The Mass Hierarchy of Neutrinos
Reimer van den Hoek
10454268
University of Amsterdam NIKHEF
July 18, 2017
Supervisor: Prof. dr. P.J.G. Mulders
Second assessor: Dr. W.J. Waalewijn
Report Bachelor Project Physics and Astronomy, size 15 EC
Conducted between 01-04–2017 and 18-07–2017
Abstract
This report will discuss how neutrino oscillations are related to the neutrino mass hierarchy and what this could mean for future experiments. This is done by first constructing transition probabilities that describe the neutrino oscillations and then comparing the various possible oscillations. From this an optimal value for the distance travelled can be determined for every flavour change. Only the more interesting differences in transition probabilities are shown. The probabilities that were found differ from available simulations which indicates that not all necessary effects were taken into account. This report also tries to simplify the formulae in this problem.
Contents
I Introduction 2
II Oscillation with two flavour states 3
III Oscillations with Three flavours 6
IV Mass hierarchy 10
V Current experiments 13
VI Conclusion and Discussion 15
I.
Introduction
This report will try to investigate the different properties of neutrinos, with a focus on neutrino oscillations. Neutrino oscillations are a process where a neutrino of one flavour changes into a neutrino of another flavour. This happens because the flavour states are mixtures of mass states. As they travel this will lead to one flavour state becoming another.
Neutrinos are particles with spin 1/2. They interact with other particles through the weak interaction, every flavour can interact with one corresponding charged lepton. Neutrinos where thought to be massless until the phenomenon of neutrino oscillations was noticed which required neutrinos to have a small mass, three slightly different masses in fact. This allows them to be affected by gravity but also causes the different mass states the propagate at a different rate.
Since these different flavour states do not directly correspond to a mass state, but rather to a superposition of the three mass states, the flavour "oscillates" from one state to the other as the neutrinos travel. The shape of this oscillation is determined by the mixing angle and how the CP-symmetry, the combined particle-antiparticle symmetry and inversion symmetry, is violated. Furthermore it is influenced by the order of the three different masses, the mass hierarchy. This mass hierarchy of neutrinos is such that either one mass state is much heavier than the other two (normal hierarchy) or much lighter (inverted hierarchy). The mixing angle appears just as a parameter without known physical meaning and CP-symmetry violation manifests itself in the fact that transitions are not the same for particles and anti-particles.
hope to answer is: How can neutrinos oscillations be used to determine the mass hierarchy of neutrinos?
II.
Oscillation with two flavour states
Oscillations with two flavour states (and two mass states) are easier to understand than those with three. As such this is the process we use to illustrate the principle of neutrino oscillations.We use the flavour states as an orthonormal basis which gives: νe =1 0 , νµ = 0 1 . (1)
We assume every mass state to be a linear combination of the flavour states. ν1 =c1νe+c2νµ , ν2 =c3νe+c4νµ. (2) The mass eigenstates should also be orthonormal and as such: ν1·ν2 =0 which
implies c1∗c∗3 = −c2∗c4∗, ν1·ν1 =1 which implies c1c∗1+c2c2∗ =1 and ν2·ν2 =1
which implies c3c3∗+c4c4∗ =1 and since we already know the flavour states to be
orthonormal c∗1∗c2 = −c3∗∗c4, c1c∗1+c3c∗3 =1 and c2c2∗+c4c4∗ =1
A mixing matrix that achieves this is the rotation matrix with potentially an extra phase difference which can always occur. This is a unitary matrix. So we can write: ν1 ν2 =
cos(θ) −eiδsin θ e−iδsin θ cos θ
νe
νµ
, (3)
since the mixing matrix is unitary:
U =
cos(θ) −eiδsin θ e−iδsin θ cos θ
with UU† =U†U = I, (4)
We can invert the equation to express flavour states as a superposition of the mass states: νe νµ =
cos θ eiδsin θ −e−iδsin θ cos θ
ν1
ν2
. (5)
These equations can be further simplified by absorbing the δ phase in one of the states. Take a flavour (or mass) basis and shift it by a phase:
ν 0 e ν 0 µ ! → e iδa ν 0 e eiδbν 0 µ ! (6)
and as a result:
cos θ −eiδsin θeiδ e−iδsin θ cos θ
ν 0 e ν 0 µ ! →
eiδacos θ −ei(δb+δ)sin θ ei(δa−δ)sin θ eiδbcos θ
ν 0 e ν 0 µ ! . (7) For a=1/2 and b= −1/2 this gives:
eiδ2 cos θ −eiδ2 sin θ e−iδ2sin θ e−iδ2 cos θ ! ν 0 e ν 0 µ ! . (8)
This is just a general phase which we can absorb: ν1 ν2 =cos θ −sin θ sin θ cos θ νe νµ , (9)
Be aware that these phases can no longer be completely ignored later when looking at three flavours as we cannot redefine the states in such a way that all the phases of the 3 state system can be absorbed.
Now to make the process of mixing useful we need to look at states as they are actually created and measured, namely as neutrinos with a single flavour.
A neutrino that starts as an electron neutrino can be described by the following wave function: Ψe(~x= ~0, t=0) E =|νei = cos θ|ν1i +sin θ|ν2i. (10)
Since the mass eigenstates propagate as plane waves the wave function at an arbitrary time and place is given by:
|Ψe(~x, t)i =cos θ|ν1iei( ~p1·~x−E1t)+sin θ|ν2iei( ~p2·~x−E2t). (11)
Where(E1,~p1) and(E2,~p2) are the energy and three-momenta of their respective
mass state. At a specific time T and distance L this simplifies to:
|Ψe(L, T)i =cos θ|ν1iei(p1L−E1T)+sin θ|ν2iei(p2L−E2T). (12)
Since ν1 and ν2 are not measurable we need to express |Ψe(L, T)i in terms of νe
and νµ. This is done by using function 9 and gives: |Ψe(L, T)i =cos θ(cos θ|νei −sin θ
νµ
)ei(p1L−E1T)
+sin θ(sin θ|νei +cos θ
νµ
From here we can isolate the probability of νe changing to νµ by calculating νµ Ψe(L, T) 2 =P(νe →νµ). This gives: P(νe →νµ) = (sin θ cos θe
−i(p2L−E2T)−cos θ sin θe−i(p1L−E1T))
∗(sin θ cos θei(p2L−E2T)−cos θ sin θei(p1L−E1T))
=sin2(2θ))sin2((p1L−E1T) − (p2L−E2T)
2 ).
(14)
The energy of a neutrino is given by E =p p2+m2 but since neutrinos are
ultra-relativistic particles m E and the energy can be approximated by E = p+m2/(2p). The momenta of the different mass states are the same as they start as the same particle and using p≈ Eν/c and Tc ≈ L this gives:
P(νe →νµ) ≈sin2(2θ))sin2( (m22−m21)T 4p ) ≈sin2(2θ))sin2((m 2 2−m21)L 4Eν ). (15)
Using ¯hc =1.9732705∗10−19 GeV km we rewrite L from GeV−1 to km:
sin2(2θ))sin2((m
2
2[GeV2] −m21[GeV2])L[GeV−1]
4Eν[GeV] ) ≈sin2(2θ))sin2(1.27(m 2 2[eV2] −m21[eV2])L[km] 4Eν[GeV] ). (16)
Taking θ to be 0.576361759 and m22−m21= ±7.37∗10−5 eV2 as was found in the literature.From ref. [3] We can plot this probability as a function of L/Eν.
50 000 100 000 150 000 200 000 L Eν 0.2 0.4 0.6 0.8 P
Figure 1: Two state probability P(νe→νµ)plotted against EL
ν
Most of this section was based on the book Modern Particle Physics by Mark Thomson. Ref. [5]
III.
Oscillations with Three flavours
The next step is to expand this idea to three flavours. When constructing the three flavour mixing matrix the product of the three two flavour matrices is needed. However since not all of the phases can be absorbed a phase δCP will remain,
which is the CP-violation phase.
U = 1 0 0 0 cos(θ23) sin(θ23) 0 − sin(θ23) cos(θ23) · cos(θ13) 0 sin(θ13)e−i·δCP 0 1 0 − sin(θ13)ei·δCP 0 cos(θ13) · cos(θ12) sin(θ12) 0 − sin(θ12) cos(θ12) 0 0 0 1 =
cos(θ12)cos(θ13) sin(θ12)cos(θ13) sin(θ13)e−i·δCP − sin(θ12)cos(θ23) −cos(θ12)sin(θ23)sin(θ13)ei·δCP cos(θ12)cos(θ23) −sin(θ12)sin(θ23)sin(θ13)ei·δCP sin(θ23)cos(θ13)
sin(θ12)sin(θ23) −cos(θ12)cos(θ23)sin(θ13)ei·δCP − cos(θ12)sin(θ23) −sin(θ12)cos(θ23)sin(θ13)ei·δCP cos(θ23)cos(θ13)
. (17)
The reason why this extra phase causes CP-violation is that when looking at anti-neutrinos the mixing matrix has to be conjugated. As such a neutrino has a slightly different chance for getting from a certain flavour regular neutrino to an anti neutrino than it has the other way around.
cumbersome but as of today there is a reasonable estimate for most of them. A recent compilation was made by F. Capozzi et al.From ref. [3]. They combined data from multiple recent neutrino experiments.
Figure 2: Results for the different mixing angles and the cp-phase as found by F. Capozzi et al.
From ref. [3] It also show the values for the mass differences as δm2 = m22−m21 and ∆m2 =m2
3− (m22+m21)/2
In the case of normal mass hierarchy and with EiT−piL simplified to φi this
would give us the following wave functions:
|Ψe(L, T)i = ((0.6880)e−iφ1+ (0.2906)e−iφ2+ (0.0214)e−iφ3)|ei
+ ((0.3521−0.0253i)e−iφ2− (0.0434−0.0852i)e−iφ3+ (−0.3086−0.05992i)e−iφ1)|µi
+ ((0.3335−0.0680i)e−iφ1− (0.2842+0.0287i)e−iφ2− (0.0493−0.0967i)e−iφ3)|τi,
(18)
Ψµ(L, T)
= ((−0.3086+0.0599i)e−iφ1+ (0.3521+0.0253i)e−iφ2− (0.0434+0.0852i)e−iφ3)|ei
+ ((0.4287)e−iφ2+ (0.4276)e−iφ3+ (0.1437)e−iφ1)|µi
+ ((−0.1437+0.0596i)e−iφ1− (0.3417+0.0596i)e−iφ2+ (0.4854)e−iφ3)|τi,
(19)
|Ψτ(L, T)i = ((0.3335 + 0.0680i)e
−iφ1− (0.2842 − 0.0287i)e−iφ2− (0.0493 + 0.0967i)e−iφ3)|ei
+ (−(0.3417 − 0.0596i)e−iφ2+ (0.4854)e−iφ3− (0.1437 + 0.0596i)e−iφ1)|µi
+ ((0.1684)e−iφ1+ (0.2807)e−iφ2+ (0.5510)e−iφ3)|τi .
(20) The probability P(νi →νj) = νj Ψi(L, T) 2
can now be calculated in the same way as with the two flavour case. Plotting the probability against the distance L/Eν will give some insight into which distances give higher and lower probabilities of finding a certain flavour neutrino if the starting neutrino is known.
0 50 000 100 000 150 000 200 000 L Eν 0.2 0.4 0.6 0.8 1.0 P
Figure 3: This figure shows: P(νe→ ντ)in blue, P(νe→ νµ)in red and P(νe→ νe)in green. With the probability being on the y-axis and the x-axis signifying the the distance travelled for a particle of a certain energy.
0 50 000 100 000 150 000 200 000 L Eν 0.2 0.4 0.6 0.8 1.0P
Figure 4: This figure shows: P(νµ→ντ)in blue, P(νµ→νµ)in red and P(νµ →νe)in green. With the probability being on the y-axis and the x-axis signifying the the distance travelled for a particle of a certain energy.
0 50 000 100 000 150 000 200 000 L Eν 0.2 0.4 0.6 0.8 1.0P
Figure 5: This figure shows: P(ντ →ντ)in blue, P(ντ →νµ)in red and P(ντ →νe)in green. With the probability being on the y-axis and the x-axis signifying the the distance travelled for a particle of a certain energy.
IV.
Mass hierarchy
The subject we are most interested in exploring is the hierarchy of neutrinos. Which mass states are heavier and which are lighter. We want to look at the probabilities, consider a few possible measurements and deduce something meaningful about this hierarchy. In order to do so we should first explore what the options are.
There are two possible mass hierarchies. One of which is called the normal mass hierarchy and the other is called inverted mass hierarchy.
Figure 6: Illustration of normal and inverted mass hierarchy
The fact that there are two possible mass hierarchies arises because all probabilities have the same general shape. When assuming δCP to be zero, all probabilities
consist of three contributing functions involving a cosine:
P= a+b cos (m 2 1−m22)L 4Eν ! +c cos (m 2 1−m23)L 4Eν ! +d cos (m 2 2−m23)L 4Eν ! . (21)
However, negative or positive values inside a cosine never give a different outcome and as such it is impossible to find the mass hierarchy from this equation. We can determine the absolute difference between two of the masses squared but not the sign of this difference. However, this changes in case of CP-violation. When δCP
is not zero three sine terms are added to equation 21 giving us:
P =a+b cos (m 2 1−m22)L 4Eν ! +c cos (m 2 1−m23)L 4Eν ! +d cos (m 2 2−m23)L 4Eν ! ±k sin (m 2 1−m22)L 4Eν ! ∓k sin (m 2 1−m23)L 4Eν ! ±k sin (m 2 2−m23)L 4Eν ! . (22) From this it becomes possible to determine the mass hierarchy since sine functions are affected by the sign of their phase. But this is not easy because as you can see the sine of (m21−m23)L/(4Eν) has the same prefactor as the sine of (m22−m23)L/(4Eν) only negative. Because of this the difference in hierarchy only slightly affects the overall probability. As the neutrinos have travelled further, however, the effect does become visible. Some examples are shown in figures 7 and 8 for P(νe →ντ). 0 20 000 40 000 60 000 80 000 100 000 120 000 140 000 L Eν 0.2 0.4 0.6 0.8 1.0P
20 000 40 000 60 000 80 000 100 000 120 000 140 000 L Eν -0.3 -0.2 -0.1 0.1 0.2 ΔP
Figure 8: P(νe→ντ)normal−P(νe→ντ)inverted
Figure 8 shows us that the biggest differences for P(νe → ντ) can be found at a L/Eν of a little under 80000 km/GeV. The diameter of the earth is at most 12742 km so this data would suggest that for any experiments set entirely on earth Eν should be smaller than 0.16 GeV. Some other transitions differ most for lower values of L/Eν. The biggest difference for lower values of L/Eν can be found in the νµ → ντ transition. (And in νe → νe which shows similar values.)
50 000 100 000 150 000 200 000 250 000 L Eν -0.4 -0.2 0.2 0.4 ΔP
Figure 9: P(νµ →ντ)normal−P(νµ→ντ)inverted
To get to the first peak at about 40000 km/GeV neutrinos of the still low energy of 0.3 GeV would suffice at the widest point on earth. All these energies seem relatively low so producing your own neutrinos is most likely not the best option.
An alternative could be to us solar neutrino which can travel much larger distances but they have their own problems, the distance they have travelled is not exactly known for example.
V.
Current experiments
The easiest experiments to compare with data are accelerator experiments since they allow for a fixed distance which results in only one remaining variable. These experiments use an accelerator to shoot neutrinos at a detector over large distances. The most interesting accelerator experiment to look at seems to be the upcoming Deep Underground Neutrino Experiment. A lot of the details about this experiment are not easily found in the public domain but what is known is that a proton beam is used to create neutrinos which will be mostly muon-flavoured and that these neutrinos will travel a distance of 1300 km to reach the detector. From this we can make a few comparisons with our data to see what kind of results are to be expected. As the distance is now known we can set L to 1300 km and focus on the values of Eν. In the DUNE experiment the neutrino energy Eν can vary between 0.25 GeV and 10 GeV.From ref. [1]
2 4 6 8 10 Eν -0.10 -0.05 0.05 0.10 ΔP
Figure 10: P(νµ→ντ)normal−P(νµ →ντ)invertedfor L=1300 km and 0.25 GeV<Eν <10 GeV
2 4 6 8 10 Eν -0.10 -0.05 0.05 0.10 ΔP
Figure 11: P(νµ→νe)normal−P(νµ →νe)invertedfor L=1300 km and 0.25 GeV<Eν <10 GeV 2 4 6 8 10 Eν -0.10 -0.05 0.05 0.10 ΔP
Figure 12: P(νµ→νµ)normal−P(νµ →νµ)invertedfor L=1300 km and 0.25 GeV<Eν <10 GeV
All of these oscillations show three clear peaks for the difference in probability between the two hierarchies with the peaks for P(νµ → νe) being slightly smaller than the other two. This leads us to believe that the best flavour to measure is the τ-neutrino at the Eν value of the lowest peak. Since this is very close to the lower limit value for Eν it might be better to look at energies around the second peak for both τ and µ. Similar experiments to dune have already taken place. NOνA is one example. In the NOνA experiment the distance between detector
and production is 810 km.From ref. [2] It looked mainly at the νµ →νe transition. Yet another current accelerator experiment is T2K which makes neutrinos travel a distance of 295 km.From ref. [4]
VI.
Conclusion and Discussion
The data that was created can be used to determine the optimal distance from the source at which to measure neutrinos when determining the difference between inverted and normal mass hierarchy. An optimum was found in the νµ → ντ transition but also in νe →νe. However these optima are at such large distances
that the neutrino energy needs to be very small for the travel distance to come close to the diameter of earth. Which makes using these peaks in accelerator experiments on earth, like DUNE, which spans about 101 of the diameter of earth, highly unrealistic.
Perhaps these observations could be used when measuring solar neutrinos but even then the measurements need to be able to determine the neutrino energy very accurately since a small difference in neutrino energy could lead to very large differences in probability and even then there is the problem of not knowing the exact distance that the neutrinos travelled.
The calculations also did not account for other effects that might take place. It was assumed that between the production of a neutrino and the detection of a neutrino there cannot be any other interactions. This is of course invalid since neutrinos take part in weak interactions and since electrons, muons and tauons do not exist in the same numbers in nature these interactions are more or less common for the different neutrino flavours. This also starts to matter when measuring the neutrinos since electron neutrinos might be easier to measure and this will make it easier to get a statistically significant number of measurements.
This can also be seen from the fact that the NOνA experiment used the νµ →νe transition even though at 810 km other transitions such as νµ →ντ had a larger difference in probability according to our estimates. However, it also seems evident that there are effects of which the origin falls outside the scope of this paper as more research is required to determine which effects were and were not accounted for. This can be illustrated by a few graphs for oscillation probabilities as shown in figure 13 which show that in the range from 0 to 2500 km/GeV there should be three peaks while the same range only shows little over half a peak according to the data from this paper as can be seen when comparing figure 13 to figure 14.
Figure 13: Transition probabilities from NOνA that ignore matter effects with the vertical axis
showing the probability and the horizontal axis showing L/Eν. From ref. [2]
500 1000 1500 2000 2500 L Eν 0.005 0.010 0.015 0.020 0.025 P
Figure 14: Transition probability as expected by the estimates made in this report in the same
This report has attempted to clarify how the mass hierarchy of neutrinos can be extracted from measurements on neutrino oscillations. A few interesting properties have been found. An example of this is the different contributions in the probability function as shown in equation 22, from this one can pinpoint the key to finding the mass hierarchy.
The construction of the different probability functions and high and low points in their differences are also an example of this. These findings illustrate the sensitivity and interplay of distances, energy, and flavours and are a necessary step to determining the mass hierarchy.
References
[1] Valentina De Romeri, Enrique Fernandez-Martinez, and Michel Sorel (2016). Neutrino oscillations at DUNE with improved energy reconstruction arXiv:1607.00293v1 [hep-ph]
[2] https://en.wikipedia.org/wiki/NOνA
[3] F. Capozzia, E. Lisic, A. Marroned, D. Montaninoe, A. Palazzod (2016). Neutrino masses and mixings: Status of known and unknown 3ν parameters arXiv:1601.07777v1 [hep-ph]
[4] https://en.wikipedia.org/wiki/T2K_experiment
Populaire nederlandse samenvatting
Neutrinos komen voor in drie varianten (smaken genoemd) die echter niet één op één met een bepaalde massa corresponderen. Elke smaaktoestand is een superpositie van de drie massatoestanden en omgekeerd. Deze massatoestanden hebben in de praktijk dezelfde impuls maar net andere energieën.Hierdoor kan de ene smaaktoestand, bij het afleggen van een bepaalde afstand, overgaan in een andere smaaktoestand.
De kans op zo een overgang wordt beschreven door een aantal oscillerende functies, vandaar de benaming neutrino oscillaties. De fase van deze oscillatie is afhankelijk van de verschillen tussen het kwadraat van de verschillende massa’s. De absolute massa’s zijn niet bekend maar een van de verscillen zijn veel groter dan de derde. Dit zorgt voor een bepaalde massa hiërarchie die of "normaal" (één zware en twee lichte massatoestanden) of "geïnverteerd" (één lichte en twee zware massatoestanden) kan zijn. Het verschil tussen deze twee massa hiërarchieën zorgt voor verschillen in de kansen op bepaalde overgangen. Dit verschil is klein en in dit verslag wordt onderzocht op welke afstanden en bij welke energieën je het best metingen kunt doen als je wil bepalen welke hiërarchie daadwerkelijk voorkomt in de natuur. 50 000 100 000 150 000 200 000 250 000 L Eν -0.4 -0.2 0.2 0.4 ΔP
Figure 15: P(νµ→ντ)normal−P(νµ→ντ)invertedwaarbij P de kans is op een overgang van een µ-neutrino en een τ-neutrino
In figuur 15 is het verschil tussen de twee hiërarchieën bij een bepaalde overgang (van muon- naar tau-neutrino) te zien. Bij de verwachte neutrino energieën
zijn dit echter geen afstanden die op aarde te vinden zijn en moeten we dus genoegen nemen met minder duidelijke verschillen. Wel is in figuur 15 te zien dat binnen de afstanden die wel op aarde te overbruggen zijn een grotere aftand de voorkeur heeft, in ieder geval voor de νµ → ντ overgang. De resultaten worden ook vergeleken met bestaande experimenten maar aangezien er een aantal effecten niet zijn meegenomen in onze aanpak levert dat niet een hele duidelijk overeenkomst op.
