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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
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 – manycomplex 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:
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.
Frontiers of Lattice Gauge Theory – p. 5
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
Frontiers of Lattice Gauge Theory – p. 6
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)
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 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 !).
Frontiers of Lattice Gauge Theory – p. 8
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.
Frontiers of Lattice Gauge Theory – p. 9
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
Frontiers of Lattice Gauge Theory – p. 11
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)
Frontiers of Lattice Gauge Theory – p. 12
Lattice QCD
(1980–) Successes:
QCD confines quarks Vqq̄ ∼ σr
Spectrum hadrons
Decays, matrix elements, heavy quarks
Frontiers of Lattice Gauge Theory – p. 13
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 ?
...
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 YMP1Gflop,10M$->104 $/Mflop
2005,PC cluster3Tflop,1M$->1$/Mflop
2008,Video card1Tflop,10−3M$->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 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
Frontiers of Lattice Gauge Theory – p. 19
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
Frontiers of Lattice Gauge Theory – p. 20
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, ...
Frontiers of Lattice Gauge Theory – p. 21
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.
Frontiers of Lattice Gauge Theory – p. 22
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 ?
Frontiers of Lattice Gauge Theory – p. 23
Technipion mass squared
Two regimes: β < βc: Goldstone behavior. Coincides withjump in gluon actionβ > βc: Restoration of chiral symmetry (?). Gluon actionsmooth.
Frontiers of Lattice Gauge Theory – p. 24
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
Frontiers of Lattice Gauge Theory – p. 25
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
Frontiers of Lattice Gauge Theory – p. 26
Possible scenario
Simplest picture for new CFP.οο
οο−οο
ma
β
0
0
O
OO1st Order
UVFP
IRFP
UVFP
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 mattercosmology
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:
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, ...
Frontiers of Lattice Gauge Theory – p. 30
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
Frontiers of Lattice Gauge Theory – p. 31
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 ?
Frontiers of Lattice Gauge Theory – p. 33
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!
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 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 – p. 37