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CaSToRCCaSToRC
Hadron structure from lattice QCD
Giannis Koutsou Computation-based Science and Technology Research Centre (CaSToRC)
The Cyprus Institute
EINN2015, 5th Nov. 2015, Pafos
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Outline
Short introduction to lattice calculations ‣ Challenges and current landscape
Nucleon charges Nucleon sigma-terms Electromagnetic matrix elements ‣ Form factors and radii ‣ Strange form factor
New directions: ‣ Neutron EDM ‣ Direct calculation of PDFs
Summary and outlook
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Lattice QCD — ab initio simulation of QCD – Freedom in choice of:
– quark masses (heavier is cheaper) – lattice spacing a (larger is cheaper) – lattice volume L3×T (smaller is cheaper)
– Choice of discretisation scheme e.g. Clover, Twisted Mass, Staggered, Overlap, Domain Wall Trade — offs and advantages for each differ
Eventually, all schemes must agree:
– At the continuum limit: a → 0
– At infinite volume limit L → ∞ – At physical quark mass
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Selected lattice simulation points used for hadron structure – Multiple collabs. simulating at physical pion mass – Size of points indicates mπL
Simulations landscape
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Sources of uncertainty – Statistical error: , with MC
samples – Correlation functions:
exponentially decay with time-separation
– Disconnected contributions: stochastic error
1pN
– Systematic uncertainties – Extrapolations
a, L, mπ – Contamination from higher
energy states
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Sources of uncertainty – Statistical error: , with MC
samples – Correlation functions:
exponentially decay with time-separation
– Disconnected contributions: stochastic error
1pN
– Systematic uncertainties – Extrapolations
a, L, mπ – Contamination from higher
energy states
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Indicative computer time requirements for nucleon structure
Multi-petascale to exa-scale requirements
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Reproduction of light baryon masses – Agreement between lattice
discretisation schemes – Reproduction of experiment
Prediction of yet-to-be-observed charmed baryons – Confidence through agreement
between lattice schemes – Nucleon structure…
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Axial charge – Agreement towards experiment – Simulations very close to or at the physical quark mass
Benchmark
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Scalar and tensor charges – Isovector charges – General agreement between calculations – Different dependence on excited states between lattice actions
gS = hN |uu� dd|Ni gT = h1i�u��d
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Axial charge — light disconnected – Required for individual u- and d- contributions – Requires dedicated calculations for “disconnected quark loop” – Large statistical fluctuations in correlation functions – Sign is negative: brings connected result down – About 10% of connected value
u�5�µu+ d�5�µd
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Axial charge — strange contribution – Contribution exclusively by “disconnected quark loop”
More by M. Constantinou’s winning poster talk: Orbital and gluon contribution to proton spin
s�5�µs
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Nucleon sigma — terms
• Pion nucleon σ-term:
• Strange σ-term: • Enter super-symmetric candidate particle scattering cross
sections with nucleon (e.g. neutralino through Higgs)
1.Direct calculation of matrix elements
�⇡N = mudhN |uu+ dd|Ni�s = mshN |ss|Ni
2.Through Feynman - Hellmann theorem: �⇡N = mud@mN
@mud�s = ms
@mN
@ms
Involves disconnected contributions
• Reliance on effective theories for dependence on mπ
• Weak dependence on ms
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Nucleon sigma — terms
• More results coming from the lattice using direct evaluation of the matrix element
• First results from simulations directly at the physical point
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Electromagnetic form factors
Dirac (F1) and Pauli (F2):
hN(p0, s0)|jµ|N(p, s)i =s
M2N
EN (p0)EN (p)u(p0, s0)Oµu(p, s)
Oµ = �µF1(q2) +
i�µ⌫q⌫
2MNF2(q
2), q = p0 � p
Isovector and isoscalar combinations:
F p � Fn =Fu � F d
F p + Fn =1
3(Fu + F d)
assuming flavour SU(2) isospin symmetry, i.e.: p ⟷ n when u ⟷ d
jsµ = u�µu+ d�µdjvµ = u�µu� d�µd,
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Electromagnetic form factors – General agreement between calculations – Two physical point lattices:
• Twisted Mass PoS LATTICE2014 (2015) 148
• Clover improved Phys.Rev. D90 (2014) 074507
Jµ = u�µu� d�µd
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EM form factors – Radii from fits to dipole form
F1(Q2) =
1
(1 +Q2/M21 )
2
F2(Q2) =
F2(0)
(1 +Q2/M22 )
2
hr2i i =12
M2i
– Curvature towards physical point – Increasing sink-source separation – Need 1% error for contact to
experiment
m [GeV]
0.10 0.15 0.20 0.25 0.30 0.35 0.40m [GeV]
0.0
0.2
0.4
0.6
0.8
1.0
r2 2V
[fm
2 ]
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EM form factors – New methods for “disconnected fermion
loops” — hierarchical probing [arXiv:1302.4018] – Sampling of the fermion propagator using site
colouring schemes J. Green et al., Phys.Rev. D92 (2015) 3, 031501, arXiv:1505.01803
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nEDM
• θ new bare parameter of action
• Gives rise to dipole form factor:
• Lattice:
1. Re-weight lattice configs. with expanded to leading order in θ
2. Include an external electric field
3. Simulate with modified action which includes a Wick rotated
|~dN | = ✓|F3(0)|2mN
leading order in θ
ei✓Q
ei✓Q
• CP-breaking term in action:
�✓1
2MNF3(q
2)u(p0)�µ⌫�5u(p)Fµ⌫
Topological charge
C. Alexandrou et al., arXiv:1509.04259
/ i✓✏µ⌫�⇢Tr[Gµ⌫G�⇢]
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nEDM from the lattice
• F3(0) from
• External
• θ in action
O(✓)
~E
Using latest experimental bound:
✓ . O(10�12 � 10�11)
Poster by A. Athenodorou
• Phys. Rev. D 78, 014503 (2008) • Phys. Rev. Lett. 115, 6, 062001 (2015) • arXiv:1510.05823 [hep-lat] • Phys. Lett. B 687 42 (2010), [arXiv:0911.3981]
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Direct calculation of PDFs on the lattice
Proposal by X. Ji [Phys.Rev.Lett. 110 (2013) 26] for the calculation of quasi-PDFs on a Euclidean lattice:
q(x, P3) =
Zdz
4⇡e
�ixP3zhN(P3)| (z)�zAz
(z, 0) (0)|N(P3)i
Would in principle require higher order Mellin moments, calculated on the lattice:
hxniq =
Z 1
�1x
nq(x)dx
For larger n: • Noisier correlation functions • Complicated renormalisation • Mixing at high derivatives
• Three-point correlation function • Boosted nucleon • Line of glue connecting quark lines • Match using PT
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Direct calculation of PDFs on the lattice
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-1 -0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-1 -0.5 0 0.5 1
u−d
x
q
q
q(0)
MSTW
CJ12
ABM11
x
q
q
q(0)
MSTW
CJ12
ABM11
Two pilot, promising calculations in lattice QCD
Talk by Fernanda Steffens, Tue.
Phys. Rev. D91 (2015) 054510
Phys.Rev. D92 (2015) 014502
2+1+1 Clover on HISQ, mπ=310 MeV
2+1+1 twisted mass, mπ=370 MeV
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Algebraic multi-grid: A. Frommer et al. SIAM J. Sci. Comput. 36 (2014) A1581-A1608
Exact eigenvalue deflation
At physical quark masses – Mathematical algorithms for alleviating “critical slowing down” – Multiple right-hand-side methods for efficient multiplication of
statistics
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Summary and outlook
Lattice QCD converging on benchmark quantities • Physical pion mass simulations from a number of collaborations • Systematic uncertainties coming under control • Huge effort in algorithmic improvements for boosting statistical
accuracy Exciting new directions • Reliable calculation of disconnected contributions • Individual quark contributions to nucleon structure • Contact to new physics searches: σ-terms, scalar and tensor
charges • Initial, promising pilot calculations of PDFs on the lattice
Future directions/challenges • Effects of isospin breaking, Nf=1+1+1(+1)
• Electromagnetic effect