Degeneracies in long-baseline neutrino experiments from nonstandard interactions

0.98 MB·

by  Jiajun Liao

Degeneracies in long-baseline neutrino experiments from nonstandard interactions
Jiajun Liao,1 Danny Marfatia,1 and Kerry Whisnant2
1Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822, USA
2Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
We study parameter degeneracies that can occur in long-baseline neutrino appearance experi-
mentsduetononstandardinteractions(NSI)inneutrinopropagation. Forasingleoff-diagonalNSI
parameter, andneutrinoandantineutrinomeasurementsat asingle L/E, thereexists acontinuous
four-folddegeneracy(relatedtothemasshierarchyandθ23 octant)thatrendersthemasshierarchy,
octant, and CP phase unknowable. Even with a combination of NOνA and T2K data, which in
principle can resolve the degeneracy, both NSI and the CP phase remain unconstrained because of
6 experimental uncertainties. A wide-band beam experiment like DUNEwill resolve this degeneracy
1 if the nonzero off-diagonal NSI parameter is ǫeµ. If ǫeτ is nonzero, or the diagonal NSI parameter
0 ǫee is O(1), a wrong determination of the mass hierarchy and of CP violation can occur at DUNE.
2 The octant degeneracy can be furthercomplicated byǫeτ, butis not affected byǫee.

y
a
M After decades of neutrino oscillation experiments, the of neutrinos in matter,
standard model (SM) with massive neutrinos is well es-
2 tablished and the study of neutrino oscillations has en- 1+ǫee ǫeµeiφeµ ǫeτeiφeτ
 
1 tered the precision era [1]. Next generation neutrino V =A ǫeµe−iφeµ ǫµµ ǫµτeiφµτ . (3)
oscillation experiments will be sensitive to physics be-  ǫeτe−iφeτ ǫµτe−iφµτ ǫττ 
] yond the SM, which is often described in a model-
h
-p ifnordeapernedceenntt mreavnienwersbeyenRoenfs.ta[2n]d.arIdningteenrearcatli,onNsS(INSnIo)t; Here, A ≡ 2√2GFNeE and ǫαβeiφαβ ≡ fP,CǫfαCβNNef, with
p only affect neutrino propagation in matter via neutral- Nf the number density of fermion f. The ǫαβ are real,
e current interactions, but also affect neutrino production and φαβ = 0 for α = β. The unit contribution to the
h
[ and detection via charged-current interactions. Model- ee element of the matrix is due to the standard charged-
independentboundsontheproductionanddetectionNSI current interaction.
2 are generally an order of magnitude stronger than the Expanding the νµ νe oscillation probability for the
v matterNSI[3]. Therefore,weneglectproductionandde- normalhierarchy(NH→)inthesmallquantitiess13,r,and
7
2 tectionNSIinthiswork,andfocusonmatterNSI,which the nondiagonal ǫ, we find (with cjk ≡ cosθjk, sjk ≡
9 canbedescribedbydimension-sixfour-fermionoperators sinθjk)
0 of the form [4] 2 2 2 2
P(ν ν )=x f +2xyfgcos(∆+δ)+y g
1.0 LNSI =2√2GFǫfαCβ[ναγρPLνβ](cid:2)¯fγρPCf(cid:3)+h.c., (1) +4Aˆǫµeµ→xfe[s223fcos(φeµ+δ)+c223gcos(∆+δ+φeµ)]
(cid:8)
0 where α,β = e,µ,τ, C = L,R, f = u,d,e, and ǫfC are +yg[c2 gcosφ +s2 fcos(∆ φ )]
6 αβ 23 eµ 23 − eµ (cid:9)
dimensionless parameters that quantify the strength of
1 the new interaction relative to the SM. Since neutral- +4Aˆǫeτs23c23{xf[fcos(φeτ +δ)−gcos(∆+δ+φeτ)]
:
v current interactions affect neutrino propagation coher- yg[gcosφeτ fcos(∆ φeτ)]
− − − }
Xi ently, the matter NSI potentially have a large effect +4Aˆ2g2c223 c23ǫeµ s23ǫeτ 2+4Aˆ2f2s223 s23ǫeµ+c23ǫeτ 2
ar Ton2Kth[e5],loNngO-bνaAse[l6in],eannedutDriUnoNEos[c7il]l.atiPorneveixoupsersimtuednitess, +8Aˆ2fgs23|c23(cid:8)c23−cos∆(cid:2)s|23(ǫ2eµ−ǫ2eτ)| |
of matter NSI in these experiments can be found in +2c23ǫeµǫeτ cos(φeµ φeτ)]
−
Refs. [8, 9]. ǫ ǫ cos(∆ φ +φ )
eµ eτ eµ eτ
− − }
In this paper, we use bi-probability plots and numeri- 2 2 3
+ (s13ǫ,s13ǫ ,ǫ ), (4)
calsimulationstoanalyzeparameterdegeneraciesinlong- O
baselineneutrinoappearanceexperimentsthatarisefrom where following Ref. [10],
matter NSI. We specifically study how well the NOνA,
2 2
T2KandDUNE experiments willresolvethese degenera- x 2s13s23, y 2rs12c12c23, r = δm21/δm31 ,
≡ ≡ | |
cies for the cases of one and two off-diagonal NSI, and sin[∆(1 Aˆ(1+ǫ ))] sin(Aˆ(1+ǫ )∆)
diagonal NSI. f, f¯ ∓ ee , g ee ,
≡ (1 Aˆ(1+ǫ )) ≡ Aˆ(1+ǫ )
Oscillation probabilities. The Hamiltonianforneu- ∓ ee ee
2
trino propagationin the flavor basis may be written as ∆ (cid:12)δm31L(cid:12), Aˆ (cid:12) A (cid:12) . (5)
≡(cid:12) 4E (cid:12) ≡(cid:12)δm2 (cid:12)
H = 1 Udiag(0,δm221,δm231)U†+V , (2) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) 31(cid:12)(cid:12)
2E (cid:2) (cid:3) Note that our definitions of f, f¯, and g here include ǫ ,
ee
whereU isthePMNSmixingmatrix[1],δm2 =m2 m2, which is not treated as a small parameter. We have set
ij i− j
and V represents the potential arising from interactions c13 1, which is accurate up to first order in θ13. The
→
2
0.10
ç ∆=0
æ ∆=90°
0.08 ææ á ∆=180°
IááHΘ23>45° à ∆=270°
ææ çç
L®Νe 0.06 áá àà
ΝΜ IHΘ23<45°çç
HP 0.04 àà ææ ææ NççHΘ23>45°
áá
çç
áá àà
0.02
NHΘ23<45° àà
0.00
0.00 0.02 0.04 0.06 0.08 0.10
PHΝΜ®ΝeL
FIG.1. Bi-probabilityplots(P(νµ →νe)versusP(νµ→νe))
for L = 1300 km and E = 3 GeV. The solid curves show
the SM values for fixed mixing angles and varying δ (one
curve for each combination of hierarchy and θ23 octant); the
dashed(dotted)[dotdashed]curvesshowthevaluesassuming
NSI with ǫeµ = 0.05 (0.10) [0.15], and varying φeµ for the
NH and first octant with δ′ = 0. The mixing angles and
mass-squared differences are set at their best-fit values from
′
Ref. [12]. FIG. 2. Values of a single nonzero ǫ as a function of δ that
give P(νµ → νe) and P(νµ → νe) degenerate with the SM
with δ=0andNH,at NOνA(L=810 km,E =2GeV)and
T2K (L = 295 km, E = 0.6 GeV). The mixing angles and
antineutrino probability P P(ν ν ), is given by
≡ e → µ mass-squared differencesare thebest-fit valuesin Ref. [12].
Eq. (4) with Aˆ Aˆ (and hence f f¯), δ δ, and
→ − → →−
φ φ . Forthe invertedhierarchy(IH), ∆ ∆,
αβ αβ
→− →−
y →−y,Aˆ→−Aˆ(i.e.,f ↔−f¯,andg →−g). Sinces13 plot, which shows P versus P (see Fig. 1). The solid
andr aresmall,soarex( 0.2)andy ( 0.02). Further-
≈ ≈ ellipses in Fig. 1 represent possible P and P values for
more, Aˆ < 0.3 for L 1300 km. Our result agrees with the SM with fixed mixing angles and varying δ. The
the O(ǫ)∼expressiono≤f Ref. [11]. The 90% C.L. limits on four ellipses are for NH and first θ23 octant, NH and
the NSI parameters that appear in Eq. (4) are ǫee <4.2, second octant, IH and first octant, and, IH and second
ǫeµ < 0.33, and ǫeτ < 3.0 [3]. The other ǫαβ do not ap- octant, corresponding to the usual four-fold degeneracy
pear in Eq. (4) up to second order in ǫ. Since Eq. (4) is that remains now that the oscillation amplitudes and
onlyvalidforsmallnondiagonalǫ,inoursimulationsthe mass-squareddifferences have been measured.
oscillation probabilities are evaluated numerically with-
The non-solid contours in Fig. 1 represent the proba-
out approximation.
bilities for the same mixing angles for the NH and first
One off-diagonal NSI. If ǫee = 0, and only one off- octant including NSI with δ′ = 0, fixed ǫ and varying
diagonalNSIinEq.(4)(i.e.,ǫeµ orǫeτ)isnonzero,anda φ. The center of the non-solid contours is located at the
measurementoftheneutrinoandantineutrinooscillation δ = 0 point of the NH-first octant solid ellipse. Clearly
probability is made at one particular L and E (which is any point on any of the solid ellipses (i.e., any δ, either
approximatelytrueforanarrow-bandbeamexperiment), hierarchy, and either octant) can be obtained by center-
then a parameter degeneracy between the SM and a ing the non-solidcontours on any other point of the NH-
modelwithNSIwilloccurwhenPSM(δ)=PNSI(δ′,ǫ,φ) firstoctantsolidellipse (i.e., anyδ′)with anappropriate
and PSM(δ) = PNSI(δ′,ǫ,φ), where δ′ is the Dirac CP valueofǫandφ. Inthelimitwheretheǫ2termscanbeig-
phase in the model with NSI. Assuming the three mix- nored,thenon-solidcontoursaresimpleellipses,thesizes
ing angles and two mass-squared differences are well- of which are determined by the magnitude ǫ; when ǫ is
measured by other experiments, for each value of δ in larger,the ellipses become distorted, their range increas-
the SM there are three unknowns to be determined that ing with ǫ. NSI with the IH and/or second octant can
giveanoff-diagonalNSI degeneracy: δ′, the magnitude ǫ beobtainedsimilarlybycenteringthenon-solidcontours
and the phase φ. With only two constraining equations, on the points of the corresponding solid ellipse.
ingeneraltherewillbeacontinuumofsolutionsthatcan Figure 2 shows the values of ǫ versus δ′ that have a
give a parameter degeneracy; i.e., for each value of δ, a degeneracy with the SM with δ = 0 and NH, for either
solution for ǫ and φ will exist for any value of δ′. ǫ or ǫ , for the baseline and approximate position of
eµ eτ
Therefore, in the context of one off-diagonal NSI, any the spectrum peak in the NOνA and T2K experiments,
CP phase value is allowed with a single measurement of assuming NSI solutions with a NH or IH. For NOνA,
P andP. Thismaybedemonstratedviaabi-probability these values are within the corresponding experimental
3
constraintsforallδ′. The valuesofǫ thatgivea degener-
acyarelargerinT2Ksincetherelativesizeofthematter
effect is smaller there due to the lower average density
and shorter distance. Similar curves exist for other val-
ues of δ for the NH; in all cases, ǫ = 0 when δ′ = δ,
but ǫ > 0 when δ′ = δ, i.e., degenerate NSI solutions
only exist when δ′ =6 δ. For the IH, the values of ǫ that
give a degeneracy a6t δ′ =270◦ are smaller than those at
δ′ =90◦. Thiscanbeunderstoodfromthebi-probability
plot in Fig. 1: the IH with δ =270◦ is closest to the NH.
Approximate formulasfor the curvesin the left panels
ofFig.2canbeobtainedbydroppingtherǫandǫ2 terms
in Eq. (4):
s2 (f f¯)
tan(φ +δ′)= 23 −
eµ 2c2 gsin∆
23
cos∆(cosδ cosδ′)[2c2 gcos∆+s2 (f +f¯)]
23 23
+ − , (6)
2c2 gsin2∆(sinδ sinδ′)
23
−
yg[cos(∆+δ) cos(∆+δ′)]
ǫ = − ,
eµ 2Aˆ[c2 gcos(∆+φ +δ′)+s2 fcos(φ +δ′)]
23 eµ 23 eµ
and FIG. 3. 1σ, 2σ and 3σ allowed regions for ǫeµ and ǫeτ (when
onlyoneofthemisnonzero) from combinedNOνAand T2K
(f¯ f)
tan(φ +δ′)= − data, and from DUNE.
eτ
2gsin∆
cos∆(cosδ cosδ′)[2gcos∆ f f¯]
+ − − − , (7)
2gsin2∆(sinδ sinδ′) precisioniscomparabletotheprojectedresultsinRef.[7],
yg[cos(∆−+δ) cos(∆+δ′)] and the expected sensitivity for the mixing angles and
ǫeτ = 2Aˆs23c23−[gcos(∆+φeτ +−δ′) fcos(φeτ +δ′)]. mofaRsse-sf.qu[1a6r]e.d dTiffheerePnrceelismininaoruyr sRimefuerlaetnicoenEmaarttchhMFiogd.e8l
−
ǫ and ǫ are obtained by first solving Eqs. (6) and (7) density profile [17] is implemented in GLoBES, and we
eµ eτ
for φ and φ . Similar equations exist for each of the use a conservative 5% uncertainty for the matter den-
eµ eτ
other possibilities, i.e., NSI with any δ′, either hierarchy sity [18]. Also, to be conservative, we adopt the central
and either octant can mimic the SM with any δ, either valuesandpriorsonthe mixinganglesandmass-squared
hierarchy and either octant. differences, and the sizes of the NSI parameters in our
To remove the NSI degeneracies, additional measure- simulation from the global-fit with NSI in Ref. [12].
ments must be made. Since there are degeneracies InFig.3weshowtheexpectedallowedregions(defined
throughout the two-dimensional δ δ′ space, one addi- by ∆χ2 for two degrees of freedom assuming the param-
tional measurement (such as a neut−rino probability at a eters are Gaussian distributed) in the δ′ ǫ space and
eµ
different L and/or E) will reduce the dimensionality of δ′ ǫ spacefromtheneutrinoandantin−eutrinoappear-
eτ
thedegeneraciesbyone,i.e.,tolinesinδ δ′ space. Thus an−cechannelsfromNOνA andT2K combined,andfrom
for each value of δ there will only be on−e δ′ that will be DUNE. We assumethatthe dataareconsistentwith the
degenerate (or perhaps a finite number of δ′ if there are SMwithδ =0,thefirstoctant,andtheNH.Wescanover
multiple solutions). Two additional measurements at a all octant and hierarchy combinations. Even with the
different L and/or E will remove the degeneracies; the NOνA and T2K data combined, there are regions near
only solutions then are δ′ =δ and ǫ=0. δ′ = 0 or 180◦ where NSI solutions are allowed at less
Simulations. In practice, even narrow-band beams than 1σ; see the left panels of Fig. 3. So, although the-
havea spreadofenergies,althoughthe energyresolution oretically the degeneracies with NSI solutions should be
and uncertainties may not allow the degeneracies to be resolved,theexperimentaluncertaintiesarelargeenough
resolved. We simulate the long-baseline experiments us- that large NSI regions are not excluded. Using the wide-
ing the GLoBES software [13], supplemented with the band beam at DUNE, which effectively measures proba-
newphysicstoolsdevelopedinRefs.[11,14]. We usethe bilitiesatavarietyofenergies,putssevererestrictionson
experimentalsetupforNOνAandT2KasinRef.[15],in NSI: δ′ <50◦ andǫ<0.1at3σforǫ ;seethetop-right
eµ
which T2K collects data for 5 years each in the neutrino panel|of|∼Fig. 3. Howe∼ver, some small degenerate regions
and antineutrino mode, while NOνA runs for 3 years in in δ′ ǫ space are allowed at less than 2σ due to the
eτ
−
each mode. For DUNE, we consider a 34 kiloton liquid mass hierarchy degeneracy; see the bottom-right panel
argon detector with a 1.2 MW beam, and running for 3 of Fig. 3. This can be understood from the oscillation
years in each mode. We checked that our measurement probabilities shown in Fig. 4. The dashed curves almost
4
0.10 0.10
SM,∆=0,NH SM,∆=0,NH
0.08 ΕeΤ=0.36,ΦeΤ=315°,∆'=234°,IH 0.08 ΕeΤ=0.36,ΦeΤ=315°,∆'=234°,IH
L®Νe 0.06 Εee=-1.7,∆'=204°,IH L®Νe 0.06 Εee=-1.7,∆'=204°,IH
HΝPΜ 0.04 HΝPΜ 0.04
0.02 0.02
0.00 0.00
0 2 4 6 8 10 0 2 4 6 8 10
E@GeVD E@GeVD
FIG. 4. νµ → νe and νµ → νe oscillation probabilities at
DUNE. The mixing angles and mass-squared differences are
thebest-fit values in Ref. [12].
FIG. 6. Same as Fig. 3, except for ǫee.
One diagonal NSI (ǫ ). In this case, the oscilla-
ee
tionprobabilitytosecondorderis simply the firstline of
Eq. (4). Since φ = 0, there is a simple two-fold degen-
ee
eracy between the SM and NSI, i.e., for measurement of
P andP atoneL/E,eachvalueofδ isdegeneratewitha
point (δ′,ǫ ) in NSI space (although in some cases, due
ee
to the nonlinear nature ofthe equations,there are multi-
ple, but finite number of degenerate solutions). In prin-
ciple, a narrow-bandbeam experiment should be able to
pinpoint both the SM value of δ and the degenerate NSI
valuesofδ′ andǫ . Inpractice,forNOνAandT2K,the
ee
uncertainties are too large and no value of δ′ is strongly
preferred;seeFig.6. DUNEwillputstrongerconstraints
onNSIandδ′. TheislandsintherightpanelofFig.6can
be understood from the dotted curves in Fig. 4. DUNE
alone cannot resolve the mass hierarchy degeneracy,and
it could also lead to a wrong determination of CP vio-
lation if ǫ is (1). Note that ǫ does not affect the
ee ee
O
octant degeneracy.
In summary, we studied parameter degeneracies that
FIG. 5. Same as Fig. 3, except both ǫeµ and ǫeτ are nonzero. occur in long-baseline neutrino appearance experiments
due to matter NSI. We derived the oscillation probabili-
tiesfortheappearancechannelstosecondorderinǫ. We
overlapthe SM curves in both the neutrino and antineu- found that there is a continuous four-folddegeneracyfor
trinochannels. Thiscouldleadtoawrongdetermination an off-diagonal NSI in narrow-band beam experiments
of the mass hierarchy and CP phase. Also, in parts of like NOνA and T2K. A combination of their data would
theallowedparameter-space,θ23liesinthesecondoctant in principle break the degeneracy, but in practice, large
whichcould leadto a wrongdetermination ofthe octant. regions of NSI parameter space remain allowed due to
WeseethatDUNEalonecannotcompletelyresolvethese large experimental uncertainties. We also discussed de-
degeneracies for nonzero ǫeτ. generaciesthatoccurfordiagonalNSI,andformorethan
Two off-diagonal NSI. If both ǫ and ǫ are one off-diagonalNSI at a time. While the DUNE experi-
eµ eτ
nonzero,then there are five free NSI parameters: δ′, two mentcanresolvemostofthedegeneracies,fornonzeroǫ
eτ
magnitudes, and two phases. Therefore, two ǫ’s and P orǫ ,there aresomeparameterregionsinwhichDUNE
ee
and P measurements at two different L and E combina- couldleadtoawrongdeterminationofthemasshierarchy
tions(fourequationsandfiveunknowns)willhavepoten- and of CP violation. Additionally, for nonzero ǫ an in-
eτ
tial NSI degeneraciesfor any δ andδ′ (just aswith one ǫ correctconclusionaboutthe octantofθ23 maybedrawn.
andP andP measurementsatjustoneL/E). Thecorre- Nonzero ǫ does not impact a resolution of the octant
ee
sponding allowed regions of ǫ and ǫ from combining degeneracy. We conclude that DUNE alone cannot re-
eµ eτ
theNOνAandT2KexperimentsareshowninFig.5;any solve all the degeneracies arising from NSI. (We did not
value of δ′ is allowed at less than 2σ. Therefore P and consider the possibility of diagonal and off-diagonal NSI
P measurementsata third L/E areinprinciple required parametersbeing nonzerosimultaneously,whichleadsto
to resolve the degeneracies. Alternatively, a wide-band degeneracies between NSI parameters [9]. Clearly, this
beam experiment can be used. Figure 5 shows expected willfurther hinder the interpretationofDUNE data.) In
allowedregionsfromDUNE.Asexpected,DUNEcannot thisworkwefocusedonhowNSImaymimictheSMwith
resolve all the degeneracies. CP conservation. In future work we consider degenera-
5
cies as a function of the CP phase [19]. of this work. This research was supported by the U.S.
DOE under Grant No. DE-SC0010504.
Acknowledgments. KW thanks the University of
Hawaii at Manoa for its hospitality in the initial stages
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