JJanuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 1 Theoretical Aspects of the Quantum Neutrino
Stephen Parke ∗ Chicago, USA ∗ E-mail: [email protected]
In this summary of my talk I will review the following the following three theoreticalaspects of the quantum neutrino: current status, why we need precision measurementsand neutrino oscillations amplitudes.
Keywords : Neutrino, Theory, Quantum
1. Current Status:
Circa 2017, it is now well established that neutrinos have mass and the that theflavor or interactions states ν e , ν µ and ν τ are mixtures of the the mass eigenstatesor propagations states, unimaginatively labelled ν , ν and ν . The interaction andpropagation states are related by an unitary matrix, the PMNS matrix, as follows: ν e ν µ ν τ = U ( θ , U ( θ , − δ ) U ( θ , ν ν ν . (1)where the U ’s are the usual complex rotation matrices given by[ U mn ( ξ, η )] ij = [1 + ( c ξ − δ im + δ in )] δ ij + ( s ξ e iη ) δ im δ jn − ( s ξ e − iη ) δ in δ jm . January 29, 2018 9:54 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 1 Theoretical Aspects of the Quantum Neutrino
Stephen Parke ⇤ Chicago, USA ⇤ E-mail: [email protected]
In this summary of my talk I will review the following the following three theoreticalaspects of the quantum neutrino: current status, why we need precision measurementsand neutrino oscillations amplitudes.
Keywords : Neutrino, Theory, Quantum
1. Current Status:
Circa 2017, it is now well established that neutrinos have mass and the that theflavor or interactions states ⌫ e , ⌫ µ and ⌫ ⌧ are mixtures of the the mass eigenstatesor propagations states, unimaginatively labelled ⌫ , ⌫ and ⌫ . The interaction andpropagation states are related by an unitary matrix, the PMNS matrix, as follows: ⌫ e ⌫ µ ⌫ ⌧ = U ( ✓ , U ( ✓ , ) U ( ✓ , ⌫ ⌫ ⌫ . (1)where the U ’s are the usual complex rotation matrices given by[ U mn ( ⇠, ⌘ )] ij = [1 + ( c ⇠ im + in )] ij + ( s ⇠ e i⌘ ) im jn ( s ⇠ e i⌘ ) in jm . – Typeset by Foil TEX – 1 – Typeset by Foil TEX – 1 – Typeset by Foil TEX – 1 – Typeset by Foil TEX – 1 – Typeset by Foil TEX – 1 – Typeset by Foil TEX – 1 – Typeset by Foil TEX – 1 m i – Typeset by Foil TEX – 1 (Dialog) In[185]:= nue = PieChart3D [{ } ,ChartStyle %{ Blue,Blue,Blue } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nue = PieChart3D [{ } ,ChartStyle %{ GrayLevel [ ]} ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] num = PieChart3D [{ } ,ChartStyle %{ Cyan } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nut = PieChart3D [{ } ,ChartStyle %{ Red,Red,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nu3 = PieChart3D [{ } ,ChartStyle %{ Cyan,Blue,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nu2 = PieChart3D [{ } ,ChartStyle %{ Cyan,Blue,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nu1 = PieChart3D [{ } ,ChartStyle %{ Cyan,Blue,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] (Dialog) Out[185]= massive_neutrinos.nb W + e + e W + µ + µ W + + provided L/E . km / MeV = 500 km / GeV !!! e = µ = = – Typeset by Foil TEX – 1 (Dialog) Out[186]=(Dialog) Out[187]=(Dialog) Out[188]= massive_neutrinos.nb W + e + e W + µ + µ W + + provided L/E . km / MeV = 500 km / GeV !!! e = µ = = – Typeset by Foil TEX – 1 (Dialog) Out[186]=(Dialog) Out[187]=(Dialog) Out[188]= massive_neutrinos.nb W + e + e W + µ + µ W + + provided L/E . km / MeV = 500 km / GeV !!! e = µ = = – Typeset by Foil TEX – 1 (Dialog) In[185]:= nue = PieChart3D [{ } ,ChartStyle %{ Blue,Blue,Blue } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nue = PieChart3D [{ } ,ChartStyle %{ GrayLevel [ ]} ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] num = PieChart3D [{ } ,ChartStyle %{ Cyan } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nut = PieChart3D [{ } ,ChartStyle %{ Red,Red,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nu3 = PieChart3D [{ } ,ChartStyle %{ Cyan,Blue,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nu2 = PieChart3D [{ } ,ChartStyle %{ Cyan,Blue,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] nu1 = PieChart3D [{ } ,ChartStyle %{ Cyan,Blue,Red } ,PlotTheme % "Business",SectorOrigin %{{(' Pi . + ) ,"Clockwise" } ,0 }] (Dialog) Out[185]= massive_neutrinos.nb W + e + e W + µ + µ W + + provided L/E . km / MeV = 500 km / GeV !!! e = µ = = – Typeset by Foil TEX – 1
Fig. 1. The identity of the neutrino mass eigenstates or propagation states is determined by their ⌫ e content: ⌫ has the largest ( ⇠ ⌫ is in the middle ( ⇠ ⌫ has the least ( ⇠ ⌫ µ and ⌫ ⌧ but it’s the ⌫ e fraction that definesthe identity of these states. Some, use the masses to label these states, but since we don’t knowboth the mass orderings at this stage, using the electron flavor content is simpler. Fig. 1. The identity of the neutrino mass eigenstates or propagation states is determined by their ν e content: ν has the largest ( ∼ ν is in the middle ( ∼ ν has the least ( ∼ ν µ and ν τ but it’s the ν e fraction that definesthe identity of these states. Some, use the masses to label these states, but since we don’t knowboth the mass orderings at this stage, using the electron flavor content is simpler. a r X i v : . [ h e p - ph ] J a n anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 2 Placing the CP violating phase in the U matrix is customary, however oscilla-tion physics is unaffected by placing this phase in U or U since U ( θ , U ( θ , − δ ) U ( θ ,
0) :=: U ( θ , δ ) U ( θ , U ( θ , U ( θ , U ( θ , U ( θ , δ )where :=: means equal after multiplying by a diagonal phase matrix on the leftand/or right hand side.At the current time, it is most convenient to label the neutrino mass eigenstatesaccording to the size of ν e fraction as is shown in Fig. 1. With this choice for themass eigenstates, it is natural to choose the order of the U αj matrices as in Eq, 1 sothat the first row and third column are simply, since these elements are most easilymeasured.Since the neutrino oscillations with a ∆ m ≈ . × − eV has only smallamounts of ν e , it is ν , that is has a | ∆ m | ≈ . × − eV . The SNO experimentdetermined the mass ordering of ν and ν , m > m , see Fig. 2.The remaining mass ordering, whether m is larger or smaller than m and m ,is shown in Fig. 3. Current and future experiments such as NO ν A, JUNO, DUNE,T2HKK are designed to determine this mass ordering.
Fig. 2. The solar mass ordering of ν and ν was determined by the SNO experiment using thematter effects in the solar interior. The mass of ν is larger than the mass of ν with ∆ m ≡ m − m ≈ . × − eV . Here, I have assumed that the ν µ fraction is equal to ν τ fraction forboth mass eigenstates. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 3 ν is larger or smaller than ν , ν . | ∆ m | ≈ | ∆ m | ≈ . × − eV .Here, I have assumed that the ν µ fraction is equal to ν τ fraction for all three mass eigenstates.Fig. 4. Is the dominant flavor of the mass eigenstate ν , ν µ or ν τ ? If ν τ dominates then theparameter θ < π/
4, whereas if ν µ dominates then θ > π/
4. This is often referred to as theoctant of θ puzzle. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 4 The octant of θ , determines which flavor state dominates ν , see Fig. 4.Whereas the range of ν µ and ν τ components of ν and ν are determined not onlyby the allowed range of θ but also by the CP violating phase δ (cos δ to beprecise). There is significant depends on δ for the magnitude of U µ , U µ , U τ and U τ elements of the PMNS matrix. This fact is quite different than in the quarksector ! Fig. 5. The ν µ and ν τ fractions of ν and ν depends sensitively on both the value of θ andthe CP phase δ . The flavor fractions vary the most for ν , between δ = 0, sin θ = 0 . δ = π , sin θ = 0 . ν , the most variation is between δ = π ,sin θ = 0 . δ = 0, sin θ = 0 . δ = ± π/
2, sin θ = 0 . ν µ and ν τ content is the same for all three mass eigenstates. In the quark sector only the magnitude of U CKMtd has any significant dependson δ CKM . This occurs because of the hierarchy in the sizes of the mixing angles inthe CKM matrix: θ ∼ λ , θ ∼ λ and θ ∼ λ where λ ≈ . anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 5
2. Why we need Precision Measurements
Four reasons for performing precision measurements:
For Discovery of New Physics
An experiment like ICECUBE can discover new physics in the flavor ratios of theirPeV neutrinos, if precise values of the predictions for the ν SM are known, see Fig.6, from . John Beacom, The Ohio State University Neutrino University Seminar, Fermilab, July 2017 32
Flavor Composition of Mass Eigenstates
Bustamante, Beacom, Winter (2015, PRL)
Neutrino mixing angles are all large, well known(and similar forboth hierarchies)
Fig. 6. Variation of flavor ratios using our current uncertainty on δ (left panel) and θ ij , δ (rightpanel), from Bustamante, Beacom and Winter. Stress Test Three Neutrino paradigm
Compared to the Quark sector the unitarity of the PMNS matrix has only beentested at the 10% level . -2 -1 | U α U β ∗ + U α U β ∗ + U α U β ∗ | or | U ei U ej ∗ + U µi U µj ∗ + U τi U τj ∗ | Rows Columns ∆ χ σ σ σ eµ Unitarity Triangle Closures
Rows α,β = e,µα,β = e,τα,β = µ,τ Columns i,j =1 , i,j =1 , i,j =2 , -2 -1 − ( | U α | + | U α | + | U α | ) or − ( | U ei | + | U µi | + | U τi | ) Rows Columns ∆ χ σ σ σ µ e Normalisations
Rows α = eα = µα = τ Columns i =1 i =2 i =3 Fig. 7. The current triangle (left) and normalization (right) unitarity constraints on the elementsof the PMNS matrix form Parke and Ross-Lonergan. There is still plenty of room for new physics,such as a light sterile neutrino. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 6 Test Theoretical Neutrino Models
Fig. 8 shows how improvements on the measurements of the mixing angles and CPviolating phase can used to distinguish various models that could possibly explainthe mixings of the neutrinos.
Stephen Pa r ke Lepton-Photon , G uangzhou Precision NeutrinoMeasurements: CP , ✓ .... Stress TestNeutrino paradigmsearch for new physicsConnection toLeptogenesisUnderstanding UniverseTest TheoreticalNeutrino ModelsDetermine flavorfractions of neutrinomass states – Typeset by Foil
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WHY MEASURE THESE PARAMETERS? ➤ Lepton mixing allows for a new source of CP violation that can be studied with neutrinos ➤ CPV through δ cp may be sufficient source for leptogenesis (Nucl. Phys. B774 (2007) 1) ➤ Neutrino masses indicate new physics beyond the standard model and electroweak scale ➤ Precise values of the mixing parameters may indicate or disfavor models of flavor symmetries
Predictions from flavor symmetry forms with current measurement precision
Nucl. Phys. B, Vol. 894, 733-768 (2015) Predictions from cos δ sum rules for discrete symmetries: Predictions of flavor symmetry forms with projected measurement precision W + ! e + ⌫ e W + ! µ + ⌫ µ W + ! ⌧ + ⌫ ⌧ provided L/E ⌧ . km / MeV = 500 km / GeV !!! ⇠ picosecond in Neutrino rest frame !!! ⇡ Age of Universe / ✓ , ✓ cos ⌫ e = ⌫ µ = ⌫ ⌧ = Propagator ⌫ ↵ ! ⌫ = ↵ e i E ⌫ t most ⌫ e least ⌫ e – Typeset by Foil TEX – 1
WHY MEASURE THESE PARAMETERS? ➤ Lepton mixing allows for a new source of CP violation that can be studied with neutrinos ➤ CPV through δ cp may be sufficient source for leptogenesis (Nucl. Phys. B774 (2007) 1) ➤ Neutrino masses indicate new physics beyond the standard model and electroweak scale ➤ Precise values of the mixing parameters may indicate or disfavor models of flavor symmetries
Predictions from flavor symmetry forms with current measurement precision
Nucl. Phys. B, Vol. 894, 733-768 (2015) Predictions from cos δ sum rules for discrete symmetries: Predictions of flavor symmetry forms with projected measurement precision W + ! e + ⌫ e W + ! µ + ⌫ µ W + ! ⌧ + ⌫ ⌧ provided L/E ⌧ . km / MeV = 500 km / GeV !!! ⇠ picosecond in Neutrino rest frame !!! ⇡ Age of Universe / ✓ , ✓ cos ⌫ e = ⌫ µ = ⌫ ⌧ = Propagator ⌫ ↵ ! ⌫ = ↵ e i E ⌫ t most ⌫ e least ⌫ e –TypesetbyFoil TEX – 1
WHY MEASURE THESE PARAMETERS? ➤ Lepton mixing allows for a new source of CP violation that can be studied with neutrinos ➤ CPV through δ cp may be sufficient source for leptogenesis (Nucl. Phys. B774 (2007) 1) ➤ Neutrino masses indicate new physics beyond the standard model and electroweak scale ➤ Precise values of the mixing parameters may indicate or disfavor models of flavor symmetries
Predictions from flavor symmetry forms with current measurement precision
Nucl. Phys. B, Vol. 894, 733-768 (2015) Predictions from cos δ sum rules for discrete symmetries: Predictions of flavor symmetry forms with projected measurement precision
Girardi, Petcov, Titov, arXiv:1410.8056
Fig. 8. Current status (left) and future status (right) of the labelled models predictions for cos δ from Girardi, Petcov and Titov . Connection to Leptogenesis Understanding Universe
Fig. 9 gives the allowed region for a model with the measured value of the Baryonasymmetry of the universe on the neutrino parameters S tephen P arke Lepton- P hoton , G uangzhou Precision NeutrinoMeasurements: CP , ✓ .... Stress TestNeutrino paradigmsearch for new physicsConnection toLeptogenesisUnderstanding UniverseTest TheoreticalNeutrino ModelsDetermine flavorfractions of neutrinomass states – Typeset by Foil
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Aug/2017 Pedro A. N. Machado | Recent highlights from neutrino theory [email protected]
The CP phase and θ octant Insights on the generation of the matter anti-matter asymmetryInsights on the flavor puzzle
Chen Fallbacher Mahanthappa Ratz Trautner 2014Chen Mahanthappa 2009 for the Jarlskog invariant, J ≡ Im( V ud V cb V ∗ ub V ∗ cd ) = 2 . × − , in the quark sector also agreeswith the current global fit value.) Potential direct measurements for these parameters at the LHCbcan test our predictions.As a result of the GJ relations, our model predicts the sum rule [8, 17] between the solar neutrinomixing angle and the Cabibbo angle in the quark sector, tan θ ⊙ ≃ tan θ ⊙ , TBM + θ c cos δ ℓ , with δ ℓ being the leptonic Dirac CP phase in the standard parametrization. In addition, our modelpredicts θ ∼ θ c / √
2. Numerically, the diagonalization matrix for the charged lepton mass matrixcombined with U T BM gives the PMNS matrix, ⎛⎜⎜⎜⎝ . e − i o . e − i o . e i o . e − i . o . e − i o . e i . o . e i o .
577 0 . ⎞⎟⎟⎟⎠ , (22)which gives sin θ atm = 1, tan θ ⊙ = 0 .
420 and | U e | = 0 . u = − . ξ = 0 . m atm = 2 . × − eV and ∆ m ⊙ = 8 . × − eV . As the three masses aregiven in terms of two VEV’s, there exists a mass sum rule, m − m = 2 m , leading to normal masshierarchy, ∆ m atm > J ℓ = − . δ ℓ = 227 o . With such δ ℓ , the correction from the charged lepton sectorcan account for the difference between the TBM prediction and the current best fit value for θ ⊙ .Our model predicts ( m , m , m ) = (0 . , − . , . α = π and α = 0.Our model has nine input parameters, predicting a total of twenty-two physical quantities:12 masses, 6 mixing angles, 2 Dirac CP violating phases and 2 Majorana phases. Our model istestable by more precise experimental values for θ , tan θ ⊙ and γ in the near future. δ ℓ is theonly non-vanishing leptonic CP violating phase in our model and it gives rise to lepton numberasymmetry, ϵ ℓ ∼ − . By virtue of leptogenesis, this gives the right sign and magnitude of thematter-antimatter asymmetry [18]. Conclusion. —We propose the complex group theoretical CG coefficients as a novel origin of CPviolation. This is manifest in our model based on SU(5) combined with the double tetrahedralgroup, T ′ . Due to the presence of the doublet representations in T ′ , there exist complex CGcoefficients, leading to explicit CP violation in the model, while having real Yukawa couplings andscalar VEVs. The predicted CP violation measures in the quark sector are consistent with thecurrent experimental data. The leptonic Dirac CP violating phase is predicted to be δ ℓ ∼ o ,which gives the cosmological matter asymmetry.8 Ma 2016, Ma 2017
Ballet King Pascoli Prouse Wang 2016
GUTs typically predict:
Majorana neutrinos Normal mass ordering θ in first octant“large” θ if θ and θ are largeNo light sterile neutrino For a certain class of flavor groups:1) δ CP is related to the Clebsch-Gordan coefficients2) Dependence on group and fermion representationsSome predictions Fig. 9. Consistency of the model by Ballet, King, Pascoli, Prouse and Wang which has themeasured value of the Baryon asymmetry, through Leptogenesis, with the measured values of theparameters θ , ∆ m and ∆ m . anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 7
3. Neutrino Amplitudes:
The neutrino oscillation probability can be written as P ( ν α → ν β ) = |A αβ | where A αβ = (cid:88) j U ∗ αj U βj e − im j L/ E (2)For two flavors we simple obtain A αα = 1 + 2 i s θ e + i ∆ sin ∆ and A αβ = (2 s θ c θ ) sin ∆where ∆ jk = ∆ m jk L/ E .For three flavors, there are an infinite number of ways of writing these amplitudeswhich all give the same result. So we need an organizational principle. It is knownthat the ν τ oscillation amplitudes can be obtained from the ν µ oscillation amplitudesby making the following replacements s ↔ c and δ → δ + π . This follows fromthe form of the U matrix on the left hand side of the PMNS matrix, eqn. 1.Similarly, if one considers the rotation matrix on the right hand side of thePMNS matrix, U , there is a symmetry that leaves the amplitudes invariant: thesymmetry is m ↔ m , s ↔ c and δ → δ + π which follows from U ( θ , δ ) (cid:18) ν ν (cid:19) = U ( π/ θ , δ ) (cid:18) − e iδ ν e − iδ ν (cid:19) = U ( π/ − θ , δ ± π ) (cid:18) e iδ ν − e − iδ ν (cid:19) . To maintain this symmetry, use unitarity to remove the U ∗ α U β term in eqn. 2,giving A αβ = δ αβ + (2 i ) (cid:88) j =(1 , U ∗ αj U βj e i ∆ j sin ∆ j , (3)then a A ee = 1 + (2 i ) c [ c e i ∆ sin ∆ + s e i ∆ sin ∆ ] A µµ = 1 + (2 i ) [ ( c c + s s s ) e i ∆ sin ∆ + ( c s + s c s ) e i ∆ sin ∆ + ( s s c s c cos δ ) e i (∆ +∆ ) sin ∆ ] (4) A µτ = (2 c s ) [ ( s − s c ) e i ∆ sin ∆ + ( c − s s ) e i ∆ sin ∆ ] − (2 s s c ) [ c e iδ − s e − iδ ] e i (∆ +∆ ) sin ∆ A µe = (2 s s c ) [ c e i ∆ sin ∆ + s e i ∆ sin ∆ ]+ (2 c c s c ) e i (∆ +∆ + δ ) sin ∆ a Since the overall phase of an amplitude is arbitrary, there are arbitrary choices for the overallphase of each amplitude. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 8 No approximation has been used to obtain these amplitudes and all of these am-plitudes explicitly satisfy the 1 ↔ e ± iδ as they al-ways appear multiplied by sin ∆ ( note sin ∆ = e − i ∆ sin ∆ − e − i ∆ sin ∆ ).If one uses the approximation that ∆ ≈ ∆ then we can rewrite A µe ≈ (2 s s c ) sin ∆ + (2 c c s c ) e i ( δ +∆ ) sin ∆ and then it’s simple to see that the CP violating term is given by∆ P CP = 8 ( s s c ) ( c c s c ) sin δ sin ∆ sin ∆ sin ∆ (5)In Fig. 10 we give a graphical representation of the amplitude for ν µ → ν e and theassociated bi-probability plot. Fig. 10. The amplitude for ν µ → ν e as well as ¯ ν µ → ¯ ν e and the associated bi-probability plot. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 9
4. Perturbative Approximation to the Neutrino OscillationProbabilites in Matter
In this section, a simple and accurate way to evaluate oscillation probabilities, re-cently shown Denton, Minakata and Parke , is given. Details as to the why’s andhow’s of this method are contained in these papers.The mixing angles in matter, which we denote by a (cid:101) θ and (cid:101) θ here, can alsobe calculated in the following way, using ∆ m ee ≡ cos θ ∆ m + sin θ ∆ m , asfollows, see Addendum ? :cos 2 (cid:101) θ = (cos 2 θ − a/ ∆ m ee ) (cid:113) (cos 2 θ − a/ ∆ m ee ) + sin θ , (6)where a ≡ √ G F N e E ν is the standard matter potential, andcos 2 (cid:101) θ = (cos 2 θ − a (cid:48) / ∆ m ) (cid:113) (cos 2 θ − a (cid:48) / ∆ m ) + sin θ cos ( (cid:101) θ − θ ) , (7)where a (cid:48) ≡ a cos (cid:101) θ + ∆ m ee sin ( (cid:101) θ − θ ) is the θ -modified matter poten-tial for the 1-2 sector. In these two flavor rotations, both (cid:101) θ and (cid:101) θ are in range[0 , π/ θ and δ are unchanged in matter for this approximation.From the neutrino mass squared eigenvalues in matter, given by (cid:102) m = ∆ m + ( a − a (cid:48) ) , (cid:102) m = 12 (∆ m + ∆ (cid:102) m + a (cid:48) ) , (8) (cid:102) m = 12 (∆ m − ∆ (cid:102) m + a (cid:48) ) , it is simple to obtain the neutrino mass squared differences in matter, i.e. the ∆ m jk in matter, which we denote by ∆ (cid:102) m jk , which are given by∆ (cid:102) m = ∆ m (cid:113) (cos 2 θ − a (cid:48) / ∆ m ) + sin θ cos ( (cid:101) θ − θ ) , ∆ (cid:102) m = ∆ m + ( a − a (cid:48) ) + 12 (cid:16) ∆ (cid:102) m − ∆ m (cid:17) , (9)∆ (cid:102) m = ∆ (cid:102) m − ∆ (cid:102) m . To see these expressions have the correct asymptotic forms, use the fact that(∆ (cid:102) m − ∆ m ) = | a (cid:48) | + O (∆ m ), for | a | (cid:29) ∆ m . Plots of the matter mixingangles and mass squared differences are given in Fig. 11. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 10 a and a (cid:48) , top right, sinesquared of mixing angles in matter, sin (cid:101) θ jk , bottom left, the mass squared eigenvalues in matter, (cid:102) m j , and bottom right, the mass squared differences in matter, ∆ (cid:102) m jk . E ν ≥ E ν ≤
0) is forneutrinos (anti-neutrinos). E ν = 0 is the vacuum values for both neutrinos and anti-neutrinos. To calculate the oscillation probabilities, to 0th order, use the above ∆ (cid:102) m jk instead of ∆ m jk and replace the vacuum MNS matrix as follows U MNS ≡ U ( θ ) U ( θ , δ ) U ( θ ) ⇒ U MMNS ≡ U ( θ ) U ( (cid:101) θ , δ ) U ( (cid:101) θ ) . That is, replace ∆ m jk → ∆ (cid:102) m jk θ → (cid:101) θ θ → (cid:101) θ , (10) θ and δ remain unchanged, it is that simple. We call this the 0th order DMPapproximation. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 11 These expressions are valid for both NO, ∆ m ee > m ee <
0. Foranti-neutrinos, just change the sign of a and δ . Our expansion parameter is (cid:12)(cid:12)(cid:12)(cid:12) sin( (cid:101) θ − θ ) sin θ cos θ ∆ m ∆ m ee (cid:12)(cid:12)(cid:12)(cid:12) ≤ . , (11)which is small and vanishes in vacuum, so that our perturbation theory reproducesthe vacuum oscillation probabilities exactly.If A ν α → ν β (∆ m , ∆ m , θ , θ , θ , δ ) is the oscillation amplitude in vacuum,see eq. 4, then A ν α → ν β (∆ (cid:102) m , ∆ (cid:102) m , (cid:101) θ , (cid:101) θ , θ , δ ) is the oscillation probabilityin matter, i.e. use the same function but replace the mass squared differences andmixing angles with the matter values given in eq. 6 - 9. The resulting oscilla-tion probabilities are identical to the zeroth order approximation given in Denton,Minakata and Parke .In Fig. 12, I have given the exact and approximate oscillation probabilities forthe ν e appearance channel for T2K and T2HK , NOvA , T2HKK and DUNE .
5. Conclusions
To summarize: • from Nu1998 to now, tremendous experimental progress on Neutrino SM:more at Nu2018 ! • LSND Sterile Nus neither confirmed or ruled out at acceptable CL: - ultrashort baseline reactor experiments. • Great Theoretical progress on understand many aspects of Quantum Neu-trino Physics: Oscillations, Decoherence, Oscillations Probabilities in Mat-ter, Leptogenesis. • Still searching for convincing model of Neutrino masses and mixings: withtestable and confirmed predictions !
6. Acknowledgements
I would like to thank the organizers, and especially Prof. Wang Wei, for the won-derful hospitality so that I could attend this great conference.This project has received funding/support from the European Union’s Horizon2020 research and innovation programme under the Marie Sklodowska-Curie grantagreement No 690575. This project has received funding/support from the Euro-pean Union’s Horizon 2020 research and innovation programme under the MarieSklodowska-Curie grant agreement No 674896.Finally, I would like to thank my collaborator Prof. Xu Zhan from TsinghuaUniversity for a wonderful collaboration and friendship. We meet at the Interna-tional Symposium on Particle and Nuclear Physics, Beijing, China, Sep 2-7, 1985and wrote a beautiful and significant paper that was a pioneering paper in thefield of “Amplitudes”. anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 12 Top panel: P exmat , P thappx and P vac Middle and bottom panels: black dotted lines red solid lines magenta solid lines P = | P exmat P vac | P = | P exmat P thappx | P = | P exmat P stappx | P = ( P exmat + P vac ) P = ( P exmat + P thappx ) P = ( P exmat + P stappx ) P = | P exmat P vac | P = | P exmat P thappx | P = | P exmat P stappx | P = ( P exmat + P vac ) P = ( P exmat + P thappx ) P = ( P exmat + P stappx ) T2HKK (1050 km)T2K & T2HK (295 km) – Typeset by Foil
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Fig. 12. For normal ordering (NO), ν µ → ν e appearance: Top Left figure is for T2K , Top Rightfigure is NOvA, Bottom Left figure is T2HKK, and Bottom Right is DUNE. In each figure, thetop panel is exact oscillation probability in matter, P exmat (blue dashes) from , the zeroth orderDMP approximation, P thappx (red dashes) from and the vacuum oscillation probability, P vac (blackdots). The Middle panel is difference between exact oscillation probabilities in matter and vacuum(black dots), and the difference between exact and 0th DMP approximation (solid red) and exactand 1st DMP approximation (solid magenta) approximations. Bottom panel is similar to middlepanel but plotting the fractional differences, ∆ P/P . anuary 30, 2018 1:23 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in lp2017˙arxiv˙parke page 13 References
1. M. Bustamante, J. F. Beacom and W. Winter,“Theoretically palatable flavor combinations of astrophysical neutrinos,”Phys. Rev. Lett. , no. 16, 161302 (2015) doi:10.1103/PhysRevLett.115.161302[arXiv:1506.02645 [astro-ph.HE]].2. S. Parke and M. Ross-Lonergan,“Unitarity and the three flavor neutrino mixing matrix,”Phys. Rev. D , no. 11, 113009 (2016) doi:10.1103/PhysRevD.93.113009[arXiv:1508.05095 [hep-ph]].3. I. Girardi, S. T. Petcov and A. V. Titov,“Determining the Dirac CP Violation Phase in the Neutrino Mixing Matrix from SumRules,”Nucl. Phys. B , 733 (2015) doi:10.1016/j.nuclphysb.2015.03.026 [arXiv:1410.8056[hep-ph]].4. P. Ballett, S. F. King, S. Pascoli, N. W. Prouse and T. Wang,“Precision neutrino experiments vs the Littlest Seesaw,”JHEP , 110 (2017) doi:10.1007/JHEP03(2017)110 [arXiv:1612.01999 [hep-ph]].5. P. B. Denton, H. Minakata and S. J. Parke,“Compact Perturbative Expressions For Neutrino Oscillations in Matter,”JHEP , 051 (2016) doi:10.1007/JHEP06(2016)051 [arXiv:1604.08167 [hep-ph]].P. B. Denton, H. Minakata and S. J. Parke,“Addendum to ’Compact Perturbative Expressions for Neutrino Oscillations in Mat-ter”’ arXiv:1801.06514 [hep-ph].6. H. W. Zaglauer and K. H. Schwarzer,“The Mixing Angles in Matter for Three Generations of Neutrinos and the MSWMechanism,”Z. Phys. C (1988) 273.7. K. Abe et al. [T2K Collaboration],“The T2K Experiment,”Nucl. Instrum. Meth. A , 106 (2011)doi:10.1016/j.nima.2011.06.067 [arXiv:1106.1238 [physics.ins-det]].8. K. Abe et al. [Hyper-Kamiokande Proto- Collaboration],“Physics potential of a long-baseline neutrino oscillation experiment using a J-PARCneutrino beam and Hyper-Kamiokande,”PTEP , 053C02 (2015) doi:10.1093/ptep/ptv061 [arXiv:1502.05199 [hep-ex]].9. D. S. Ayres et al. [NOvA Collaboration],“NOvA: Proposal to Build a 30 Kiloton Off-Axis Detector to Study ν µ → ν e Oscilla-tions in the NuMI Beamline,”hep-ex/0503053.10. K. Abe et al. [Hyper-Kamiokande proto- Collaboration],“Physics Potentials with the Second Hyper-Kamiokande Detector in Korea,”arXiv:1611.06118 [hep-ex].11. R. Acciarri et al. [DUNE Collaboration],“Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experi-ment (DUNE) : Volume 2: The Physics Program for DUNE at LBNF,”arXiv:1512.06148 [physics.ins-det].12. M. L. Mangano, S. J. Parke and Z. Xu, “Duality and Multi - Gluon Scattering,” Nucl.Phys. B298