Lattice Wess-Zumino model with Ginsparg-Wilson fermions:

One-loop results and GPU simulations

Joel Giedt, Rensselaer Polytechnic Institute

withwith

Chen Chen and Eric Dzienkowski

Student research assistants

Chen Chen (3rd yr 2010-2011)•Lattice Wess-Zumino model: one-loop counterterms•Lattice Wess-Zumino model: Pfaffian sign problem?

i i d l di f •Lattice Wess-Zumino model: 4-divergence of supercurrent

Eric Dzienkowski [M.S. May 2010; UCSB Fall 2010]Eric Dzienkowski [M.S. May 2010; UCSB Fall 2010]•LatticeWess-Zumino model: GPU code development•Twisted N=4 SYM Lattice SUSY counterterms (w/ Catterall, Syracuse)

Lattice SUSY: Emerging FieldLattice SUSY: Emerging Field

Computational resources now allow dynamical fermions

Lattice SUSY

20

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cles

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Lattice formulations that reduce/eliminate fine-tuning

Why not more?Why not more? After all, the MSSM is still the favorite for new physics

Dynamical SUSY breaking requires new strong dynamics in a hidden sector

M d l l d Mediation may also involve new strong dynamics

Doesn’t it make sense to develop the tools to understand how the soft terms are generated?how the soft terms are generated?

The spectrum and dynamics of the hidden sector may also play a role in dark matterp y

The lattice SUSY problemThe lattice SUSY problem Lattice breaks translation invariance to discrete subgroup

Two supersymmetry transformations give infinitesmalgenerator of translation

R l i P b di t diff t f il b

{Qα, Q̄α̇} = 2σμαα̇Pμ

Replacing Pμ by discrete difference operator fails because Leibnitz rule does not hold

Interacting theories have O(a) violation classicallyInteracting theories have O(a) violation, classically

δS = a²αXα

Dangerous amplificationDangerous amplification In the Ward-Takahashi identity we have

hδS Oi = a²αhXαOi

A linear divergence is enough to cause violation of SUSY in the continuum limit

CountertermsCounterterms Another way to view the problem

Most general long distance effective action consistent with lattice symmetries is not SUSY

If l h b k SUSY If non-irrelevant operators that break SUSY

Could just be a mismatch, like mass of fermion and scalar not being equal in long distance effective actionbeing equal in long distance effective action

Since regulator breaks SUSY, no reason why they should receive equal renormalizationsq

Generally must fine-tune parameters of most general short distance action in order to get desired long distance theory

Lattice symmetriesLattice symmetries Lattice symmetries can be used to restrict renormalizations

Can be used to reduce number of parameters that must be fine-tuned

W ll l h U(1) f l We will see an example where a U(1) symmetry significantly reduces the number of counterterms that must be adjusted

It is called an R symmetry because it does not commute with It is called an R-symmetry because it does not commute with SUSY

Lattice Wess Zumino modelLattice Wess-Zumino model With Ginsparg-Wilson fermions [Fujikawa, Ishibashi 01;

Fujikawa 02; Bonini, Feo 04-05; Kikukawa, Suzuki 04]

Preserves U(1)R symmetry (limits counterterms) in masslesslimitlimit

GPU simulation code

Early results collected in [1005 3276] Early results collected in [1005.3276]

Lattice derivative operatorsLattice derivative operators

1 1A = 1− aDW , DW =

1

2γμ(∂

∗μ + ∂μ) +

1

2a∂∗μ∂μ

D1 =1γ (∂∗ + ∂ )(A†A)−1/2D1 =2γμ(∂μ + ∂μ)(A A)

D2 =1

a

·1−

µ1 +

1

2a2∂∗μ∂μ

¶(A†A)−1/2

¸a

· µ2

¶ ¸D = D1 +D2 =

1

a

³1−A(A†A)−1/2

´Bonini, Feo 2004

Lattice action without fine tuningLattice action without fine-tuning

S = a4Xx

½− 1

2χTCMχ+

2

aφ∗D2φ+ (m

∗φ∗ + g∗φ∗2)(1− a

2D2)(mφ+ gφ2)

¾M /D P ∗P 2 φP 2 ∗φ∗P /D (1

aD ) 1DM = /D +mP+ +m

∗P− + 2gφP+ + 2g∗φ∗P−, /D = (1−

2D2)

−1D1

δχ = iαγ5χ, δφ = −2iαφU(1)R symmetry at m=0

6D 6Dγ5 6D = − 6Dγ5

Exact chiral symmetry?Exact chiral symmetry? The lattice Dirac operator diverges for would-be doubler

modes

Hence they are nonpropagating

W l b d BC f l T We regulate by antiperiodic BCs in time, finite lattice T.

As T ∞, divergent spectrum of Dirac operator appears.

Thi i h th NN th i d d This is how the NN theorem is evaded

Raises question of locality, since spectrum is unbounded

Range of operatorRange of operator

We follow the approach of [Hernandez, Jansen, Luscher 99]. We introduce aWe follow the approach of [Hernandez, Jansen, Luscher 99]. We introduce aunit point source η at the origin x = 0 and then compute

ψ(x) =

·³1− a

D2

´−1D1η

¸(x) (1)ψ(x)

·³1

2D2

´D1η

¸(x) (1)

The “taxi-driver distance”

r = ||x||1 =Xμ

(|xμ| or |Lμ − xμ|) (2)

t th i i i d fi d th h t t l th i l t d i h “ ” t t tto the origin is defined; the shortest length is selected in each “or” statement,where Lμ/a is the number of lattice sites in the μ direction.

Range of operatorRange of operator

One then computes the norm ||ψ(x)|| in spinor index space at site x andobtains the function

f(r) = max{||ψ(x)|| | ||x||1 = r}

Range of operatorRange of operator

Auxiliary formulationAuxiliary formulationS = −a4

X½1

2χTCDχ+ φ∗D2

1φ+ F∗F + FD2φ+ F

∗D2φ∗X

x

½2χ χ φ 1φ 2φ 2φ

−1aXTCX − 2

a(FΦ+ F∗Φ∗)

+1

2χ̃TC

³mP+ +m

∗P− + 2gφ̃P+ + 2g∗φ̃∗P−

´χ̃

+F̃ ∗(m∗φ̃∗ + g∗φ̃∗2) + F̃ (mφ̃+ gφ̃2)

¾(1)+F (m φ + g φ ) + F (mφ+ gφ )

¾(1)

φ̃ = φ+ Φ, χ̃ = χ+X, F̃ = F + F

When written with auxiliary fields, all terms involve exponentially local operators

Lattice action with fine-tuning ( i )(massive)

½1 2

S = a4Xx

½− 1

2χTC( /D +m1P+ +m

∗1P−)χ+

2

aφ∗D2φ

+m22|φ|2 + λ1|φ|4 +

¡m23φ2 + g1φ

3 + g2φφ∗2 + λ2φ

4 + λ3φφ∗3 + h.c.

¢2| | | |

¡3

¢−χTC(y1φP+ + y∗1φ∗P−)χ− χTC(y2φP− + y

∗2φ∗P+)χ

¾

•Impose CP, gives only real parameters•Fix m1, y1•Still have 8 parameters to fine-tune

Lattice action with fine-tuning ( l )(massless)

In the limit m1 → 0 we can impose the U(1)R symmetry. This restricts theaction to

S = a4X½

− 1

2χTC /Dχ+

2

aφ∗D2φ+m

22|φ|2 + λ1|φ|4

Xx

½2 a

−χTC(y1φP+ + y∗1φ∗P−)χ¾

Fix y1, fine-tune two parametersp

One loop calculationOne-loop calculation No mass counterterms, no coupling counterterms, only

wavefunction renormalization

Zφ − 1 = limp→0 Σφ(p)/p2

φ p→0 φ(p)/p

Zχ − 1 = lim6p→0 Σχ(p)/ 6p

( )ZF − 1 = limp→0 ΣF (p)

Example: scalar self energyExample: scalar self energy

ma (Zφ − 1)(∞)/|g|2 (Zφ − 1)(8)/|g|2ma (Zφ 1)(∞)/|g| (Zφ 1)(8)/|g|0.5 -0.00589 -0.0063550.25 -0.00884 -0.0095320.125 -0.01205 -0.0133580.0625 -0.01573 -0.0175410.03125 -0.01986 -0.0218370.015625 -0.02573 -0.0262050.0078125 -0.02956 -0.0305830.00390625 -0.03370 –0.001953125 -0.03671 –

Table 1: Wavefunction counterterm from self-energy for the scalar, for L = ∞and mL = 8.

W h fit th L i l d t ith ≤ 0 125 tWe have fit the L =∞ numerical data with ma ≤ 0.125 to

f(ma) = c0 ln(ma) + c1ma+ c2(ma)2

Giving the data points equal weight, the fit for the scalar self energy is

c0 = 0.00604(7), c1 = 0.024(20), c2 = −0.15(17)

while for the fermion the fit is

c0 = 0.00597(5), c1 = 0.061(15), c2 = −0.39(12)c0 0.00597(5), c1 0.061(15), c2 0.39(12)

Thus we see that at L → ∞ the log divergences of the scalar Zφ − 1 and thefermion Zχ − 1 match. For the auxiliary field we obtain instead

c0 = −0.0261(5), c1 = 0.87(11), c2 = −3.9(1.0)

so that its ZF − 1 does not match the scalar and fermion. This is consistentso that its ZF 1 does not match the scalar and fermion. This is consistentwith the results found in [Kikukawa, Suzuki 04].

One loop summaryOne-loop summary Nonrenormalization of superpotential due to cancellation of

diagrams

Only wavefunction renormalization, with

C i t t ith i t di

Zφ = Zχ 6= ZF

Consistent with previous studies

Monte Carlo simulationMonte Carlo simulation In addition to the perturbative analysis, we have developed

simulation code to run on graphics processing units (GPUs) that are CUDA enabled

What we have is RHMC code that works with scalars and What we have is RHMC code that works with scalars and spinors that reside on the card

We have benchmarks for the code and have run various checks

Serious simulations and analysis still needs to be done

Nvidia GTX 285Nvidia GTX 285

240 cores, $450, 200W

4 of the 30 MPUs

Each MPU has 8 single preccores and 1 double prec core

http://www.nvidia.com/object/product_geforce_gtx_285_us.html

“The most powerful single GPU on the planet for gaming and beyond.”

GPU BenchmarksGPU Benchmarks

Lattice CPU GPU (CUDA) GPU (Ours) CPU GPU (CUDA) GPU (Ours)single single single double double double

83 32 1 1 6 9 24 0 94 4 4 1183 × 32 1.1 6.9 24 0.94 4.4 11163 × 32 0.88 14 71 0.69 10 35323 × 64 0.11 20 – 0.085 10 –

Table 1: Comparison of timing, Gflop/s, for fast fourier transform of the spinorfield. Both single and double precision results are given.

PreconditioningPreconditioning

At weak couplings, we expect that preconditioning by the inverse of the freetheory fermion matrix M0 will improve convergence of the conjugate gradientsolver. For this purpose, we re-express the problem

M †M bM †Mx = b

as follows:(M−1†

0 M †)(MM−10 )(M0x) = (M

−1†0 b)(M0 M )(MM0 )(M0x) (M0 b)

Large speed upLarge speed-up

Lattice Precision NPC secs. NPC iters. Gflop/s PC secs. PC iters. Gflop/s speed-up83 × 32 single 1.3 830 14 0.038 8 17 3483 × 32 double 4.9 1600 7.2 0.13 15 8.1 38163 × 32 single 7.1 870 25 0.20 8 31 36163 × 32 double 31 1800 12 0.74 15 13 42323 × 64 single 420 1600 14 6.8 8 15 62323 64 d bl 2200 3900 6 5 25 15 7 0 86323 × 64 double 2200 3900 6.5 25 15 7.0 86

Table 1: Timing benchmarks at m = 1, g = 1/5. PC indicates preconditioningwhereas NPC is the inversion without preconditioning The time in secondswhereas NPC is the inversion without preconditioning. The time in secondsand the number of iterations (iters.) is for convergence. The criterion forconvergence is that the relative residual is less than 1×10−6 for single precisionand less than 1× 10−12 for double precision. The speed-up is the ratio of NPCp p ptime to PC time.

MeasurementMeasurement Simplest SUSYWard-Takahashi identity is vanishing of

auxiliary field vacuum expectation value

We evaluate F in terms of φ using the equations of motion

F 1/2 d 1/10 f d ll l For ma=1/2 and g=1/10 we find a very small value:

1V

PxhF (x)i = O(10−3)

We can interpret this small violation of SUSY as coming from two-loop and higher results

V

two loop and higher results

Since at one-loop <F> vanishes

ConclusionsConclusions Reasonable fine-tuning in massless limit

Working, fairly fast GPU code

Large speed-up from preconditioning

Next: parameter space scans

m22, λ1

to obtain h∂μSμ(x)O(0)i ≈ 0

and degeneracy in fermion/boson effective mass

Finding difficulty with autocorrelation in massless limit