Table Of ContentTwo day’s worth of KATRIN data
Robert Ehrlich1
1George Mason University, Fairfax, VA 22030∗
(Dated: February 28, 2017)
TheKATRINexperimentplanstotakedatafor3to5yearssoastoachieveaten-foldimprovement
onpreviousexperimentsinthevaluefortheelectronantineutrinoeffectivemass. Asshownelsewhere
thisexperimentcanalsotesta3+3modeloftheneutrinomassstateswiththreeunconventionally
large masses. It is shown here that only two day’s worth of data should be more than sufficient
to exclude or validate this model. It is further shown that while final state distributions probably
cannot explain the good fits this model gave to previous direct mass data, they would alter the
predictions of the 3+3 model for KATRIN, without making them less observable. Four further
considerations are discussed which strengthen the case for taking the unconventional model more
seriously.
7
1
PACSnumbers: 95.35.+d
0
2
b I. INTRODUCTION Another use of Eq. 3 was made by the author in ref. [8]
e in testing his 3 +3 model of the neutrino mass states
F The KATRIN tritium β−decay experiment expects to consisting of three L-R doublets having unconvention-
7 set an improved upper limit of 0.2 eV on the ν¯ effective ally large masses. In that paper it was shown that this
e
2 mass, m (eff) after 3 years of data taking, or it might model gave excellent fits to the data from three existing
ν
h] deVis.co[v1e]rInthseinagclteuβal−vdaelcuaeyatthae5eσffelcetvievleifmmasνs(ecffan) >be0d.3e5- hTirgohitpskre,cMisiaoinnztraitnidumLiβve−rmdeocraeygerxopueprsimweitnhtosudtonneecbeysstahre-
p fined in terms of the separate mass states having masses ily being in conflict with the cosmological constraint on
- m contributing to the ν¯ state: [2] Σm or oscillation data [8]. Interestingly, those good fits
p j e to the model were obtained despite the 95% C.L. upper
e
h m2ν(eff)=Σj|Ue,j|2m2j (1) limit mν(eff)< 2 eV [2] two of those experiments have
set on ν¯.
[ e
To find a value for mν(eff) from the tritium β− spec-
5
trum one can do a fit of the distribution of the electron
v
kineticenergies,E,neartheendpointE ,usingthephase II. THE 3+3 MODEL
8 0
8 space term, i.e., the square of the Kurie function K(E)
4 where: [3] The3+3modelpostulatesthreeνL,νRdoubletshaving
0 masses that were originally inferred from an analysis of
0 SN1987A data [9–11]. The model was given subsequent
(cid:113)
1. K2(E)=(E −E) R((E −E)2−m (eff)2) (2) support by dark matter fits to the Milky Way galaxy
0 0 ν
0 and to four galaxy clusters. [12] The masses of the three
7 and where R is the Ramp function defined by R(x) = x doublets are: m1 ≈ m2 = 4.0±0.5 eV, m3 ≈ m4 =
:1 for x ≥ 0, and R(x) = 0 for x < 0. Note that here E0 is 2(cid:112)1.2±1.2 eV, and m25 ≈m26 =−0.2 keV2, or µ5 ≈µ6 =
v understoodtobegivenbythedecayQ-value,whichisthe −m2 = 447 eV. Each of the three masses would, in
5
i truespectrumendpointonlywhenoneignoresfinalstate accordance with Eq. 3, create a specific feature in the
X
distributions, and the acceptance of an experiment for spectrum.
r
a energies close to E0. Moreover, fits using Eq. 2 are only Thethreefeaturesinclude: (a)akinkatE0−E =m1,
appropriate if the separate m are not distinguishable (b) a second kink at E − E = m , and (c) a linear
j 0 3
experimentally. Otherwise, theproperprocedureistofit variation of K2(E) ∝ µ (E −E) between that second
5 0
the spectrum near E using K2(E) defined as: [3] kink and E . As noted in ref. [8] the only detectable
0 0
effect on the spectral shape given the limited statis-
tics of existing data sets is the first kink occuring at
(cid:88) (cid:113)
K2(E)=(E0−E) |Uej|2 R((E0−E)2−m2j) (3) E0−E = m3 = 21.2 eV. In fact, this feature was found
to exist in all three data sets examined with a common
whichessentiallydescribesasumofspectraforeachof spectral weight U324 ≡|Ue23|+|Ue24|=0.5.
the m with weight |U |2. Eq. 3 has been used, for ex-
j ej
ample, in searches for a possible sterile mass state. [4–7]
A. Why revisit the model here?
The purpose of the present paper is to consider six
∗Electronicaddress: [email protected] topicsthatwerenotaddressedpreviouslyinref.[8],each
2
of which will be seen to strengthen the case for taking
the unconventional 3+3 model more seriously. Thus, it 100
is shown here that: P(%) (a)
10
1. The good fits the model gave to three existing tri-
tiumβ−decaydatasetsareverylikelynotanarti-
1
factresultingfromfinalstatedistributions(FSD’s).
0.1
2. These FSD’s alter the predicted 3 + 3 model re-
sults for KATRIN without making them any less
0.01
observable.
0 10 20 30 40 50
3. A less radical alternative explanation to the good p 0.3
fits the 3+3 model gave to three tritium β−decay (b)
experiments, i.e., an m≈20 eV sterile neutrino, is
0.2
implausible.
4. While the Troitsk collaboration has retracted its 0.1
controversial “step” in the spectrum [13], thereby
raising serious questions about its inclusion in
0
ref.[8],thereareneverthelessgroundsforregarding
0 10 20 30 40 50
the report of its demise as premature. p
0.5
5. The need for a precise understanding of the value
0.4 (c)
ofall systematicuncertainties (vitalforKATRIN’s
intended goal) is significantly less important if the 0.3
data should be in accord with the 3+3 model. 0.2
6. KATRIN will have the sensitivity to look for all 0.1
threepartsofthe3+3modelsignature, andinfact 0
will be able it to validate or reject the model in a 0 10 20 30 40 50
mere two days of data-taking. E (eV)
FIG. 1: Final state distributions (FSD’s) (a) shows the
distribution of molecular energy levels in tritium gas accord-
B. How FSD’s might create a kink near 20 eV ing to Saenz et al. [15], (b) shows the f1(E) distribution of
energy losses by electrons in single inelastic scattering, and
(c) shows the F (E) distribution of energy losses for any
Sincepublicationofref.[8],itwassuggestedtotheau- Loss
number of scatterings expected in the KATRIN experiment.
thor[14]thatthegoodfitsofthe3+3modeltoβ−decay
data might be explained due to the presence of FSD’s
thatwilldistorttheobservedspectrum. Thetwotypesof because the probabilities P for n scatterings depend on
n
finalstatedistributionsconsideredhereincludemolecular the average column density of radioactive tritium gas,
excitations, and scattering processes in the source which i.e., the number of mean free paths, N according to the
lead to β−electron energy loss. Both processes lead to a Poisson distribution:
modification of the endpoint region of the spectrum, but
Nne−N
they have a very different nature. The former includes
P = (4)
possible final states of a diatomic molecule containing n n!
tritium, i.e., T ,HT,DT going to HeT+,HHe+,DHe+,
2 The sharp peaks in Figs. 1(b) and (c) at E ≈ 12
Loss
which can be populated before beta decay, and which
eV, and the much smaller one at E ≈ 25 eV in fig.
Loss
change the emitted β electron energy distribution from
1(c) are due to single and double inelastic scatterings,
what is found when assuming a decay Q-value for a T
2 respectively. The presence of the small 25 eV peak and
molecule in its ground state.
the broader one in fig. 1(a) at a slightly higher energy,
The latter process lowers the observed energy of elec-
suggestthepossibilityofoneorbothofthesepeaksbeing
trons after emission, and the distribution of losses de-
the true cause of the good fits reported in ref. [8] rather
pends on the number of scatterings. The distribution of
thanthe3+3model.[14]Inordertotestthispossibility
electron energy loss in single inelastic scattering f (E)
1 wehavedoneasimulationandgeneratedspectraforboth
is the same for all experiments using gaseous tritium –
the3+3modelandthestandardallm =0orm (eff)=0
j ν
see fig. 1 (b). However, the distribution of losses for
case using the following 5-step procedure:
any number of scatterings, i.e., F (E) = ΣP f (E)
Loss n n
showninfig. 1(c)forKATRINisexperimentdependent 1. Using the expected value of N for a particular ex-
3
spectrum F (E).
Counts FSD
per 0.5 eV-h 5. Do a convolution of F (E) with a Gaussian-
FSD
distributedresolutionfunctionhavinganappropri-
40000 ate FWHM to yield a final spectrum that incorpo-
rates the effect of FSD’s and resolution.
35000
C. Simulation results for KATRIN
30000
The results of this simulation are shown in two figures
inordertodisplaybetterthedistortionsofthespectrum
25000 expected in two spectral regions surrounding each kink.
Itcaneasilybeseeninfig. 2thattheimpactofFSD’sand
resolutiononthespectrumistoroundofftheabruptness
20000
of a kink, without making it less observable. The long
dashed curve in Fig. 2 (the FSD-modified spectrum for
the all m = 0 case) shows that while FSD’s clearly do
15000 j
alter the spectrum shape near 20 eV they do not create
a kink in the spectrum there when one is not originally
10000 present, i.e., when the initial spectrum is parabolic. In
fact the FSD-modified long dashed spectrum shows no
observableinflectionatthepositionofthekinkanditcan
5000 be approximated for the last 30 eV of the spectrum by a
single power law of the form dN/dE ∝ (E −E−1.0)γ
0
with γ ≈2.8
0
In order to carry out steps 1-5 of the simulation one
-28 -26 -24 -22 -20 -18 -16 needsavalueofNforagivenexperiment,whichdepends
in part on whether a magnetic trap is used. In such a
E-E (eV) case only electrons which start with a certain range of
0
pitch angles can pass the trap. [16]. This is less of a
FIG.2: SimulatedKATRINdatabasedon3+3model problem in KATRIN than some previous experiments in
foradayofdata-taking,i.e.,onehourofdata-takingforeach lightofitsgoodangularacceptance–seeFig. 9inref.[1],
of 24 energy bins of 0.5 eV width. The solid curve (with which shows transmissions as high as a constant 37%
10Xinflatederrorbarsforclarity)istheexpectedcountrate for electron energies more than 1 eV from the spectrum
above background. The long dashed curve, which has the endpoint. KATRIN also has the advantages over earlier
same count rate at 20 eV, shows shows the “kink-free” all experiments of about four times better energy resolution
m = 0 case after FSD’s and energy resolution have been
j ( 0.93 eV) and a very low background rate close to the
included. The short dashed curve shown with slightly larger
endpoint. The expected background (<10−2 cps), [1] is
normalization(forclarity)isthe3+3modelpredictionbefore
also a significant improvement over earlier experiments
including FSD’s and energy resolution. The thick grey curve
suchasMainzbyfactorsof1.5to2.5.[17]Finally,wenote
represents a failed attempt to mimic the 3+3 model results
based on an anomalous energy loss distribution including a that KATRIN has a far more sophisticated method of
large amplitude δ-function, as discussed in the text. findingtheimpactofelectronenergylossonthespectrum
thanusedhere. Bydoingathree-dimensionalsimulation
of the motion of individual electrons emitted at various
periment, find the P(cid:48)s using Eq. 4 points in the gas with various directions and energies,
n
KATRIN can find the number of mean free paths N for
2. Do a convolution of the original spectrum defined them to reach the detector, assuming they make it past
by Eq. 3 with the T2 excitation probability to find the trap.
an “excitation modified” β− spectrum.
3. Find the inelastic scattering distributions, f (E)
n D. What sort of FSD might mimic the 3+3 model?
corresponding to n scatterings by convolving the
single scattering function with itself n times,
and then find the resultant energy loss function FindinganaverageNinagivenexperimentwhichuses
F (E) for n≥0 scatterings a magnetic trap is difficult without doing the kind of
Loss
three-dimensional simulation just described. Thus, in-
4. Do a convolution of the excitation modified spec- stead of doing unrealistic simulations for the three ex-
trum with F (E) to yield the FSD-modified periments examined in ref. [8], we instead consider what
Loss
4
conceivableF (E)mightcreateakinkinthespectrum
Loss Counts
near20eVroughlyresemblingthatproducedbythe3+3
model,andhowthoseresultscomparewiththoseforKA- per 0.5 eV-h
TRIN. 3000
It is clear that given the existence of a sharp peak
in fig. 1 (c) at E = 0, a F (E) distribution with a 300
Loss
second sharp peak near 20 eV will result in a kink of
somemagnitudeinthefinalconvoluteddistributionnear
E − E = 20 eV. Let us therefore assume an energy
0
loss curve having a delta function of height A at 20 eV 200
atop the E (E) curve in fig. 1(c), and consider what
Loss 2000
valuesofAcouldproduceakinkofthemagnitudecreated
by the 3+3 model simulation. The result of such an
100
exerciseisshownbythethicksemitransparentgreycurve
in fig. 2, where a detectable kink occurs at −20 eV for
a suitably large A value. However, not only is the kink
lessnoticeabletheoneforthe3+3model,buttheshape
0
of the spectrum to the left of it is quite different, being
concave up rather than down as that model yields. In 1000 -5 -4 -3 -2 -1 0
addition, the value of A used to create the small kink is
equivalent to an impossibly large P = 220× the height
2
of the double scattering peak or 7.7 (i.e., 770%), at least
when using KATRIN values for the P(cid:48)s.
n
Inlightofthisresultonemightquestionwhetherusing
P(cid:48)sfromotherexperimentsmightalsorequireanimpos-
n
sibly large double scattering peak, i.e, one large enough 0
to mimic that of the 3 + 3 model. The magnitude of
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
the spectral kink produced from the two delta functions
obviously would be proportional to the height of each
E-E (eV)
one, and hence the product, P P . We may therefore use 0
0 2
d(P0P2) =0,tofindN whichyieldsamaximumpossi-
dN max
bleamplitudekink. TheresultisNmax =1,whichyields FIG.3: Simulated KATRIN data based on 3+3 model
P0 =2P2 =e−1 =0.37(37%), and (P0P2)max =0.069. foraseconddayofdata-takingintheregionE0−E <10eV
Using the KATRIN value P =0.43 [18], we find N = foronehourofdata-takingforeachof24energybinsof0.5eV
0
0.84,P = 0.15 so that (P P ) = 0.066. In other width. The solid curve (with non-inflated error bars) is the
2 0 2 KAT
words, we have shown (P P ) ≈ (P P ) . Since expected count rate above background for one hour of data-
0 2 KAT 0 2 max
theKATRINN-valueissoclosetoyieldingthemaximum takingforeachof24energybinsof0.5eVwidth. Ithasbeen
normalized to yield the expected count rate at E −E = 20
amplitude kink based on the two delta function model, 0
eV.Thedashedcurvewiththesamenormalizationshowsthe
we can infer that what is true for that experiment is
“kink-free”allm =0caseafterFSD’sandenergyresolution
also true for others. Namely, if an impossibly large P j
2 have been included. The insert shows the last 5 eV with an
is required to produce a kink that mimicks of the 3+3
expanded vertical scale.
model for KATRIN, then the same would be true for
other experiments.
E. Two day’s worth of KATRIN data
Of course, taking data only for E −E <20 eV would
0
KATRIN will measure the number of counts N(E) in exclude the energy where the first and most prominent
some time interval ∆t integrated over the energy region 3+3 model kink is predicted. It is therefore important
fromE toE . Thus,twosuchintegralmeasurementsare to note that based on the simulation it should be pos-
0
neededtofindthedifferentialintensity∆N/∆Eforasin- sible to either confirm or reject the model using a mere
gleenergybin∆E.Experimentersmustthereforechoose two day’s worth of data. That period would include 24
carefully what region of the spectrum to measure, and hoursforeachofabout24energybins∆E =0.5eVwide
how much data to take for each energy bin. Apparently centered on each of the two expected kinks. In generat-
KATRIN intends to take data mainly over last 20 eV in ingsimulatedKATRINdatawehaveassumedanenergy
light of the desire to minimize the effects of FSD’s, [14] resolution of 0.93 eV, a P = 43%, and a count rate of
0
and because this is where the effects of a finite m (eff) .01Hz forthelast1eV,[1,18]whichyieldsabout11,500
ν
are most important. counts per hour in a 0.5 eV wide bin at E−E =20 eV.
0
5
(cid:113)
F. The m2 <0 spectral feature in the last 4 eV i.e., σ = σ2 +σ2 . The ignoring of systematic
tot syst stat
uncertainties is justified here because if σ ≈σ for
stat syst
The most prominent features of the simulated 3+3 3 y of data taking then σ >> σ when the data
stat syst
model results for KATRIN are the two rounded kinks taking period is only 2 d. As a result it can be said that
at −21 eV and −4 eV. In addition, fig. 3 also shows if the data closely resembles the 3+3 simulation results
the feature associated with the m2 < 0 mass state. We for 2 d of data taking, the null hypothesis (all very small
assumed earlier that Eq. 3 applies regardless of the sign m ) could be rejected with very high confidence.
j
of m2. In that case for the m2 < 0 state Eq. 3 yields
a linear fall of the spectrum beyond the 4 eV kink, i.e.,
K2(E) ∝ µ (E −E), whose slope should be in accord
5 0 H. A sterile ν alternative to the 3+3 model?
(cid:112)
with the value of µ = −m2 specified by the 3 + 3
5 5
model.
The existence of a possible sterile neutrino ν of mass
However, there is a different way the m2 < 0 mass 4
m≈20−30eVwouldseemtobeamuchlessextremeal-
state might reveal itself in the event that Eq. 3 is inap-
ternativetothe3+3model. However,whilea20eVster-
propriatefortachyons. CiborowskiandRembielinski[19]
ileneutrinocouldexplainthegoodfitsobservedforthree
(C-R)andmorerecentlyRadzikowski[20]havedeveloped
tritiumdecayexeriments,aswellastheearlierSN1987A
causaltheoriesoftachyonswhichrequiretheexistenceof
data and the dark matter fits, there are some serious ob-
a preferred frame of reference, perhaps that of the cos-
stacles to this alternative:
mic neutrino background (CNB). Let us assume the lab
frame is close to being that preferred reference frame, 1. The three fits to tritium experiments required a
which is certainly true for the CNB frame assumed to large degree of mixing (≈ 50%) of the 20 eV state
be the same as the CMB, for which v/c = 0.00123. In with the other states, and this would be ruled out
that case C-R predict that the m2 < 0 component of by oscillation data, assuming a standard mass hi-
the spectrum would have virtually constant height close erarchy.
to the spectrum endpoint. [19] Clearly, either possibility
for the m2 < 0 state, i.e., a linear fall-off as depicted or 2. A 20 eV sterile neutrino could conceivably yield
a constant height beyond 3.5 eV as C-R predict should results similar to the 3+3 model results in Fig. 2,
be readily distinguishable from the quadratic fall off ex- but of course it would not agree with those in Fig.
pected for the all m = 0 case (the dashed curve) – see 3, with the kink near 4 eV should it be observed.
j
fig. 3. There is, however, an interesting difference be-
tween the two possibilities, should one of them be ob- 3. Based on Eq. 1 the 2 eV upper limit on the ν¯e
served (thereby confirming the existence of a tachyonic effective mass, and a much smaller limit from cos-
mass state). Finding that the fall-off is linear would si- mology on the sumof the flavor state masses, only
multaneouslyrevealtheactualmassofthestateanddis- the existence of a m2 <0 mass state could keep ν¯e
prove the C-R causal theory, while finding that there is effective mass very small. [8]
nofall-offwouldvalidatethattheorybutnotyieldamass
If the actual KATRIN data should closely resemble
value.
the simulated results for the 3+3 model in figs. 2 and
3, experimenters would be severely challenged to explain
them in terms of FSD’s or other systematic effects and
G. What about systematic uncertainties?
the standard view of small neutrino masses.
KATRIN has made a major effort to understand all
possible sources of systematic uncertainty. Thus, in
ref. [1] they group sources of systematic uncertainty in
Acknowledgment
four broad categories, having mostly to do with the tri-
tium source. They further estimate the total system-
The author is indebted to Hamish Robertson for
atic uncertainty in the effective mass squared of ν¯ as
e
σ ≤ 0.017 eV2, or about the same as the statisti- calling attention to the possibility that the good fits
syst
reported in ref. [8] might be due to FSD’s rather than
cal uncertainty after 3 y of data taking. [1] Let us here
the 3 + 3 model, and for his critical comments on an
assume that the KATRIN results closely resemble those
earlier version of this manuscript.
of the 3 + 3 model in figs. 2 and 3 after two days of
data taking. If one does a one parameter fit of those
results to the standard all m = 0 curve, adjusting
j
only its normalization one finds χ2 = 6880 with 45 dof,
thereby allowing this null hypothsis to be rejected with Appendix A: Is the Troitsk anomaly dead?
extremeconfidence. Thisassertiondependsontheuseof
√
σ = n,anditignoressystematicuncertaintiesσ , The inclusion of the 1999 Troitsk data in ref. [8] with
stat syst
which are normally combined in quadrature with σ , its controversial “step” might be challenged, because in
stat
6
a 2012 reassessment [13], after failing to explain this fea- plitude step than originally seen, the grounds for drop-
ture for over a decade, the Troitsk authors upon drop- ping this feature are not as clear as Troitsk suggests. As
pingsomeuncalibratedruns,concludedthat“withinthe canbeseeninFig. 8ofthe2012reassessment[13],given
existing statistical errors, there are no reasons for intro- the error bars at various times of the year the average
ducing such a feature.” [13] In fact, one of the Troitsk stepamplitudecanbecomputedas2.18±0.48mHz.This
authors has gone even further. V. S. Pantuev, in a pri- result may fall short of the 5σ criterion for a discovery,
vate communication, has noted that Troitsk no longer whichperhapswouldjustifyTroitskriddingthemselvesof
botherstodisplaytheactualspectrumbecause: “In[the] theirunexplainedanomalyonstatisticalgrounds. Never-
newanalysisthisstepisabsolutelyinvisiblebyeyeinthe theless,a2.18/0.48=4.5σstepinthespectrumishardly
spectrum.” [21]. Can this really be the case? While the insignificant, and it would be very surprising if such an
reanalyzed Troitsk data did result in a 25% smaller am- effect could not be seen in the spectrum.
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http://arxiv.org/pdf/1111.0502
