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Die Mathematisch-Naturwissenschaftliche Fakultät und
das Helmholtz-Institut für Strahlen-und Kernphysik, Abteilung Theorie
laden ein zum
Gedenk-Kolloquium für Dr. Klaus Erkelenz und zur Gründung der Dr. Klaus Erkelenz Stiftung
Programm 16:15 Einführung Professor Dr. Ulf-G. Meißner Dekan der Mathematisch-Naturwissenschaftlichen Fakultät an der Universität Bonn 16:30 Vortrag Professor Dr. Ruprecht Machleidt University of Idaho, Moscow, USA Klaus Erkelenz and the Bonn Potential 17:15 Vortrag Professor Dr. Ulf-G. Meißner Direktor der Abteilung Theorie des Helmholtz-Instituts für Strahlen- und Kernphysik der Universität Bonn Nuclear Theory: A Modern Perspective 18:00 Professor e.m. Dr. Peter David Erinnerungen an Dr. Klaus Erkelenz
Prof. R. Machleidt, Univ. of Idaho, USA Klaus Erkelenz and the Bonn Potential Dr. Klaus Erkelenz was a research associate at the Institute for Theoretical Nuclear Physics of the Bonn University until November 1973, when he died untimely at the age of 42. Even though his career was cut short, he made substantial contributions to nuclear physics. His main focus was the meson theory of nuclear forces. In particular, he worked on a consistent relativistic derivation of the nucleon-nucleon interaction based upon meson field theory. His initiatives inspired a decade of further work on the subject at the Bonn Institute. The resulting nuclear force models have become known in the international community as the „Bonn Potentials“.
Prof. U.-G. Meißner, Univ. Bonn & FZ Jülich Nuclear Theory: A Modern Perspective Effective Field Theory is a modern tool in many branches of theoretical physics. In particular, the problem of the forces between two and three nucleons has gained renewed interest based on this framework. I discuss the essentials of this approach and show how it ties to the meson field theory potentials like the one set up by Klaus Erkelenz and others. In addition, the use of high performance computers allows for ab initio calculations of the properties of atomic nuclei and gives new insight into the fine-tuning underlying nucleosynthesis in the Big Bang and in stars.
Freitag, 15. November 2013, 16.00 Uhr c.t. im Hörsaal I des Physikalischen Instituts, Nussallee 12, 53115 Bonn
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
2
Nuclear Theory:A Modern Perspective
Ulf-G. Meißner, Univ. Bonn & FZ Julich
Supported by DFG, SFB/TR-16 and by DFG, SFB/TR-110 and by EU, I3HP EPOS and by BMBF 06BN9006 and by HGF VIQCD VH-VI-417
Nuclear Astrophysics Virtual Institute
NLEFT
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
3
CONTENTS
• Some basic facts
• Ab initio calculation of atomic nuclei
• The fate of carbon-based life
• Towards medium-mass nuclei
• Summary & outlook
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
4
Some basicfacts
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
5
STRUCTURE FORMATION in QCD
• The strong interactions are described by QCD
• Quarks and gluons are confined within hadrons
• Protons and neutrons form atomic nuclei
⇒ This requires the inclusion of electromagnetism
⇒ Atomic nuclei make up the visible matter in the Universe
• up and down quarks are very light, a few MeV
So how are these strongly interacting composites generated?
How sensitive are they to changes in the fundamental parametersof QCD+QED?
QCDO(α )
245 MeV
181 MeV
ΛMS(5) α (Μ )s Z
0.1210
0.1156
0.1
0.2
0.3
0.4
0.5
αs(Q)
1 10 100Q [GeV]
Heavy QuarkoniaHadron Collisionse+e- AnnihilationDeep Inelastic Scattering
NL
O
NN
LO
TheoryData L
attic
e
211 MeV 0.1183s4
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
6THE NUCLEAR LANDSCAPE: AIMS & METHODS
• Theoretical methods:
− Lattice QCD: A = 0,1,2, . . .
− NCSM, Faddeev-Yakubowsky, GFMC, ... :A = 3 - 16
− coupled cluster, . . .: A = 16 -100
− density functional theory, . . .: A ≥ 100
• Chiral EFT:
− provides accurate NN and 3N forces
− successfully applied in light nucleiwith A = 2, 3, 4
− combine with simulations to get to larger A
⇒ Nuclear Lattice Simulations
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
7
Ab initio calculationsof atomic nuclei
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
8Ingredients
• Nuclear binding is shallow: E/A ≤ 8 MeV
⇒ Nuclei can be calculated from the A-body Schrodinger equation: HΨA = EΨA
• Forces are of (dominant) two- and (subdominant) three-body nature:V = VNN + VNNN
⇒ can be calculated systematically and to high-precisionWeinberg, van Kolck, Epelbaum, M., Entem, Machleidt, . . .
⇒ fit all parameters in VNN + VNNN from 2- and 3-body data
⇒ exact calc’s of systems with A ≤ 4 using Faddeev-Yakubowsky machinerysee fig.
⇒ connection to boson-exchange models (as in Machleidt’s talk)?→ slide
But how about ab initio calculations for systems with A ≥ 5?
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
9
Examples
0 60 120 180
θ [deg]
12
16
20
dσ/d
Ω [
mb/
sr]
EGM N3LOEM N3LOCD Bonn 2000Gross, Stadler
Nijmegen PWA
60 120 180
θ [deg]
0
0,1
0,2
0,3
Ay
10
100
dσ/dΩ [mb/sr]
10 MeV
1
10
100
dσ/dΩ [mb/sr]
65 MeV
-0,1
0
T20
-0,4
-0,2
0
0,2T20
0 60 120θ [deg]
-0,08
-0,03
0,02
0,07 T21
0 60 120 180θ [deg]
-0,2
-0,1
0
T21
0 60 120 180-0.4
0.0
0.4
0.8K
X X’ (N
)
0 60 120 180
0.4
0.6
0.8
1.0
K Y Y
’ (N)
0 60 120 180CM [deg]
-0.6
-0.4
-0.2
0.0
K Z X
’ (N)
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
10CONNECTION to BOSON–EXCHANGE MODELSEpelbaum, M., Glockle, Elster, Phys. Rev. C65 (2002) 044001 [nucl-th/0106007]
• Basic idea:
Expand meson-exchange in powers of t/M2R
and map on 4N operators
...M
= + +
ε1
CD-BonnNijm-93
Bonn-B Nijm-INijm-IIAV-18
C C C C C C C C1S0 1S0 3S1 3S1 1P1 3P0 3P1 C3P2
0.5
1.0
1.5
-0.5
0.0
g2
t−M2R
= − g2
M2R− g2 t
M4R
+ . . .
with • ∝ t0 ∝ ~∇0
∝ t ∝ ~∇2
• works amazingly well
all LECs of natural size
(in units of 1/Fnπ Λmχ )
NLO
NNLO
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
11NUCLEAR LATTICE SIMULATIONSFrank, Brockmann (1992), Koonin, Muller, Seki, van Kolck (2000) , Lee, Schafer (2004), . . .
Borasoy, Krebs, Lee, UGM, Nucl. Phys. A768 (2006) 179; Borasoy, Epelbaum, Krebs, Lee, UGM, Eur. Phys. J. A31 (2007) 105
• new method to tackle the nuclear many-body problem
• discretize space-time V = Ls × Ls × Ls × Lt:nucleons are point-like fields on the sites
• discretized chiral potential w/ pion exchangesand contact interactions
• typical lattice parameters
Λ =π
a' 300 MeV [UV cutoff]
p
p
n
n a
~ 2 fm
• strong suppression of sign oscillations due to approximate Wigner SU(4) symmetryJ. W. Chen, D. Lee and T. Schafer, Phys. Rev. Lett. 93 (2004) 242302
• hybrid Monte Carlo & transfer matrix (similar to LQCD)
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
12
CONFIGURATIONS
⇒ all possible configurations are sampled⇒ clustering emerges naturally⇒ perform ab initio calculations using only VNN and VNNN as input⇒ grand challenge: the spectrum of 12C
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
13SPECTRUM OF 12C & the HOYLE STATE
Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 106 (2011) 192501Epelbaum, Krebs, Lahde, Lee, UGM, Phys. Rev. Lett. 109 (2012) 252501Viewpoint: Hjorth-Jensen, Physics 4 (2011) 38
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
14THE TRIPLE-ALPHA PROCESS→ MOVIE
c©ANU
• the 8Be nucleus is instable, long lifetime→ 3 alphas must meet
• the Hoyle state sits just above the continuum threshold→ most of the excited carbon nuclei decay
(about 4 out of 10000 decays produce stable carbon)
• carbon is further turned into oxygen but w/o a resonant condition
⇒a triple wonder !
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
15RESULTS
• some groundstate energies and differences
E [MeV] NLEFT Exp.3He -3H 0.78(5) 0.764He −28.3(6) −28.38Be −55(2) −56.512C −92(3) −92.2
0.5
0.6
0.7
0.8
0.9
1
5 10 15 20 25
E3H
e −
Etr
iton (
MeV
)
L (fm)
latticephysical (infinite volume)
• promising results
• excited states more difficult
⇒ new projection MC method
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
16The SPECTRUM of CARBON-12
• After 8 · 106 hrs JUGENE/JUQUEEN (and “some” human work)
2+
Exp Th
−92
Hoyle
2+
0−84
−86
−88
−90
−87.72
−84.51 −85(3)
−82
0 +
−92(3)
−88(2)
2+−82.6(1)
2 +
0 + +−83(3)
E [
MeV
]
0 +
−92.16
⇒ First ab initio calculationof the Hoyle state
√
Structure of the Hoyle state:
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
17
The fate of carbon-based lifeas a function of the fundamental
parameters of QCD+QED
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
18FINE-TUNING of FUNDAMENTAL PARAMETERS
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
19
EARLIER STUDIES of the ANTHROPIC PRINCIPLE
• rate of the 3α-process: r3α ∼(Nα
kT
)3
Γγ exp
(−
∆E
kT
)∆E = E?12 − 3Eα = 379.47(18) keV
• how much can ∆E be changedso that there is still enough12C and 16O?
⇒ |∆E| . 100 keV
Oberhummer et al., Science 289 (2000) 88
Csoto et al., Nucl. Phys. A 688 (2001) 560Schlattl et al., Astrophys. Space Sci. 291 (2004) 27[Livio et al., Nature 340 (1989) 281]
too few 16O too few 12C
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
20FINE-TUNING: MONTE-CARLO ANALYSISEpelbaum, Krebs, Lahde, Lee, UGM, PRL 110 (2013) 112502; Eur.Phys.J. A49 (2013) 82
• consider first QCD only→ calculate ∂∆E/∂Mπ
• relevant quantities (energy differences)
∆Eh ≡ E∗12 − E8 − E4, ∆Eb ≡ E8 − 2E4
• energy differences depend on parameters of QCD (LO analysis)
Ei = Ei
(MOPEπ ,mN(Mπ), gπN(Mπ), C0(Mπ), CI(Mπ)
)gπN ≡
gA2Fπ
• remember: M2π± ∼ (mu +md) Gell-Mann–Oakes–Renner (1968)
⇒ quark mass dependence ≡ pion mass dependence
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
21PION MASS VARIATIONS
• consider pion mass changes as small perturbations
∂Ei∂Mπ
∣∣∣∣Mphysπ
=∂Ei
∂MOPEπ
∣∣∣∣Mphysπ
+ x1
∂Ei∂mN
∣∣∣∣mphysN
+ x2
∂Ei∂gπN
∣∣∣∣gphysπN
+ x3
∂Ei∂C0
∣∣∣∣Cphys
0
+ x4
∂Ei∂CI
∣∣∣∣CphysI
with
x1 ≡∂mN
∂Mπ
∣∣∣∣Mphysπ
, x2 ≡∂gπN∂Mπ
∣∣∣∣Mphysπ
, x3 ≡∂C0
∂Mπ
∣∣∣∣Mphysπ
, x4 ≡∂CI∂Mπ
∣∣∣∣Mphysπ
⇒ problem reduces to the calculation of the various derivativesusing AFQMC and the determination of the xi
• x1 and x2 can be obtained from LQCD plus CHPT
• x3 and x4 can be obtained from two-body scattering and its Mπ-dependence
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
22
VISUALIZATION of the PION MASS VARIATIONS
• At LO, one-pion exchange and 4N contact terms
N (M )πnucleon mass m
four−nucleoncouplings C(M )π
pion propagator2
pion−nucleon
coupling g(M )π
π1/(q − M )2As,t ≡
∂a−1s,t
∂Mπ
∣∣∣∣M
physπ
m
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
23CORRELATIONS
• vary the quark mass derivatives of a−1s,t within −1, . . . ,+1:
-6 -4 -2 0 2 4 6K
E4
π
-600
-400
-200
0
200
400
600
K∆Ebπ
K∆Ehπ
K∆Eh+bπ
• clear correlations: α-particle BE and the energies/energy differences
⇒ anthropic or non-anthropic scenario depends on whether the 4He BE moves!
∆Eb = E(8Be)− 2E(4He)
∆Eh = E(12C∗)− E(8Be)− E(4He)
∆Eh+b = E(12C∗)− 3E(4He)
∂OH
∂Mπ
= KπH
OH
Mπ
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
24THE END-OF-THE-WORLD PLOT• |δ(∆Eh+b)| < 100 keV
→∣∣∣∣(0.571(14)As + 0.934(11)At − 0.069(6)
)δmq
mq
∣∣∣∣ < 0.0015
As,t ≡∂a
−1s,t
∂Mπ
∣∣∣∣M
physπ
The light quark massis fine-tuned to' 2−3 %
Similarly:αEM is fine-tunedto ' 2.5%
Berengut et al.,Phys. Rev. D 87 (2013) 085018
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
25
Towards medium-mass nuclei
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
26GOING up the ALPHA CHAIN
• Consider the α ladder 12C, 16O, 20Ne, 24Mg, 28Si as tCPU ∼ A2
• Improved “multi-state” technique to extract g.s. energies
⇒ higher A, better accuracy
⇒ overbinding at LO beyond A = 12 persists up to NNLO
-155
-150
-145
-140
-135
-130
-125
-120
-115
0 2 4 6 8 10 12 14
E16
(LO
) [M
eV]
Nt
-8.0e-5-6.5e-5-5.5e-5-4.5e-5
-210
-200
-190
-180
-170
-160
-150
0 2 4 6 8 10 12 14
E20
(LO
) [M
eV]
Nt
-8.0e-5-6.5e-5-5.5e-5-4.5e-5
-270
-260
-250
-240
-230
-220
-210
-200
0 2 4 6 8 10 12 14
E24
(LO
) [M
eV]
Nt
-8.0e-5-6.5e-5-5.5e-5-4.5e-5
-360
-340
-320
-300
-280
-260
0 2 4 6 8 10 12 14
E28
(LO
) [M
eV]
Nt
-5.5e-5-4.5e-5-3.5e-5
E = −131.3(5) E = −165.9(9) E = −232(2) E = −308(3)
[−127.62] [−160.64] [−198.26] [−236.54]
16O 20Ne 24Mg 28Si
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
27REMOVING the OVERBINDINGLahde et al., arXiv:1311.0477 [nucl-th]
• Overbinding is due to four α clusters in close proximity
⇒ remove this by an effective 4N operator [long term: N3LO]
V (4Neff ) = D(4Neff )∑
1≤(~ni−~nj)2≤2
ρ(~n1)ρ(~n2)ρ(~n3)ρ(~n4)
• fix the coefficient D(4Neff ) from the BE of 24Mg
⇒ excellent description of the g.s. energies
A 12 16 20 24 28Th −90.3(2) −131.3(5) −165.9(9) −198(2) −233(3)
Exp −92.16 −127.62 −160.64 −198.26 −236.54
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
28GROUND STATE ENERGIES
28Si
24Mg
20Ne
16O
12C
8Be
4He
-400 -350 -300 -250 -200 -150 -100 -50 0
E (MeV)
ExperimentNNLO
NNLO + 4Neff
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
29
SUMMARY & OUTLOOK
• Nuclear forces can be calculated accurately and systematically in chiral EFT
• Connection to boson-exchange models can be made
• Nuclear lattice simulations as a new quantum many-body approach
• Fix parameters in few-nucleon systems→ predictions(ab initio calculations)
• 12C spectrum at NNLO→ Hoyle state and its structure
• Fine-tuning of mquark and αEM→ viability of life
⇒ changes in mquark of about 2% and in αEM of about 2.5% are allowed
• First ab initio results for medium mass nuclei
⇒ the strong interactions remain a challenge
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
30
SPARES
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
31
The RELEVANT QUESTIONDate: Sat, 25 Dec 2010 20:03:42 -0600From: Steven Weinberg 〈weinberg@zippy.ph.utexas.edu〉To: Ulf-G. Meissner 〈meissner@hiskp.uni-bonn.de〉Subject: Re: Hoyle state in 12C
Dear Professor Meissner,Thanks for the colorful graph. It makes a nice Christmas
card. But I have a detailed question. Suppose you calculate not only the energyof the Hoyle state in C12, but also of the ground states of He4 and Be8. Howsensitive is the result that the energy of the Hoyle state is near the sum of therest energies of He4 and Be8 to the parameters of the theory? I ask because Isuspect that for a pretty broad range of parameters, the Hoyle state can be wellrepresented as a nearly bound state of Be8 and He4.
All best,Steve Weinberg
• How does the Hoyle state relative to the 4He+8Be threshold,if we change the fundamental parameters of QCD+QED?
• not possible in nature, but on a high-performance computer!
-110
-100
-90
-80
-70
-60
LO NLO EM+IB NNLO Exper.
E (
MeV
)
JP = 0
+1
JP = 0
+2
Jz = 0, J
P = 2
+1
Jz = 2, J
P = 2
+1
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
32The NON-ANTHROPIC SCENARIO
•Weinberg’s assumption: The Hoyle state stays close to the 4He+8Be threshold
g g+ ∆gg ∆g−
4He+8BeHoyle
−84.80
−84.51
fundamental parameter
en
erg
y d
iffe
ren
ce
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
33The ANTHROPIC SCENARIO
•The AP strikes back: The Hoyle state moves away from the 4He+8Be threshold
g g+ ∆gg ∆g−
4He+8BeHoyle
−84.80
−84.51
en
erg
y d
iffe
ren
ce
fundamental parameter
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
34EARLIER STUDIES of the AP
• By hand modification of the energy diff. & network calcs in massive starsLivio et al., Nature 340 (1989) 281
→ a 60 keV increase does not significantly alter carbon production
→ a 60 keV decrease roughly doubles the carbon production rate
→ a ±277 keV change leaves essentially no carbon (just oxygen)
→ weak conclusion: the strong AP might be in trouble
• Changing NN and em interactions in a microscopic model & network calcsOberhummer et al., Science 289 (2000) 88
→ modified NN strength & fine structure constant in [0.996, 1.004]
→ no influence on the width but on the relative position of the Hoyle state
→ use up-to-date stellar evolution model
→ more than 0.5[4]% in the strong coupling [αQED] would destroyall carbon (oxygen) in stars
→ “should be of interest to AP considerations”
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
35
Introduction II:Effective Field Theoryfor Nuclear Physics
only a brief reminder→ details in
E. Epelbaum, H.-W. Hammer, UGM, Rev. Mod. Phys. 81 (2009) 1773[arXiv:0811.1338 [nucl-th]]
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
36CHIRAL EFT FOR FEW-NUCLEON SYSTEMSGasser, Leutwyler, Weinberg, van Kolck, Epelbaum, Bernard, Kaiser, UGM, . . .
• Scales in nuclear physics:
Natural: λπ = 1/Mπ ' 1.5 fm (Yukawa 1935)
Unnatural: |anp(1S0)| = 23.8 fm , anp(3S1) = 5.4 fm 1/Mπ
• this can be analyzed in a suitable EFT based on
LQCD → LEFF = Lππ + LπN + LNN + . . .
• pion and pion-nucleon sectors are perturbative in Q/Λχ → chiral perturbation th’y
• LNN collects short-distance contact terms, to be fitted
• NN interaction requires non-perturbative resummation
→ chirally expand VNN(N), use in regularized LS/FY equation
repulsive
core
CD Bonn
Reid93
AV18
0 0.5 1 1.5 2 2.5
300
200
100
0
-100
Vc
(r)
[ Me
V]
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
37
CHIRAL POTENTIAL and NUCLEAR FORCES
• explains naturally the observed hierarchy of nuclear forces
• MANY successfull tests in few-nucleon systems (continuum calc’s)
O((Q/Λχ)0)
O((Q/Λχ)2)
O((Q/Λχ)3)
O((Q/Λχ)4)
2 LECs
7 LECs
15 LECs
2 LECs
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
38
Nuclear lattice simulations– Formalism –
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
39NUCLEAR LATTICE SIMULATIONSFrank, Brockmann (1992), Koonin, Muller, Seki, van Kolck (2000) , Lee, Schafer (2004), . . .
Borasoy, Krebs, Lee, UGM, Nucl. Phys. A768 (2006) 179; Borasoy, Epelbaum, Krebs, Lee, UGM, Eur. Phys. J. A31 (2007) 105
• new method to tackle the nuclear many-body problem
• discretize space-time V = Ls × Ls × Ls × Lt:nucleons are point-like fields on the sites
• discretized chiral potential w/ pion exchangesand contact interactions
• typical lattice parameters
Λ =π
a' 300 MeV [UV cutoff]
p
p
n
n a
~ 2 fm
• strong suppression of sign oscillations due to approximate Wigner SU(4) symmetryJ. W. Chen, D. Lee and T. Schafer, Phys. Rev. Lett. 93 (2004) 242302
• hybrid Monte Carlo & transfer matrix (similar to LQCD)
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
40
CONFIGURATIONS
⇒ all possible configurations are sampled⇒ clustering emerges naturally
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
41
TRANSFER MATRIX METHOD
• Correlation–function for A nucleons: ZA(t) = 〈ΨA| exp(−tH)|ΨA〉
with ΨA a Slater determinant for A free nucleons
• Ground state energy from the time derivative of the correlator
EA(t) = −d
dtlnZA(t)
→ ground state filtered out at large times: E0A = lim
t→∞EA(t)
• Expectation value of any normal–ordered operator O
ZOA = 〈ΨA| exp(−tH/2)O exp(−tH/2) |ΨA〉
limt→∞
ZOA (t)
ZA(t)= 〈ΨA|O |ΨA〉
Euclidean time
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
42
TRANSFER MATRIX CALCULATION
• Expectation value of any normal–ordered operator O
〈ΨA|O |ΨA〉 = limt→∞
〈ΨA| exp(−tH/2)O exp(−tH/2) |ΨA〉〈ΨA| exp(−tH)|ΨA〉
• Anatomy of the transfer matrix
OΨfree
Ψfree
02Lto
+ Lti
SU(4) π
Lto
+ Lti/2 L
toL
to+ L
ti
full LO full LO SU(4) π
operator insertion forexpectation value
Z,N Z,N
inexpensive filter inexpensive filter
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
43
PROJECTION MONTE CARLO TECHNIQUE
• Insert clusters of nucleons at initial/final states (spread over some time interval)→ allows for all type of wave functions (shell model, clusters, . . .)→ removes directional bias
• Example: two basic configurations in the spectrum of 12C
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
44
MONTE CARLO with AUXILIARY FILEDS
• Contact interactions represented by auxiliary fields s, sI
exp(ρ2/2) ∝∫ +∞
−∞ds exp(−s2/2− sρ) , ρ ∼ N†N
• Correlation function = path-integral over pions & auxiliary fields
p
p
⇒
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
45
COMPUTATIONAL EQUIPMENT
• Past = JUGENE (BlueGene/P)• Present = JUQUEEN (BlueGene/Q)
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
46
Nuclear lattice simulations– Results –
nuclei neutron matter
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
47
FIXING PARAMETERS & FIRST PREDICTIONS
• work at NNLO including strong and em isospin breaking
• 9 NN LECs from np scattering and Qd
• 2 LECs for isospin-breaking (np, pp, nn)
• 2 LECs D,E related to the leading 3NF
⇒ make predictions 0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140
pCM (MeV)
1S0
δ(1S
0)
(deg
rees
)
LO3NLO3
PWA93 (np)
40
60
80
100
120
140
160
180
0 20 40 60 80 100 120 140
pCM (MeV)
3S1
δ(3S
1)
(deg
rees
)
LO3NLO3
PWA93 (np)
40
60
80
100
120
140
160
180
0 20 40 60 80 100 120 140
pCM (MeV)
3S1
δ(3S
1)
(deg
rees
)
LO3NLO3
PWA93 (np)
• pp vs np scattering
• nd spin-3/2quartet channel
• . . .
0 0.10 0.20 0.30 0.40
p2 (fm-2)
0
-0.05
0.05
0.10
-0.10
-0.15
p c
ot
δ (
fm-1
)
0.15
n-d (exp.)
NNLO
NLO
LO
p-d (exp.)
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140 160
pCM (MeV)
1S0
δ(1S
0)
(deg
rees
)
LONLO + IB + EM
PWA93 (pp)
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
48
Ground statesEpelbaum, Krebs, Lahde, Lee, UGM, arxiv:1208.1328
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
49PREDICTIONS: TRITON & HELIUM-3Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 104 (2010) 142501; Eur. Phys. J. A 45 (2010) 335
• binding energies of 3N systems: E(L) = B.E.−a
Lexp(−bL)
see also Hammer, Kreuzer (2011)
⇒ predict the energy difference E(3He)− E(3H)
0.5
0.6
0.7
0.8
0.9
1
5 10 15 20 25
E3H
e −
Etr
iton (
MeV
)
L (fm)
latticephysical (infinite volume)
0.76 MeV [exp.]0.78(5) MeV
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
50
Ground state of 4He
L = 11.8 fm
-30
-28
-26
-24
-22
-20
0 0.02 0.04 0.06 0.08 0.1 0.12
E(t
) (M
eV)
t (MeV-1)
(A)
LO
-6
-4
-2
0
2
4
6
0 0.02 0.04 0.06 0.08 0.1 0.12
t (MeV-1)
(B)
∆NLO-IS∆IB + ∆EM
∆NNLO
LO (O(Q0)) −28.0(3) MeVNLO (O(Q2)) −24.9(5) MeV
NNLO (O(Q3)) −28.3(6) MeVExp. −28.3 MeV
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
51
Ground state of 8Be
L = 11.8 fm
-65
-60
-55
-50
-45
-40
0 0.02 0.04 0.06 0.08 0.1 0.12
E(t
) (M
eV)
t (MeV-1)
(A)
LO
-10
-5
0
5
10
15
0 0.02 0.04 0.06 0.08 0.1 0.12
t (MeV-1)
(B)
∆NLO-IS∆IB + ∆EM
∆NNLO
LO (O(Q0)) −57(2) MeVNLO (O(Q2)) −47(2) MeV
NNLO (O(Q3)) −55(2) MeVExp. −56.5 MeV
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
52
Ground state of 12C
L = 11.8 fm
-120
-110
-100
-90
-80
-70
-60
0 0.02 0.04 0.06 0.08 0.1 0.12
E(t
) (M
eV)
t (MeV-1)
(A)
LO (1)LO (2)
-20
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1 0.12
t (MeV-1)
(B)
∆NLO-IS (1)∆NLO-IS (2)
∆IB + ∆EM (1)∆IB + ∆EM (2)
∆NNLO (1)∆NNLO (2)
LO (O(Q0)) −96(2) MeVNLO (O(Q2)) −77(3) MeV
NNLO (O(Q3)) −92(3) MeVExp. −92.2 MeV
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
53
Ground state of 16O
L = 11.8 fm
-170
-160
-150
-140
-130
-120
-110
0 0.02 0.04 0.06 0.08 0.1
E(t
) (M
eV)
t (MeV-1)
(A)
LO (1)LO (2)
-30
-20
-10
0
10
20
30
40
50
60
0 0.02 0.04 0.06 0.08 0.1
t (MeV-1)
(B)
∆NLO-IS (1)∆NLO-IS (2)
∆IB + ∆EM (1)∆IB + ∆EM (2)
∆NNLO (1)∆NNLO (2)
LO (O(Q0)) −144(4) MeVNLO (O(Q2)) −116(6) MeV
NNLO (O(Q3)) −135(6) MeVExp. −127.6 MeV
to be published
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
54
EXCITED STATES of 12C
• Lowest excited state is 2+1 (as in nature)
E(2+1 ) = −89(3) MeV
[−87.7 MeV]
-4
-3
-2
-1
0
1
0 0.02 0.04 0.06 0.08 0.1
log
[Z(t
)/Z
0+ 1(t)]
t (MeV-1)
(A)
LO
-20
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1E
(t)
(MeV
)t (MeV-1)
(B)
∆NLO-IS∆IB + ∆EM
∆NNLO
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
55THE HOYLE STATE (0+2 )
• energy: E(0+2 ) = −85(3) MeV
• close to E(4He) + E(8Be) = −83.3(2.0) MeV
• structure: “bent” alpha-chain like (not “BEC”)
-110
-100
-90
-80
-70
-60
-50
0 0.02 0.04 0.06 0.08 0.1 0.12
E(t
) (M
eV)
t (MeV-1)
(A)
LO (1)LO (2)
-20
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1 0.12
t (MeV-1)
(B)
∆NLO-IS (1)∆IB + ∆EM (1)
∆NNLO (1)
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
56A HOYLE STATE EXCITATION (2+2 )
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.02 0.04 0.06 0.08 0.1
log
[Z(t
)/Z
0+ 2(t)]
t (MeV-1)
(A)
LO
-20
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1
E(t
) (M
eV)
t (MeV-1)
(B)
∆NLO-IS∆IB + ∆EM
∆NNLO• a 2+ state 2 MeV above the Hoyle state
• interpretation:a rotational band of the Hoyle stategenerated from excitations of the alpha-chain
• what’s in the data ?a 2+ state 3.51 MeV above the Hoyle state seen in 11B(d, n)12Cnot included in the level scheme! Ajzenberg-Selove, Nucl. Phys. A506 (1990) 1
a 2+ state 3.8(4) MeV above the Hoyle state seen in 12C(α,α)12CBency John et al., Phys. Rev. C 68 (2003) 014305
• and much more, see next slide and:→ talk by Henry Weller
⇒ ab initio prediction requires experimental confirmation
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
57SPECTRUM OF 12C• Summarizing the results for carbon-12:
0+1 2+
1 0+2 2+
2
LO −96(2) MeV −94(2) MeV −89(2) MeV −88(2) MeVNLO −77(3) MeV −74(3) MeV −72(3) MeV −70(3) MeV
NNLO −92(3) MeV −89(3) MeV −85(3) MeV −83(3) MeV−82.6(1) MeV [1,2]
Exp. −92.16 MeV −87.72 MeV −84.51 MeV −82.32(6) MeV [3]−81.1(3) MeV [4]−82.13(11) MeV [5]
[1] Freer et al., Phys. Rev. C 80 (2009) 041303[2] Zimmermann et al., Phys. Rev. C 84 (2011) 027304[3] Hyldegaard et al., Phys. Rev. C 81 (2010) 024303[4] Itoh et al., Phys. Rev. C 84 (2011) 054308[5] Weller et al., in preparation
• importance of consistent 2N & 3N forces
• good agreement w/ experiment, can be improved
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
58
Testing theAnthropic Principle
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
59
MC ANALYSIS of the AP
• consider QCD only→ calculate ∂∆E/∂Mπ
• relevant quantities (energy differences)
∆Eh ≡ E∗12 − E8 − E4, ∆Eb ≡ E8 − 2E4 ∆Ec ≡ E∗12 − E12
• energy differences depend on parameters of QCD (LO analysis)
Ei = Ei
(MOPEπ ,mN(Mπ), gπN(Mπ), C0(Mπ), CI(Mπ)
)gπN ≡
gA2Fπ
• remember: M2π± ∼ (mu +md)
⇒ quark mass dependence ≡ pion mass dependence
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
60PION MASS VARIATIONS
• consider pion mass changes as small perturbations
∂Ei∂Mπ
∣∣∣∣Mphysπ
=∂Ei
∂MOPEπ
∣∣∣∣Mphysπ
+ x1
∂Ei∂mN
∣∣∣∣mphysN
+ x2
∂Ei∂gπN
∣∣∣∣gphysπN
+ x3
∂Ei∂C0
∣∣∣∣Cphys
0
+ x4
∂Ei∂CI
∣∣∣∣CphysI
with
x1 ≡∂mN
∂Mπ
∣∣∣∣Mphysπ
, x2 ≡∂gπN∂Mπ
∣∣∣∣Mphysπ
, x3 ≡∂C0
∂Mπ
∣∣∣∣Mphysπ
, x4 ≡∂CI∂Mπ
∣∣∣∣Mphysπ
⇒ problem reduces to the calculation of the various derivativesusing AFQMC and the determination of the xi
• x1 and x2 can be obtained from LQCD plus CHPT
• x3 and x4 can be obtained from two-body scattering and its Mπ-dependence
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
61
AFQMC RESUTS for the DERIVATIVES
• 4He • 12C(0+2 )
1.1
1.15
1.2
1.25
1.3
1.35
1.4
2 4 6 8 10 12 14 16
fitdE/dc11 [l.u.]not fitted
-1.7
-1.65
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
2 4 6 8 10 12 14 16
fitdE/dcii [l.u.]
not fitted
-0.046
-0.045
-0.044
-0.043
-0.042
-0.041
-0.04
-0.039
2 4 6 8 10 12 14 16
fitdE/dm
π (OPE)
not fitted
-0.21
-0.205
-0.2
-0.195
-0.19
-0.185
-0.18
-0.175
2 4 6 8 10 12 14 16
fitdE pion dM [MeV]
not fitted
-0.14
-0.138
-0.136
-0.134
-0.132
-0.13
-0.128
-0.126
-0.124
-0.122
-0.12
-0.118
2 4 6 8 10 12 14 16
fitdE pion IB [MeV]
not fitted
-0.068
-0.066
-0.064
-0.062
-0.06
-0.058
-0.056
-0.054
-0.052
2 4 6 8 10 12 14 16
fitdE/dmN
not fitted
0.255
0.26
0.265
0.27
0.275
0.28
0.285
0.29
0.295
0.3
0.305
2 4 6 8 10 12 14 16
fitdE/dg
πιN [l.u.]not fitted
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.4
2 4 6 8 10 12 14 16
fitdE dCpp [MeV]not fitted
0.54
0.545
0.55
0.555
0.56
0.565
0.57
0.575
0.58
0.585
2 4 6 8 10 12 14 16
fitdE dCoul [MeV]not fitted
-29
-28.5
-28
-27.5
-27
-26.5
-26
-25.5
-25
2 4 6 8 10 12 14 16
fitold dataE [MeV]
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
2 4 6 8 10 12 14 16
fitdE/dc11 [l.u.]
gnd state
-8.2
-8.1
-8
-7.9
-7.8
-7.7
-7.6
-7.5
2 4 6 8 10 12 14 16
fitdE/dcii [l.u.]
gnd state
-0.18
-0.178
-0.176
-0.174
-0.172
-0.17
-0.168
-0.166
2 4 6 8 10 12 14 16
fitdE/dm
π(OPE)
gnd state
-0.82
-0.815
-0.81
-0.805
-0.8
-0.795
-0.79
-0.785
-0.78
-0.775
-0.77
-0.765
2 4 6 8 10 12 14 16
fitdE pion dM [MeV]gnd state
-0.545
-0.54
-0.535
-0.53
-0.525
-0.52
-0.515
-0.51
-0.505
2 4 6 8 10 12 14 16
fitdE pion IB [MeV]gnd state
-0.44
-0.435
-0.43
-0.425
-0.42
-0.415
-0.41
-0.405
-0.4
-0.395
2 4 6 8 10 12 14 16
fitdE/dmN
gnd state
1.27
1.28
1.29
1.3
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
2 4 6 8 10 12 14 16
fitdE/dg
πιN [l.u.]gnd state
1.96
1.98
2
2.02
2.04
2.06
2.08
2.1
2.12
2.14
2 4 6 8 10 12 14 16
fitdE dCpp [MeV]
gnd state
5.55
5.6
5.65
5.7
5.75
5.8
5.85
2 4 6 8 10 12 14 16
fitdE dCoul [MeV]
gnd state
-95
-90
-85
-80
-75
-70
-65
-60
2 4 6 8 10 12 14 16
fitold data, fitted
E [MeV], Hoyle state
-100
-95
-90
-85
-80
-75
2 4 6 8 10 12 14 16
fitE [MeV], gnd state
E(Nt) = E(∞) + const exp(−Nt/τ )
Nt Nt
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
62DETERMINATION of the xi
• x1 from the quark mass expansion of the nucleon mass: x1 ' 0.8± 0.2
• x2 from the quark mass expansion of the pion decay constantand the nucleon axial-vector constant: x2 ' −0.056 . . . 0.008
• x3 and x4 can be obtained from a two-nucleon scattering analysis& can be deduced from:
−∂a−1
∂Mπ
≡A
aMπ
=1
πLS′(η)
∂η
∂Mπ
, η ≡ mNE
(L
2π
)2
⇒ while this can straightforwardly be computed, we prefer to use a representationthat substitutes x3 and x4 by:
∂a−1s
∂Mπ
∣∣∣∣∣Mphysπ
,∂a−1
t
∂Mπ
∣∣∣∣∣Mphysπ
⇒ we are ready to study the pertinent energy differences
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
63RESULTS
• putting pieces together:
∂∆Eh∂Mπ
∣∣∣∣Mphysπ
= −0.455(35)∂a−1
s
∂Mπ
∣∣∣∣∣Mphysπ
− 0.744(24)∂a−1
t
∂Mπ
∣∣∣∣∣Mphysπ
+ 0.056(10)
∂∆Eb∂Mπ
∣∣∣∣Mphysπ
= −0.117(34)∂a−1
s
∂Mπ
∣∣∣∣∣Mphysπ
− 0.189(24)∂a−1
t
∂Mπ
∣∣∣∣∣Mphysπ
+ 0.012(9)
∂∆Ec∂Mπ
∣∣∣∣Mphysπ
= − 0.07(3)∂a−1
s
∂Mπ
∣∣∣∣∣Mphysπ
− 0.14(2)∂a−1
t
∂Mπ
∣∣∣∣∣Mphysπ
+ 0.017(9)
• x1 and x2 only affect the small constant terms
• also calculated the shifts of the individual energies (not shown here)
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
64INTERPRETATION
• (∂∆Eh/∂Mπ)/(∂∆Eb/∂Mπ) ' 4
⇒ ∆Eh and ∆Eb cannot be independently fine-tuned
•Within error bars, ∂∆Eh/∂Mπ & ∂∆Eb/∂Mπ appear unaffectedby the choice of x1 and x2→ indication for α-clustering
• For ∆Eh & ∆Eb, the dependence on Mπ is small when
∂a−1s /∂Mπ ' −1.6× ∂a−1
t /∂Mπ
• the triple alpha process is controlled by :∆Eh+b ≡ ∆Eh + ∆Eb = E?12 − 3E4
∂∆Eh+b
∂Mπ
∣∣∣∣Mphysπ
= −0.571(14)∂a−1
s
∂Mπ
∣∣∣∣∣Mphysπ
− 0.934(11)∂a−1
t
∂Mπ
∣∣∣∣∣Mphysπ
+ 0.069(6)
⇒ so what can we say about the quark mass dependence of the scattering lengths?
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
65CONSTRAINTS on the SCATTERING LENGTHS
• Quark mass dependence of hadron properties:δOH
δmf
≡ KfH
OH
mf
, f = u, d, s
• NN scattering lengths as a function ofMπ: −∂a−1
s,t
∂Mπ
≡As,t
as,tMπ
, As,t ≡Kqas,t
Kqπ
• earlier determinations from chiral EFT at NLOBeane, Savage (2003), Epelbaum, Glockle, UGM (2003)
• new determination at NNLO: Epelbaum et al. (2012)
Kqas
= 2.3+1.9−1.8 , K
qat
= 0.32+0.17−0.18 →
∂a−1t
∂Mπ
= −0.18+0.10−0.10 ,
∂a−1s
∂Mπ
= 0.29+0.25−0.23
• note the magical central value:∂a−1
s /∂Mπ
∂a−1t /∂Mπ
' −1.6+1.0−1.7
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
66CORRELATIONS
• vary the quark mass derivatives of a−1s,t within −1, . . . ,+1:
-6 -4 -2 0 2 4 6K
E4
π
-6
-4
-2
0
2
4
6
KE*12π
KE12π
KE8π
K∆Ecπ
-6 -4 -2 0 2 4 6K
E4
π
-600
-400
-200
0
200
400
600
K∆Ebπ
K∆Ehπ
K∆Eh+bπ
• clear correlations: α-particle BE and the energies/energy differences
⇒ anthropic or non-anthropic scenario depends on whether the 4He BE moves!
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
67THE END-OF-THE-WORLD PLOT• |δ(∆Eh+b)| < 100 keV
→∣∣∣∣(0.571(14)As + 0.934(11)At − 0.069(6)
)δmq
mq
∣∣∣∣ < 0.0015
As,t ≡∂a
−1s,t
∂Mπ
∣∣∣∣M
physπ
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •
68
Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •