±1.0MeV (15b)
M − M = 4.6±0.6MeV. (15c)
Ξ− Ξ0
3.1. Statisticalandsystematicuncertainties
Theerrorsquotedaretheresultofacompleteerroranalysis,takingintoaccountthecorrelated
uncertainties arising from all of the fit parameters, as well as propagating the quoted uncertainty
in R. We estimate the systematic error in our result by considering variations of the regulator and
allowing for deviation of the phenomenologically set parameters f, F, C and δ from their central
valuesby±10%.
Monopole, dipole, Gaussian and sharp cutoff regulators u(k) are considered in our analysis.
The variation of our final results as u(k) is changed is of order 1% of our determined mass dif-
ferences, and is included in the quoted error. The deviation as the parameters f, F, C and δ are
perturbed is similarly small, and the statistical uncertainty arising from the fit to lattice data is
smallerstill.
In fact, the dominant uncertainty by an order of magnitude is that that arising from the quoted
error band on R, the light quark mass ratio. It is clear that better estimates of this value will
allow our results to be greatly improved in precision, without the need for further lattice data.
Conversely,aprecisedeterminationoftheelectromagneticcontributiontothen−pmassdifference
couldpossiblyfacilitateanimprovedestimateofRbythismethod.
4. ApplicationtoQCDSF-UKQCDlatticeresults
The method described above can be applied equally to other sets of lattice data for the octet
baryon masses. In particular, we consider recent 2 + 1-flavor QCDSF-UKQCD lattice simula-
tions [12], which follow a significantly different trajectory in the light-strange quark mass plane
tothePACS-CSsimulations. Insteadofholdingthestrangequarkmassfixedalongthesimulation
trajectory,theQCDSF-UKQCDCollaborationholdsthesingletquarkmass(m2 +m2/2)fixed.
K π
We use simulation results from this collaboration which lie both along the ‘singlet’ line and
along the SU(3) symmetric line [12]. Precisely as was done in our analysis of the PACS-CS
lattice data, we calculate small finite-volume corrections. The lattice spacing a = 0.072(1) fm
is determined by fixing X = (1/3)(M + M + M ) to the experimental value at physical quark
N N Σ Ξ
masses. This a is somewhat lower than that quoted by the QCDSF-UKQCD Collaboration as
we account for chiral curvature. The quality of our fit to the isospin-averaged results is clearly
excellent, with a χ2/dof of 0.6 and regulator mass Λ = 1.0 ± 0.1. The fit is shown in Figures 1,
2 and 3, and the values of the octet baryon masses extrapolated to the physical point are given in
Table6. Thesearelargelyconsistentwiththephysicalvalues.
ApplyingthemethoddescribedinprevioussectionstotheQCDSF-UKQCDlatticedatagives
resultsforthestrongcontributiontotheoctetbaryonmasssplittingswhichareconsistentwithour
7
B Mass(GeV) Experimental
N 0.966(16)(10) 0.939
Λ 1.112(13)(5) 1.116
Σ 1.193(12)(4) 1.193
Ξ 1.307(11)(0) 1.318
Table6: OctetbaryonmassesbasedonachiralextrapolationoftheQCDSF-UKQCDdataset. Thefirstuncertainty
quoted is statistical and the second results from the variation of various chiral parameters and the form of the UV
regulatorasdescribedinthetext. Theexperimentalmassesareshownforcomparison.
fittothePACS-CSCollaborationsimulationresults. Wefind
M − M = (ω/m2 )(16.6±1.2)MeV (16a)
n p π(phys)
MΣ− − MΣ+ = (ω/m2π(phys))(48.9±1.7)MeV (16b)
M − M = (ω/m2 )(32.2±1.6)MeV, (16c)
Ξ− Ξ0 π(phys)
correspondingto
M − M = 2.4±0.3MeV (17a)
n p
MΣ− − MΣ+ = 7.0±0.9MeV (17b)
M − M = 4.6±0.6MeV, (17c)
Ξ− Ξ0
whenRissettothephysicalvaluegiveninEquation14.
The QCDSF-UKQCD Collaboration has also recently presented a determination of the strong
contributiontothebaryonicmasssplittings,basedontheselatticesimulations[6]. Thatcalculation
usedalinearandquadraticSU(3)flavorsymmetrybreakingexpansioninthequarkmasses. Asthe
meson and baryon octet expansion coefficients depend only on the average quark mass, provided
these are kept constant this method allows for an estimation of the baryon mass splittings at the
physicalpointusingtheavailable2+1-flavorlatticeresults. Theresultsofthisapproachare:
M − M = 3.13±0.15±0.53MeV (18a)
n p
MΣ− − MΣ+ = 8.10±0.14±1.35MeV (18b)
M − M = 4.98±0.10±0.84MeV. (18c)
Ξ− Ξ0
The first uncertainty quoted in Equation 18 is statistical, while the second allows for violations of
Dashen’stheorem.
5. Discussion
Wehavecarriedoutananalysisofisospin-averagedlatticesimulationsforoctetbaryonmasses
using a formal chiral expansion based on broken SU(3) symmetry. Using the resulting expansion
coefficients one can evaluate the strong contribution to the baryon mass splittings. Our results,
8
basedonanalysesofPACS-CSandQCDSF-UKQCDlatticedatasets,aresummarizedinTable7.
Both calculations yield compatible values, despite significant differences between the two lattice
simulations,includinginparticulardifferentlatticesizes,latticespacings,anddifferentmethodsof
determiningthesespacings. Ofcourse,asemphasizedpreviously,thetwosimulationsalsofollow
quitedifferenttrajectoriesinm −m space.
π K
Furthermore, we note that the results of the QCDSF-UKQCD Collaboration analysis of their
dataareareentirelyconsistentwithourvalues. WhiletheapproachtakenbytheQCDSF-UKQCD
group makes use of only that lattice data calculated along a trajectory which holds the average
quark mass constant, we have also included in our fit the data from that collaboration which lies
away from this line. This contributes to our reduced uncertainties. We also point out that both
methods require some theoretical input: we input the up-down quark mass ratio R, while the
the Horsely et al. calculation uses Dashen’s theorem (with some uncertainty) to estimate ‘pure
QCD’ meson masses at the physical point. The clear consistency between the two independent
calculationsisencouraging.
∆md−mu (MeV) Mn − Mp MΣ− − MΣ+ MΞ− − MΞ0
Chiral(PACS-CS) 2.9(4) 7.5(10) 4.6(6)
Chiral(QCDSF-UKQCD) 2.4(3) 7.0(9) 4.6(6)
QCDSF-UKQCD 3.13(55) 8.10(136) 4.98(86)
Exp. &EM(traditional) 2.05(30) 7.91(30) 5.99(30)
Exp. &EM(WLCM) 2.6(6)
Table 7: Up-down quark mass contribution to octet baryon mass splittings. Lines 1 and 2 show the results of our
chiralextrapolationofPACS-CSandQCDSF-UKQCDlatticedatarespectively. Line3showstheQCDSF-UKQCD
Collaborationanalysisoftheirdataasdescribedintheprevioussection, whilelines4and5giveestimatesdeduced
fromthetotalmasssplittingsandelectromagneticcontributions,asdeterminedbyGasserandLeutwyler(traditional)
orWalker-Loudetal. (WLCM).
WhileourresultsandthoseoftheQCDSF-UKQCDCollaborationareconsistentwithboththe
traditional and Walker-Loud et al. (WLCM) determinations of the strong contribution from the
electromagneticcomponent(seeTable7),wenotethatthereisasmalldiscrepancybetweenourre-
sultsandthetraditionalvalueforthecascadebaryonmasssplitting. Clearly,determinationsofall
octet baryon electromagnetic mass splittings using the WLCM analysis would be of considerable
interest.
While more lattice data for isospin-averaged octet baryon masses, on larger lattice volumes,
would allow the uncertainties of our calculation to be somewhat reduced, we emphasize that the
dominantuncertaintyinourcalculationarisesfromtheup-downquarkmassratio. Amoreprecise
value of m /m could reduce the uncertainty of our determination by as much as an order of
u d
magnitude. Conversely, direct lattice determinations of the electromagnetic contributions to the
mass splittings, with the analysis presented here, may act to significantly improve the value of
m /m .
u d
9
Acknowledgements
This work was supported by the University of Adelaide and the Australian Research Council
throughtheARCCentreofExcellenceforParticlePhysicsattheTerascaleandgrantsFL0992247
(AWT)andDP110101265(RDY).
References
References
[1] P.J.Mohr,B.N.Taylor,D.B.Newell, CODATARecommendedValuesoftheFundamentalPhysicalConstants:
2010(2012).
[2] J.Beringer,others[ParticleDataGroup], (ParticleDataGroup), Phys.Rev.D86(2012)010001.
[3] S. R. Beane, K. Orginos, M. J. Savage, Strong-isospin violation in the neutron-proton mass difference from
fully-dynamicallatticeQCDandPQQCD, Nucl.Phys.B768(2007)38–50.
[4] T. Blum, et al., Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor
latticeQCD+QED, Phys.Rev.D82(2010)094508.
[5] G.M.deDivitiis,etal., Isospinbreakingeffectsduetotheup-downmassdifferenceinLatticeQCD(2011).
[6] R.Horsley,etal., Isospinbreakinginoctetbaryonmasssplittings(2012).
[7] A.Duncan,E.Eichten,H.Thacker, ElectromagneticSplittingsandLightQuarkMassesinLatticeQCD, Phys.
Rev.Lett.76(1996)3894–3897.
[8] S. Basak, et al., Electromagnetic splittings of hadrons from improved staggered quarks in full QCD, PoS
LATTICE127(2008).
[9] A. Portelli, Systematic errors in partially-quenched QCD plus QED lattice simulations, PoS LATTICE 136
(2011).
[10] B.Glaessle,G.S.Bali,Electromagneticcorrectionstopseudoscalardecayconstants,PoSLATTICE282(2011).
[11] S.Aoki,etal., 2+1FlavorLatticeQCDtowardthePhysicalPoint, Phys.Rev.D79(2009)034503.
[12] W. Bietenholz, et al., Flavor blindness and patterns of flavor symmetry breaking in lattice simulations of up,
down,andstrangequarks, Phys.Rev.D84(2011)054509.
[13] J.Gasser,H.Leutwyler, QuarkMasses, Phys.Rept.87(1982)77–169.
[14] A.Walker-Loud,C.E.Carlson,G.A.Miller, TheElectromagneticSelf-EnergyContributionto M −M and
p n
theIsovectorNucleonMagneticPolarizability(2012).
[15] R.D.Young, A.W.Thomas, Octetbaryonmassesandsigmatermsfromansu(3)chiralextrapolation, Phys.
Rev.D81(2010)014503.
[16] P.E.Shanahan,A.W.Thomas,R.D.Young, MassoftheH-dibaryon, Phys.Rev.Lett.107(2011)092004.
[17] P. E. Shanahan, A. W. Thomas, R. D. Young, Sigma terms from an SU(3) chiral extrapolation (2012).
[arXiv:1205.5365].
[18] R.E.Stuckey,M.C.Birse, Baryonmassesinachiralexpansionwithmeson-baryonformfactors, J.Phys.G
23(1997)29.
[19] J.F.Donoghue,etal., SU(3)baryonchiralperturbationtheoryandlongdistanceregularization, Phys.Rev.D
59(1999)036002.
[20] R. D. Young, D. B. Leinweber, A. W. Thomas, S. V. Wright, Chiral Analysis of Quenched Baryon Masses,
Phys.Rev.D66(2002)094507.
[21] R.D.Young,D.B.Leinweber,A.W.Thomas, ConvergenceofChiralEffectiveFieldTheory, Prog.Part.Nucl.
Phys.50(2003)399–417.
[22] D.B.Leinweber,A.W.Thomas,R.D.Young, PhysicalNucleonPropertiesfromLatticeQCD, Phys.Rev.Lett.
92(2004)242002.
[23] G. Amoros, J. Bijnens, P. Talavera, QCD isospin breaking in meson masses, decay constants and quark mass
ratios, Nucl.Phys.B602(2001)87–108.
[24] H.Leutwyler, TheRatiosoftheLightQuarkMasses, Phys.Lett.B378(1996)313–318.
10