Introduction to Lattice QCD in Hadronic Physics

Home , Lattice QCD

Giannis Koutsou

Computational-based Science and Technology Research Center (CaSToRC), The Cyprus Institute

EINN, 28th October 2013, Paphos

CaSToRC

...... Outline

■ Introduction to Lattice QCD − Discretization - fermions and gluons − Simulation - the path integral formulation

■ Hadrons on the Lattice − Hadron spectrum on the lattice − Hadron matrix elements on the lattice

■ Techniques for Hadron matrix elements − Systematic eﬀects − Noise reduction techniques

CaSToRC ...... Lattice Quantum Chromodynamics

■ Numerical solution of QCD − Ab initio simulation of QCD, i.e. of the underlying fundamental theory − Well-established method for non-purterbative region of QCD − Access non-purterbative phenomena (conﬁnement, quark-gluon plasma)

Quark Conﬁnement Quark Gluon Plasma CaSToRC ...... Discretization

■ Lattice QCD allows direct simulation, from the QCD Lagrangian − The QCD Lagrangian: 1 L = ψ¯(iD/ − m)ψ − Fa Fµν, QCD 4 µν a where: a a − a − b c Fµν = ∂µGν ∂νGµ gfabcGµGν − Deﬁne link variables ∫ C c −ig dxG (x)Tc Uµ(x1; x0) ≡ e µ

CaSToRC ...... Discretization

■ Discretizing the gluon action − 4D space-time lattice, with lattice spacing a − Gluon link variables "live" between sites

c −iagG (xn)Tc Uµ(xn) ≡ e µ

− Simplest choice for gluon action: ∫ { } 1 ∑ ∑ 1 S [U] = − d4xFc Fµν → β ℜ Tr [1 − P (x )] G 4 µν c 3 µν n n µ<ν

† † Plaquette: Pµν(xn) = Uµ(xn)Uν(xn +µ ˆ)Uµ(xn +ν ˆ)Uν(xn) CaSToRC ...... Discretization

■ Discretizing the fermion action − Fermion variables "live" on sites − Various ways to deﬁne the fermion action on the lattice − All choices recover the continuum fermion action as a → 0

■ Naive:

∑ 4 1 µ S = a γ [ψ¯(x)Uµ(x)ψ(x + aµˆ) − ψ¯(x)U−µ(x)ψ(x − aµˆ)] + mψ¯(x)ψ(x) F 2 x,µ a

Introduces "fermion doublers"

CaSToRC ...... Discretization

■ Wilson fermions: − Add a Laplacian term which "lifts" the doubler's mass as a → 0 ∑ 4 1 S = − a [ψ¯(x)Uµ(x)(r − γµ)ψ(x + aµˆ) + ψ¯(x)(r + γµ)U−µ(x)ψ(x − aµˆ)] F 2 x,µ a ∑ 4 + a4 (m + r)ψ¯(x)ψ(x) Explicitly breaks chiral symmetry x a

− Additive renormalization to ̸ quark mass (amcrit = 0) − Local fermion action → nearest-neighbor coupling

CaSToRC ...... Discretization

■ O(a)-improvement for Wilson fermions − In today's calculations Wilson fermions are combined with a prescription for removing O(a) discretiztion eﬀects ▶ Twisted Mass [ ] u √1 Change of variables: ψ = to χ = (1 − iγ5τ3)ψ d 2 Tune bare mass to critical value and exchange for twisted-mass µ: (mcrit + iγ5µτ3)¯χχ ▶ Clover fermions ¯ µν Adds an irrelevant: cSWψσµν F ψ term Tune cSW for O(a)-improvement − Recent simulations exist which combine both Clover improvement and twisted mass

CaSToRC ...... Discretization

■ Chiral Symmetry is tricky − No-go theorem (Nielsen and Ninomiya) − On the lattice, need to either ▶ Break locality of operator ▶ Introduce doublers

■ Actions which preserve Chiral symmetry − Staggered (allows for doublers) − Domain Wall (introduce 5th dimension) − Overlap (non-local operator)

CaSToRC ...... Simulation

■ Lattice QCD simulations via the Feynmann path integral formulation: ∫ ¯ D[U]D[ψ¯]D[ψ]O(U, ψ, ψ¯)eiS[U,ψ,ψ] ⟨O⟩ = ∫ D[U]D[ψ¯]D[ψ]eiS[U,ψ,ψ¯] ψ, ψ¯: Grassmann variables, no way to represent on computers ⇒ integrate over the fermion ﬁelds analytically:

∫ ( ) −1 iS[U] ( ) D[U]O M , Uµ(z) det(M)e ⟨O ψ(x), ψ¯(y), Uµ(z) ⟩ = ∫ D[U] det(M)eiS[U]

where: M = iD/ − m CaSToRC ...... Simulation

■ Wick Rotation to imaginary time: ∫ 1 − ⟨O⟩ = D[U]Oe SG[U]+ln(det(M[U])) Z

■ Introduce pseudo-fermion ﬁelds for the determinant calculation (e.g. for two degenerate ﬂavors) ∫ 1 − − ¯ + −1 ⟨O⟩ = D[U]D[ϕ¯]D[ϕ]Oe SG[U] ϕ(M M) ϕ Z

■ Compute quark correlation functions via Wick's theorem − − ⟨ψ ψ ...ψ¯ ψ¯ ...⟩ = ⟨(−1)pM 1 M 1 ... + all permutations⟩ j1 j2 i1 i2 U i1j1 i2j2 U

p: number of permutations CaSToRC ...... Simulation

■ Monte Carlo simulation ∫ 1 − − ¯ † −1 ⟨O⟩ = D[U, ϕ,¯ ϕ]Oe SG[U] ϕ(M M) ϕ Z − Generate representative conﬁgurations of the gauge ﬁeld U ¯ † −1 with weight −SG[U] − ϕ(M M) ϕ − − O depends on M 1[U] − ϕ¯ and ϕ integrated out ■ Boils down to Mx = b † − − Need ϕ¯(M M) 1ϕ for update − − Need M 1 for observables − Repeated applications of M − Iterative solvers employed for inversions: CG, GMRES, BiCG, etc. CaSToRC ...... Simulation

■ Typical parameter space requirements − Need at least 3 lattice spacings to take a → 0 limit − Need several mq to take mq → 0 limit − Need at least two volumes sizes to check for finite volume effects − Need O(1000) configurations per ensemble ■ However, simulations with quark masses at the physical point have become possible

CaSToRC ...... Computational requirements

■ Typical problem sizes − Box length L ∼5 fm − Lattice spacing a ∼0.1 fm ■ Typical lattice size > 483×96 = O(107) sites − Link variables are color matrices Uµ(x): 3×3 complex matrix, 1 per 4 directions ⇒ 36 complex numbers per grid-point − − Resulting quark propagators are color-spinors M 1b: 3×4 complex vector ⇒ 12 complex numbers per grid-point Data-sets of typically several GBytes per gauge-field configuration ■ Cost of 1000 gauge configurations at physical quark mass is currently of O(10) TFlop×year

CaSToRC ...... Computational requirements

■ One of the most computationally demanding problems on today's supercomputers − 4D grid means problem scales with L4 where L the box-length − Depending on the quantity under investigation, the required statistics vary from O(103) to O(106) ■ However it is also an extremely "computer friendly" problem − Regular, nearest-neighbor communications allow for perfect scaling − Simple, deterministic compute kernels mean code optimization is more eﬀective ■ Advances in algorithms and numerical techniques, have opened the way for precision lattice QCD calculations of hadronic matrix elements CaSToRC ...... − Multi-nucleon physics require escallops sustained

Computational requirements

− Precision single-hadron measurements require petaﬂops sustained resources

CaSToRC ...... Computational requirements

− Precision single-hadron measurements require petaﬂops sustained resources − Multi-nucleon physics require escallops sustained

CaSToRC ...... Hadron spectrum

■ Hadron masses are computed by calculating an appropriate two-point correlation function − An interpolating operator with the quantum numbers of the required hadronic state serves as the trial state, e.g.: + ▶ π ¯ A pion interpolating operator: χ (x) = d(x)γ5u(x) ▶ p abc a b c A proton interpolating operator: χσ(x) = ϵ uσ(x)[u (x)Cγ5d (x)] − The hadronic two-point correlation function is expressed in terms of quark propagators after the appropriate Wick contractions, e.g. for the π+: ⟨ π+ π+ ⟩ ⟨ | | ⟩ χ (x)¯χ (0) = Ω γ5Gu(x; 0)γ5Gd(0; x) Ω

where Gu (Gd) is the up- (down-) quark propagator:

ab a b Gµν(y; x) = qµ(y)q¯ν(x), q = u, d CaSToRC ...... Hadron spectrum

■ Hadron masses are computed by calculating an appropriate two-point correlation function − The quark propagator is computed as the inverse of the descretized Dirac matrix ▶ It is computationally unfeasible to compute the complete inverse (from all elements to all elements) due to computational cost as well as storage requirements − Rather, the propagator multiplying an appropriately deﬁned spinor is computed: ∑ −1 ′ ′ G(y; x) = x′ M (y; x )bx(x ), where b the source vector ▶ ′ ′ E.g. when bx(x ) = δ(x − x ) (a point source) then G is a column of the propagator (a point-to-all lattice-sites propagator)

− The lattice Dirac matrix obeys γ5-hermiticity allowing the reversal of the propagator coordinates without extra inversions, e.g.: ▶ † For Wilson: Gq(y; x) = γ5Gq(x; y)γ5 ▶ † For Twisted Mass: Gu(y; x) = γ5Gd(x; y)γ5

CaSToRC ...... Hadron spectrum

■ Hadron masses are computed by calculating an appropriate two-point correlation function

− At the hadronic level, the two-point correlation function is expressed as a sum of exponentials over all states that overlap with the interpolating operator: ∑ + + + −ˆ ⃗ˆ⃗ + + − ⃗ ⃗⃗ ⟨χπ (x)¯χπ (0)⟩ = ⟨Ω|χπ e Hteipxχ¯π |Ω⟩ = |⟨Ω|χπ |⃗p, n⟩|2e En(p)teipx ⃗p,n

after inserting a complete set of states ∑ ˆ 1 = |⃗p, n⟩⟨⃗p, n|, H|⃗p, n⟩ = En(⃗p)|⃗p, n⟩ ⃗p,n CaSToRC ...... Hadron spectrum

■ Hadron masses are computed by calculating an appropriate two-point correlation function

− At large time separations, the smallest energy exponential dominates, such that the ground state mass can be extracted ∑ + + ≫ + − ⟨χπ (x)¯χπ (0)⟩ −−→t |⟨Ω|χπ |π+(⃗p)⟩|2e E(⃗p)tei⃗p⃗x ⃗p

− Project to any momentum via a discreet Fourier Transform ∑ − + + ≫ + − e i⃗q⃗x⟨χπ (x)¯χπ (0)⟩ −−→|⟨t Ω|χπ |π+(⃗q)⟩|2e E(⃗q)tei⃗q⃗x ⃗x

CaSToRC ...... Hadron spectrum

■ Nucleon and pion two-point correlation functions 101 0.6 ± 0 10 + 0 0.5 p /n 10 1 0.4 10 2 ) t ( ) f f

t 3 e ( 10 0.3 C m 10 4 a 0.2 10 5

± 0.1 10 6 p + /n0 10 7 0.0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 ts /a ts /a − Signal drops exponentially with mass while statistical errors constant − Heavier hadrons get "lost in noise" at smaller time-separations − 1 C(t−1) Eﬀective mass: ameff(t) = 2 log[ C(t+1) ] becomes constant when ground state dominates (plateau region)

CaSToRC ...... Hadron spectrum

■ Setting the scale − Obtaining masses in physical units requires setting the scale, i.e. determining the lattice spacing − This can only be done after calculating an observable on the lattice of which the physical value is known − Examples of quantities typically used are the pion decay constant fπ and the nucleon mass mN

1.4

1.3 − From eight ensembles of Twisted 1.2 Mass gauge-ﬁeld conﬁgurations 1.1 β=1.90, L/a=32, L=3.0fm − (GeV) β=1.90, L/a=24, L=2.2fm Four lattice spacings are let to vary N β=1.90, L/a=20, L=1.9fm m 1 β=1.95, L/a=32, L=2.6fm β=1.95, L/a=24, L=2.0fm such that the curve passes through 0.9 β=2.10, L/a=48, L=3.1fm β=2.10, L/a=32, L=2.1fm the physical point β=2.10, CSW=1.57551, L/a=48, L=4.5fm 0.8 0 0.05 0.1 0.15 0.2 0.25 − Curve is drawn from HBχPT 2 2 mπ (GeV )

CaSToRC ...... Hadron spectrum

■ What about excited states? − As opposed to low-lying states, calculation of excited states remains a challenge in lattice QCD − This is usually done using a variational method where diﬀerent interpolating ﬁelds are use as a basis − A Generalized Eigenvalue Problem (GEVP) is then solved:

C(t)vn(t, t0) = λn(t, t0)C(t0)vn(t, t0), n = 1,..., N, t > t0

where C is now an N × N matrix of two-point correlation functions − The eﬀective energy of the nth state is given by the nth eigenvalue λn: En = limt→∞ −∂t log λn(t, t0)

CaSToRC ...... Hadron spectrum

■ What about excited states? − A basis of initial/ﬁnal states can also be built-up by smearing the source vectors − Gauge-invariant Gaussian smearing of the quark sources qsm = (1 + αH)ns q with:

∑3 H(⃗x,⃗y; U) = [Ui(⃗x, t)δ⃗x,⃗y−aˆı + U−i(⃗x, t)δ⃗x,⃗y+aˆı] i=1

allows for the creation of Gaussian sources with diﬀerent widths, altering the overlap of the trial state with the physical state

CaSToRC ...... Hadron spectrum

■ What about excited states? − An example for the nucleon

n=0 (3x3) n=1 (3x3) n=2 (3x3) n eff 1 aE

0 2 4 6 8 10 12 14 t/a − Increased statistical noise for higher states − Identiﬁcation of multi-particle states non-trivial ▶ Multiple volumes are required to identify between resonance states and mutli-particle states CaSToRC ...... Hadron matrix elements

■ Hadron matrix elements require the calculation of an appropriate three-point function

′ ⟨H|J|H ⟩ → ⟨χH(x2)|J(x1)|χ¯H′ (0)⟩

■ Access to − Charge radii, magnetic moments, axial charges − Form factors − Parton properties (momentum fraction, spin fraction) − Access to new physics (coupling to scalars) − Decay widths

CaSToRC ...... Hadron matrix elements

− Of interest is the matrix element at a given momentum transfer − The sink and insertion positions are Fourier transformed

∑ ′ ′ − ′ ⟨ ⃗ | ⃗ | ⃗ ⟩ ∝ ⟨ α | | β ⟩ i⃗x1⃗q i⃗x2⃗p H (p ) J(q) H(p) Γβα χH(x2) J(x1) χ¯H′ (0) e e ⃗x1,⃗x2

with ⃗p′ the momentum of the ﬁnal state, ⃗q the momentum of the current insertion and ⃗p = ⃗p′ − ⃗q the initial state momentum from momentum conservation CaSToRC ...... Hadron matrix elements

− Expanding in terms of quark propagators:

∑ [ ] J J Tr G(x2; x1)O G(x1; 0)[...G(x2; 0)...] , with J(x) = q¯(x)O q(x)

⃗x1,⃗x2

− G(x2; x1) is being summed on both ends ⇒ requires knowledge of the all-to-all propagator

CaSToRC ...... Hadron matrix elements

■ Sequential (or extended) propagator techniques − Observe that during an inversion, the propagator is being summed over one coordinate ∑ −1 ′ ′ G(y; x) = M (y; x )bx(x ) x′ − OJ i⃗x⃗q By deﬁning an appropriate source: b⃗q,0(⃗x, t) = G(x; 0)e δt,t1 with support only on t1, the inversion yields the quark-line with the insertion current in: ∑ ∑ OJ ≡ −1 OJ F( ;⃗q; x2; 0) M (x2; x)b⃗q,0(x) = G(x2; x1) G(x1; 0)

x ⃗x1

− So-called sequential inversion through insertion or ﬁxed insertion method CaSToRC ...... Hadron matrix elements

■ Fixed insertion ■ Fixed sink

− Fix insertion current − Fix final state momentum − Fix final state time − Fix insertion current time − Free insertion current − Free final state momentum − Free final state time − Free insertion current time

■ Choice depends on the problem at hand

CaSToRC ...... Hadron matrix elements

■ Hadronic level

− After inserting two sets of energy/momentum states:

∑ ′ α β i⃗x1⃗q −i⃗x2⃗p ⟨χ ( )| ( )|χ¯ ′ (0)⟩ = H x2 J x1 H e e ⃗x1,⃗x2 ∑ ′ ′ − − ′ − ′ ′ ⟨ | ⟩⟨ | ⟩ (t2 t1)E ′ (⃗p ) t1En(⃗p)⟨ | | ⟩ χH ⃗p , n ⃗p, n χ¯H′ e n e ⃗p , n J ⃗p, n n,n′

− Need to take large time separations for ground state domination

′ t2−t1≫ ′ ′ − − ⃗ − ⃗ ′ ′ −−−−−→ ⟨ | ⟩⟨ | ⟩ (t2 t1)EH(p ) t1E ′ (p)⟨ | | ⟩ χH H(⃗p ) H (⃗p) χ¯H′ e e H H(⃗p ) J H (⃗p) t1≫ CaSToRC ...... Hadron matrix elements

■ Hadronic level

′ ′ ′ − − ⃗ − ′ ⃗ ′ ′ (t2 t1)EH(p ) t1EH (p) ⟨χH|H(⃗p )⟩⟨H (⃗p)|χ¯H′ ⟩e e ⟨H(⃗p )|J|H (⃗p)⟩ − Cancel unknown overlaps and exponentials need to isolate matrix element − Use two-point functions, optimally with as short separations as possible − Optimum ratio: √ ′ HJH ′ H′H′ HH ′ HH ′ C (⃗p,⃗p ; t2, t1) C (t2 − t1,⃗p)C (t1,⃗p )C (t2,⃗p ) HH ′ HH ′ H′H′ H′H′ C (t2,⃗p ) C (t2 − t1,⃗p )C (t1,⃗p)C (t,⃗p)

t −t ≫1 ′ ′ ′ −−−−−−→2 1 Π(⃗p,⃗p ; J) ∝ ⟨H(p )|J|H (p)⟩ t1≫1 Plateau region CaSToRC ...... Hadron matrix elements

■ Example: the momentum − Fixed sink method: new inversion for every ts fraction − Vary tins for plateau identiﬁcation 1.4 t s =10a − ≃ t s =12a a 0.087 fm, plateau around 1.2 t s =14a t s =16a 1.2 fm t =18a u ±d 1.0 s − Lattice matrix elements need to be 0.8 renormalized ) → x s

,t 0.6 ▶

ins Renormalization constants

R(t 0.4 typically evaluated in separate 0.2 calculations ▶ 0.0 Various methods for calculation 10 5 0 5 10 (tins ts/2)/a (perturbative, non-perturbative)

CaSToRC ...... Hadron matrix elements

■ Nucleon electromagnetic form factors

− Various sources for Electric and Magnetic form factors − Current calculations are running at physical mπ − Discreet values of momentum transfer − No direct calculation of magnetic FF at Q2 = 0

CaSToRC ...... Hadron matrix elements

■ Disconnected diagrams

2pt J C (x2)Tr[G(x1; x1)O ] − Fermion quark loops arise as contributions to three-point functions − Sum over diagonal of a all-to-all propagator − In certain cases they can be omitted based on symmetry ▶ E.g. in isovector current combinations: u¯OJu − d¯OJd they cancel assuming degenerate u- and d-quarks − In other cases they are the only contributions ▶ E.g. strange-quark content of a nucleon where no connected contributions arise

CaSToRC ...... Hadron matrix elements

■ Disconnected diagrams

2pt J C (x2)Tr[G(x1; x1)O ] − Crucial for evaluating key hadronic quantities, e.g.: ▶ Individual quark spin contributions to nucleon spin ▶ Individual quark angular momentum contributions to nucleon spin ▶ Individual proton and neutron form factors − Very challenging to evaluate; have only lately started to be included in lattice QCD calculations

CaSToRC ...... Hadron matrix elements

■ Disconnected diagrams

2pt J C (x2)Tr[G(x1; x1)O ]

■ Hopping parameter expansion − Another way is to expand the inverse around the inverse quark mass

− M ∝ (1 − κH) ⇒ M 1 ∝ 1 + κH + κ2H2 + ...

where κ is a proxy for the bare quark mass κ = 1/(8 + 2am) − In the sum over diagonal elements only even powers of the hopping term contribute − Usually used in combination with stochastic method CaSToRC ...... Hadron matrix elements

■ Disconnected diagrams

2pt J C (x2)Tr[G(x1; x1)O ]

■ Stochastic evaluation − a Deﬁne a set of Nr source vectors ηµ(x)r which obey: √ √ ⟨ a b ⟩ − O ⟨ a ⟩ O ηµ(x)ην (y) r = δabδµν δ(x y)+ (1/ Nr) and ηµ(x) r = (1/ Nr)

− A stochastic estimate of the all to all can be evaluated via: ∑ 1 ′ ′ ab ⟨ a b† ⟩ O a −1 ab ′ b ′ Gµν (y; x) = ϕµ(y)ην (x) r+ ( ) with ϕµ(y) = (M )µν′ (y; x )ην′ (x ) Nr ′ x

CaSToRC ...... Hadron matrix elements

■ Disconnected diagrams

2pt J C (x2)Tr[G(x1; x1)O ]

■ Stochastic evaluation − Notoriously diﬃcult to evaluate due to increased computational cost

− Nr strongly depends on matrix element under study − Furthermore, statistical errors are usually large for disconnected quark loops − Need at least O(104) measurements as opposed to O(103) for connected

CaSToRC ...... Hadron matrix elements

■ An example: the nucleon axial charge gA (isoscalar)

− Connected diagram − Disconnected diagram

0.8 0.05

0.7 0.00 0.6 [MeV]

u +d A 0.5 0.05 u +d A

) → g 0.4 s ,t ) → g 0.10 s ins

0.3 ,t

ins ts =8a R(t

0.2 ts =10a

R(t 0.15 ts =10a ts =16a ts =12a ts =12a ts =18a 0.1 ts =14a ts =14a 0.20 0.0 8 6 4 2 0 2 4 6 8 10 5 0 5 10 (tins ts /2)/a (tins ts/2)/a − 1,200 statistics − 150,000 statistics − ∼3% relative error − ∼13% relative error CaSToRC ...... Hadron matrix elements

■ An example: quark intrinsic spin contributions to the nucleon

0.4 1 u 2ΔΣ

0.2

0 1 d 2ΔΣ

-0.2 Contribution to nucleon spin

0 0.05 0.1 0.15 0.2 0.25 2 2 mπ(GeV ) − Pion mass dependence of quark contributions to nucleon spin − Linear combinations of isovector and isoscalar − Disconnected fermion loops included in one measurement

CaSToRC ...... Advanced methods for hadron matrix elements

CaSToRC ...... Deﬂating the lattice Dirac Matrix

■ It is common to require multiple inversions for the same gauge-field configuration − Propagators are computed as point-to-all − Better use of lattice gauge configurations can be used by computing multiple two- or three-point functions from different positions − Knowledge of the eigenvalues of the Dirac matrix can accelerate propagator calculations on the same configuration − These techniques are known as deflation

CaSToRC ...... Deﬂating the lattice Dirac Matrix

■ Approximate deﬂation − Iterative methods for the solution of Mx = b require an initial guess x0 and iterate for x such that the residual r = Mx − b reaches the desired threshold − The number of iterations is a power of the ratio of largest to smallest eigenvalue − With knowledge of approximate eigenvectors of the smallest eigenvalues of M, form a guess for the initial vector x0 = Vd ▶ V = (v1, v2, ...vn) the matrix of eigenvectors ▶ d = (V†MV)−1V†b − Projects-out the smallest eigenvalues, reducing the number of iterations required to solution

CaSToRC ...... Deﬂating the lattice Dirac Matrix

■ Incremental deflation − Incrementally construct V during the solver iteration − Most efficient when large number of inversions per configuration are required

25 5

20 4

15 3 ×1000 T [sec]

10 iters 2 N

5 1 CG Eig-CG 0 0 0 10 20 30 40 50 0 10 20 30 40 50 Nrhs Nrhs − Improvement of ×3 in computer time CaSToRC ...... Summation method

■ In matrix element calculations, it is important to ensure excited state effects are under control − In either fixed-sink or fixed insertion methods, one source-sink separation is fixed − For multiple source-sink separations with the fixed-sink method, all data can be combined via the summation method

CaSToRC ...... Summation method

■ Excited state eﬀects in three-point functions − After optimum ratio has been taken

′ −∆(⃗p)t1 −∆(⃗p )(t2−t1) R(t2; t1) = M[1 + e + e + ...]

▶ M ground state matrix element of interest ▶ ∆(⃗p) = E1(⃗p) − E0(⃗p) − Sum over insertion ∑ ′ −∆(⃗p)t2 −∆(⃗p )t2 R(t2; t1) = Const. + M[t2 + O(e ) + O(e ) + ...]

− Elimination of powers of t1 ⇒ larger suppression of excited states CaSToRC ...... Summation method

■ Example of matrix element with large excited state contamination: Nucleon σ-term (connected) 3.0

0.20 2.5

0.15 2.0 ) s n i t (

[GeV] 1.5 R s N

0.10 n i t X

1.0

0.05 0.5 Summation ts =14a

ts =10a ts =16a

ts =12a ts =18a 0.00 0.0 10 5 0 5 10 4 6 8 10 12 14 16 18 (tins ts/2)/a ts/a − Check for agreement between summation method and plateau − Calculation of multiple source-sink separations can be accelerated with deﬂation CaSToRC ...... Mixed precision techniques

■ Accuracy to which propagator is required is dependant on matrix element − Define an approximate correlation function which fluctuates closely with exact r = Corr[Cappx(t), C(t)] − Then define transformations g on correlation function which are statistically equivalent, e.g. changing the position of the source − Approximation can be relaxed solver precision − Improved estimator: 1 ∑ Cimp(t) = C(t) − Cappx(t) + Cappx(t) N g g g √ − Error scales with N : 2(1 − r) + 1 assuming r ≃ 1 g Ng CaSToRC ...... Mixed precision techniques

■ Covariant approximation averaging − Need a small number O(1) high-precision (HP) correlators − Need to choose low-precision (LP) such that ▶ LP correlators are precise enough to correlate strongly with HP (r ≃ 1) ▶ LP correlator precision is relaxed enough to be much cheaper than HP

▶ Nucleon two-point correlation function ▶ Correlation between LP and HP largest at smaller time-separations ▶ Can use deflation to speed up further inversion of consecutive LP propagators on same gauge-field configuration

CaSToRC ...... Mixed precision techniques

■ Covariant approximation averaging

0.36 16 2 EigCG, |r| <1e 02 2 14 EigCG, |r| <1e 03 2 EigCG, |r| <1e 04 0.35 2 12 EigCG, |r| <1e 05 2 EigCG, |r| <1e 06 ]

3 10 EigCG, HP 0 0.34

1 CG, HP × N [

m s 8 a h r / s

r 0.33 e t

i 6 N

4 0.32 Full ensemble (from arXiv:1303.5797) 2 Nhp =1

Nhp =12 0 0.31 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 rhs Nlp

− Test for statistical error on nucleon eﬀective mass − LP more than ×10 faster√ to compute when also using deﬂation − Error scales with 1/ NLP

CaSToRC ...... Conclusions and summary

■ Low-lying hadron masses at the percent level

Baryon Spectrum 1.8 Ω Ξ* − Milestone calculation in lattice Σ* Ξ Δ Σ QCD Λ − N Conﬁrmation of hadron masses M (GeV) from (lattice) QCD

K Exp. results BMW collaboration, Science 322 (2008) Width Input ETMC results BMW results ETM collaboration Phys. Rev. D78 (2008) 0.0 0.3 0.6 0.9 1.2 1.5 0 1 2 3 Strangeness

CaSToRC ...... Conclusions and summary

■ Hadron matrix elements: new era for lattice QCD 2.0 0.30

0.25 1.5 0.20 d u A 1.0 ®

g 0.15 x 0.10 DWF, Nf =2 +1 TMF, Nf =2 +1 +1

0.5 Clover (QCDSF), Nf =2 Clover (CLS), Nf =2 +1 Clover (QCDSF), Nf =2 TMF, Nf =2 +1 +1 HISQ TMF, Nf =2 0.05 DWF, Nf =2 +1 TMF, Nf =2 Hybrid (DWF/Stag.), Nf =2 +1 TMF w/ Clover, Nf =2 Hybrid (DWF/Stag.), Nf =2 +1 Clover (MIT/BMW), Nf =2 +1

Clover (MIT/BMW), Nf =2 +1 TMF w/ Clover, Nf =2 0.0 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 m2 [GeV2] m2 [GeV2] − New simulations at the physical quark mass − New methods for eﬃcient increase of statistical accuracy − Techniques for assessing excited state eﬀects and disconnected diagrams CaSToRC ...... Thank you, questions?

CaSToRC ......