Tutorials on Monte Carlo Integration
with BASES
Junichi Kanzaki (Kavli IPMU)
Iwate Collider School 2023
@ ANA Crowne Plaza Resort Appi Kogen
February 28, 2023

Purpose of this tutorial• We already have a great tool:

MadGraph5_aMC@NLO which provides- the computations of cross-sections,- the event generations, and- the use of tools for data manipulation andanalysis.

- You should only give physics process informationand some parameters.

-> This tutorial aims to help you to understandwhat MG5 is doing internally by experiencing
calculations by yourselves.
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Environment and Plan
• Environment:- Essential: Development with gfortran, make, ...,
for the installation of BASES- Optional (recommended): Python3 with matplotlib
for data visualization• Plan:- Monte Carlo integration with BASES

simple 1-dim., multi-dim. integration with
examples of plotting.

- Example of calculating cross-section of a physics
process

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Numerical Integration• The computation of physics processes often reduces
to the calculation of multivariable functions.

- Physics quantities (ex. Cross-sections):

f = f(xi,i=1~N)xi; four-vectors, helicities, etc.

- If you want to observe some quantities,
xj,j=1~n (pT of lepton or jet), you must
integrate other unobserved variables.

• Monte Carlo integration using random numbers isparticularly useful for higher-dimensional integrals,
and Monte Carlo randomly chooses points at which
the integrand is evaluated.
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Monte Carlo Integration (ref.)•We already have a good lecture by Olivier thismorning.

•VEGAS: G.P. Lepage, J. Comput. Phys. 27(1978) 192–203.

•BASES: S. Kawabata, Comput. Phys. Commun.

88, 309 (1995) - Integration/Event generation•MG5: J. Alwall, M. Herquet, F. Maltoni, O.

Mattelaer, T. Stelzer, JHEP06 (2011) 128 Integration/Event generation
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Random number•Recent random number generators haveenough performance in uniformity.

•For the integrations of multi-dimensional
functions, the quality of random number
generation becomes essential.

•The tutorial example of BASES includes a

generator based on the Mersenne Twister:
https://en.wikipedia.org/wiki/
Mersenne_Twister
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-0• We assume the development environment with
gfortran.

• Install python with "matplotlib" (optional) to
experience plotting.

• Preparations: install libraries, bases50, helasv43,
and sm• bases50: BASES library for the MC
integration• helasv43: Helicity amplitude library (example4)• sm: Utility subroutines for the SM procces
Presented by J. Kanzaki at HOW2019 in March 19, 2023

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Installation of examples• After downloading the example package, bsexamples.tar.gz,
please follow the README instruction.
1) Make bases libraries
for BASES:
cd bases50
make installfor HELAS:
cd helasv43
make installfor other utilities:
cd sm
make install
The install commands above generate the following threelibraries under ./lib: libbases50.a, libhelasv43.a and libsm.
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Installation of examples (cont'd)2) Make examples
cd example1
make./example1
The program generates a histogram dump file: example1.dump
3) Test plotting (requiers matplotlib)./plot.py
This script reads contents of "example1.dump" and generates
a plot.

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorials with BASES•Prepared samples are written in Fortran 90•Examples are composed of the following three
files:•mainb.f90 - main program•bfunc.f90 - integrand function•user_modules.f90 - initialization etc.

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorials with BASES (cont'd)•BASES: two program phases•Grid optimizations•Data accumulations•Parameters of BASES:•ncall - the number of function calls periteration•it1, it2 - the maximum iterations for eachprogram phase (it1-optimization,it2accumulation).

•acc1, acc2 - the accuracy to stop iterations
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorials with BASES (cont'd)• In "mainb.f90" the following subroutines arecalled:call userin: initialization
call bases: execute integration with BASES
call bsinfo: display integration results
call bhplot: display histograms
call xhdump: generate histogram dump file

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-1
The integration of 1-dim. function:pf (x) = 2 (1<latexit sha1_base64="SICz/VzBHuP6BBDEbXCp9hFa16I=">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</latexit>

x2 )between -1 < x < 1.

1. Make "example1" and execute it. Checkthe answer
2. Increase "ncall" and check the accuracy.

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-1 (cont'd)
1. Learn how to define histograms in "userin."call xhinit(id,xmin,xmax,nbins,title)
2. Learn how to fill histograms in "bfunc":
call xhfill(id,val,func)
3. The program generates a histogram dump
file, "example.dump." If you have"matplotlib," execute the script: plot.py,
which reads the file and generates a plot.

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-2•Another example of the integration of multidimensional function:<latexit sha1_base64="GmIVjI1MkcIvFJAQGDCB3ztaKdY=">AAACknicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvopmlUaCrYKsQUFOWnxFdn2hrWxlXnxcfkJpZkFOVWp2Tm1tYqaBgpVADlajXjBZQN9AzAQAGTYQhlKDNAQUC+wHKGGIYUhnyGZIZShlyGVIY8hhIgO4chkaEYCKMZDBkMGAqAYrEM1UCxIiArEyyfylDLwAXUWwpUlQpUkQgUzQaS6UBeNFQ0D8gHmVkM1p0MtCUHiIuAOhUYVA2uGqw0+GxwwmC1wUuDPzjNqgabAXJLJZBOguhNLYjn75II/k5QVy6QLmHIQOjC6+YShjQGC7BbM4FuLwCLgHyRDNFfVjX9c7BVkGq1msEig9dA9y80uGlwGOiDvLIvyUsDU4NmM3ABI8AQPbgxGWFGeoZmeiaBJsoOTtCo4GCQZlBi0ACGtzmDA4MHQwBDKNDeyQw7GQ4xHGYSYbJicmRyhihlYoTqEWZAAUw+AHzkmpM=</latexit>

f (x) =

nY
dim

(2xi )

i=1•Try to change integration parameters.
•Try to prepare "plot.py" for the dump file,"example2.dump."

Presented by J. Kanzaki at Iwate Collider School in February 28, 202315

2 次元正規分布
Tutorial-3

2 次元正規分布

次元正規分布(のうちY integration
が独立な簡単なケース
exampleof X,
theof a 2- )•2Anotherdimensional Gaussian function:
fXY (x, y) = !

(x−µX )2
−
2σ 2
X

1
2
2πσX

e

×!

1
2
2πσY

2
E[X]
=
µ
,
E[Y
]
=
µ
,
V[X]
=
σ
X
Y
and histograms.
X,•Check resultse(y−µY )2
−
2σ 2
Y

.

2
V[Y ] = σY,

母共分散
CXY =and(独立なので) = 0,
Learn
howto
definefill
2-dimensional•
Cov[X, Y ]histograms.
"
母相関係数 rXY = "
=0

V[X]

V[Y ]

•Test "plot.py" and "plot_2d.py."
2 変量データ

確率統計☆演習 I(2015)L04

同時分布

確率統計☆演習 I(201108Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-4
Calculate cross-sections of the Standard
Model physics processes: e e+ ! ZH
<latexit sha1_base64="LEjF5+rdcBR16lebsXdorsZJuMA=">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</latexit><latexit sha1_base64="IlDXFoQUoMo+KBDU+NIjZ2aLdvE=">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</latexit>

=

=

1
2s
1 1
⇥
2 2
helicities of external particles=

amplitude of i-th channel=

phase space

flux factor=

spin average

=

Mi

Z X X
1 11
|
Mi | 2 d
2s 2 2iPresented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Generation of phase space
Before the calculation of amplitude of physics
process, we have to generate phase space:e e+ ! ZH<latexit sha1_base64="LEjF5+rdcBR16lebsXdorsZJuMA=">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</latexit>

W: total collision energy
mZ: Z boson mass
mH: Higgs boson mass-> calculate four-vectors of external
particles: pe-, pe+, pZ, pH

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Generation of phase space
Two body phase space:
Z 2⇡2
2 Z +11
m1 m2
d cos ✓
d
d 2=
(
,
)
8⇡
s
s
2
2⇡
1
0
Total energy: W(=\sqrt{s}) -> P1(m1) + P2(m2)<latexit sha1_base64="YJLdOzN0ml27A7G+DEkxmtLkCEc=">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</latexit>

1.Determine energies of P1 and P2 (E1 and E2)
from total energy, W
2. Determine directions (\theta and \phi) offinal particles with two random numbers andcalculate four momenta for P1 and P2.
\theta: -1 < cos(\theta) < +1
\phi: 0 < \phi < 2 \piPresented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Generation of phase space
Then go to three body phase space:<latexit sha1_base64="uUlSytL0+NpRmPcPkb9sq90FTzo=">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</latexit>d

3 =Z (ps m3 )2(m1 +m2 )2

2 Z

dq
2⇡

dq 2

d cos ✓3

d 3

1 1
1
=
2 32⇡ 2 32⇡ 2

⇥

Z +1
1

2 (P = q + p3)

Z (p s m 3 )2<latexit sha1_base64="wa9SuH7c/PQmoukiLZLM/927Dlk=">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</latexit><latexit sha1_base64="YVLuioBkFOOcxHlaBgddKd9+vWg=">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</latexit>d(m1 +m2 )2

Z 2⇡
0

Z

d 2(q = p1 + p2 )

✓ 2
◆ ✓ 2
◆
2
2
q m3
m1 m2
,
, 2
2
s s
q
q

Z +1
1

d cos ✓ˆ1

Z 2⇡

d ˆ1

0

W(=\sqrt{s}) -> p1(m1) + p2(m3) + p3(m3)-> Combination of two two-body phase space:PS(p3,q=p1+p2) + PS(p1,p2)
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

20

Generation of phase space
1.Determine the invariant mass of the system of
p1+p2, q2, with a random number:
m1+m2 < q2 < W-m3
2. Generate four momenta of p3 and p1+p2 withtwo body phase space.
3. Then generate p10 and p20 at the rest frameof p1+p2 with q2.
4. Rotate&Boost generated p10 and p20 to thelab. Frame, p1 and p2.

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-4 (cont'd)• Requires two more libraries in addition tobases50:- helasv43: new helicity amplitude library- sm: utility library for the SM processes• Four program components:- mainb.f90- bfunc.f90, matrix.f90, user_modules.f90

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-4 (cont'd)• Generate a physics process with MG5 and save
results with the "output standalone" command.
• Generate a physics process with MG5 and save
results with the "output" command. Extract thefunction "matrix" part starting from:
REAL*8 FUNCTION MATRIX(P,NHEL,IC)and copy it as "matrix.f90." Then, modify thecharacter "C" at the beginning of lines to "!."
Replace "INCLUDE 'coupl.inc'" with "usesmcupl."Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-4 (cont'd)• bfunc.f90:- generate two-body phase space: ph2bdy- select helicity combination as "nhel"• As initialization in "userin":- call sminit - initialize SM parameters- call hlmode(1) - initialize the helicity
amplitude library- initialize initial state particles

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-4 (cont'd)
1. Learn how to prepare the SM parameters.
2. Learn phase space generations.
3. Try other physics processes by yourself.
You will learn more about how to use "MG5"!

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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Tutorial-5
Calculate cross-sections of the Standard
Model physics processes with decays:

e e+ ! Z(! µ µ+ )H(! bb)<latexit sha1_base64="qYHdnPhZNAbm7kkP9N5EGB621e4=">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</latexit>

1. The procedure is almost the same as"Tutorial-4."
2. Try the four-body phase space withdecays.
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

26

Generation of phase space
Phase space generation for the decay of aresonance:
for the generation of the Breit-Wigner
resonant shape the following variabletransformation is effective.

x = tan ✓<latexit sha1_base64="XMwKyMY0mJ9EUbs8YrAxLyMwzeo=">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</latexit>

12
dx = d tan ✓ =
d✓
=
(1
+
tan
✓) d✓
2
cos ✓
<latexit sha1_base64="SY16SnXKPsvaxo5wug9YAiywqF4=">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</latexit>dx!
= d✓ = d arctan x2
1+x<latexit sha1_base64="6nOkHUy9KFA/TJ+h+3PkdN15Zrc=">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</latexit>

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

27

Generation of phase space
Phase space generation for the decay of aresonance:
dq 2
dq 2
1
=
✓
◆2
2
2
2
2
2
(q
m ) + (m )(m )q 2 m2+1m<latexit sha1_base64="2Y0nxAg0Dn09J3pE/3xK9tbIy+A=">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</latexit>

q2<latexit sha1_base64="45m9+pPu+uQP+4DRtOyXZ8SyW2I=">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</latexit>

m2

d m
1
1
q 2 m2==d arctan✓◆2mm
m
q 2 m2
+1
m

(q 2

2 2
2
2
2
m
)
+
(m
)
q
m
dq 2 =
d arctanmm(q 2 m2 )2 + (m )2=d✓m<latexit sha1_base64="9KDbaAHIZgyF2ydc0J/mts5OXtY=">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</latexit><latexit sha1_base64="MXzmgQbOyKqCAQJGh/ICAyziKxg=">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</latexit>

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

28

Generation of phase space
Phase space generation for the decay of aresonance:<latexit sha1_base64="9KDbaAHIZgyF2ydc0J/mts5OXtY=">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</latexit>

dq 2 =<latexit sha1_base64="MXzmgQbOyKqCAQJGh/ICAyziKxg=">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</latexit>

=

(q 2

(q 2

m2 )2 + (m )2
q 2 m2d arctanmmm2 )2 + (m )2d✓m<latexit sha1_base64="uKJd+VXHNneApbj7XEwVcyh+KC0=">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</latexit>✓ = arctan

q2

m2
m

q 2 = m tan ✓ + m2
<latexit sha1_base64="jwrVCPBj09blWfPbmhUPOGj28AA=">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</latexit>

2
2
qmin
< q 2 < qmax
) ✓min < ✓ < ✓max
<latexit sha1_base64="saxyQ/P8BrJvX3G8twTccvxDKy0=">AAAHzXiclVNLb9NAEN4CISU82sIFiYtFFeQobbVOeAkRqYID3OiDPqRuYzn2trHqV+xNSTHmisQf4MAJJIQQP4MLJ24c+hMQx1biAojZ9RqnD9KyUbzj2fm++WZm3QocO2IYbw+dOHmqcLo4fKZ09tz5CyOjYxcXI78bmnTB9B0/XG4ZEXVsjy4wmzl0OQip4bYcutTauM/PlzZpGNm+95htBXTVNdY9e802DQYufaywXrbITNvW6w1ie0yPVVfXqq5eqzRrSTNWSdQJWRwlk65e5y5irYWGGVsdsOMaCeyEw5SUo6bONDrVoF7JfTW10wiAMQDGUrmhSLwG4CS365ypeajnOKK4GuJDmbwLMWlRZiTEoWtMlYRCbZRMyFeApQ4S2uttVhmIhdQ8mFPk+FrmkgylMmG2S6NU7qQGKqtaYhHTjwhrA6deF0e4mTbNIgHvuXJYfNtgcQpKdG0fSpwBFE5KZcXqKQ1oMzO8NB7eSNZATtXkTU6ZeBg3AKVAmKpVOSwPqJAJGaEQwoPSwoww9J9IUquXxFq1xzFJmlfmtIgRmsCm9JRSmUvKFUFfsvvCYbEq9kmXPyv8UVVd8sBwXaOS8R6A7AkgE9k5r1HMKXMQRnssYlsOVfqyJP1+SZVNTQjQEp6S19zXvSxQtCtbUtd/Z9vLIpaUPpDqMNQxSjmsjHxERH4PfYrzWtOW983gqHH1Y2WGAQnEzcrUHZs6v7n5JT9eqqwa6ey7lkpVEYBSR49J6Cqu7SUi+G46Ab7LE6OXnpC5/HsoyW/6LxbiJXFm5GguBZY+Oo6nsFjKQUOTxjiSa8YffY8IspCPTNRFLqLIQwxsBxkogt8K0hBGAfhWUQy+ECxbnFOUoBJguxBFIcIA7wY81+FtRXo9eOeckUCbkMWBfwhIBZXxV/wB7+DP+CP+hn/+kysWHFzLFuytFEsDfeTl5fkfR6Jc2Blq56iBmhlaQ7eFVhu0B8LDqzBT/ObTVzvzd+bK8TX8Fn8H/W/wNv4EFXibu+a7WTr3GvEBaPvbfdBYrE1pN6duzF4fn74nRzGMrqCrSIV+30LT6CGaQQvILHwp7BZ+FX4XHxW7xWfF52noiSGJuYT2rOKLP1dXcus=</latexit>

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

29

Tutorial-5 (cont'd)• bfunc.f90:- generate two-body phase space with decays:
ph2dcyThe implementation of the variabletransformations in the previous pages is
included in this subroutine. Please check thedetails.
- You can include kinematical cuts in"bfunc.f90", which simulate the detector
acceptance.
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

30

Summary•We already have many excellent software tools
for physics studies. They have much automaticfunctionality, which helps your analysis.
•Still, I want you to understand what
procedures/technologies such software uses, notjust using it•I hope my examples help you experience the MC
integration and the calculation of cross-sections
and understand the calculation processes.

Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

31

Summary•If possible, try to modify these codes and test
them by yourself:- check Ncall dependence of integration
accuracies,- apply kinematical cuts to final stateparticles,- try other physics processes which you havesome interest- etc.
Presented by J. Kanzaki at Iwate Collider School in February 28, 2023

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