View source for Automatic calculation of particle interaction or decay

{{Short description|Application of computational physics}}
{{Expert needed|physics|date=September 2023}}

The '''automatic calculation of particle interaction or decay''' is part of the [[computational particle physics]] branch. It refers to computing tools that help calculating the complex particle interactions  as studied in [[high-energy physics]], [[astroparticle physics]] and [[cosmology]]. The goal of the automation is to handle the full sequence of calculations in an automatic (programmed) way: from the [[Lagrangian (field theory)|Lagrangian]] expression describing the physics model up to the [[Cross section (physics)|cross-section]]s values and to the event generator software.

==Overview==
Particle accelerators or colliders produce collisions (interactions) of particles (like the [[electron]] or the [[proton]]). The colliding particles form the ''Initial State''. In the collision, particles can be annihilated or/and exchanged producing possibly different sets of particles, the ''Final States''. The Initial and Final States of the interaction relate through the so-called scattering matrix ([[S-matrix]]).

For example, at [[LEP]], {{nowrap|{{Subatomic particle|Positron}} + {{Subatomic particle|Electron}} &rarr; {{Subatomic particle|Antielectron}} + {{Subatomic particle|Electron}}}},  or {{nowrap|{{Subatomic particle|Antielectron}} + {{Subatomic particle|Electron}} &rarr; {{Subatomic particle|Antimuon}} + {{Subatomic particle|muon}}}} are processes where the ''initial state'' is an electron and a positron colliding to produce an electron and a positron or two muons of opposite charge: the ''final states''. In these simple cases, no automatic packages are needed and [[cross section (physics)|cross-section]] analytical expressions can be easily derived at least for the lowest approximation: the [[Born approximation]] also called the leading order or the tree level (as [[One-loop Feynman diagram|Feynman diagram]]s have only trunk and branches, no loops).

But particle physics is now requiring much more complex calculations like at [[LHC]] <math>p p \rarr n_\text{jets}</math> where <math>p</math> are protons and <math>n_\text{jets}</math> is the number of [[particle jet|jets of particle]]s initiated by proton constituents ([[quarks]] and [[gluons]]). The number of subprocesses describing a given process is so large that automatic tools have been developed to mitigate the burden of hand calculations.

'''Interactions''' '''at''' '''HighahEnergih''' open a large spectrum of possible final states and consequently increase the number of processes to compute.

High precision experiments impose the calculation of [[higher order terms|higher order calculation]], namely the inclusion of subprocesses where more than one [[virtual particle]] can be created and annihilated during the interaction lapse creating so-called ''[[loop integral|loop]]s'' which induce much more involved calculations.

Finally new theoretical models like the [[supersymmetry]] model ([[Minimal Supersymmetric Model|MSSM]] in its minimal version) predict a flurry of new processes.

The automatic packages, once seen as mere teaching support, have become, this last 10 years an essential component of the data simulation and analysis suite for all experiments.
They help constructing [[event generator]]s and are sometimes viewed as ''generators of event generators'' or ''Meta-generators''.

A particle physics model is essentially described by its [[Lagrangian (field theory)|Lagrangian]]. To simulate the production of events through [[event generator]]s, 3 steps have to be taken. The Automatic Calculation project is to create the tools to make those steps as automatic (or programmed) as possible:

'''I''' Feynman rules, coupling and mass generation
* [http://theory.sinp.msu.ru/~semenov/lanhep.html LanHEP] is an example of [[Feynman rules]] generation.
* Some model needs an additional step to compute, based on some parameters, the mass and coupling of new predicted particles.

'''II''' Matrix element code generation: Various [[#Matrix element calculation methods|methods]] are used to automatically produce the [[Matrix element (physics)|matrix element]] expression in a computer language ([[Fortran]], [[C (programming language)|C/C++]]). They use values (i.e. for the masses) or expressions (i.e. for the couplings) produced by step '''I''' or model specific libraries constructed ''by hands'' (usually heavily relying on [[Computer algebra]] languages). When this expression is integrated (usually numerically) over the internal degrees of freedom it will provide the total and differential cross-sections for a given set of initial parameters like the ''initial state'' particle energies and [[vacuum polarization|polarization]].

'''III''' Event generator code generation: This code must them be interfaced to other packages to fully provide the actual ''final state''. The various effects or phenomenon that need to be implemeted are:
* [[Initial state radiation]] and [[beamstrahlung]] for {{Subatomic particle|Antielectron}}{{Subatomic particle|Electron}} initial states.
* [[Parton distribution functions]] describing the actual content in terms of [[gluons]] and [[quarks]] of the p or p-bar initial state particles
* [[Parton showering]] describing the way final state quarks or gluons due to the [[Colour confinement|QCD confinement]] generate additional quark/gluon pairs generating a so-called shower of [[partons]] before transforming into hadrons.
* [[Hadronization]] describing how the final quark pairs/triplets form the visible and detectable hadrons.
* [[Underlying event]] takes care of the way the rest, in term of constituent, of the initial protons also contribute to any given event.

The interplay or ''matching'' of the precise matrix element calculation and the approximations resulting from the simulation of the ''parton shower'' gives rise to further complications, either within a given level of precision like at [[Leading-order term|leading order (LO)]] for the production of n jets or between two levels of precision when tempting to connect matrix element computed at [[Leading-order term#Next-to-leading order|next-to-leading]] (NLO) (1-loop) or next-to-next-leading order (NNLO) (2-loops) with LO partons shower package.

Several methods have been developed for this matching, including: [[Subtraction methods]].

But the only correct way is to match packages at the same level theoretical accuracy like the NLO matrix element calculation with NLO parton shower packages. This is currently in development.

== History ==
The idea of automation of the calculations in high-energy physics is not new. It dates back to the 1960s when packages such as [[Schoonschip|SCHOONSCHIP]] and then [[reduce (computer algebra system)|REDUCE]] had been developed.

These are symbolic manipulation codes that automatize the algebraic parts of a matrix element evaluation, like traces on Dirac matrices and contraction of Lorentz indices. Such codes have evolved quite a lot with applications not only optimized for high-energy physics like [[FORM (symbolic manipulation system)|FORM]] but also more general purpose programs like [[Mathematica]] and [[Maple (software)|Maple]].

Generation of QED Feynman graphs at any order in the [[coupling constant]] was automatized in the late 70's[15]. One of the first major application of these early developments in this field was the calculation of the anomalous magnetic moments of the electron and the muon[16]. The first automatic system incorporating all the steps for the calculation of a cross section, from Feynman graph generation, amplitude generation through a REDUCE source code that produces a FORTRAN code, phase space integration and event generation with BASES/SPRING[17] is GRAND[18]. It was limited to tree-level processes in QED. In the early nineties, a few groups started to develop packages aiming at the automation in the SM[19].<ref>
{{cite conference
 |last=Kaneko 
 |first=T. 
 |year=1990 
 |title=Automatic calculation of Feynman amplitudes 
 |url=http://www.slac.stanford.edu/spires/find/hep?key=5471150
 |archive-url=https://archive.today/20121211161254/http://www.slac.stanford.edu/spires/find/hep?key=5471150
 |url-status=dead
 |archive-date=2012-12-11
 |book-title=New computing techniques in physics research 
 |pages=555 
}}</ref><ref>
{{cite journal
 |last=Boos |first=E.E
 |year= 1994
 |title=Automatic calculation in high-energy physics by Grace/Chanel and CompHEP.
 |journal=[[International Journal of Modern Physics C]]
 |volume=5 |issue=4 |page=615
 |doi=10.1142/S0129183194000787
|bibcode = 1994IJMPC...5..615B |display-authors=etal}}</ref><ref>
{{cite journal
 |last=Wang |first=J.-X.
 |year= 1993
 |title=Automatic calculation of Feynman loop-diagrams I. Generation of a simplified form of the amplitude
 |journal=[[Computer Physics Communications]]
 |volume=77 |issue=2 |page=263
 |doi=10.1016/0010-4655(93)90010-A
|bibcode = 1993CoPhC..77..263W }}</ref><ref>
{{cite conference
 |last=Kaneko 
 |first=T. 
 |author2=Nakazawa, N. 
 |year=1995 
 |title=Automatic calculation of two loop weak corrections to muon anomalous magnetic moment 
 |url=http://www.slac.stanford.edu/spires/find/hep?key=3147525
 |archive-url=https://archive.today/20121210100247/http://www.slac.stanford.edu/spires/find/hep?key=3147525
 |url-status=dead
 |archive-date=2012-12-10
 |book-title=New computing techniques in physics research 
 |pages=173 
 |arxiv=hep-ph/9505278 
|bibcode = 1995hep.ph....5278K }}</ref><ref>
{{cite conference
 |last=Jimbo 
 |first=M. 
 |author2=(Minami-Tateya Collaboration) 
 |year=1995 
 |title=Automatic calculation of SUSY particle production 
 |url=http://inspirehep.net/record/406343
 |book-title=High energy physics and quantum field theory 
 |page=155 
 |arxiv=hep-ph/9605414 
 |display-authors=etal 
|bibcode = 1996hep.ph....5414J }}</ref><ref>
{{cite journal
 |last=Franzkowski |first=J.
 |year=1997
 |title=Automatic calculation of massive two-loop self-energies with XLOOPS
 |journal=[[Nuclear Instruments and Methods in Physics Research A]]
 |volume=389 |issue=1–2 |pages=339–342
 |doi=10.1016/S0168-9002(97)00121-6
 |arxiv=hep-ph/9611378|bibcode = 1997NIMPA.389..333F }}</ref><ref>
{{cite arXiv
 |last=Brucher|first=L.
 |year=2000
 |title=Automatic Feynman diagram calculation with xloops: A Short overview
 |eprint=hep-ph/0002028
}}</ref><ref>
{{cite conference
 |last=Perret-Gallix 
 |first=D. 
 |year=1999 
 |title=Automatic amplitude calculation and event generation for collider physics: GRACE and CompHEP 
 |url=http://www.slac.stanford.edu/spires/find/hep?key=4515617
 |archive-url=https://archive.today/20121211101138/http://www.slac.stanford.edu/spires/find/hep?key=4515617
 |url-status=dead
 |archive-date=2012-12-11
 |book-title=High energy physics and quantum field theory 
 |pages=270 
}}</ref><ref>
{{cite journal
 |last=Belanger |first=G.
 |year=2006
 |title=Automatic calculations in high energy physics and GRACE at one-loop
 |journal=[[Physics Reports]]
 |volume= 430 |issue=3 |pages=117–209
 |doi=10.1016/j.physrep.2006.02.001
 |arxiv=hep-ph/0308080|bibcode = 2006PhR...430..117B |s2cid=7049291
 |display-authors=etal}}</ref><ref>
{{cite journal
 |last=Fujimoto |first=J.
 |year=2004
 |title=Automatic one-loop calculation of MSSM processes with GRACE
 |journal=[[Nuclear Instruments and Methods in Physics Research A]]
 |volume=534 |issue=1–2 |page=246
 |doi=10.1016/j.nima.2004.07.095
 |arxiv=hep-ph/0402145|bibcode = 2004NIMPA.534..246F |s2cid=7717301
 |display-authors=etal}}</ref>

== Matrix element calculation methods ==
{{Main|Feynman diagram}}

=== Helicity amplitude ===
Feynman amplitudes are written in terms of spinor products of wave functions for massless fermions, and
then evaluated numerically before the amplitudes are squared. Taking into account fermion masses implies that Feynman amplitudes are decomposed into vertex amplitudes by splitting the internal lines into wave function of fermions and polarization vectors of gauge bosons.

All helicity configuration can be computed independently.

=== Helicity amplitude squared ===
The method is similar to the previous one, but the numerical calculation is performed after squaring the Feynman Amplitude. The final expression is shorter and therefore faster to compute, but independent helicity information are not anymore available.

=== Dyson-Schwinger recursive equations ===
The scattering amplitude is evaluated recursively through a set of [[Dyson-Schwinger equation]]s. The computational cost of this algorithm grows asymptotically as 3<sup>n</sup>, where n is the number of particles involved in the process, compared to n! in the traditional Feynman graphs approach. Unitary gauge is used and mass effects are available as well. Additionally, the color and helicity structures are appropriately transformed so the usual summation is replaced by the Monte Carlo techniques.<ref>
{{cite journal
 |last1=Kanaki |first1=A. |last2=Papadopoulos |first2=C.G.
 |year=2000
 |title=HELAC: A Package to compute electroweak helicity amplitudes
 |journal=[[Computer Physics Communications]]
 |volume=132 |pages=306–315
 |doi=10.1016/S0010-4655(00)00151-X
 |arxiv=hep-ph/0002082|bibcode = 2000CoPhC.132..306K
 |issue=3 |s2cid=14533093 }}</ref>

== Higher order calculations ==

<ref>
{{cite journal
 |last=Belanger |first=G.
 |year=2006
 |title=Automatic calculations in high energy physics and Grace at one-loop
 |journal=[[Physics Reports]]
 |volume=430 |issue= 3|pages=117–209
 |doi= 10.1016/j.physrep.2006.02.001
 |arxiv=hep-ph/0308080|bibcode = 2006PhR...430..117B |s2cid=7049291
 |display-authors=etal}}</ref>
{{Empty section|date=August 2022}}

== Additional package for Event generation ==
The integration of the "matrix element" over the multidimensional internal parameters phase space provides the total and differential cross-sections. Each point of this phase space is associated to an event probability. This is used to randomly generate events closely mimicking experimental data. This is called event generation, the first step in the complete chain of event simulation. The initial and final state particles can be elementary particles like electrons, muons, or photons but also partons ([[protons]] and [[neutrons]]).

More effects must then be implemented to reproduce real life events as those detected at the colliders.

The initial electron or positron may undergo radiation before they actually interact: initial state radiation and beamstrahlung.

The bare partons that do not exist in nature (they are confined inside the hadrons) must be so to say dressed so that they form the known hadrons or mesons. They are made in two steps: parton shower and hadronization.

When the initial state particles are protons at high energy, it is only their constituents which interact. Therefore, the specific parton that will experience the "hard interaction" has to be selected. Structure functions must therefore be implemented. The other parton may interact "softly", and must also be simulated as they contribute to the complexity of the event: the ''underlying event''.

=== Initial state radiation and beamstrahlung ===
{{Empty section|date=August 2015|section=}}

=== Parton shower and Hadronization ===
{{Empty section|date=August 2015|section=}}

==== At leading Order (LO) ====
{{Empty section|date=August 2015|section=}}

==== At Next-to-Leading order (NLO) ====
{{Empty section|date=August 2015|section=}}

=== Structure and Fragmentation Functions ===
The [[fragmentation function]] (F.F.) is a probability distribution function. It is used to find the density function of fragmented mesons in hadron -hadron collision.

The [[structure function]], like the fragmentation function, is also a probability density function. It is analogous to the [[structure factor]] in solid-state physics.

=== Underlying event ===
{{Empty section|date=August 2015|section=}}

== Model specific packages ==
{{Empty section|date=August 2015|section=}}

=== SM ===
=== MSSM ===
Automatic software packages can be useful in exploring a number of [[Beyond the Standard Model]] (BSM) theories, such as the [[Minimal Supersymmetric Standard Model]] (MSSM), to predict and understand possible particle interactions in future physics experiments.

== Related computational issues ==
Several computation issues need to be considered for automatic calculations. For example one scenario is the fact that special  functions often need to be calculated in these software packages, both/either algebraically and/or numerically. For  algebraic calculations, symbolic packages e.g. Maple, Mathematica often need to consider [[abstract structure|abstract]], [[mathematical structure]]s in subatomic particle collisions and emissions.

=== Multi-dimensional integrators ===
{{further|Curse of dimensionality}}{{Empty section|date=August 2015|section=}}

=== Ultra-High Precision Numerical computation ===
{{further|Arbitrary precision arithmetic|extended precision}}{{Empty section|date=August 2015|section=}}

== Existing Packages ==

=== Feynman rules generators ===
* [https://server06.fynu.ucl.ac.be/projects/feynrules FeynRules]{{dead link|date=October 2016 |bot=InternetArchiveBot |fix-attempted=yes }}
* [http://theory.sinp.msu.ru/~semenov/lanhep.html LanHEP]

=== Tree Level Packages ===
{| class="wikitable"
|-
! Name
! Model
! Max FS
! Tested FS
! Short description
! Publication
! Method
! Output
! Status
|-
| [https://launchpad.net/madgraph5 MadGraph5]
| Any Model
| 1/2->n
| 2->8
| complete, massive, helicity, color, decay chain
| [https://web.archive.org/web/20110706132400/https://server06.fynu.ucl.ac.be/projects/madgraph what is MG5]
| HA (automatic generation) 
| Output
| [https://launchpad.net/madgraph5 PD]
|-
| [[Grace (particle physics)|Grace]]
| SM/MSSM
| 2->n
| 2->6
| complete,massive,helicity,color
| [http://minami-home.kek.jp/grace/gracedoc.ps Manual v2.0]
| HA
| Output
| [http://minami-home.kek.jp PD]
|-
| [[CompHEP]]
| Model
| Max FS
| Tested FS
| Short description
| Publication
| method
| Output
| Status
|-
| [https://launchpad.net/calchep CalcHEP]
| Model
| Max FS
| Tested FS
| Short description
|  [https://theory.sinp.msu.ru/~pukhov/calchep.html Publication]
| Method
| Output
| Status
|-
| Sherpa
| SM/MSSM
| 2->n
| 2->8
| massive
| [https://archive.today/20121215033929/http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+EPRINT+HEP-PH/0311263 publication]
| HA/DS
| Output
| [http://www.sherpa-mc.de PD]
|-
| GenEva
| Model
| Max FS
| Tested FS
| Short description
| Publication
| Method
| Output
| Status
|-
| [[HELAC]]
| Model
| Max FS
| Tested FS
| Short description
| Publication
| Method
| Output
| Status
|-
| Name
| Model
| Max FS
| Tested FS
| Short description
| Publication
| Method
| Output
| Status
|}
'''Status''': ''PD'': Public Domain, <br />
'''Model''': ''SM'': [[Standard Model]], ''MSSM'': [[Minimal Supersymmetric Standard Model]] <br />
'''Method''': ''HA'': [[#Helicity amplitude|Helicity Amplitude]], ''DS'': [[#Dyson Schwinger|Dyson Schwinger]] <br />
'''Output''': ''ME'': Matrix Element, ''CS'': Cross-Sections, ''PEG'': Parton level Event Generation, ''FEG'': Full particle level Event Generation

=== Higher-order Packages ===
{| class="wikitable"
|-
! Name
! Model
! Order tested
! Max FS
! Tested FS
! Short description
! Publication
! Method
! Status
|-
| Grace L-1
| SM/MSSM
| 1-loop
| 2->n
| 2->4
| complete,massive,helicity,color
| NA
| Method
| NA
|-
| Name
| Order
| Model
| Max FS
| Tested FS
| Short description
| Publication
| Method
| Status
|}

=== Additional package for Event generation ===

== References ==
<references/>

{{DEFAULTSORT:Automatic Calculation Of Particle Interaction Or Decay}}
[[Category:Computational particle physics]]