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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
25
30
35
cles
0
5
10
15Art
ic
0
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