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

PDF

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