Frontiers of Lattice Gauge Theory · Frontiers of Lattice Gauge Theory Simon Catterall Syracuse...

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 forwardboundaries of elementary particle physics


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)


QCD is hard

g >> 1 - perturbative expansion breaks down – manycomplex graphs ...

Try to solve problem by discretization on grid:lattice in spacetime


What is a LGT ? 1.

Recipe: Quantum amplitude for some process:

A ∼

∫

DAµDΨe−1

~S(Aµ,Ψ)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 colorsUµ is 3 × 3 unitary matrix.


What is a LGT ? 2.

Force Uµ on links Matter Ψ(x) on sites.

S(Uµ) =1

g2

∑

squares

Tr(

U1(x)U2(x + 1)U †1(x + 2)U †

2(x))

SF (Uµ, Ψ) =∑

sites

Ψ̄(x)U1(x)Ψ(x + 1) + . . . = Ψ̄MΨ

Gauge symmetry Uµ(x) → G(x)Uµ(x)G†(x + µ); Ψ → G(x)Ψ


What is LGT ? 3.

Lattice gauge theories look like 4D stat mech lattice modelswith local interactions

Well known Monte Carlo algorithms exist to generateP (Uµ) = e−S(U)

But ... Pauli. Represent by anticommuting fields.Computer...?

Integrate out – gaussian -> detM(U) Matrix Mx,x′ islarge and sparse.

S = S(U) + Tr ln (M)


Why fermions are hard ..

Actions now non-local

Traditional algorithms eg. Metropolis impossibly slow –>quenched approx

Last 5+ years new algorithm - Rational Hybrid MonteCarlo RHMC

Exact, efficient. Any number of quarks

Still hard (eg. QCD simulations with 324 lattices andlight dynamical fermions still take O(1) year on 1 Tflopmachine !).


First steps .. get rid of fermions

Replace fermions by pseudofermions F

det(M(U)) =

∫

DFDF̄ e−F̄ (M†M)−12 F̄

Fractional power ? Partial fractions

x− p

q ∼ α0 +N

∑

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.


RHMC

Ex. simple system: q(t) with action S(q)

Step 1. Introduce momentum p and hamiltonianH = 1

2p2 + S(q)

Step 2. Evolve system in auxiliary time τ using leapfrog

δp = −δτ

2

δH

δq

δq = δτp

δp = −δτ

2

δH

δq

Step 3. Accept/reject entire classical trajectory usingMetropolis 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 onfields with high probability of acceptance

For non-local pseudofermion action force given bysolving sparse linear system

CPU ∼ V α to update V dof ! best case α ∼ 1


For dynamical fermions ...

δS

δU= −

N∑

i=1

αiχiδ(M †M)

δUχi

where(M †M + βi)χi = F

Remarkably, using multimass CG-solver can iterativelysolve all N large, sparse linear equations for price of one!

Technical remarks: δSδU covariant derivative

Don’t hold full M(U) in store – just non-zeroes (sparse)


Lattice QCD

(1980–) Successes:

QCD confines quarks Vqq̄ ∼ σr

Spectrum hadrons

Decays, matrix elements, heavy quarks


Future ?

Precision studies of QCD – matrix elements needed totest 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 isorigin of mass ? Is there really a Higgs ? eg technicolor

Supersymmetry – Is Nature supersymmetric, how issupersymmetry broken, String theory ?

...


Large Hadron Collider LHC

27 km, ECM=14TeV 15m high, 40TB/s->100


Hardware

1992,Cray YMP1Gflop,10M$->104 $/Mflop

2005,PC cluster3Tflop,1M$->1$/Mflop

2008,Video card1Tflop,10−3M$->0.1c/Mflop


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 ...


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

..?


Technicolor

Assume scalar Higgs is condensate of fermions formedby new strong techniforce

c.f Landau-Ginzburg theory of superconductivityH ∼< q̄q >

Attractive and conservative scenario – BUT precisiondata from LEP rules it out ...

If new force like QCDIt may not be !

One attractive possibility: theory should lie close toconformal fixed point in theory space


What is it and where is it

CFT: Scale invariant, massless, field theory whose longdistance dynamics independent of bare coupling ...

For technicolor model: need theory that is almostconformal ....

Need g(E) walk in ΛEW < E < ΛETC

Known supersymmetric examples. But analytic calcsfail in non-SUSY case.

Can we use lattice to search for such theories?Vary Nf , Ncolors, symmetry transformation


One example

2 techniquarks with 2 colors transforming in adjoint repnof symmetry group (Ψ → GΨG†)

Near conformality suggested by approx calcs (N cf ∼ 2)

Simulation 83 × 16 lattice using Wilson fermions.

2D parameter space (β = 4g2 ,ma) – 100+ points (70

Gflop sustained over 3 months on BG/L)

Phases ? Continuum limit requires critical line/ptsmq → 0 ? Gluon energy, meson masses, ...


Mean gluon action

-2 -1.5 -1 -0.5 0 0.5ma

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Plaq

uette

beta=1.50beta=1.75beta=1.90beta=1.95beta=2.00beta=2.05beta=2.10beta=2.25beta=2.35beta=2.40beta=2.50beta=2.75beta=3.00

Discontinuity at small β: 1st order transition.


Latent Heat

1.5 1.6 1.7 1.8 1.9 2 2.1beta

0

0.05

0.1

0.15

0.2

0.25

Lat

ent H

eat

Critical end point β = βc ∼ 2.0 ?


Technipion mass squared

Two regimes: β < βc: Goldstone behavior. Coincides withjump in gluon actionβ > βc: Restoration of chiral symmetry (?). Gluon actionsmooth.


Masses along “critical” line

1 1.5 2 2.5 3beta

0

0.5

1

1.5

2

pi, r

ho m

asse

s al

ong

the

criti

cal l

ine

pirho

Notice: masses degenerate for β > βc

All data consistent with theory in conformal phase forβ > βc ∼ 2.0


Caveats

Continuum limit requires careful tuning of g with a toavoid finite size effects

If running g(a) slow expect extreme sensitivity in inversea = a(g). Modest decreases in g correspond to hugedecreases in lattice spacing and large potential finitesize effects.

Eg. Physical box size so small system deconfines andlooks quasi-free.

Alternatively: confinement scale Λa will be small andrequire large lattices to see long distance behavior.

Need large lattices to resolve between these scenarios


Possible scenario

Simplest picture for new CFP.οο

οο−οο

ma

β

0

0

O

OO1st Order

UVFP

IRFP

UVFP


Supersymmetry SUSY - why ?

Scalar fields natural in SUSY theories

String theory demands SUSY

Dark matter ? Grand Unification


Unknowns in SUSY

Is Nature supersymmetric ? How is SUSY broken ?

Spectrum of SUSY theories; LHC, dark mattercosmology

AdSCFT: Gravity as a gauge theory ?5D (super)gravity = gauge theory on 4D boundary(Maldacena 1998)

Hard analytically .. .lattice ?


Lattice and SUSY incompatible ?

SUSY theories have new kind of symmetry:

bosonδSUSY→ fermion

Unfortunately: 2 SUSYs give translation in spacetimebroken by lattice

What to do ? Recently, new lattice formulations: preservesubset 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, ...


SUSY lattices

Matter distributed over links in latticeDetails of lattice action tightly constrained by:

gauge invariance, SUSY and no fermion doubling

Exact SUSY: δ2Q|field >= 0

No. exact SUSYs = no. site fermions


Simulations

In D = 4 one solution to constraints – N = 4 SYMDeveloped parallel code to simulate using tools/algs LGTFirst thing: check SUSY:

< SB > −SexactB N = 4 in D = 2

L δSB

SB

2 0.014(1)3 0.007(2)4 0.006(2)

0 100 200 300 400 500

Monte Carlo time0

200

400

600

800

Bosonic actionPseudofermion action

Q=16 D=4 SU(2)g

2=0.5, L=2

N = 4 in D = 4Frontiers of Lattice Gauge Theory – p. 32

Blackholes from gauge theory

Conjectured duality between N = 4 SYM reduced to D = 1and string theory in D = 5!At low T, gravity theory contains a black hole

0 2 4 6 8 10 12

T

0

5

10

15

20

(1/N

2 )E/T

SU(5) quenchedSU(3)SU(5)SU(8)black hole

Smoothly connected to stringy phase at high T ?


Conclusions/Outlook

Special time in history LGT: LHC offers promise of newphysics.

LGT is now mature – Improvements in algs/hardwaremake exploratory calcs of non-QCD like theoriespossible

Plenty of theoretical ideas: conformal gauge theories,supersymmetry, strings, ...)

Lots to do!


The End


Code issues

C++ code allows easy to debug programs. Map classesto basic mathematical objects eg.Gauge_Field iscollection of complex matrices accessed using a latticeposition vector;

U.get(x,mu)

where x is object type Lattice_Vector

Operator overloading for compact easily debuggedcode:

G.set(x,mu,kappa*U1.get(x,mu)*U2.get(x,mu))

Optimize linear solver (99% CPU )

MC time series analyzed via binning/bootstraptechniques – Mean and error.

Parallel code necessary.Frontiers of Lattice Gauge Theory – p. 36

Parallel code

Utilize existing communication libraries where possibleeg MDP (matrix distributed processing, part ofFermiQCD).

Decides how lattice mapped to physical processors.

Hides MPI calls. Allows for simple port from scalar toparallel.

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 · Frontiers of Lattice Gauge Theory Simon Catterall Syracuse...

Documents

Transcript of Frontiers of Lattice Gauge Theory · Frontiers of Lattice Gauge Theory Simon Catterall Syracuse...