Frontiers of Lattice Gauge Theory Simon Catterall Syracuse University

Frontiers of Lattice Gauge Theory – p. 1

Talk What is lattice gauge theory LGT? Successes to date New problems - Physics Beyond Standard Model BSM New ideas/developments - theoretical, computational, experimental Key idea: New theories, better algorithms, much faster hardware – LGT can play a important role in pushing forward boundaries of elementary particle physics

Frontiers of Lattice Gauge Theory – p. 2

Particle Physics Basic picture: matter described by fermion fields Ψ(x, t) (quarks, electrons, ...) Forces carried by gauge boson fields eg Aµ (x, t) of EM (photon, gluons - strong nuclear force QCD)

Frontiers of Lattice Gauge Theory – p. 3

QCD is hard

g >> 1 - perturbative expansion breaks down – many complex graphs ... Try to solve problem by discretization on grid: lattice in spacetime

Frontiers of Lattice Gauge Theory – p. 4

What is a LGT ? 1. Recipe: Quantum amplitude for some process: Z − ~1 S(Aµ ,Ψ) A ∼ DAµ DΨe O(Aµ , Ψ) Continuum ill-defined.. Lattice - (large) multiple integral Looks like thermal system with T → ~ (also need t → iτ ) Discrete model Aµ (x) → Uµ (x) = eiAµ (x) Wilson .. QCD: quarks come in 3 colors Uµ is 3 × 3 unitary matrix.

Frontiers of Lattice Gauge Theory – p. 5

What is a LGT ? 2.

Matter Ψ(x) on sites.

Force Uµ on links

  1 X Tr U1 (x)U2 (x + 1)U1† (x + 2)U2† (x) S(Uµ ) = 2 g squares SF (Uµ , Ψ) =

X

¯ ¯ Ψ(x)U 1 (x)Ψ(x + 1) + . . . = ΨM Ψ

sites

Gauge symmetry Uµ (x) → G(x)Uµ (x)G† (x + µ); Ψ → G(x)Ψ Frontiers of Lattice Gauge Theory – p. 6

What is LGT ? 3. Lattice gauge theories look like 4D stat mech lattice models with local interactions Well known Monte Carlo algorithms exist to generate P (Uµ ) = e−S(U ) But ... Pauli. Represent by anticommuting fields. Computer...? Integrate out – gaussian -> detM (U ) Matrix Mx,x′ is large and sparse. S = S(U ) + Tr ln (M )

Frontiers of Lattice Gauge Theory – p. 7

Why fermions are hard .. Actions now non-local Traditional algorithms eg. Metropolis impossibly slow –> quenched approx Last 5+ years new algorithm - Rational Hybrid Monte Carlo RHMC Exact, efficient. Any number of quarks Still hard (eg. QCD simulations with 324 lattices and light dynamical fermions still take O(1) year on 1 Tflop machine !).

Frontiers of Lattice Gauge Theory – p. 8

First steps .. get rid of fermions Replace fermions by pseudofermions F Z † −1 ¯ ¯ 2F − F (M M ) det(M (U )) = DF DF¯ e Fractional power ? Partial fractions x

− pq

∼ α0 +

N X i=1

αi M † M + βi

Optimal {αi , βi } determined offline by Remez alg. Minimises fractional error over some interval ǫ → 1. N = O(10) and ǫ = 10−7 yields error = 10−8 .

Frontiers of Lattice Gauge Theory – p. 9

RHMC Ex. simple system: q(t) with action S(q) Step 1. Introduce momentum p and hamiltonian H = 21 p2 + S(q) Step 2. Evolve system in auxiliary time τ using leapfrog δτ δp = − 2 δq = δτ p δτ δp = − 2

δH δq δH δq

Step 3. Accept/reject entire classical trajectory using Metropolis test e−δH Restart with new p drawn from gaussian distribution. Frontiers of Lattice Gauge Theory – p. 10

Requirements Requirements: leapfrog is symplectic and reversible -> detailed balance. Refresh trajectory -> ergodic Classical dynamics used to propose global move on fields with high probability of acceptance For non-local pseudofermion action force given by solving sparse linear system CP U ∼ V α to update V dof ! best case α ∼ 1

Frontiers of Lattice Gauge Theory – p. 11

For dynamical fermions ... N

X δ(M † M ) δS αi χi =− χi δU δU i=1

where (M † M + βi )χi = F

Remarkably, using multimass CG-solver can iteratively solve all N large, sparse linear equations for price of one! δS Technical remarks: δU covariant derivative Don’t hold full M (U ) in store – just non-zeroes (sparse)

Frontiers of Lattice Gauge Theory – p. 12

Lattice QCD (1980–) Successes: QCD confines quarks Vq¯q ∼ σr Spectrum hadrons Decays, matrix elements, heavy quarks

Frontiers of Lattice Gauge Theory – p. 13

Future ? Precision studies of QCD – matrix elements needed to test Standard Model Nuclear physics via QCD – astrophysics, RHIC, ... Lattice chiral gauge theories eg. weak nuclear force (W,Z bosons) Search for new strongly interacting theories – what is origin of mass ? Is there really a Higgs ? eg technicolor Supersymmetry – Is Nature supersymmetric, how is supersymmetry broken, String theory ? ...

Frontiers of Lattice Gauge Theory – p. 14

Large Hadron Collider LHC

27 km, ECM =14TeV

15m high, 40TB/s->100

Frontiers of Lattice Gauge Theory – p. 15

Hardware

1992,Cray YMP 1Gflop,10M$->104 $/Mflop

2005,PC cluster 3Tflop,1M$->1$/Mflop

2008,Video card 1Tflop,10−3 M$->0.1c/Mflop Frontiers of Lattice Gauge Theory – p. 16

LHC – search for Higgs Vacuum is not empty – contains a non-zero Higgs field H Mass of particle: interactions with H

Main goal LHC: find Higgs boson ...

Frontiers of Lattice Gauge Theory – p. 17

Problems with SM Higgs Hard to arrange for non-zero vacuum Higgs Natural mass of Higgs in SM very large Solutions: Supersymmetry. Extra dimensions. Composite models – eg technicolor ..?

Frontiers of Lattice Gauge Theory – p. 18

Technicolor Assume scalar Higgs is condensate of fermions formed by new strong techniforce c.f Landau-Ginzburg theory of superconductivity H ∼< q¯q > Attractive and conservative scenario – BUT precision data from LEP rules it out ... If new force like QCD It may not be ! One attractive possibility: theory should lie close to conformal fixed point in theory space

Frontiers of Lattice Gauge Theory – p. 19

What is it and where is it CFT: Scale invariant, massless, field theory whose long distance dynamics independent of bare coupling ... For technicolor model: need theory that is almost conformal .... Need g(E) walk in ΛEW < E < ΛET C Known supersymmetric examples. But analytic calcs fail in non-SUSY case. Can we use lattice to search for such theories? Vary Nf , Ncolors , symmetry transformation

Frontiers of Lattice Gauge Theory – p. 20

One example 2 techniquarks with 2 colors transforming in adjoint repn of symmetry group (Ψ → GΨG† ) Near conformality suggested by approx calcs (Nfc ∼ 2) Simulation 83 × 16 lattice using Wilson fermions. 2D parameter space (β = g42 , ma) – 100+ points (70 Gflop sustained over 3 months on BG/L) Phases ? Continuum limit requires critical line/pts mq → 0 ? Gluon energy, meson masses, ...

Frontiers of Lattice Gauge Theory – p. 21

Mean gluon action 0.9 beta=1.50 beta=1.75 beta=1.90 beta=1.95 beta=2.00 beta=2.05 beta=2.10 beta=2.25 beta=2.35 beta=2.40 beta=2.50 beta=2.75 beta=3.00

0.8

Plaquette

0.7

0.6

0.5

0.4

0.3 -2

-1.5

-1

ma

-0.5

0

0.5

Discontinuity at small β : 1st order transition.

Frontiers of Lattice Gauge Theory – p. 22

Latent Heat 0.25

Latent Heat

0.2

0.15

0.1

0.05

0 1.5

1.6

1.7

1.8 beta

1.9

2

2.1

Critical end point β = βc ∼ 2.0 ?

Frontiers of Lattice Gauge Theory – p. 23

Technipion mass squared

Two regimes: β < βc : Goldstone behavior. Coincides with jump in gluon action β > βc : Restoration of chiral symmetry (?). Gluon action smooth. Frontiers of Lattice Gauge Theory – p. 24

Masses along “critical” line 2

pi, rho masses along the critical line

pi rho 1.5

1

0.5

0

1

1.5

2 beta

2.5

3

Notice: masses degenerate for β > βc All data consistent with theory in conformal phase for β > βc ∼ 2.0 Frontiers of Lattice Gauge Theory – p. 25

Caveats Continuum limit requires careful tuning of g with a to avoid finite size effects If running g(a) slow expect extreme sensitivity in inverse a = a(g). Modest decreases in g correspond to huge decreases in lattice spacing and large potential finite size effects. Eg. Physical box size so small system deconfines and looks quasi-free. Alternatively: confinement scale Λa will be small and require large lattices to see long distance behavior. Need large lattices to resolve between these scenarios

Frontiers of Lattice Gauge Theory – p. 26

Possible scenario Simplest picture for new CFP. οο

ma UVFP

0

O

IRFP

er 1st Ord O UVFP

− οο

0

O

β

οο

Frontiers of Lattice Gauge Theory – p. 27

Supersymmetry SUSY - why ? Scalar fields natural in SUSY theories String theory demands SUSY Dark matter ? Grand Unification

Frontiers of Lattice Gauge Theory – p. 28

Unknowns in SUSY Is Nature supersymmetric ? How is SUSY broken ? Spectrum of SUSY theories; LHC, dark matter cosmology AdSCFT: Gravity as a gauge theory ? 5D (super)gravity = gauge theory on 4D boundary (Maldacena 1998) Hard analytically .. .lattice ?

Frontiers of Lattice Gauge Theory – p. 29

Lattice and SUSY incompatible ? SUSY theories have new kind of symmetry: δ

boson SUSY → fermion

Unfortunately: 2 SUSYs give translation in spacetime broken by lattice What to do ? Recently, new lattice formulations: preserve subset of SUSY exactly D. B Kaplan, M. Unsal, A. Cohen, A. Katz SC, J Giedt, P. Damgaard, S. Matsuura, F. Sugino N. Kawamoto, A. d’Adda, ...

Frontiers of Lattice Gauge Theory – p. 30

SUSY lattices

Matter distributed over links in lattice Details of lattice action tightly constrained by: gauge invariance, SUSY and no fermion doubling 2 |field >= 0 Exact SUSY: δQ No. exact SUSYs = no. site fermions

Frontiers of Lattice Gauge Theory – p. 31

Simulations In D = 4 one solution to constraints – N = 4 SYM Developed parallel code to simulate using tools/algs LGT First thing: check SUSY:

< SB >

exact −SB

N = 4 in D = 2

L 2 3 4

δSB SB

0.014(1) 0.007(2) 0.006(2)

Q=16 D=4 SU(2) 2

g =0.5, L=2

800

600 Bosonic action Pseudofermion action 400

200

0

0

100

200

300

Monte Carlo time

400

500

N = 4 in D = 4

Frontiers of Lattice Gauge Theory – p. 32

Blackholes from gauge theory Conjectured duality between N = 4 SYM reduced to D = 1 and string theory in D = 5! At low T, gravity theory contains a black hole 20 SU(5) quenched SU(3) SU(5) SU(8) black hole

2

(1/N )E/T

15

10

5

0

0

2

4

6

8

10

12

T

Smoothly connected to stringy phase at high T ? Frontiers of Lattice Gauge Theory – p. 33

Conclusions/Outlook Special time in history LGT: LHC offers promise of new physics. LGT is now mature – Improvements in algs/hardware make exploratory calcs of non-QCD like theories possible Plenty of theoretical ideas: conformal gauge theories, supersymmetry, strings, ...) Lots to do!

Frontiers of Lattice Gauge Theory – p. 34

The End

Frontiers of Lattice Gauge Theory – p. 35

Code issues C++ code allows easy to debug programs. Map classes to basic mathematical objects eg.Gauge_Field is collection of complex matrices accessed using a lattice position vector; U.get(x,mu)

where x is object type Lattice_Vector Operator overloading for compact easily debugged code: G.set(x,mu,kappa*U1.get(x,mu)*U2.get(x,mu)) Optimize linear solver (99% CPU )

MC time series analyzed via binning/bootstrap techniques – Mean and error. Parallel code necessary. Frontiers of Lattice Gauge Theory – p. 36

Parallel code Utilize existing communication libraries where possible eg MDP (matrix distributed processing, part of FermiQCD). Decides how lattice mapped to physical processors. Hides MPI calls. Allows for simple port from scalar to parallel. mdp_lattice space(3,L); mdp_matrix_field phi(space,2,2); mdp_site x; ... forallsites(x) phi(x)=phi(x+1)+phi(x-1)-2*phi(x); phi.update() Frontiers of Lattice Gauge Theory – p. 37