These slides describe an example Monte Carlo simulation of a physical process

Thecodeisin:

/home/pi/physrpi/campagnari/python/muonSim.py

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Wewill“simulate”theprocessInparticularweareinterestedinthemomentumdistributionofthemuons.

z-axis

Whatreallyhappensisup-quarkfromoneprotonandanti-downquarkfromtheotherproton

proton proton

momentumP momentumPW

upquark anti-downquarkmomentumx1P momentumx2P

W

x1andx2arethefractionoftotalprotonmomentumcarriedbythequarks

pp → W+ → μ+ν

ud̄ → W+ → μ+ν

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ThePDFsforx(up-quark)andx(anti-downquark)insideaproton

Note:nobodyknowshowtocalculatethese!!!!Theyare“measured”

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upquark anti-downquarkmomentumx1P momentumx2P

W

momentum(x1-x2)PW

spin

anti-downquark upquarkmomentumx2P momentumx1P

W

momentum(x1-x2)P

Wspin

z-axis

Interestingphysics,theWispolarized.• Thisisaweakinteractionprocess• Onlylefthandedquarksandrighthandedantiquarkscontribute• Parityviolation….theworldislefthanded!

• AsaresulttheWwhichisS=1willhaveSz=±1andneverSz=0

InreallifeonewouldhavetogeneratethezcomponentoftheWmomentum,ie,pickthex’sappropriately.However1. Thisiscomplicated(therearededicatedlibrariestodothis--probablynotinpython)2. Notimportantforwhatwecareabout(moreonthislater)Solet’sworkintheWrestframe(equivalenttox1=x2) �4

W+spin

µ+

ν

α

z-axis

θ

ThedirectionofthemuonwithrespecttothespinhasPDF:

P(cosα)dcosα ∼ (1+cosα)2dcosα

Sincethespincanbeinthe+zor–zdirectionwithequalprobability,thepdfforthespaceangleθis

P(cosθ)dcosθ ∼ (1+cos2θ)dcosθ

MassofW: M=80.4GeV/c2Massofµ: m=0.105GeV/c2~0Massofν: m<10-9GeV/c2~0

InthemasslessapproximationbothmuonandneutrinohavemomentumP=½Mc=40.2GeV/c

Thereforeweshould• Pickacosθaccordingto• Givethemuonthismomentum

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Physicsaside:theang.distributionisfromtheQMrotationmatrices.Hastodowithchangingtheaxisofquantizationforangularmomentum

A complicationHeisenberguncertaintyprincipleΔEΔt~h/2π• Energyismassandmassisenergy• WbosonsliveforaveryshortΔt• Thereforethe”mass”ofaWbosoninagivencollisionisnotalways80.4GeV• Itis“broadened”accordingtotherelativisticBreitWignerfunction

WhereM0=80.4GeV/c2Γ =2.2GeV/c2Cisanormalizationconstant

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Another ingredientWeare“simulating”ppcollisionsatCERNThetwobigexperimentsAtlasandCMSareembeddedinsolenoidswithconstantB-fieldinthezdirection.

Chargedparticlesbendinthemagneticfield.Theirtrajectoriesaremeasured.KnowingtheB-fieldandthecurvature,canmeasure“transversemomentum”

PTismeasuredwithfiniteresolution.About1.5%(Itismorecomplicatedthanthat,butOK)

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Another complicationTheWactuallyhasasomePTaswell

Realdata

Thismade-upfunctionkindoflooklikeit.Sowewilluseit

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Plan of work to obtain the PT distribution of muons1. PickaWmassMfromaBreitWigner2. PickacosθforthemuonintheWrestframe3. P=½McandPT=Psinθ =½Mcsinθ4. PickatransversemomentumfortheWfromthedistributionofthepreviouspage.

CallthatQT.Thex-axiswillbethedirectionofQT

• ChoosingQTalongx-axis(ory-axis)isarbitrary.Makesalgebrain(6)atinybiteasier5. PickarandomazimuthalangleφforthemuonintheWrestframe

• Px=PTsinφ Py=PTcosφ 6. BoostthemuonmomentumfromtherestframetotheLABframe

• Theboostisinthex-direction.βγ =cQT/M

7. CalculatethePTofthemuonintheLABaftertheboost8. “Smear”itbyitsresolution9. Plotit

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What about the motion of the W in z?

• Wewantedtolookatthetransversemomentumofthemuon• CangofromaWwithPz=0toaWwithfinitePzbyperformingaboostofthesysteminthez-direction.• Butaboostinthezdirectionleavesallxandymomentumcomponentsunchanged• Therefore:itdoesnotmatterthatweworkedwithPz=0• Atleastfortheonesimpleplotwewanttomake

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CDF experiment, Fermilab, 2007

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