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WOLFRAM NOTEBOOK
Neutrino Propagation in Supernovae with Non-Standard Interactions
(beyond the Standard Model of Elementary Particles)
Neutrino Propagation in Supernovae with Non-Standard Interactions
(beyond the Standard Model of Elementary Particles)
(beyond the Standard Model of Elementary Particles)
A lesson on how Wolfram Language can be used to create a simple toy model
that can efficiently guide towards understanding general features of complex systems
that can efficiently guide towards understanding general features of complex systems
Setting up a system of interacting neutrinos and antineutrinos with two distinct types
(electron neutrino/antineutrino or non-electron neutrino/antineutrino)
Setting up a system of interacting neutrinos and antineutrinos with two distinct types
(electron neutrino/antineutrino or non-electron neutrino/antineutrino)
(electron neutrino/antineutrino or non-electron neutrino/antineutrino)
COMMENT:
In case you would like to learn more about the system, please, see: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.093007
In case you would like to learn more about the system, please, see: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.093007
◼
First we define a toy function to describe the shape of the background potential caused by neutrinos interacting with other non-neutrino background particles
(including a non-standard interaction contributions which are assumed to behave similarly as weak interactions):
(including a non-standard interaction contributions which are assumed to behave similarly as weak interactions):
In[1]:=
potBG[yshiftBG_,xshiftBG_,widthBG_,r_]:=(yshiftBG-((r-xshiftBG)/widthBG)^2);(*yshiftBGcanbeusedtoadjustashiftalongy-axis,xshiftBGcanbeusedtoadjustashigtalongx-axis,widthBGcanbbeusedtoadjustwidthofthepotential,rdescribestheradialdistancefromtheemissionlocation*)
◼
Next we define a function to describe the shape of the background potential caused by neutrinos interacting with other surrounding neutrinos and antineutrinos:
In[2]:=
potNu[scaleNu_,steepnessNu_,r_]:=scaleNuExp[-r/steepnessNu];potAnu[asymmetry_,scaleNu_,steepnessNu_,r_]:=-asymmetrypotNu[scaleNu,steepnessNu,r];(*potNu[scaleNu_,steepnessNu_,r_]describestheneutrinopotential,scaleNucanbeusedtoadjustintialscalingofthepotential,steepnessNucanbeusedtoadjuststeepnessofthepotential,potAnu[asymmetry_,scaleNu_,steepnessNu_,r_]describestheantineutrinopotential(whichisassumedtoscalelinearlywithrespecttotheneutrinopotential),asymmetrydescribestheinitialasymmetrybetweenneutrinosandantineutrinos,rdescribestheradialdistancefromtheemissionlocation*)
◼
Now we can visualize the different possible extreme configurations for the potentials with some interesting input values:
In[4]:=
yBG=400(*anexamplevalueforyshiftBG*);xBG=500(*anexamplevalueforxshiftBG*);wBG=8(*anexamplevalueforwidthBG*);scNu=25000(*anexamplevalueforscaleNu*);stNu=80(*anexamplevalueforsteepnessNu*);as=0.9(*anexamplevaluefortheinitialneutrino-antineutrinoasymmetry*);rmin=20(*initialradialevaluationpoint*);rmax=700(*finalradialevaluationpoint*);Plot[{potBG[xBG,yBG,wBG,r](*backgroundpotential*),-potBG[xBG,yBG,wBG,r](*inverseofbackgroundpotential*),potNu[scNu,stNu,r]+potAnu[as,scNu,stNu,r](*totalneutrinopotential,bothwithelectrontypeneutrinoandantineutrino*),-potNu[scNu,stNu,r]-potAnu[as,scNu,stNu,r](*totalneutrinopotential,bothwithnon-electrontypeneutrinoandantineutrino*),potNu[scNu,stNu,r]-potAnu[as,scNu,stNu,r](*totalneutrinopotentialwithelectrontypeneutrinoandnon-electrontypeantineutrino*),-potNu[scNu,stNu,r]+potAnu[as,scNu,stNu,r]}(*totalneutrinopotentialwithnon-electrontypeneutrinoandelectrontypeantineutrino*),{r,rmin,rmax},FrameTrue,PlotStyle{Cyan,{Cyan,Dashed},Red,{Red,Dashed},{Red,DotDashed},{Red,Dotted}},FrameLabel{"Radial Distance [a.u.]","Interaction Potential [a.u.]"},FrameTicksStyleThick,LabelStyleDirective[FontSize20],PlotLegends{"Background Potential","Inverse Background Potential","Neutrino Potential (both electron type)","Neutrino Potential (both non-electron type)","Neutrino Potential (electron neutrino, non-electron antineutrino)","Neutrino Potential (non-electron neutrino, electron antineutrino)"}]
Out[12]=
Simulating the evolution of neutrinos and antineutrinos
Simulating the evolution of neutrinos and antineutrinos
In this section we will set up the equations that allow us to follow the evolution of neutrinos and antineutrinos using density matrix formalism.
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First, we need to define Hamiltonian describing non-interacting neutrinos:
In[13]:=
mix=0.15(*aquantitydescribinginternalquantummixingbetweenthedifferentneutrinotypes*);Hv=0.5{{-Cos[2mix],Sin[2mix]},{Sin[2mix],Cos[2mix]}}(*Hamiltoniandescribingtwotypesofnon-interactingmixedneutrinos*);
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Next, we define the Hamiltonian describing the “non-neutrino” background contributions as described in the previous section:
In[15]:=
Hbg=potBG[xBG,yBG,wBG,r]DiagonalMatrix[{1,0}](*noticethecouplingonlytoelectroncomponent!*);
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Finally, we define the Hamiltonian describing the neutrino interactions with other neutrinos and antineutrinos using neutrino and antineutrino density matrices:
In[16]:=
rho={{r11[r],r12[r]},{r21[r],r22[r]}}(*explicitlydefinedneutrinodensitymatrixwithunkowncomponents*);rhob={{rb11[r],rb12[r]},{rb21[r],rb22[r]}}(*explicitlydefinedantineutrinodensitymatrixwithunknowncomponents*);Hnu=potNu[scNu,stNu,r](rho-asConjugate[rhob])(*totalHamiltonianduetoneutrinointeractionswithotherneutrinosandantineutrinos*);
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The total Hamiltonians for neutrinos and antineutrinos depend on all of the above contributions:
In[19]:=
Htotnu=Hv+Hbg+Hnu(*totalneutrinoHamiltonian*);Htotanu=Hv-Hbg-Conjugate[Hnu](*totalantineutrinoHamiltonian*);
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We are ready to set up the evolution equations for the density matrices using Liouville-von Neumann equations
(notice that the evolution equations are non-linear!):
(notice that the evolution equations are non-linear!):
In[21]:=
rdot=-I(Htotnu.rho-rho.Htotnu)(*right-hand-sideoftheLiouville-vonNeumannequationforneutrinos*);rbdot=-I(Htotanu.rhob-rhob.Htotanu)(*right-hand-sideoftheLiouville-vonNeumannequationforantineutrinos*);
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Now we can use NDSolve to solve the evolution equations (notice that the evolution equations for neutrinos and antineutrinos are coupled!):
In[23]:=
solevo=NDSolve[{(*neutrinoandantineutrinoevolutionequationswrittenexplictlyforallcomponents:*)r11'[r]==rdot[[1,1]],r12'[r]rdot[[1,2]],r21'[r]==rdot[[2,1]],r22'[r]==rdot[[2,2]],rb11'[r]==rbdot[[1,1]],rb12'[r]==rbdot[[1,2]],rb21'[r]==rbdot[[2,1]],rb22'[r]==rbdot[[2,2]],r11[rmin]1,r12[rmin]0,r21[rmin]0,r22[rmin]0(*initialconditionsforneutrinosassumingelectrontypeneutrinos*),rb11[rmin]1,rb12[rmin]0,rb21[rmin]0,rb22[rmin]0(*initialconditionsforantineutrinosassumingelectrontypeantineutrinos*)},{r11,r12,r21,r22,rb11,rb12,rb21,rb22}(*listofunknownvariablesforNDSolve*),{r,rmin,rmax}(*rangeofintegration(overtheradialcoordinate)*),MaxSteps10000000];nu11[r_]:=Abs[r11[r]]/.solevo[[1]](*solutionsfortheevolutionofinitiallyelectrontypeneutrinocomponent*);anu11[r_]:=Abs[rb11[r]]/.solevo[[1]](*solutionsfortheevolutionofinitiallyelectrontypeantineutrinocomponent*);
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Here is a visualization of the results comparing the simulated total neutrino potential using the above solutions to the possible extreme configurations as described in the first section:
In[26]:=
potNuTot[r_]:=potNu[scNu,stNu,r](2nu11[r]-1-as(2anu11[r]-1))(*simulatedtotalneutrinopotential*);Plot[{potBG[xBG,yBG,wBG,r](*backgroundpotential*),-potBG[xBG,yBG,wBG,r](*inverseofbackgroundpotential*),potNu[scNu,stNu,r]+potAnu[as,scNu,stNu,r](*totalneutrinopotentialwithbothelectrontypeneutrinoandantineutrino*),-potNu[scNu,stNu,r]-potAnu[as,scNu,stNu,r](*totalneutrinopotentialwithbothnon-electrontypeneutrinoandantineutrino*),potNu[scNu,stNu,r]-potAnu[as,scNu,stNu,r](*totalneutrinopotentialwithelectrontypeneutrinoandnon-electrontypeantineutrino*),-potNu[scNu,stNu,r]+potAnu[as,scNu,stNu,r](*totalneutrinopotentialwithnon-electrontypeneutrinoandelectrontypeantineutrino*),potNuTot[r](*simulatedtotalneutrinopotential*)},{r,rmin,rmax},PlotStyle{Cyan,{Cyan,Dashed},Red,{Red,Dashed},{Red,DotDashed},{Red,Dotted},Blue},FrameTrue,PlotRange{{rmin,rmax},{-1000,1000}},FrameTrue,PlotStyle{Cyan,{Cyan,Dashed},Red,{Red,Dashed},{Red,DotDashed},{Red,Dotted}},FrameLabel{"Radial Distance [a.u.]","Interaction Potential [a.u.]"},FrameTicksStyleThick,LabelStyleDirective[FontSize20],PlotLegends{"Background Potential","Inverse Background Potential","Neutrino Potential (both electron type)","Neutrino Potential (both non-electron type)","Neutrino Potential (electron neutrino, non-electron antineutrino)","Neutrino Potential (non-electron neutrino, electron antineutrino)","Simulated Neutrino Potential"}]
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Fascinating!!!
It turns out that the total neutrino potential follows the non-neutrino background contributions due to actively maintained cancellation of the potentials.
This is possible because both neutrinos and antineutrinos can change their types during the evolution. This phenomenon has been observed in the simulations of neutrino propagation in astrophysical environments (such as supernovae and neutron star/black hole mergers) only very recently and has been called as Matter-Neutrino-Resonance.
For more information, see: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.093007 (and references therein).
It turns out that the total neutrino potential follows the non-neutrino background contributions due to actively maintained cancellation of the potentials.
This is possible because both neutrinos and antineutrinos can change their types during the evolution. This phenomenon has been observed in the simulations of neutrino propagation in astrophysical environments (such as supernovae and neutron star/black hole mergers) only very recently and has been called as Matter-Neutrino-Resonance.
For more information, see: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.093007 (and references therein).
Authorship information
Daavid Vaananen
June 19, 2017
djvaanan@ncsu.edu