Determination of mass hierarchy with \nu_{\mu}\to\nu_{\tau} appearance and the effect of nonstandard interactions

# Determination of mass hierarchy with νμ→ντ appearance and the effect of nonstandard interactions

Ahmed Rashed111E-mail:amrashed@go.olemiss.edu, and Alakabha Datta222E-mail:datta@phy.olemiss.edu Department of Physics and Astronomy, University of Mississippi, Lewis Hall, University, Mississippi, 38677 USA
Department of Physics, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt
Center for Fundamental Physics, Zewail City of Science and Technology, Giza 12588, Egypt
The Abdus Salam ICTP, Strada Costiera 11, 34014 Trieste, Italy
###### Abstract

Crucial developments in neutrino physics would be the determination of the mass hierarchy (MH) and measurement of the CP phase in the leptonic sector. The patterns of the transition probabilities and are sensitive to these oscillation parameters. An asymmetry parameter can be defined as the difference of these two probabilities normalized to their sum. The profile of the asymmetry parameter gives a clear signal of the mass ordering as it is found to be positive for inverted hierarchy and negative for normal hierarchy. The asymmetry parameter is also sensitive to the CP phase. We consider the effects of non-standard neutrino interactions (NSI) on the determination of the mass hierarchy. Since we assume the largest new physics effects involve the sector only, we ignore NSI in production and study the NSI effects in detection as well as along propagation. We find that the NSI effects can significantly modify the prediction of the asymmetry parameter though the MH can still be resolved.

UMISS-HEP-2016-02

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After ruling out the zero value of the smallest mixing angle in the lepton sector with C.L. around An:2012eh (), the main scope of the future experiments is to answer some open questions such as the absolute mass scale, mass hierarchy, and CP asymmetry in the lepton sector.

Knowledge of the mass hierarchy has an impact on determining the neutrino absolute mass scale, CP asymmetry in the lepton sector, and the nature of the neutrino to be either Dirac or Majorana. Once the ordering of the neutrino mass states is determined, the uncertainty on the measurement of the CP-violating phase, , is significantly reduced. Measuring the mass ordering can cut down the domain for observation of a signal in the neutrinoless double beta decay experiments. Cosmological measurements are sensitive to the sum of neutrino masses, thus, knowledge of the mass hierarchy could help in determining the absolute neutrino mass scale.

The mass hierarchy (MH) can be determined using different techniques. The transition probability from a neutrino flavor to another, in the presence of matter effect, is sensitive to the mass hierarchy. The shape of the oscillation profile can be used to infer the sign of thereby indicating whether we have normal hierarchy (NH) or the inverted hierarchy (IH). The standard proposal is to use the appearance channel to measure MH. Determination of mass hierarchy is in the scope of several future experiments such as DUNE Strait:2016mof (); Acciarri:2016crz (); Acciarri:2016ooe (); Acciarri:2015uup (), Hyper-Kamiokande Abe:2014oxa (); Abe:2011ts (), LBNO ::2013kaa (); Agarwalla:2014tca (), and INO Athar:2006yb (). A variety of other experiments has some sensitivity to the mass hierarchy such as the reactor neutrino experiments JUNO (formerly known as Daya Bay II) and RENO as well as PINGU Akhmedov:2012ah () at IceCube. The CP asymmetry can be measured in very long base line neutrino experiments such as LBNO (2300 km baseline length) and DUNE (1300 km baseline length) as well as Hyper-K (295 km baseline length). The existence of neutrino masses and mixing require physics beyond the standard model (SM). Hence it is not unexpected that neutrinos could have non-standard interactions (NSI). An important question is how this NSI impact the MH determination or the measurement of the CP violating phase NSIref (). Even if NSI does not significantly impact the MH determination it will be useful to have alternate channels to confirm the results from the standard channel.

In this work we want to explore MH in the channel. Compared to the standard channel, has certain advantages. The transition probability for is proportional to sine of the atmospheric mixing angle, while is suppressed by small oscillation parameters, such as and . We are going to focus, in this paper, on the long baseline DUNE and LBNO experiments. We will also consider NSI effects in our analysis.

There are several reasons to consider NSI involving the sector. First, the third generation may be more sensitive to new physics effects because of their larger masses. As an example, in certain versions of the two Higgs doublet models (2HDM) the couplings of the new Higgs bosons are proportional to the masses and so new physics effects are more pronounced for the third generation. Second, the constraints on new physics (NP) involving the third generation leptons are somewhat weaker allowing for larger new physics effects.

A key property of the SM gauge interactions is that they are lepton flavor universal. Evidence for violation of this property would be a clear sign of new physics (NP) beyond the SM. Interestingly, there have been some reports of non universality in the lepton sector from experiments. In the charged current sector the decays , have been measured by the BaBar RD_BaBar (), Belle RD_Belle () and LHCb RD_LHCb () Collaborations. It is found that the values of the ratios () deviate from the SM predictions RDtheory () and this could be indication of lepton non universal new physicsdatta (). Specifically, RDRK_Isidori ()

 RD ≡ B(¯B→D+τ−¯ντ)expt/B(¯B→D+τ−¯ντ)SMB(¯B→D+ℓ−¯νℓ)expt/B(¯B→D+ℓ−¯νℓ)SM=1.37±0.18 , RD∗ ≡ B(¯B→D∗+τ−¯ντ)expt/B(¯B→D∗+τ−¯ντ)SMB(¯B→D∗+ℓ−¯νℓ)expt/B(¯B→D∗+ℓ−¯νℓ)SM=1.28±0.08 . (1)

The measured values of and represent deviations from the SM of 2.0 and 3.8, respectively. There also appears to be violation of lepton universality in coupling though it is difficult to explain Wtaun ().

There has been another recent hint of lepton non-universality in the neutral current sector. The LHCb Collaboration measured the ratio of decay rates for () in the dilepton invariant mass-squared range 1 GeV GeV RKexpt (), and found

 RK ≡ B(B+→K+μ+μ−)B(B+→K+e+e−) (2) = 0.745+0.090−0.074 (stat)±0.036 (syst) .

This differs from the SM prediction of RKtheory () by .

These measurements might be hinting towards lepton non universal new physics with the largest effects involving the third generation leptons GGL (). The new physics could arise in the third generation and feed down to other generation through mixing effects and so in this picture we expect the largest NSI to involve the third generation neutrino. In our analysis, therefore, we will assume NSI only involving the third generation leptons.

The tau-neutrino appearance channel is relevant to the Long Baseline Neutrino Oscillation Experiment (LBNO) which has an access to both transitions and . The experiment consists of a near detector at CERN in addition to a far detector situated at Pyhäsalmi in Finland 2300 km away from CERN, where the source of neutrino beam is located. The muon- neutrino and anti-neutrino fluxes fall in the energy range of GeV where it peaks at 5 GeV ::2013kaa (). This means that the quasi-elastic neutrino interaction is dominant in the energy range of the experiment.

An upcoming experiment is the Deep Underground Neutrino Experiment (DUNE) experiment which has a program to make precise measurements of the mixing between the neutrinos, CP violation, and the ordering of neutrino masses. The two main oscillation channels are and , but access to and modes is possible. The baseline of DUNE is 1300 km and the flux of the neutrino beam ranges from 0-10 GeV Acciarri:2015uup ().

In table 3 in Ref. ::2013kaa (), one can find a comparison between LBNO and DUNE. In the case of LBNO, the expected number of events in the channel that comes from charged current interactions is 215/239 for NH/IH, while the number for is anticipated to be 98/99 for NH/IH in 2.5 years of data-taking. The DUNE will observe less number of events in these channels.

The pattern of the transition probability of depends on the sign of and the CP violating phase , so one can extract information on MH and the CP phase from this probability. An asymmetry parameter can be defined as the difference of the two transition probabilities and normalized to their sum. The MH can be sensitive to the sign of the asymmetry and the size of the asymmetry can carry some sensitivity to the CP violating phase .

We also consider the NSI effects on the determination of MH. The NSI effects can arise at the source, along propagation and at detection. Assuming significant NSI only involving the third generation we will ignore NSI at the source. In any new physics model NSI in propagation and detection are connected. However, we will not use specific models and instead will adopt an effective Lagrangian description of the new physics effects. The NSI effects are parameterized by some co-efficients that depend on the parameters in the effective Lagrangian and we will use experiments to constrain the size of these effects. Along propagation we will consider the effects of the NSI parameters to the transition probability pattern (Adamson:2013ovz (); Mitsuka:2011ty (); Choubey:2015xha ()). For simplicity we will assume the NSI parameters to be real. Discussion on the impact of NSI parameters (moduli and phases) for CP violation measurement using for the DUNE experiment can be found in Ref. Masud:2016bvp ().

In previous work Rashed:2012bd (); Rashed:2013dba (); Liu:2015rqa () we introduced NSI at detection and considered various phenomenology connected to neutrino physics. For NSI at detection we use the following picture. The measurement of the transition probability can be expressed as relationship ():

 N(ντ)=P(νμ→ντ)×Φ(νμ)×σSM(ντ), (3)

where is the number of observed events, is the flux of muon neutrinos at the detector, is the total cross section of tau neutrino interactions with nucleons in the SM at the detector, and is the probability for the flavor transition in the presence of matter effect. In the presence of NSI at the detector, Eq. 3 is modified as

 N(ντ)=Ptot(νμ→ντ)×Φ(νμ)×σtot(ντ), (4)

with , where refers to the additional terms to the SM contribution towards the total cross section. Hence, includes contributions from both the SM and NP interference amplitudes, and the pure NP amplitude. From Eqs. (3, 4)

 Ptot(νμ→ντ)=P(νμ→ντ)σSM(ντ)σtot(ντ). (5)

Moving on to the transition probabilities, we define the asymmetry parameter as the difference between the neutrino and anti-neutrino transition probabilities normalized to their sum

 A=Ptot(νμ→ντ)−Ptot(¯νμ→¯ντ)Ptot(νμ→ντ)+Ptot(¯νμ→¯ντ). (6)

In the limit where matter effects are neglected is just a measure of CP violation.

The transition probability of the appearance channel in the presence of matter effect and NSI along propagation is given as Kikuchi:2008vq (); Meloni:2009ia (); Ohlsson:2012kf (); Agarwalla:2015cta ()

 P(να→νβ;εeμ,εeτ,εμμ,εμτ,εττ) = P(να→νβ;2 flavor in vacuum) (7) + P(να→νβ;εeμ,εeτ) + P(να→νβ;εμμ,εμτ,εττ),

where and denote one of and , and ’s are the NSI parameters. The first term in Eq. 7 has a form that it appears in the two flavor oscillation in vacuum:

 P(νμ→ντ;2 flavor in vacuum) = 4c223s223sin2Δm231L4E, (8)

where and . The second and third terms in the oscillation probability in the channel are given by

 P(νμ→ντ;εeμ,εeτ) (9) = 4c223s223|Ξ|2(aL4E)sinΔm231L2E−8c223s223|Ξ|2sinaL4EsinΔm231L4EcosΔm231−a4EL + 4c223s223|Θ±|2(aΔm231−a)(aL4E)sinΔm231L2E − 8c223s223|Θ±|2(aΔm231−a)2cosaL4EsinΔm231L4EsinΔm231−a4EL + 8c23s23(c223−s223)|Ξ||Θ±|cos(ξ−θ±)(aΔm231−a)(aΔm231)sin2Δm231L4E + 8c23s23|Ξ||Θ±|(aΔm231−a)sinaL4EsinΔm231L4E ×[s223cos(ξ−θ±−Δm231−a4EL)−c223cos(ξ−θ±+Δm231−a4EL)],

and

 P(νμ→ντ;εμμ,εμτ,εττ) (10) = −2c223s223(s213Δm231a−S1)(aL2E)sinΔm231L2E+c223s223S21(aL2E)2cosΔm231L2E − 8c23s23(c223−s223)[c12s12s13cosδ(Δm221a)−|E|cosϕ](aΔm231)sin2Δm231L4E + 4c23s23(c223−s223)S1|E|cosϕ(aΔm231)[(aL2E)sinΔm231L2E−2(aΔm231)sin2Δm231L4E] + 4c223s223|E|2(aΔm231aL2E)sinΔm231L2E + 4|E|2[(c223−s223)2−4c223s223cos2ϕ](aΔm231)2sin2Δm231L4E.

The subscript in these equations denote the normal and the inverted mass hierarchies, which corresponds to the positive and negative values of . The simplified notations which involve ’s in the sector are as follows:

 Θ± ≡ s13Δm231a+(s23εeμ+c23εeτ)eiδ≡|Θ±|eiθ±, Ξ ≡ (c12s12Δm221a+c23εeμ−s23εeτ)eiδ≡|Ξ|eiξ, E ≡ c23s23(εμμ−εττ)+c223εμτ−s223ε∗μτ≡|E|eiϕ, S1 ≡ (c223−s223)(εττ−εμμ)+2c23s23(εμτ+ε∗μτ)−c212Δm221a. (11)

We also note that , , and are complex numbers while is real. The matter potential is given by

 a = 2√2GFNeE (12) = 7.6324×10−5eV2ρgcm−3EGeV.

Using the Preliminary Reference Earth Model (PREM) PREM:1981 (), the line-averaged constant matter density is for the LBNO baseline of which corresponds to the distance between CERN and Pyhäsalmi Agarwalla:2011hh (); Stahl:2012exa (); Agarwalla:2014tca (). For the DUNE experiment, we use the standard value of the matter density . In matter, the probability for T conjugate channels is obtained by the replacement and those for CP conjugate channels are obtained by and .

In Fig. 1 we show and its CP conjugate channel in the LBNO energy range. Here, we consider no NSI along propagation (top panel) and with experimental upper bound of (bottom panel) (Adamson:2013ovz (); Mitsuka:2011ty (); Choubey:2015xha ()). Other NSI parameters are taken to be zero. In Fig. 2, we show the asymmetry parameter which is positive for IH and decreases with energy, while it is negative for NH and increases with energy for . In the presence of NSI , changes sign for both the hierarchies. The asymmetry profile for the two hierarchies has a crossing point where the MH cannot be resolved. The shape of the parameter in this case can therefore resolve the MH ( except at the crossing point) and provides clear evidence of NSI. One can notice that that parameter is sensitive to the CP phase. The same plots for the energy range and baseline relevant to DUNE experiment are shown in Figs. (3, 4). Compared to LBNO results, one can find that the asymmetry parameter has smaller values with considering no NSI along propagation and so it will be difficult to resolve the MH. In this case, , does not flip sign, in the desired energy range, when NSI along propagation is included. However, at larger energies in the presence of NSI, is substantially different for the two hierarchies so that the MH can be resolved.

Now, we study the effects of new physics contributions to the tau-neutrino interactions at the detector on the pattern of the asymmetry parameter in the energy range relevant to LBNO and DUNE. Adopting an effective Hamiltonian approach we include generic vector axial-vector, scalar, and tensor interactions. In the LBNO and DUNE energy range, the quasielastic tau neutrino scattering is dominant.

In the presence of NP, the effective Hamiltonian for the scattering process can be written in the form ccLag (),

 Heff = 4GFVud√2[(1+VL)[¯uγμPLd] [¯lγμPLνl]+VR[¯uγμPRd] [¯lγμPLνl] (13) +SL[¯uPLd][¯lPLνl]+SR [¯uPRd] [¯lPLνl]+TL[¯uσμνPLd] [¯lσμνPLνl]],

where is the Fermi coupling constant, is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, are the projectors of negative/positive chiralities. We assume the neutrino to be always left chiral. The Hamiltonian can be written as,

 Heff = GFVud√2{[¯uγμ(1−γ5)d][¯lγμ(1−γ5)νl]+[¯u(AS+BSγ5)d][¯l(1−γ5)νl] (14) + [¯uγμ(AV+BVγ5)d][¯lγμ(1−γ5)νl]+TL[¯uσμν(1−γ5)d][¯lσμν(1−γ5)νl]},

where , , and with and are the left and right handed scalar couplings, and are the left and right handed vector couplings and is the tensor coupling. The operators that describe the process can be obtained from the hermitian conjugate of the above Hamiltonian. The co-effecients in the effective Hamiltonian are fixed by low energy observables such as decays Rashed:2012bd (); Rashed:2013dba (); Liu:2015rqa () .

In the hadronic effects are described in terms of form factors. We define the charged hadronic current for the process in the SM as

 ⟨p(p′)∣∣J+μ|n(p)⟩ = ⟨p(p′)∣∣(Vμ−Aμ)|n(p)⟩ (15) = Vud¯p(p′)Γμn(p),

with

 Γμ = [FV1(t)γμ+FV2(t)iσμνqν2M+FA(t)γμγ5+FP(t)γ5qμM]. (16)

Here are the hadronic form factors which are functions of the squared momentum transfer . The expressions for the vector and axial-vector hadronic currents in Eq. 15 are

 ⟨p(p′)∣∣Vμ|n(p)⟩ = Vud¯p(p′)[γμFV1(t)+i2MσμνqνFV2(t)]n(p), −⟨p(p′)∣∣Aμ|n(p)⟩ = Vud¯p(p′)[γμFA(t)+qμMFP(t)]γ5n(p). (17)

Similarly, in the presence of ,

 ⟨p(p′)∣∣J′+μ|n(p)⟩ = ⟨p(p′)∣∣(AVVμ+BVAμ)|n(p)⟩, (18)

with

 ⟨p(p′)∣∣AVVμ|n(p)⟩ = VudAV¯p(p′)[γμFV1(t)+i2MσμνqνFV2(t)]n(p), ⟨p(p′)∣∣BVAμ|n(p)⟩ = −VudBV¯p(p′)[γμFA(t)+qμMFP(t)]γ5n(p). (19)

The scalar current for the process can be parametrized as follows

 ⟨p(p′)|J+|n(p)⟩ = ⟨p(p′)|¯u(AS+BSγ5)d|n(p)⟩ (20) = Vud¯p(p′)(ASGS+BSGPγ5)n(p).

Using the equation of motion,

 GS(t) = rNFV1(t),withrN=Mn−Mpmd−mu∼O(1), GP(t) = −(FA(t)(Mn−Mpmd−mu)+FP(t)md+muM), (21)

with MeV and MeV Agashe:2014kda ().

In the presence of tensor state, the tensor current can be parametrized as follows

 ⟨p(p′)|Jμν|n(p)⟩ = ⟨p(p′)|¯uσμν(1−γ5)d|n(p)⟩ (22) = iVudKS,P¯p(p′)(Γμ~Γν−~ΓμΓν)n(p) = i2MVud¯p(p′)(KSΠμν1−KPΠμν2γ5)n(p),

with defined as

 ~Γμ(p,p′) = γ0Γμ†(p′,p)γ0, (23) = [FV1(t)γμ−FV2(t)iσμνqν2M+FA(t)γμγ5−FP(t)γ5qμM],

and

 KS = −M4t(M2p−M2n)−(m2u−m2d)(Mp−Mn)GSFAFP, KP = −M4t(M2p−M2n)−(m2u−m2d)(Mp+Mn)GPF1FP, (24)

with

 Πμν1 = F1F2(γμγν⧸q−2γμ⧸qγν+⧸qγμγν)−4FAFP(γνqμ+γμqν), Πμν2 = FAF2(γμγν⧸q−2γμ⧸qγν+⧸qγμγν)−4F1FP(γνqμ+γμqν). (25)

The total differential cross section is

 dσtot(ν)dt = G2Fcos2θc32πE2νM2[Atot+Btot(s−u)+Ctot(s−u)2], (26)

with

 Atot = 16M4(xt−xl)[AV±A+ASr+AT+AV±A−Sr+AV±A−T], Btot = 8M2[BV±A+BV±A−Sr+BV±A−T+BSr−T+BT], Ctot = CV±A+CT, (27)

where

 AV±A = (1+AV)2[F21(1+xl+xt)+F22(xl+x2t+xt)+2F1F2(xl+2xt)] + (1−BV)2[F2A(−1+xl+xt)+4F2Pxlxt+4FAFPxl], ASr = A2SG2S(xt−1)+B2SG2Pxt, AT = 64T2LF22(F21K2S(xt−1)(xl+xt)+F2AK2P(xlxt+xl+x2t)), AV±A−Sr = −2(1−BV)BSGP√xl(FA+2FPxt)(1−4xtM2M2W), AV±A−T = −32(1−BV)TLKSF1F2FA√xl(xt−1)+16(1+AV)F2FAKPTL√xl(2F1xt+F1+3F2xt),
 BV±A = 2(1+AV)(1−BV)xtFA(F1+F2), BV±A−Sr = (1+AV)ASGS√xl(F1+F2xt), BV±A−T = −16(1+AV)TLKSxt√xlF1F2(F1+F2)+16(1−BV)TLKPxt√xlF2F2A, BSr−T = 8ASGSF2xtFAKPTL, BT = −128T2LxtxlF1F22FAKSKP, (29)
 CV±A = (1+AV)2(F21−xtF2