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


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)


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)


Elastic ππ Scattering Phase Shifts

Outline

1 Motivation


3 Spectra from GEVP






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


Spectra from GEVP

Outline

1 Motivation


3 Spectra from GEVP






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]


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.


Computational Details

Outline

1 Motivation


3 Spectra from GEVP







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)



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.


Momenta Combinatorics

Outline

1 Motivation


3 Spectra from GEVP







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.



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


3 Spectra from GEVP






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:


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)


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


Dispersive(0) Dispersive(1) Dispersive(2) Energy(0) Energy(1) Energy(2)


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


Dispersive(0) Dispersive(1) Dispersive(2) Dispersive(3) Energy(0) Energy(1) Energy(2) Energy(3)


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)


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


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)


Summary and Outlook

Outline

1 Motivation


3 Spectra from GEVP






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


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


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].


https://doi.org/10.1016/0550-3213(91)90366-6

https://doi.org/10.1088/1126-6708/2009/04/094

http://arxiv.org/abs/0902.1265

https://doi.org/10.1007/JHEP01(2012)140

http://arxiv.org/abs/1108.3774

https://doi.org/10.1016/j.cpc.2005.06.008

http://arxiv.org/abs/hep-lat/0505023

http://arxiv.org/abs/hep-lat/0505023

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]


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〉:


Summary and Outlook

Isospin (cont’d)

〈I = 1|I = 1〉:


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


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)


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.


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)


Pion-pion Scattering with Physical Quark Masses (from ...€¦ · valued quantities taken modulo...

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