Pion-pion Scattering with Physical Quark Masses (from ...€¦ · valued quantities taken modulo...
Transcript of Pion-pion Scattering with Physical Quark Masses (from ...€¦ · valued quantities taken modulo...
Pion-pion Scattering with Physical Quark Masses (fromLattice QCD)
Dan Hoying
(RBC & UKQCD Collaboration)
July 25, 2018
Lattice 2018
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 1 / 34
Motivation
Outline
1 Motivation
2 Elastic ππ Scattering Phase Shifts
3 Spectra from GEVP
4 Computational Details
5 Momenta Combinatorics
6 Preliminary Results
7 Summary and Outlook
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 2 / 34
Motivation
Motivation: K → ππ
Direct comparison of low energy QCD - experiment vs. lattice
K → ππ - an important decay to understand CP violation
Lattice calculation difficult, Gparity calculation already done, butneeds a check
See talks tomorrow for more information (Weak decay session, T.Wang, C. Kelly)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 3 / 34
Motivation
G-parity vs. Periodic Boundary Conditions
We compute using spatially periodic boundary conditions (antiperiodic intime)
G-parity - charge conjugation + 180 degree isospin rotation
G-parity eliminates (kinematically) unphysical stationary ππ state
G-parity and Periodic boundary conditions have different finite volumeerrors, useful check
Periodic lattices ready for use, evolution cost amortized
Good exercise for K → ππ (same physics is present, strong phasecommon to both)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 4 / 34
Elastic ππ Scattering Phase Shifts
Outline
1 Motivation
2 Elastic ππ Scattering Phase Shifts
3 Spectra from GEVP
4 Computational Details
5 Momenta Combinatorics
6 Preliminary Results
7 Summary and Outlook
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 5 / 34
Elastic ππ Scattering Phase Shifts
Luscher Method
Luscher gives a quantization condition which maps lattice spectra ontoinfinite volume scattering phase shifts.Luscher’s formula[1]:
δ(p) = −φ(k) + πn, n ∈ Z
tan φ(k) =π3/2κ
Z00(1;κ2)
κ =pL
2π
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 6 / 34
Spectra from GEVP
Outline
1 Motivation
2 Elastic ππ Scattering Phase Shifts
3 Spectra from GEVP
4 Computational Details
5 Momenta Combinatorics
6 Preliminary Results
7 Summary and Outlook
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 7 / 34
Spectra from GEVP
Extracting Spectra from Correlation Functions
We can define a generalized eigenvalue problem (GEVP) from an NxNmatrix of correlators we compute on the lattice
Cij ≡⟨
0|Oi (t)O(0)†j |0⟩
=∞∑
n=1
(e−Ent + e−En(Lt−(t−t0))
)ψniψ
∗nj
ψ∗ni =⟨
0|Oi |n⟩
⇒ C (t)vn(t, t0) = λn(t, t0)C (t0)vn(t, t0)
λn(t, t0) = e−En(t−t0) + e−En(Lt−(t−t0))
Important points:
Systematic error in nth energy state: εn ∼ e−(EN+1−En)t if t0 ≥ t/2[2]
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 8 / 34
Spectra from GEVP
Extracting Spectra (cont’d)
Operator basis is composed of single particle operators qq, qγµq andtwo particle operator qγ5q, with various momentum combinationsand non-zero ~pCM up to ±(1, 1, 1).
Operators are projected onto A1 irrep and definite isospin (0, 1, 2).
t0 is arbitrary, we fix it to be either⌈
t2
⌉or t − 1 (if later times aren’t
very noisy)
GEVP also exists for matrix elements [3]. We plan to use this forK → ππ.
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 9 / 34
Computational Details
Outline
1 Motivation
2 Elastic ππ Scattering Phase Shifts
3 Spectra from GEVP
4 Computational Details
5 Momenta Combinatorics
6 Preliminary Results
7 Summary and Outlook
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 10 / 34
Computational Details
Lattice Details
V = 243 ID lattice, 1.015 GeV lattice spacing
Lt = 64
2 + 1 flavor Mobius Domain Wall Fermions (generated byRBC/UKQCD)
∼ 5 fm box
Physical quark mass, unquenched (no chiral extrapolation needed)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 11 / 34
Computational Details
Timing Summary
For our 243 run, we compute on 32 KNL nodes (64 cores, 192 GB memory)for 20 hours (current wall time). Timing breakdown (of expensive steps)
Lanczos - low mode eigenvectors (amortized, not in an individualconfig run) - 6 hours
400 iterations of Zmobius Split CG (ls = 12) 1.3 hours(split-MADWF= 6 hours)
π meson field computation (all-to-all propagators[4]) 1 hour
ππ → ππ contractions 10 hours
σ meson field 1 hour
ρ meson field 3 hours (3 polarizations, projected)
σ (scalar) contractions 1 hour
ρ (vector) contractions 2 hour
Our time is dominated not by inversions, but contractions. We are stillimproving this number.
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 12 / 34
Momenta Combinatorics
Outline
1 Motivation
2 Elastic ππ Scattering Phase Shifts
3 Spectra from GEVP
4 Computational Details
5 Momenta Combinatorics
6 Preliminary Results
7 Summary and Outlook
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 13 / 34
Momenta Combinatorics
Momentum Combinations
Compute pions with plat,max = ±(1, 1, 1) (also permutations) in units of2πL .
27 possible individual pion momenta.
Cube this if you want to get a rough estimate of the combinationsallowed
Using momenta for irreps we care about (e.g. A1) and crossingsymmetry, we get 13890 separate ππ → ππ correlators.
qq, qγµq correlators are not included in this number.
Initial run did not include enough of the correlators (recent bug foundin crossing symmetry code) → spin contamination, but this is nowfixed.
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 14 / 34
Momenta Combinatorics
Auxiliary Diagrams
We have large number of momentum possibilities we could compute, butare all of them necessary? It turns out we can reduce this number byexactly a factor of 2 via a labeling symmetry between source and sink. Ifwe swap these, we will get the exact same correlation function (with timevalued quantities taken modulo the box size in time) up to an overallphase. Stated again, we have
Rule of Thumb:
Two lattice correlation functions are bit-by-bit identical (up to timereordering, phase) if they are related via a swap of source and sink labels.
We refer to the diagram derived (at the analysis stage) via this auxiliarysymmetry as the auxiliary diagram1. In practice, our data set is not fullysymmetric under auxiliary symmetry, so our (projected) gain is only 3/2.
1Hoying 2017, unpub.Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 15 / 34
Preliminary Results
Outline
1 Motivation
2 Elastic ππ Scattering Phase Shifts
3 Spectra from GEVP
4 Computational Details
5 Momenta Combinatorics
6 Preliminary Results
7 Summary and Outlook
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 16 / 34
Preliminary Results
Caveat Emptor
We must still calculate the correction from Zmobius→Mobius (thesecorrections are expected to be small).
We go beyond 4π threshold, but experimental amplitude is small, sowe are probably safe to neglect until higher energies (Luscher formuladoes not exist for inelastic processes)
Unmeasured systematic error due to having only one latticespacing/size.
Now, results:
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 17 / 34
Preliminary Results
I = 2, ~pCM = 0
4 5 6 7 8 9 10 11t/a
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
amef
f(t)
Energy[0] = 0.28085+/-0.00104Energy[1] = 0.60727+/-0.00105Energy[2] = 0.80874+/-0.00377Energy[3] = 0.96825+/-0.00786
2/dof=1.0137, dof=5Double jackknife fit.4x4 GEVP, I2, , pCM = 000 (zmobius) matdt3 37 configs
Dispersive(0) Dispersive(1) Dispersive(2) Dispersive(3) Energy(0) Energy(1) Energy(2) Energy(3)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 18 / 34
Preliminary Results
I = 2, ~pCM = 001
4 6 8 10 12 14t/a
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
amef
f(t)
Energy[0] = 0.43664+/-0.00121Energy[1] = 0.69959+/-0.00182Energy[2] = 0.85019+/-0.00509
2/dof=1.1026, dof=8Double jackknife fit.3x3 GEVP, I2, , pCM = 100 (zmobius) matdt3 37 configs
Dispersive(0) Dispersive(1) Dispersive(2) Energy(0) Energy(1) Energy(2)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 19 / 34
Preliminary Results
I = 2, ~pCM = 011
4 5 6 7 8 9 10 11t/a
0.5
0.6
0.7
0.8
0.9
amef
f(t)
Energy[0] = 0.53518+/-0.00188Energy[1] = 0.59396+/-0.00134Energy[2] = 0.77001+/-0.004Energy[3] = 0.7981+/-0.00263
2/dof=1.3186, dof=10Double jackknife fit.4x4 GEVP, I2, , pCM = 011 (zmobius) matdt3 37 configs
Dispersive(0) Dispersive(1) Dispersive(2) Dispersive(3) Energy(0) Energy(1) Energy(2) Energy(3)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 20 / 34
Preliminary Results
I = 2, ~pCM = 111
4 5 6 7 8 9 10 11 12t/a
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95am
eff(t
)
Energy[0] = 0.61391+/-0.00777Energy[1] = 0.69497+/-0.00356
2/dof=2.0972e-01, dof=4Double jackknife fit.2x2 GEVP, I2, , pCM = 111 (zmobius) matdt3 37 configs
Dispersive(0) Dispersive(1) Energy(0) Energy(1)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 21 / 34
Preliminary Results
I = 2 phase shifts
300 400 500 600 700 800 900 1000s
40
30
20
10
0
10
20
Phas
e Shif
t (de
gree
s)I = 2, s vs. Phase Shift
phenop0p1p11p111
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 22 / 34
Preliminary Results
I = 0, 3x3, Spin ContaminatedIgnore the tiny error bar green points past t = 8 and yellow past t = 12
2 4 6 8 10 12 14 16t/a
2
0
2
4
6
8
amef
f(t)
Dispersive energy[0] = 0.59353Dispersive energy[1] = 0.2795Dispersive energy[2] = 0.79147
Energy[0] = 0.27315+/-0.000268Energy[1] = 0.51762+/-0.00474
2/dof=9.7473e-01, dof=12Double jackknife fit.3x3 GEVP, , , momtotal000 exact matdt2 cumulative
Dispersive(0) Dispersive(1) Dispersive(2) Energy(0) Energy(1) Energy(2)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 23 / 34
Summary and Outlook
Outline
1 Motivation
2 Elastic ππ Scattering Phase Shifts
3 Spectra from GEVP
4 Computational Details
5 Momenta Combinatorics
6 Preliminary Results
7 Summary and Outlook
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 24 / 34
Summary and Outlook
Summary and Outlook
We have run on 170 gauge configurations and generated promising partialresults. This is the first calculation of pipi scattering spectra from thelattice at physical pion mass.
We have also run on 30+ configs of new, complete data andgenerated improved preliminary results, (I = 1 soon).K → ππ periodic code ready to be tested statisticallyWe will run next (fiscal) year on 323 lattice (1, 1.4 GeV latticespacing) to test continuum limitDistillation study proposed as wellMuch of the analysis code and production code for these present andfuture runs is finished (but under-tested).O(200)+ configs likely needed to resolve I = 0, more a2a noisesamples may be needed for σ.
Pipi data is available on request in standard binary format hdf5. Myanalysis code is also publicly available on github:github.com/goracle/lattice-fitter
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 25 / 34
Summary and Outlook
Thanks!
The RBC & UKQCD collaborations
BNL and BNL/RBRC
Ziyuan BaiNorman ChristDuo GuoChristopher KellyBob MawhinneyMasaaki TomiiJiqun TuBigeng Wang
University of Connecticut
Peter BoyleGuido CossuLuigi Del DebbioTadeusz JanowskiRichard KenwayJulia KettleFionn O'haiganBrian PendletonAntonin PortelliTobias TsangAzusa Yamaguchi
Nicolas Garron
Jonathan FlynnVera GuelpersJames HarrisonAndreas JuettnerJames RichingsChris Sachrajda
Julien Frison
Xu Feng
Tianle WangEvan WickendenYidi Zhao
UC Boulder
Renwick Hudspith
Mattia BrunoTaku IzubuchiYong-Chull JangChulwoo JungChristoph LehnerMeifeng LinAaron MeyerHiroshi OhkiShigemi Ohta (KEK)Amarjit Soni
Oliver Witzel
Columbia University
Tom BlumDan Hoying (BNL)Luchang Jin (RBRC)Cheng Tu
Edinburgh University
York University (Toronto)
University of Southampton
Peking University
University of Liverpool
KEK
Stony Brook University
Jun-Sik YooSergey Syritsyn (RBRC)
MIT
David Murphy
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 26 / 34
Summary and Outlook
Martin Luscher. “Two particle states on a torus and their relation tothe scattering matrix”. In: Nucl. Phys. B354 (1991), pp. 531–578.doi: 10.1016/0550-3213(91)90366-6.
Benoit Blossier et al. “On the generalized eigenvalue method forenergies and matrix elements in lattice field theory”. In: JHEP 04(2009), p. 094. doi: 10.1088/1126-6708/2009/04/094. arXiv:0902.1265 [hep-lat].
John Bulava, Michael Donnellan, and Rainer Sommer. “On thecomputation of hadron-to-hadron transition matrix elements in latticeQCD”. In: JHEP 01 (2012), p. 140. doi:10.1007/JHEP01(2012)140. arXiv: 1108.3774 [hep-lat].
Justin Foley et al. “Practical all-to-all propagators for lattice QCD”.In: Comput. Phys. Commun. 172 (2005), pp. 145–162. doi:10.1016/j.cpc.2005.06.008. arXiv: hep-lat/0505023[hep-lat].
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 27 / 34
Summary and Outlook
All-to-All Propagators
D−1A2A ≡
Nl−1∑l=0
|φl〉1
λl〈φl |+
Nh−1∑h=0
(D−1 −
Nl−1∑l=0
|φl〉1
λl〈φl |
)︸ ︷︷ ︸
D−1Defl
|ηh〉 〈ηh|
I = limNh→∞
|ηh〉 〈ηh| , ⇒ limNh→∞
D−1A2A = D−1
Deflate with 2000 low modes |φl〉, Zmobius eigenvectors
In practice, set Nh = 1 (more hits improve excited state noise. Gaugenoise dominates lower energy states.)
12 ∗ Lt ∗Nh high modes (spin, color, time diluted) → 768 high modes.
We obtain the exact point-to-point propagator in this stochasticlimit.[4]
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 28 / 34
Summary and Outlook
Operator Construction: Isospin Projection
On our lattices (2 + 1), isospin is a good symmetry. We want to know thespectra of the different isospin channels I = 0, 1, 2 Examples below. Notethe disconnected diagram in I = 0 which is very noisy, but important(needs large statistics).〈I = 0|I = 0〉:
〈I = 2|I = 2〉:
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 29 / 34
Summary and Outlook
Isospin (cont’d)
〈I = 1|I = 1〉:
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 30 / 34
Summary and Outlook
Operator Construction:Spin Irrep Projection
We would like to project our operator set onto the lowest spin in eachisospin channel (K is spin 0).
Continuum spin states have known correspondence to irreps of thegroup of allowed lattice rotations O (we can project continuumrepresentations to lattice irreps)
Each irrep in general corresponds to a tower of spin states
We project onto irreps with the lowest spins → easier to resolve
Bose symmetry ⇒ I = 1 needs p-wave irrep: T1
I = 0, 2 needs s-wave irrep: A1
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 31 / 34
Summary and Outlook
Motivation: K → ππ
Likely explanation for matter/antimatter asymmetry in Universe,baryogenesis, requires violation of CP.
Amount of CPV in Standard Model appears too low to describemeasured M/AM asymmetry: tantalizing hint of new physics.
Direct CPV first observed in late 90s at CERN (NA31/NA48) andFermilab (KTeV) in K 0 → ππ:
η00 =A(KL → π0π0)
A(KS → π0π0)η± =
A(KL → π+π−)
A(KS → π+π−)
Re(ε′
ε) =
1
6
(1−
∣∣∣∣η00
η±
∣∣∣∣2)
= 1.66(23) x 10−3(Experiment)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 32 / 34
Summary and Outlook
K → ππ (cont’d.)
In terms of isospin states,
∆I = 3/2 decays to I = 2 final states, amplitude A2
∆I = 1/2 decays to I = 0 final states, amplitude A0
A(K 0 → π+π−) =
√2
3A0e
iδ0 +
√1
3A2e
iδ2
A(K 0 → π0π0) =
√2
3A0e
iδ0 − 2
√1
3A2e
iδ2
⇒ ε′ =iωe i(δ2−δ0)
√2
(Im A2
Re A2− Im A0
Re A0
)ω =
ReA2
ReA0
Small size of ε′ makes it particularly sensitive to new direct-CPVintroduced by most BSM models.
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 33 / 34
Summary and Outlook
Infinite Volume Scattering Phase Shifts
We can then compare to experiment via phenomenological data on phaseshift vs. energy.
0.0 0.2 0.4 0.6 0.8 1.0Ea (Lattice Units, a^(-1)=1.015 GeV)
0
25
50
75
100
125
150
175 (d
egre
es)
Luscher Method, Schenk Phenomenology, Isospin=0SchenkLuscher
Figure: Phenomenology* v. Luscher, predictions for I = 0 on 24c, a−1 = 1.015 GeV. The ultimate goal is to obtainenough phase shift points to fit to a Breit-Wigner form and extract the mass and resonance width.
(*=phenomology data is outdated, useful for illustrative purposes only)
Dan Hoying (RBC/UKQCD) ππ scattering, physical quarks July 25, 2018 34 / 34