Noise in Glueball Correlators and Simple Fermion Disconnected Diagrams

Noise in glueball correlators and simple fermion disconnected diagrams.

Plymouth University

March 31, 2015

Noise in glueball correlators and simple fermion disconnected diagrams. Introduction

If I have learned one thing from lattice QCD. Small errors are good. Large errors are bad.

I report on a study of looking at the noise in glueball correlators. I attempt to use quasi random numbers to compute disconnected diagrams

Noise in glueball correlators and simple fermion disconnected diagrams. Some background to lattice QCD

To solve QCD for bound state properties “all” that is required is to compute

1 SF SG cij (t)= du dψdψO(t)i O(0)†e− − Z Z Z j The path integral is regulated by the introduction of a space-time lattice. The integral is computed in Euclidean space using Monte Carlo techniques on the computer. The Monte Carlo process produces an ensemble of gauge conﬁgurations (“ snapshot of the vacuum.”

0 m0t 1 m1t cij (t)= cij e− + cij e− . . . The result is a table of numbers corr(length in time,number samples) you average and fit exponentials to get masses, using ROOT, MINUIT or R. Then another fit with lattice spacing, volume, physical masses, to find answer. Noise in glueball correlators and simple fermion disconnected diagrams. Introduction to glueballs

Glueballs are bound states of glue. The holy grail of hadron spectroscopy is to find experimental evidence for glueballs. Unfortunately in the real world, glueballs will probably mix with quark-antiquark states with the same quantum numbers. Also in unquenched QCD the glueball will decay with the strong force, hence resonance effects need to be considered. Note that pure Yang-Mills theory (QCD with no quarks) is a consistent quantum field theory, but because it doesn’t describe reallity, it is not easy to compute the lattice spacing.

Noise in glueball correlators and simple fermion disconnected diagrams. Ed Witten on glueballs

There is a very interesting paper by Witten on gauge theory and glueballs (arXiv:0812.4512). Glueballs don’t exist in the classical gauge theory. (From a paper by Coleman). People often say to me: I don’t understand how glueballs can be massive, if the gluons are massless.

but Ed Witten says I have spent most of my career wishing that we had a really good way to quantitatively understand the mass gap in four dimensional gauge theory.

Noise in glueball correlators and simple fermion disconnected diagrams. Main diﬃculties with lattice glueball studies

The correlators have a poor signal to noise ratio. High statistics are required or better techniques. Strong lattice spacing dependence.

2 am = 0.839(28) |c | = 0.94 2 0 2 am = 1.447(55) |c0| = 0.98

1.5

eff 1 am

0.5

0 0 2 4 6 8 t/a

Figure : Eﬀective mass (a=0.09fm) 0++

Noise in glueball correlators and simple fermion disconnected diagrams. Main diﬃculties with lattice glueball studies

The correlators have a poor signal to noise ratio. High statistics are required or better techniques. Strong lattice spacing dependence.

8 1+-

2++ 6 G m 0 r

0++

2 0 0.2 0.4 0.6 0.8 2 (as/r0)

Figure : Cont. extrapolation, hep-lat/9704011

Noise in glueball correlators and simple fermion disconnected diagrams. The quenched glueball spectrum

In pure gauge theory there is rich spectrum of glueball states. From Morningstar and Peardon (hep-lat/9901004). Perhaps the famous graph from lattice QCD. Statistical errors below error from setting lattice spacing.

12 0+−

+− 3−− 10 2 −− *−+ 2 4 2 −− ++ 1 3 0*−+ 3+− 8 2−+ 1+− 3 *++ 0 −+ G 0 6 2++ m (GeV) 0 G r

2 m 4 0++

1 2

0 0 ++ −+ +− −− PC

Noise in glueball correlators and simple fermion disconnected diagrams. Searching for 0++ glueball ✞ ☎ Chen et al. hep-lat/0510074, M0++ = 1710(50)(80) MeV ✝ ✆ Meson M MeV Γ MeV f0(400) (σ) 400 - 550 400 - 700 f0(980) 980 40 - 100 f0(1370) 1200-1500 200-500 f0(1500) 1505 109 f0(1710) 1724 137 Table : Light 0++ mesons from PDG

Need glueball, qq and two meson interpolating operators, at zero and non-zero momentum. How many exponentials needed to see glueball degrees of freedom. High statistics required.

Noise in glueball correlators and simple fermion disconnected diagrams. Recent unquenched glueball calculation

A long time ago we (a group in Liverpool) wanted to study ++ + ﬂavour singlet mesons with quantum numbers: 0 and 0− with our allocation on an old supercomputer called the QCDOC. The ASQTAD improved staggered fermion action with the one loop tadpole improved gauge action was used. (Same as MILC collaboration) The strange quark mass was not very well tuned. The parameters were chosen to be part of a larger (but with lower statistics) set of calculations performed by the MILC collaboration (arXiv:0903.3598 for an overview).

a fm mπ MeV L fm Number of conﬁgs 0.12 280 2.9 5000 0.09 360 2.9 3000 The higher statistics allowed to ﬁnd a signal for the pseudoscalar + 0− glueball. Noise in glueball correlators and simple fermion disconnected diagrams. Glueball versus expt, (plot from quark model review in PDG, 1005.2473)

Solid black are quenched, blue unquenched and expt bursts. Unquenching eﬀects seem small – but still need continuum limit First time a signal seen for pseudo-scalar glueball in unquenched QCD.

η (2S) c χ c2(1P)

η 3 c Mass GeV

fJ(2220) 2

f0(1710) η f (1525) f (1500) (1475) 2 0 η(1405) f0(1370) η (1295) f2(1270) 1 ++ -+ ++ 0 0 2

Noise in glueball correlators and simple fermion disconnected diagrams. Extending the variational basis

Lucini, Rago and Rinaldi developed a glueball variational code and applied it to large Nc (1007.3879). Much bigger basis of glueball operators used. Variational basis included two body glueball states. Basis included bi-torelon operators (products of two loops winding in a compact spatial direction). Lucini et al. ran their code on the “UKQCD” staggered conﬁgurations with a = 0.09 fm. The ﬁnal publication was 1208.1858.

Noise in glueball correlators and simple fermion disconnected diagrams. Summary plot (1208.1858)

A comparison of lattice results with PDG and results from Crystal Barrel collaboration. Need 550,000 conﬁgurations to get 50 MeV + errors on the spin exotic 0 − glueball.

Noise in glueball correlators and simple fermion disconnected diagrams. Improvements to the glueball calculation

Some combination of Myself (Plymouth) Biagio Lucini (Swansea) Antonio Rago (Plymouth) wanted to improve the previous glueball calculation. Reduce errors on glueball and ﬂavour singlet correlators Introduce fermionic operators.

Noise in glueball correlators and simple fermion disconnected diagrams. Open boundary conditions

The basis of our numerical lattice QCD calculations is the Monte Carlo process. It is important that the sample size is large enough. Different obserables require a different number of Monte Carlo samples (long autocorrelation times). People worry the most about the topology of the gauge fields. It is not clear that the gauge field topology is important for the masses, but people worry..... The topological charge Q is defined by 1 Q = F a F˜ a d 4x. 32π2 Z µν µν

a ˜ a in terms of the Euclidean color gauge ﬁeld Fµν and its dual Fµν,

Noise in glueball correlators and simple fermion disconnected diagrams. Topological charge as a function of trajectory

MILC (Phys.Rev.D81:114501,2010) Topological charge with the asqtad action as a function of trajectory. (Top a=0.12 fm, a=0.09fm, a=0.06fm, bottom a=0.045fm)

Noise in glueball correlators and simple fermion disconnected diagrams. Open boundary conditions

Consider a lattice with dimensions T L L L × × × Normally we use periodic boundary conditions in space and anti-periodic conditions in time for the quark ﬁelds. L¨uscher and Schaefer (CERN) proposed the use of open boundary conditions to “solve” the problem with topology ( JHEP 1107 (2011) 036).

F0k (x) x0=0= F0k (x) x0=T = 0 | | for all k=1,2,3 There are more complicated boundary conditions for the quark ﬁelds.

Noise in glueball correlators and simple fermion disconnected diagrams. First use of open boundary conditions for glueballs

arXiv:1402.7138, Chowdhury et al. Pure SU(3) simulations with Wilson gauge action. They computed the 0++ glueball masses with periodic and open boundary conditions. Also used Wilson ﬂow for smearing (not so important in my opinion). Lattice spacing was determined from the Wilson ﬂow.

Noise in glueball correlators and simple fermion disconnected diagrams. Continuum limit (1402.7138, Chowdhury et al.)

A problem with open boundary conditions is that you can’t use correlators close to time boundaries. (O1 open BC and P1 periodic

BC).

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Noise in glueball correlators and simple fermion disconnected diagrams. Continuum limit

From arXiv:1402.7138, Chowdhury et al.

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Noise in glueball correlators and simple fermion disconnected diagrams. Continuum limit

From arXiv:1402.7138, Chowdhury et al. with results from other groups. Use r0 0.48 fm for quenched QCD. ∼

Masses of 0 + + glueball 5.0

4.5

4.0

3.5 m r0 Meyer (periodic) Morningstar and Peardon (periodic) 3.0 Bali et al. (periodic) Lucini et al. (periodic) Chowdhury et al. (open) 2.5 Chowdhury et al. (periodic)

2.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 (a/r0)**2

Noise in glueball correlators and simple fermion disconnected diagrams. Why think about statistics and noise?

The accurate calculation of disconnected diagrams is still a challenge. Gregory et al. (arXiv:0709.4224, arXiv:1112.4384 ) found tails of distributions were important for signal and error of η and η′. Progress in molecules, nuclear type lattice calculations is limited by noise in the correlators. (Endres, 1112.4023).

Noise in glueball correlators and simple fermion disconnected diagrams. Lattice QCD and statistics

In lattice QCD spectroscopy calculations we measure some correlators and then average to get the central value. The law of large numbers protects us from needing to know about the detailed form of the probability distribution. The Lepage argument (from 1990) for signal to noise ratio of ρ meson. mρt c(t)= ρ(t)ρ(0) e− h i∼

2 4mπ t σ = ρ(t)ρ(t)ρ(0)ρ(0) e− h i∼ Has some support from lattice QCD calculations (NPLQCD collaboration). Plenty of physics missing, especially as the number of quarks increases.

Noise in glueball correlators and simple fermion disconnected diagrams. In an ideal world

From paper by Kaplan et al. 1112.4023 Some lattice data for unitary fermions (not QCD). Moments based analysis techinique (estimator has reduced errors, but biased)

Noise in glueball correlators and simple fermion disconnected diagrams. DeGrand tests

DeGrand (rXiv:1204.4664) looked at his BSM data and found much evidence for lognormal behaviour, but moments method of Kaplan et al. didn’t work. Below is a plot for J=7/2 baryon in SU(7).

Noise in glueball correlators and simple fermion disconnected diagrams. The log normal distribution

ln X N(a, b2) ∼ 1 1 1 p(x)= exp (ln x µ)2 √2πσ x −2σ2 −

It is the distribution for a large number of random variables multiplied together. Length of spoken words in phone conversations. Crystals in ice cream. Latency periods of diseases. Stock market data (eg closing prices). Some useful properties of the log normal distribution. µ 1 σ2 Mean of p(x) = e − 2 . 2 2 Variance of p(x) = e2µ σ (eπ 1) − − Noise in glueball correlators and simple fermion disconnected diagrams. Plot of log normal distribution

p(x)= 1 1 exp 1 (ln x µ)2 √2πσ x − 2σ2 − Fix σ = 1 and varyµ.

Plot of log normal distribution

Parameters mean=0, sigma=1 mean=1, sigma=1 mean=2, sigma=1 Probability density 0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8 10 x

Noise in glueball correlators and simple fermion disconnected diagrams. Plot of log normal distribution

p(x)= 1 1 exp 1 (ln x µ)2 √2πσ x − 2σ2 − Fix µ = 1 and varyσ.

Plot of log normal distribution

Parameters mean=1, sigma=0.2 mean=1, sigma=0.3 mean=1, sigma=0.5 Probability density 0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8 10 x

Noise in glueball correlators and simple fermion disconnected diagrams. Source of log-normal distribution

The sum of random variables distributed with Gaussian probability distribution are also distributed with the Gaussian probability distribution. The product of random variables distributed with Gaussian probability distribution are distributed with the log-normal probability distribution. The above statements can be proved by mathematics, but they can also be easily checked with a computer simulation. It is not obvious that lattice correlators will be distributed according to the lognormal distribution.

Noise in glueball correlators and simple fermion disconnected diagrams. Stable distributions

In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. Sum of variables from same probability distribution with nonﬁnite variance will tend to stable probability distribution. Stable probability distributions also have long tails.

Noise in glueball correlators and simple fermion disconnected diagrams. Example stable distributions

from package in R black is stable dist. and red line is Gaussian.

alpha = 1.9 beta = 0.3 Density 0.00 0.05 0.10 0.15 0.20 0.25 0.30

−4 −2 0 2 4 6 8

Noise in glueball correlators and simple fermion disconnected diagrams. Lognormal and long tails

Generate random numbers from the log normal distribution for a variety of sample sizes. What are the corrections to the law of large numbers? In the plot below I compare the error on the mean from using σ/√N versus the error from replicating the calculation and computing the standard deviation.

replicate large N error 0.0 0.5 1.0 1.5

100 200 300 400 500 600 700 800

Noise in glueball correlators and simple fermion disconnected diagrams. Loop through some data sets

Look at the data sets Histogram the correlators at speciﬁc timeslices and ﬁt a Guassian. Kolmogorov-Smirnov test for normality. Also compute the skewness

n 3 (xk x) γˆ = k=1 − P (n 1)s3 − where (s) standard deviation and x is the mean. The Kurtosis (good to look for tails).

n 4 (xk x) κˆ = k=1 − P (n 1)s4 −

Noise in glueball correlators and simple fermion disconnected diagrams. 164 quenched a 0.1fm Wilson 0++ ≈ 3 basis states and t = 1,2,4 (glueball correlators)

0++ glueball corr (peroidic BC in time), oper1_block1_time1 0++ glueball corr (peroidic BC in time), oper1_block1_time2 0++ glueball corr (peroidic BC in time), oper1_block1_time4 Density Density Density 0.0000 0.0005 0.0010 0.0015 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 29500 30000 30500 29200 29400 29600 29800 30000 30200 30400 29400 29600 29800 30000 30200 30400

c(t) c(t) c(t)

0++ glueball corr (peroidic BC in time), oper1_block2_time1 0++ glueball corr (peroidic BC in time), oper1_block2_time2 0++ glueball corr (peroidic BC in time), oper1_block2_time4 Density Density Density 0e+00 1e−04 2e−04 3e−04 4e−04 5e−04 6e−04 0e+00 2e−04 4e−04 6e−04 8e−04 0e+00 2e−04 4e−04 6e−04 8e−04 1e−03

38000 39000 40000 41000 42000 38000 39000 40000 41000 38500 39000 39500 40000 40500 41000 41500

c(t) c(t) c(t)

0++ glueball corr (peroidic BC in time), oper1_block3_time1 0++ glueball corr (peroidic BC in time), oper1_block3_time2 0++ glueball corr (peroidic BC in time), oper1_block3_time4 Density Density Density 0.00000 0.00005 0.00010 0.00015 0.00000 0.00005 0.00010 0.00015 0.00020 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 24000 26000 28000 30000 32000 34000 36000 38000 26000 28000 30000 32000 34000 24000 26000 28000 30000 32000 34000 36000

c(t) c(t) c(t)

Noise in glueball correlators and simple fermion disconnected diagrams. 4 + 16 quenched a 0.1fm Wilson 0− ≈ 3 basis states and t = 1,2,4 (Preliminary results with open boundary conditions were less normal.) (glueball correlators)

0−+ glueball corr (peroidic BC in time), oper1_block1_time1 0−+ glueball corr (peroidic BC in time), oper1_block1_time2 0−+ glueball corr (peroidic BC in time), oper1_block1_time4 Density Density Density 0 5 10 15 20 25 0 5 10 15 0 2 4 6 8 10 12 14

0.0 0.1 0.2 0.3 0.4 0.5 0.6 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 −0.2 −0.1 0.0 0.1 0.2

c(t) c(t) c(t)

0−+ glueball corr (peroidic BC in time), oper1_block2_time1 0−+ glueball corr (peroidic BC in time), oper1_block2_time2 0−+ glueball corr (peroidic BC in time), oper1_block2_time4 Density Density Density 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5

0 1 2 3 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0

c(t) c(t) c(t)

0−+ glueball corr (peroidic BC in time), oper1_block3_time1 0−+ glueball corr (peroidic BC in time), oper1_block3_time2 0−+ glueball corr (peroidic BC in time), oper1_block3_time4 Density Density Density 0.0 0.1 0.2 0.3 0.4 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

0 10 20 30 −10 −5 0 5 10 15 20 −20 −10 0 10

c(t) c(t) c(t)

Noise in glueball correlators and simple fermion disconnected diagrams. 164 quenched a 0.1fm Wilson 2++ ≈ 3 basis states and t = 1,2,4 (glueball correlators)

2++ glueball corr (peroidic BC in time), oper1_block1_time1 2++ glueball corr (peroidic BC in time), oper1_block1_time2 2++ glueball corr (peroidic BC in time), oper1_block1_time4 Density Density Density 0 20 40 60 80 100 0 10 20 30 40 50 0 10 20 30 40 50

0.00 0.05 0.10 0.15 −0.05 0.00 0.05 0.10 −0.05 0.00 0.05

c(t) c(t) c(t)

2++ glueball corr (peroidic BC in time), oper1_block2_time1 2++ glueball corr (peroidic BC in time), oper1_block2_time2 2++ glueball corr (peroidic BC in time), oper1_block2_time4 Density Density Density 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

0.0 0.2 0.4 0.6 0.8 1.0 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4 0.6

c(t) c(t) c(t)

2++ glueball corr (peroidic BC in time), oper1_block3_time1 2++ glueball corr (peroidic BC in time), oper1_block3_time2 2++ glueball corr (peroidic BC in time), oper1_block3_time4 Density Density Density 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 2 4 6 8 10 −2 0 2 4 6 −6 −4 −2 0 2 4

c(t) c(t) c(t)

Noise in glueball correlators and simple fermion disconnected diagrams. Unquenched iwasaki b1.95 L32T64 k0.161236 mu0.0055 + musigma0.135 mudelta0.17 0−

3 basis states and t = 1,2,4 (glueball correlators)

0.010 0.015 0.020 0.025 0.030 0.035 0.040 −0.010 −0.005 0.000 0.005 0.010 0.015 −0.010 −0.005 0.000 0.005 0.010

c(t) c(t) c(t)

0.10 0.15 0.20 0.25 0.30 −0.05 0.00 0.05 0.10 −0.10 −0.05 0.00 0.05

c(t) c(t) c(t)

0−+ glueball corr (peroidic BC in time), oper1_block3_time1 0−+ glueball corr (peroidic BC in time), oper1_block3_time2 0−+ glueball corr (peroidic BC in time), oper1_block3_time4 Density Density Density 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 −0.4 −0.2 0.0 0.2 0.4

c(t) c(t) c(t)

Noise in glueball correlators and simple fermion disconnected diagrams. 3 + Unquenched ASQTAD fermions 32 64, 0− × 3 basis states and t = 1,2,4 (glueball correlators)

0.03 0.04 0.05 0.06 0.07 −0.020 −0.015 −0.010 −0.005 0.000 0.005 0.010 0.015 −0.02 −0.01 0.00 0.01 0.02

c(t) c(t) c(t)

0.20 0.25 0.30 0.35 0.40 −0.05 0.00 0.05 0.10 −0.10 −0.05 0.00 0.05 0.10

c(t) c(t) c(t)

0.4 0.5 0.6 0.7 0.8 −0.2 −0.1 0.0 0.1 0.2 −0.2 −0.1 0.0 0.1 0.2

c(t) c(t) c(t)

Noise in glueball correlators and simple fermion disconnected diagrams. 3 + Unquenched ASQTAD fermionic loops 32 64, 0− ×

1 basis states and t = 4 (qq correlators) Eric Gregory worked on distributions in arXiv:0709.4224.

eta^P corr (peroidic BC in time) Density 0 500 1000 1500

0.0000 0.0005 0.0010 0.0015 0.0020

c(t)

Noise in glueball correlators and simple fermion disconnected diagrams. Challenge of disconnected diagrams

There are many physics quantities which require disconnected diagrams to be computed.

Physics of η and η′. Tetraquark Flavour singlet 0++ Flavour singlet form factors

Noise in glueball correlators and simple fermion disconnected diagrams. Computing traces with noise

η†ηj = δij h i i The matrix equation is solved for the noise vector ηi for i running from 1 to the number of samples Nsamp.

Mχi = ηi

The trace of the inverse of the M matrix is estimated from

Nsamp 1 1 trace(M− ) ηi†χi (1) ≈ Nsamp Xi=1 Nsamp 1 1 ηi†M− ηi (2) ≈ Nsamp Xi=1

Noise in glueball correlators and simple fermion disconnected diagrams. Speed up techniques

Normally, the error 1 error 1/2 ∼ (Nsamp)

Choice of noise Z2 versus Gaussian Variance reduction (eg. hopping parameter dilution/partition distillation Use eigenvalues, eigenvectors Use a fast inverter, such as Multigrid See review by Wilcox hep-lat/9911013, and work by Bali et al. arXiv:0910.3970 ☛ 1 ✟ Can the error decrease faster than 1/2 ? Nsamp ✡ ✠

Noise in glueball correlators and simple fermion disconnected diagrams. Introduction to quasi-random numbers

The ﬁrst algorithmic improvement of disconnected loops was the use of Z2 noise rather than Gaussian noise by Dong and Liu, hep-lat/9308015. . There was a theoretical argument (DOI: 10.1016/0010-4655(94)90004-3), which showed that Z2 noise was superior for simple loops. This motivates that perhaps other types of noise can be used. Here I will talk about quasi-random numbers. There has been recent work on quasi-random numbers for lattice QCD problems. Jansen et al. arXiv:1302.6419, and (Ammon et al.) arXiv:1503.05088

Noise in glueball correlators and simple fermion disconnected diagrams. Introduction to quasi-random numbers

Quasi-random numbers normally applied to numerical integration. Regular grids require too many function evaluations. For small dimensions the cost of quasi-random integration is O(1/Nsamp ) A picture is hopefully worth a 1000 words ... Plot pseudo-random numbers on the left and the quasi-random (Sobol) numbers on the right.

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Noise in glueball correlators and simple fermion disconnected diagrams. Quasi-random numbers

Quasi-random numbers are used in integration in a similar way to basic Monte Carlo.

Nsamp 1 1 1 f (x, y)dxdy f (xi , yi ) Z Z ≈ Nsamp 0 0 Xi=1

where (xi , yi ) are from a Quasi-random sequence, for example: Halton sequence Sobol sequence Can be used with importance and stratiﬁed sampling. Quasi-random integration methods are described in many books on ﬁnance.

Noise in glueball correlators and simple fermion disconnected diagrams. Concerns about quasi-random numbers

Quasi-random methods require a space to operate. It is not clear that quasi-random will be better than pseudo-random based methods for large spaces. First do some experiments Use matlab because this package has various quasi-random packages in it and a library of test matrices. First look at integration Use error estimate error = exact QR estimate | − | Strict quasi random methods are deterministic, so error estimate is non-trivial. Typically the quasi-random random numbers are “scrambled” (some randomness added).

Noise in glueball correlators and simple fermion disconnected diagrams. Using quasi-random numbers to integrate

1 1 1 1 4 Test on 0 0 0 0 x1x2x3x4dx1dx2dx3dx4 = (1/2) R R R R

0.02 Monte Carlo Quasi (Sobol) 0.015 Quasi (Halton)

0.01 error

0.005

0 200 300 400 500 600 N samp

Noise in glueball correlators and simple fermion disconnected diagrams. Using quasi-random numbers to integrate

1 1 1 1 1 1 1 1 8 0 0 0 0 0 0 0 0 x1x2x3x4x5x6x7x8dx1...dx8 = (1/2) R R R R R R R R

0.002 Monte Carlo random Quasi (Halton) 0.0015 Quasi (Sobov) Quasi (Halton scrambled)

0.001 Error

0.0005

0 200 300 400 500 600 Nsamp

Noise in glueball correlators and simple fermion disconnected diagrams. Introduction to quasi-random numbers

Now consider methods for computing

1 trace(M− )

Why did I consider quasi-random methods Partly because I was teaching Monte Carlo integration to students, and I just wondered whether it was possible. Monte Carlo Methods Paperback 1967 by J. M. Hammersley and D.C. Handscomb COMPUTING THE TRACE OF A FUNCTION OF A SPARSE MATRIX VIA HADAMARD-LIKE SAMPLING, Wong et al. From the numerical analysis literature. Not really a quasi-random method, but in the same spirit Some numerical success over Monte Carlo methods,

Noise in glueball correlators and simple fermion disconnected diagrams. Additional noise

η†ηj = δij h i i It is instructive to see how to construct the above equation with uniform random numbers distributed between 0 and 1.

1/2 xdx = 0 Z 1/2 − 1/2 1 x2dx = Z 1/2 12 − So the noise used in the random sources should be uniform between 1 -1/2 and 1/2. An additional normalization factor of 12 is required.

Noise in glueball correlators and simple fermion disconnected diagrams. Testing some stochastic methods

Comsider a Poisson matrix of order 1024. Compare the exact trace against calculation using pseudo-random and quasi-random numbers.

Poisson matrix order 1024 Error on trace(M^-1)

0.09 Z2 noise Uniform rand. Sobol

0.06 error

0.03

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Noise in glueball correlators and simple fermion disconnected diagrams. Testing some stochastic methods II

Comsider a Poisson matrix of order 1024. Compare the exact trace against calculation using pseudo-random and calculation using Hadamard matrices (from Wong et al.).

Poisson matrix order 1024 Error on trace(M^-1)

0.09 Uniform Hadamard matrix

0.06 error 0.03

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Noise in glueball correlators and simple fermion disconnected diagrams. Techniques not mentioned

Antithetic variates is method used in Monte Carlo methods, where some correlation is introduced in the noise to reduce the variation. It turns out that I have already used antithetic variates (but perhaps not understood it) with Chris Michael. A noise source was generated.

A second source was generated by multiplying by γi . This reduced the error, but not as much as with the hopping parameter method. A good method to use with multi-grid, where it is diﬃcult to implement the hopping parameter expansion.

Noise in glueball correlators and simple fermion disconnected diagrams. Conclusions and Summary

I have told you about noise in disconnected/glueball correlators. A clear picture has not emerged. It is not clear that the distribution of a correlator is physical (local meson operators prob. OK.) Not clear what open boundary conditions do to the glueball spectrum. I have attempted to use quasi random numbers to compute disconnected diagrams Not clear that a useful algorithm for QCD will be produced.

Noise in glueball correlators and simple fermion disconnected diagrams.