1Prof.W.FisherPhysics493/803

PHY493/803Spring2017,IntrotoElementaryParticlePhysics

Homework7,Duedate:Wednesday,April26,2017

Pleaseclearlystateanyassumptions,showallyourwork,numbertheequations,andindicatelogicalconnectionsbetweenthelines.

1. (5x20pts=100ptstotal)IntheCHOOZexperiment,aneutrinodetectorwaspositionedadistanceL≈1kmfromanuclearreactoremittingantineutrinosofmeanenergyE≈3MeV.Thenumberofantineutrinointeractionsobservedwasconsistentwiththenumberpredictedassumingnoantineutrinooscillations,givingtheresult𝑃 𝜈! → 𝜈! = 1.01 ± 0.04(theuncertaintyisreportedatthe68%ConfidenceLevel).a) Showthatneutrinooscillationsassociatedwiththe(solar)mass-squared

difference ∆𝑚!"! ≈ 7.5×10!!canbeneglectedfortheCHOOZ

experiment,andthat

𝑃 𝜈! → 𝜈! ≈ 1 − sin!2𝜃!"sin!∆!"where

∆!"≡∆𝑚!"

! 𝐿4𝐸

b) Inthelimit ∆𝑚!"! ≫ (𝐸/𝐿),explainwhyagivenmeasurement,𝑃!"#,of

thesurvivalprobability𝑃 𝜈! → 𝜈! determinestheneutrinomixingtobesin!2𝜃!" = 2(1 − 𝑃!"#).

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c) Inthelimit ∆𝑚!"! ≪ (𝐸/𝐿),showthatagivenmeasurement,𝑃!"#,of

thesurvivalprobability𝑃 𝜈! → 𝜈! determinestheneutrinomixingtobesin!2𝜃!" ∝ 1/(∆𝑚!"

! )!,withconstantofproportionality𝐶 = (1 − 𝑃!"#)(4𝐸/𝐿)!.

d) ThenullresultfromtheCHOOZexperiment,𝑃 𝜈! → 𝜈! = 1.01 ± 0.04,canbeusedtoexcludearegionofthe(sin!2𝜃!", ∆𝑚!"

! )parameterspace.Thisisconventionallypresentedastheregionwhichcanbeexcludedatthe90%ConfidenceLevel,whichfortheCHOOZexperimentencompassesallvaluesof(sin!2𝜃!", ∆𝑚!"

! )thatwouldgiveasurvivalprobability𝑃 𝜈! → 𝜈! < 0.92.Inthefiguregivenbelow,publishedbytheCHOOZcollaboration,thecurvescorrespondtothecontour 𝑃 𝜈! → 𝜈! = 0.92andtheexcludedregionliesaboveandtotherightofthecurves.(Thetwosimilarcurvescorrespondtoslightlydifferentstatisticalapproachestotheanalysisofthedata).

Usetheresultsderivedabovetojustifytheapproximateshapeandpositionoftheexclusioncontourintheregionsclosetotheinterceptswiththeupperhorizontalandright-handverticalaxes.Giveaqualitativeexplanationoftheshapeoftheremainderofthecontour.Evaluatethesurvivalprobability𝑃 𝜈! → 𝜈! fortwopointslyingoneithersideofthecontour,(sin!2𝜃!", ∆𝑚!"

! )=(0.5, 5×10!! eV!)and(0.5, 5×10!! eV!).

e) Experimentsstudyingatmosphericneutrinooscillationsindicateamass-squaredsplittingintherange ∆𝑚!"

! ≈ 2 − 3×10!! eV!.Whatconstraintscannowbeplacedontheangle𝜃!"?

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Forthoseinterested,youcanreadmoreabouttheCHOOZexperimenthere:http://arxiv.org/abs/hep-ex/0301017