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Lattice QCDand its interface to phenomenology
Stephan Durr
University of WuppertalJulich Supercomputing Center
EINN 2015Paphos – 3.11.2015
,Lattice QCD (1): combined UV/IR regulator,
QCD Lagrangian contains quarks and gluons [Fritzsch, Gell-Mann, Leutwyler (1973)]
LQCD =1
2Tr(FµνFµν) +
Nf∑i=1
q(i)(D/+m(i))q(i) + iθ1
32π2εµνρσTr(FµνFρσ)
• QCD must be regulated both in the UV and in the IR to make it well-defined.
• The lattice does the job through a>0 and V =L4<∞, but other options arepossible. Each gauge/fermion action combination is a different regulator.
• For a→0 correlation lengths diverge, but ratios ξπ/ξΩ stay finite (renormalization).The extrapolations a→0 and V →∞ are performed in dimensionless observables.
• The result is independent of the action, thanks to universality [Wilson], but someEuclidean results must be rotated to Minkowskian (ill-posed problem).
• Does this Lagrangian-regulator-extrapolation package explain phenomena such asconfinement, spontaneous/anomalous symmetry breaking, hadron spectrum ?
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 1
,Lattice QCD (2): effective masses and decay constants,A
eff(t
)=C
(t)
exp
(Meff(t
)t)
Meff(t
)=
1 2lo
g(C
(t−
1)
C(t
+1))
0 10 20 30 40 50 60
0.0515
0.052
0.0525
0.053
0.0535
0.054
meff_inft_ll
meff_cosh_ll
0 10 20 30 40 50 60
0.189
0.19
0.191
0.192
0.193
0.194
0.195
0.196
0.197
0.198
0.199
meff_inft_ls
meff_cosh_ls
0 10 20 30 40 50 60
0.05
0.0505
0.051
0.0515
0.052
0.0525
feff_inft_ll
feff_cosh_ll
0 10 20 30 40 50 60
0.0595
0.06
0.0605
0.061
0.0615
0.062
0.0625
0.063
feff_inft_ls
feff_cosh_ls
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 2
,Lattice QCD (3): scale hierarchies,
typical spacing: 0.04 fm≤a≤0.20 fm
1 GeV≤a−1≤5 GeV
typical length: 2 fm≤L≤6 fm
require (UV): amq 1
require ( IR ): MπL ≥ 4
u c (t)
d s b︷ ︸︸ ︷work near
︷ ︸︸ ︷interpolate to
︷ ︸︸ ︷extrapolate
12(mphys
u +mphysd ) mphys
s ,mphysc mb→mphys
b
In QCD with Nf flavors, 1+Nf observables utilized to set scale and quark masses.
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 3
Overview
• How to cure/assess systematic uncertainties
• T =0 topics: spectroscopy, quark masses, LECs in ChPT, hadron structure
• T =0 topics: FLAG compilations of flavor related quantities
• T >0 topics: pseudocritical Tc, EoS, transport coefficients, sign problem at µ 6=0
• Including electromagnetism and topics left out
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 4
,Systematic uncertainties (1): theory for a→ 0,
• Functional form of asymptotic effects via Symanzik analysis ∝ a, αQCD
a, a2.
• A priori no telling how close to the continuum Symanzik scaling sets in.
• A priori no knowledge about size of prefactor in Symanzik regime.
• Possibly competition with subleading order in accessible a-range, e.g.
O(αa) with O(a2) for fat clover, in such cases include both options.
J.R
olf,
S.S
int
[AL
PH
A],
JHE
P02
12(2
002)
007
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 5
,Systematic uncertainties (2): theory for L→∞,
ChPT predicts relative finite-volume effects of
MP (L), fP (L) [GasserLeutwyler 87]:
Mπ(L) = Mπ
1 +
1
2Nfξg1(MπL) +O(ξ2)
fπ(L) = fπ
1− Nf
2ξ g1(MπL) +O(ξ2)
with ξ = M2
π/(4πFπ)2 = M2π/(8π
2f2π) and
16 24 32
L/a
0.14
0.15
0.16
0.17
0.18
0.19
aM
π ignored in final analysis
MπL=4M
πL=3
Du
rret
al.[
BM
W],
1011
.271
1
g1(z) =24
zK1(z) +
48√2zK1(√
2z) +32√3zK1(√
3z) +24
2zK1(2z) + ...
K1(z) =
√π
2ze−z
1 +
3
8z− 3 · 5
2(8z)2+
3 · 5 · 21
6(8z)3− 3 · 5 · 21 · 45
24(8z)4+ ...
• Unlike in the Symanzik case, here coefficients [1/(2Nf), Nf/2] are known.
• For realistic masses/volumes this expansion converges slowly [CD 03, CDH 05].
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 6
,Systematic uncertainties (3): theory for mud→ mphysud ,
• Memo: correct values of mud or ms not known a priori, must be dialed by
adjusting M2π/X
2 and (2M2K−M2
π)/X2 to their physical values.
• Memo: X=fπ,MΩ, ... is the scale setting quantity.
• In Nf =2+... QCD, input values must be corrected for isospin-breaking effects.
0 0.01 0.02 0.03 am
0
0.02
0.04
0.06
0.08
(amπ )2
mπ∼676 MeV
484
381
294
0 0.02 0.04 0.06 0.08 (amπ)2
0
0.02
0.04
0.06
aFπ
676
484381
294
M.L
usc
her
,P
oS(L
at20
05)
002
• ChPT is standard tool to extrapolate; now simulations at Mπ'135 MeV feasible.
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 7
,Systematic uncertainties (4): histogramming method,
Include all reasonable ansaetze, and count spread towards systematic uncertainty !
1.188 1.1885 1.189 1.1895 1.19 1.1905 1.191 1.1915 1.1920
1000
2000
3000
4000
5000
6000
7000
distribution of symm in continuum
linear
quadratic
1.188 1.189 1.19 1.191 1.1920
200
400
600
800
1000
1200
1400
1600
1800
2000
distribution of symm in continuum
Mka_10
Mka_30
Mka_50
Mka_70
Mka_90
Multitude of “final” results (i.e. at a=0, L=∞,mphysud , ...) may be weighted:
• equal weight
• weighting with dof/χ2
• weighting with p-value
• weighting with AIC •S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 8
,QCD spectroscopy (1): ground states,
0
500
1000
1500
2000M
[MeV
]
p
K
r K* NLSX D
S*X*O
experiment
widthinput
QCD
After a→0, L→∞,Mπ=135 MeV agreement with experiment [S. Durr et al., Science 322, 1224 (2008)]
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 9
,QCD spectroscopy (2): anatomy of mq calculation,
1. Decide on observables to be utilized for setting scale and quark masses, e.g.Mπ,MK,MΩ in Nf = 2+1 QCD, and look up their “experimental” values in aworld without isospin splitting and without electromagnetism [Mπ=134.8(3) MeV,MK=494.2(5) MeV according to FLAG].
2. For a given bare coupling β (yields a) tune bare masses 1/κud, 1/κs such that theratios Mπ/MΩ,MK/MΩ assume their physical values.
3. Read off 1/κud,s or determine bare amud,s via AWI and convert them (perturbati-vely or non-perturbatively) to the scheme of your choice (e.g. MS at µ=3 GeV).
4. Repeat steps 2 and 3 for at least three lattice spacings and extrapolate the resultto the continuum by means of Symanzik scaling.
In practice, step 2 involves inter-/extrapolations to the physical quark masses.
In practice, step 3 may be technically involved [RI/MOM, SF renormalization].
In practice, finite-volume corrections will to be included.
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 10
,QCD spectroscopy (3): Nf =3 RI-running extrapolation for ZS,
Evolution ZRIS (µ)/ZRI
S (4 GeV) has no visible cut-off effects among three finest lattices:separate continuum limit with RRI
S (µ, 4 GeV) = limβ→∞ ZRIS,β(4 GeV)/ZRI
S,β(µ).
0 10 20 30 40
µ2[GeV
2]
0.5
0.6
0.7
0.8
0.9
Z^SR
I (µ2)
β=3.8
β=3.7
β=3.61
β=3.5
β=3.31
10 20 30 40
µ2[GeV
2]
0.8
0.9
1
1.1
1.2
ZR
I
S,n
onper
t(µ2)/
ZR
I
S,4
-loop(µ
2)
µ=
4G
eV
β=3.8
Plots taken from BMW-collaboration paper [arXiv:1011.2403,1011.2711].On the finest lattice contact within errors to 4-loop PT for µ ≥ 4 GeV.
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 11
,QCD spectroscopy (4): excited states,
Charmonium spectroscopy withvarious source/sink wave functions(including open charm states),with disconnected contributionsand GEP technology
G. Bali, S. Collins, C. Ehmann [arXiv:1110.2381]
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 12
,QCD spectroscopy (5): scattering states,
J. Dudek, R. Edwards, C. Thomas [1212.0830] I = 1 P-wave ππ at Mπ = 390 MeV •
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 13
,QCD meson structure (1): leptonic decays,
Leptonic decays of pseudoscalar mesons, with P = π,K,D,Ds, B,Bs, Bc, ..., via
Γ(P → `ν`) =G2F
8π|Vij|2f2
Pm2`MP (1−m2
`/M2P )2
where experiment determines product |Vij| fP , but lattice can determine fP .
JCCµ = (u, c, t)γµ
1
2[1−γ5]
( d′s′
b′
) d′
s′
b′
=
Vud Vus VubVcd Vcs VcbVtd Vts Vtb
︸ ︷︷ ︸
VCKM
dsb
〈0|(uγµγ5d)(x)|π−(p)〉 = ifπpµeipx
〈0|(uγµγ5s)(x)|K−(p)〉 = ifKpµeipx
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 14
,QCD meson structure (2): semi-leptonic decays,
q ≡ p−p′ momentum transfer
〈π(p′)|sγµu|K(p)〉 = f0(q2)M2K−M2
π
q2qµ + f+(q2)
[(p+p′)µ −
M2K−M2
π
q2qµ
]
• Experiment can determine |Vus|f+(0), lattice can determine f+(0) [RBC/UKQCD].
• CKM matrix element Vus determined from leptonic K decay and from semi-leptonic• K → π decay must agree, if SM is correct.
• Strong dynamics restricted to QCD matrix elements 〈0|Aµ|π〉, 〈0|Aµ|K〉 and form• factors 〈π|Vµ|K〉 with specified external currents.
• Natural generalization to BSM processes (matrix elements).
•S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 15
,FLAG effort (1): methodology and observables,
For each quantity FLAG provides [see http://itpwiki.unibe.ch/flag]:
• complete list of references
• summary of essential ingredients of each study (Nf , action, ... )
• averages for “mature” quantities (separately for each Nf)
• pressure on reader to cite original papers
1 st edition [arXiv:1011.4408, Eur.Phys.J. C71 (2011) 1695] covers:
1. light quark masses mud,ms
2. Vus and Vud via decay constants and form factors
3. chiral low-energy constants (LECs)
4. kaon bag parameter BK
2nd edition [arXiv:1310.8555, Eur.Phys.J. C74 (2014) 2890] in addition:
5. D-meson decay constants and form factors
6. B-meson decay constants, mixing parameters, and form factors
7. strong coupling constant
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 16
,FLAG effort (2): color coding in FLAG-2,
FLAG-2 [arXiv:1310.8555, Eur.Phys.J. C74 (2014) 2890] definitions as follows:
Chiral extrapolation:F Mπ,min < 200 MeV 200 MeV ≤Mπ,min ≤ 400 MeV Mπ,min > 400 MeV
Continuum extrapolation:F 3 or more lattice spacings and at least 2 points below 0.1 fm / in scaling window 2 or more lattice spacings and at least 1 point below 0.1 fm / in scaling window otherwise
Finite-volume effects:F (Mπ)minL > 4 or at least 3 volumes (Mπ)minL > 3 and at least 2 volumes otherwise
Renormalization (where applicable):F non-perturbatively or 2-loop well-convergent PT 1-loop (or higher) PT with reasonable error estimate otherwise
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 17
,FLAG effort (3): light quark masses,
2 3 4 5 6
=+
=pheno.
MeVMaltman 01Narison 06Dominguez 09PDG
CP-PACS 01JLQCD 02QCDSF/UKQCD 04SPQcdR 05QCDSF/UKQCD 06ETM 07RBC 07JLQCD/TWQCD 08AETM 10BDürr 11our estimate for =MILC 04, HPQCD/MILC/UKQCD 04HPQCD 05CP-PACS/JLQCD 07RBC/UKQCD 08PACS-CS 08MILC 09MILC 09AHPQCD 09APACS-CS 09Blum 10RBC/UKQCD 10AHPQCD 10MILC 10APACS-CS 10BMW 10ALaiho 11PACS-CS 12RBC/UKQCD 12our estimate for = +
70 80 90 100 110 120
=+
=pheno.
MeVVainshtein 78Narison 06Jamin 06Chetyrkin 06Dominguez 09PDG
CP-PACS 01JLQCD 02QCDSF/UKQCD 04ALPHA 05SPQcdR 05QCDSF/UKQCD 06ETM 07RBC 07ETM 10BDürr 11ALPHA 12our estimate for =MILC 04, HPQCD/MILC/UKQCD 04HPQCD 05CP-PACS/JLQCD 07RBC/UKQCD 08PACS-CS 08MILC 09MILC 09AHPQCD 09APACS-CS 09Blum 10RBC/UKQCD 10AHPQCD 10PACS-CS 10BMW 10ALaiho 11PACS-CS 12RBC/UKQCD 12our estimate for = +
• inconsistencies only among red results
• for each Nf all green points nicely consistent
• mphysud and mphys
s depend only mildly on Nf , ratio essentially Nf -independent
• reasonable consistency with non-lattice results
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 18
,FLAG effort (4): chiral low-energy constants,
200 250 300 350
=+
+=
+=
MeVJLQCD/TWQCD 07AJLQCD/TWQCD 08ACERN 08ETM 08ETM 09CJLQCD/TWQCD 10Bernardoni 10TWQCD 11ATWQCD 11ETM 12ETM 13Brandt 13
our estimate for =
RBC/UKQCD 08JLQCD/TWQCD 08BTWQCD 08MILC 09MILC 09A [SU(2) fit]MILC 09A [SU(3) fit]JLQCD 09RBC/UKQCD 10AJLQCD/TWQCD 10MILC 10ABorsanyi 12BMW 13
our estimate for = +
ETM 13
our estimate for = + +
/
3.5 4.0 4.5 5.0 5.5
=+
+=
+=
Gasser 84Colangelo 01
JLQCD/TWQCD 08AETM 08JLQCD/TWQCD 09ETM 09CTWQCD 11Bernardoni 11Brandt 13Gülpers 13
our estimate for =
RBC/UKQCD 08MILC 09AMILC 10ANPLQCD 11Borsanyi 12RBC/UKQCD 12BMW 13
our estimate for = +
ETM 10
our estimate for = + +
• inconsistency between RBC/UKQCD 10A and MILC 10A in case of Σ for Nf = 2+1
• discrepancy resolved by inflated FLAG error (−→ less precise than the better calc.)
• no warning through inconsistency in case of ¯4, but why non-monotonic in Nf ?
• scarcity of non-lattice results
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 19
,FLAG effort (5): fD and fDs,
180 200 220 240
=+
+=
+=
ETM 09
ETM 11A
ETM 13B
our average for =
FNAL/MILC 05
HPQCD/UKQCD 07
HPQCD 10A
PACS-CS 11
FNAL/MILC 11
HPQCD 12A
our average for = +
FNAL/MILC 12B
FNAL/MILC 13
ETM 13F
230 250 270
=+
+=
+=
MeV
ETM 09
ETM 11A
ETM 13B
our average for =
FNAL/MILC 05
HPQCD/UKQCD 07
HPQCD 10A
PACS-CS 11
FNAL/MILC 11
HPQCD 12A
our average for = +
FNAL/MILC 12B
FNAL/MILC 13
ETM 13F
1.10 1.15 1.20 1.25
=+
+=
+=
ETM 09ETM 11AETM 13B
our average for =
FNAL/MILC 05HPQCD/UKQCD 07PACS-CS 11FNAL/MILC 11HPQCD 12A
our average for = +
FNAL/MILC 12BFNAL/MILC 13ETM 13F
/
• several good-quality calculations available or underway
• roughly fDs/fD ' fK/fπ ' 1.19(1)
• so far consistent with CKM unitarity (cf. next slide)
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 20
,FLAG effort (6): D → K and D → π form factors,
0.55 0.65 0.75
=+
+
=+
=
ETM 11B
FNAL/MILC 04
HPQCD 11 / 10B
our average for = +
+ ( )
0.65 0.70 0.75 0.80
=+
+=
+=
ETM 11
FNAL/MILC 04
HPQCD 10B
our average for = +
+ ( )
0.20 0.22 0.24
=+
=
CKM unitarity
neutrino scattering
ETM 13B
our average for =
HPQCD 11/10B
FNAL/MILC 11
HPQCD 12A/10A
our average for = +
| |
0.95 1.05
=+
+=
+=
CKM unitarity
neutrino scattering
ETM 13B
our estimate for =
HPQCD 10B
FNAL/MILC 11
HPQCD 10A
our estimate for = +
| |
• semi-leptonic / leptonic comparison tests CKM paradigm
• good precision challenging, better proposal for q2 available [FNAL/MILC, HPQCD]
• so far no threat for CKM unitarity
•S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 21
,QCD baryon structure (1): lattice technology,
Lots of interesting topics [1406.4310]:
• nucleon sigma terms
• rms charge radius of proton
• transition form factors, GPDs, ...
• signal for strange/charmed baryons
→ talk by Giannis Koutsou
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 22
,QCD baryon structure (2): from quarks to nuclei,
HalQCD collaboration [1408.4892]:nuclei from quarks via Bethe-Salpeter
• NN -potential for S,D-waves• at Mπ ' 470 MeV
• Single particle levels in 16O and 40Ca• at Mπ ' 470 MeV
• E versus A of doubly magic nuclei• follows Bethe-Weizsacker (straight line)
→ talk by Takashi Inoue •S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 23
,QCD thermodynamics (1): transition temperature,
• At physical values of mud,ms, ... QCD shows crossover.
• Transition reasonably sharp, hence pseudo-critical Tc meaningful concept.
• Tc depends on observable, e.g. P , 〈ψψ〉, ..., but each one is well-defined.
! !!!!!!!!
!!!!!! ! !!!
""
""""
"
""
"""""" "
!!!!!!!
!!!!! !
! ! ! !
##
##
#
#
##!!""!!
ContinuumNt"16Nt"12Nt"10Nt"8
100 120 140 160 180 200 220
0.2
0.4
0.6
0.8
1.0
T !MeV"
#l,s
! !!!!!!!!
!!!!!!
! !!!
""
"""""
"""""""
" "
!!!!!!!
!!!! !
! !! ! !
####
##
##!!""!!
ContinuumNt"16Nt"12Nt"10Nt"8
100 120 140 160 180 200 2200.0
0.1
0.2
0.3
0.4
T!MeV"#$$%R
S.B
orsa
nyi
etal
,ar
Xiv
:100
5.35
08
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 24
,QCD thermodynamics (2): equation of state,
• Apparent discrepancy for trace anomaly• ε− 3p between Wuppertal-Budapest• and HotQCD in the past.
• Apparent discrepancy disappeared with• improved HotQCD action and analysis.
A. Bazavov et al [HotQCD], 1407.6387
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 25
,QCD thermodynamics (3): troubles with µbary 6= 0,
At non-zero baryon density (equivalent: chemicalpotential µ 6=0) the fermion determinant becomescomplex, which creates a major difficulty to theconcept of importance sampling.
A clear establishment/measurement of a second-order endpoint would be a major leap forward.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
µ (MeV)
0
20
40
60
80
100
120
140
160
180
200
T (
MeV
)
© 2012 Andreas Kronfeld/Fermi Natl Accelerator Lab.
hadrons
quark-gluon plasma
n
other phases plo
tby
A.K
ron
feld
In QCD many approaches to solve the sign problem have been tried:
• absorb phase in observable [ancient]• two-parameter reweighting from µ=0 [Fodor Katz]• work at imaginary µ and continue [Philipsen deForcrand]• compute Taylor coefficients at µ=0
In QCD-inspired models many tricks/reformulations become possible.
•
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 26
Summary
• LQCD is the full definition of QCD, requires 1+Nf input observables
• Systematics non-trivial, in particular when matching to EFT-framework
• Main topics in T = 0 studies:
spectroscopy, quark masses, CKM elements, LECs in ChPT, hadron structure, ...
FLAG compilations [e.g. arXiv:1310.8555] serve as interface to phenomenology
• Main topics in T > 0 studies:
pseudocritical Tc, EoS, transport coefficients, sign problem at µbary 6= 0, ...
• New frontier of including electromagnetic interactions
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 27
,Outlook (1): simulations with electromagnetism,
• 2002-20??:
Nf = 2+1 QCD requires 3 polished input values [e.g. Mπ,MK,MΩ in theory withmu,md → 1
2(mu+md) and e→ 0]
−→ isospin analysis suggests Mπ=134.8(3)MeV,MK=494.2(5)MeV [FLAG]
• 2010-2???:
Nf = 2+1+1 QCD requires 4 polished input values [ditto and MDs in theory withmu,md → 1
2(mu+md) and e→ 0]
−→ charm unquenched, but no conceptual change on isospin issue
• 2014-????:
Nf = 1+1+1+1 QCD requires 5 input variables [e.g. Mπ±,MK±,MK0,MDs,MΩ]
−→ requires disconnected contribution to flavor-singlet quantities−→ analysis of π0-η-η′ mixing mandatory to extract physical masses−→ QED and QCD renormalization intertwined (ms/md is RGI, mu/md is not)
−→ final word on mu?=0 [in QCD+QED] will be possible
S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 28
,Outlook (2): disentangling mass splittings,
Simulate Nf =1+1+1+1 QCD with non-compact QED. Use quark charges 2e/3 and−1e/3 with e varying between 0 and 1.41.
After algorithmic challenges are solved and finite-volume effects are under control,determine physical isospin splittings on each ensemble, and extrapolate results a→ 0,L→∞, mq → mphys
q for q ∈ d, u, s, c, αem → 1/137.036.
Compare to exp, test Coleman-Glashow relation ∆CG ≡ ∆MN −∆MΣ + ∆MΞ = 0.
∆qmMN
=+
2.52
(17)
(24)
MeV
∆emMN
=−
1.0
0(07
)(14
)M
eV
∆la
tMN
=+
1.51
(16)
(23)
MeV
∆expMN
=+
1.2
9333
21(4
)M
eV
0
2
4
6
8
10
ΔM
[MeV
]
ΔN
ΔΣ
ΔΞ
ΔD
ΔCG
ΔΞcc
experimentQCD+QEDprediction
BMW 2014 HCH BM
Wco
llab
orat
ion
,S
cien
ce34
7(2
015)
1452
•S. Durr, BUW/JSC EINN 2015, Paphos, 3.11.2015 29