High Precision Physics fromHigh Performance Computing
The 30th Anniversary Symposium of theCenter for Computational Sciences
University of Tsukuba
October 13-14, 2022

N.H. ChristColumbia University

RBC and UKQCD Collaborations

Outline• Lattice QCD and particle physics• Advances in numerical methods– Riemannian Manifold HMC• Application challenges– K → p p decay and CP violation– Muon g-2 and 4.2 s SM-expt. discrepancy• Outlook30th Anniversary CSS - 10/13/2022

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Particle Physics & Lattice QCD• The Standard Model does not predict
everything:– gm -2 anomaly– Insufficiently large CP violation– Dark matter– Dark energy….
• High energy experiments at the LHC search for new particles• High precision results at lower energies may have greater reach:
Belle II at KEK.
• Most experiments involve quarks so that the strong interactionsof the quarks must be untangled: Lattice QCD is required.
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QCD: theory of quarks and gluons• Do u, d, s, c, b, t quarks obey
standard model predictions?
• Need <1% control of theirbinding by QCD forces.
• Much more difficult thantreating atoms:– u and d quarks are masslessand move at relativistic speeds– pair creation makes even asingle proton a complex, manybody problem

B→p+m+nbu

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QCD: theory of quarks and gluons• Do u, d, s, c, b, t quarks obey
standard model predictions?
• Need <1% control of theirbinding by QCD forces.
• Much more difficult thantreating atoms:– u and d quarks are massless m
and move at relativistic speeds– pair creation makes even asingle proton a complex, manybody problem

B→p+m+nuun

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Lattice QCD

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Lattice QCD• Introduce a space-time lattice.
• Evaluate the Euclidean Feynman pathintegral.
no i– Study e - HQCD t– 1st principles non-perturbative formulation• Use Monte Carlo importance sampling with ahybrid MD/Langevin evolution.
• 963 x 192 lattice → 5 x 109 dimensional integral• ~40 samples give sub-percent errors
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Numerical methods• Use low eigenmodes to solve D Gn = hn wheremultiple right-hand sides allow reuse (deflation).
• Low eigenmodes also important for all-to-all
propagators.
• 963 x 192 volume requires 5K eigenvectors (160 TB).
• Compress using local coherence: 30x (Clark, Jung &Lehner, arXiv:1710.06884)• New directions:– Domain-decomposed HMC: use to reduce inter-nodecommunication.
– Riemannian manifold HMC.

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Riemannian Manifold HMC• Invented in 1986: [S. Duane, et al., Phys. Lett. B
176 (1986) 143; Girolami and Calderhead, Journal of
the Royal Statistical Society: Series B 73 (2011) 123]• Modify the canonical mass term

• Choose M(l) to match power spectrum ofthe HMC force• Suggestion of RMHMC 3x acceleration
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Applications

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e’Direct CP violation inK → pp decay
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K0 – K0 mixing• D S=1 weak decays allow K0 and K0 to decay tothe same p p state.
• Resulting mixing described by Wigner-Weisskopf• Decaying states are mixtures of K0 and K0

Indirect CPviolation
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CP violation• CP violating, experimental
amplitudes:• Where:Indirect: |e | = (2.228 ± 0.011) x 10-3
Direct:Re(e /e ) = (1.66 ± 0.23) x 10-3

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K → p p and CP violation• Final pp states can have I = 0 or 2.
DI = 3/2
DI = 1/2• CP symmetry requires A0 and A2 be real.
• Direct CP violation in this decay ischaracterized by:
Direct CPviolation
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Challenging calculation• Presence of vacuum implies exponentiallyfalling signal/noise ratio.

• Two-pion final state was a new challenge.
• We first attempted this calculation in 1997:• Seven generations of graduate students:– Calin Christian (2002)– Changhoan Kim (2004)– Sam Li (2008)– Matthew Lightman (2011)– Qi Liu (2012)– Daiqian Zhang (2015)– Tianle Wang (2021)

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Low Energy Effective Theory• Represent weak interactions bylocal four-quark Lagrangian•• Vqq – CKM matrix elements• zi and yi – Wilson Coefficients• Qi – four-quark operators
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Local four quark operators• Current-currentoperators• Electro-WeakPenguins• QCD Penguins

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Lattice calculation of pp |HW |K 
• The operator product d(x)s(x) easily creates akaon.
• Use finite-volume energy quantization(Lellouch-Luscher) and adjust L so that a finitevolume state obeys: Epp = MK .

• Use boundary conditions on the quarks: Epp(gnd) = MK
• For (pp )I=0 use G-parity boundary conditionso G|p  = - |p o Complex at the quark level G:

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Recent results• 2015 [Phys. Rev. Lett. 115 (2015) 212001]• I = 0 pp phase shift:d0 = 23.8(5.4)o• Dispersion theory result: d0 = 34o[Colangelo, et al.]

• Re(e /e) = (1.38 ± 5.15stat ± 4.59sys ) x 10-4

• Expt.: (16.6 ± 2.3) x 10-4 (2.1 s difference)• Introduce more
pp operators todistinguishexcited pp states• Increase statistics
216 → 741

• 2020 [Phys. Rev. D 102 (2020) 054509]• I = 0 pp phase shift:d0 = 31.7(6)°• Re(e /e) = (21.7 ± 2.6stat ± 6.2sys ± 5.0 isospin) x 10-4

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I = 0 pp scattering with three operators
pp (111)
pp (111) &pp (311)
pp (111) & s
pp (111) &s & pp (311)• d0 = 31.7(6)° vs 34° data-driven prediction (5-15
fit, statistical errors only).
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Extended results for pp phase shifts• Combine pion operators carrying various
momenta to examine Pcm  0 → varying s• We are now in good agreement with
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dispersive results.

Example of K → pp data• Examine dependence on pp HW separation
plot pp(tpp) HW (top) K(tK)  versus t = tpp-top
top – tK  

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Systematic errors
DescriptionError
Re(A0)Im(A0)

Excited state-

Unphysical kinematics

5%

Operator
renormalization4%

Wilson coefficients

12%

Finite lattice spacing

12%

Lellouch-Luscher factor

1.5%

Finite volume

7%

Parametric errors
Missing G1 operator
Total

0.3%

6%

5%
19.8%

20.7%

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e ’ – Next steps• Continuum limit (Chris Kelly) :– G-parity evolution and measurement code now optimizedfor GPUs
– Now generating 403x96 G-parity ensemble on Perlmutter• Isospin breaking (E&M + mu-md) now an importantfocus (Xu Feng, Joe Karpie)– Work in Coulomb gauge: Coulomb + transverse photons– Coulomb part may be solved• Explore periodic boundary conditions. Use GEVP
to extract matrix element of excited state. (Tom Blum,Dan Hoying, Luchang Jin, Masaaki Tomii)

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gm-2
Anomalous muon
magnetic moment
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g - 2 for the muon
• Anomalous moment: am = (gm - 2)/2
• FNAL expt E989 :xam = 11699204  5.4 x 10-10• 4.2s difference between the standard
model prediction and experiment:
am EXP - am SM = 25.1  5.9 x 10-10 (0.46 ppm)

FNAL E989• Effects of quark and gluons enter at order aEM2:

HVP
692.5(2.7) x 10-10

HLbL
10.5(2.6) x 10-10
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Hadronic vacuum polarization

BMW1%

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Time-momentum analysis(Bernecker-Meyers representation)Large time
pp states

Small time moresensitive to a2 errors

Time separation
between currents• HVP contribution to gm-2 is the area under
this curve.
• Compare middle region where lattice
precision is greatest
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``Middle Window’’ comparison

1%

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Additional Calculations• Complete calculation gm- 2 with reduced errors.
• KL-KS mass difference• Long distance part of eK• Electromagnetic corrections to K- → m- n
• Two photon exchange contribution to KL → m + m • Electromagnetic corrections to e’30th Anniversary CSS - 10/13/2022

Outlook• Faster machines carry us into new regimes where
old and new algorithmic ideas become effective,multiplying the computers’ impact.
• Exascale machines Frontier (ORNL) and Aurora(ANL) should be examples.
– Planning new 2+1+1 flavor
ensembles:

963x384, 1/a=3 GeV
1283x512, 1/a=4 GeV
1603x640, 1/a=5 GeV:
– (mcharm a)2 = (1.2 GeV / 5 GeV)2 = 6% -- should allow
accurate calculation of charm quark loops.
– We can move from 3-flavor calculation in the StandardModel to calculations which involve an active charm
quark: K0 – K0 mixing is the easiest example.

• By the 60th anniversary of CSS we will havelearned much more about the Standard Model
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and what lies beyond it!