CP violation in neutrino oscillation experiments with super-beam in presence of standard and non-standard interactions

0.23 MB·English

by  Zini Rahman

Discovery reach of CP violation in neutrino oscillation experiments with
standard and non-standard interactions
Zini Rahman,1,∗ Arnab Dasgupta,1,† and Rathin Adhikari1,‡
1Centre for Theoretical Physics, Jamia Millia Islamia (Central University), Jamia Nagar, New Delhi-110025, India
Considering DayaBay experimental valueof sinθ13 we find that for standard interaction of neu-
trinos with matter the betterdiscovery reach of CP violation is possible in short baseline neutrino
oscillation experiment. However, this changes in presence of non-standard interactions (NSIs). For
small non-standard interactions (small ε ) long baseline is found to be more suitable, in general,
αβ
forbetterdiscoveryreachofCP violation. EvenforlargevalueofNSIεeµ longbaselineisfoundto
2 be better. We have also discussed discovery reach of hierarchy and the discovery reach of different
1 NSIsfor longer baseline.
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∗ [email protected]
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‡ [email protected]
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I. INTRODUCTION
Among the various neutrino oscillation parameters the values of two mixing angles θ and θ have been
12 23
provided by experiments with certain accuracy. The magnitude of the mass square differences ∆m2 and
| 31|
∆m2 are also known but the sign of ∆m2 is still not known i.e. we still do not know exactly if neutrino
21 31
massesfollownormalheirarchy(NH)orinvertedhierarchy(IH). Apartfromthatwestilldonotknowabout
the CP violating phase δ. Earlier,the third mixing angle sin22θ wasalso not knownaccuratelyexcept its
13
upper bound but recently the Daya Bay experiment [1] have predicted the value of this parameter with 5σ
confidence level. Now the only unknowns in the neutrino mixing matrix - so called PMNS matrix are CP
violating phase δ and the hierarchy of neutrino masses.
In this work we consider neutrino superbeam (which mainly contains ν and ν¯ ) coming from CERN
µ µ
travellingabaselinelengthof2300KmreachesPyha¨salmi(Finland)whereaLiquidArgondetectorisplaced.
We also consider another baseline of 130 Km with a superbeam source at CERN and Water Cherenkov
detector placed at Fr´ejus (France). We do a comparative study in the discovery potentials of the CP
violating phase δ for these two baselines in presence of both standard and non-standard interactions.
Thispaperisorganisedasfollows: InsectionIIwediscusshowtheprobabilityofoscillationforν ν for
µ e
→
short and long baseline differ when NSIs are taken into account. In section III we discuss the experimental
setup and assumptions in doing the numerical analysis using GLoBES [2]. In section IV we discuss the
discovery reaches of CP violation, hierarchy and NSIs. Finally in section V we conclude with remarks on
the prospects of short and long baselines neutrino oscillation experiments.
II. NEUTRINO OCILLATION PROBABILITIES WITH NSI
In addition to the Standard Model (SM) Lagrangian density we consider the following non-standard
fermion-neutrino interaction in matter defined by the Lagrangian:
M = 2√2G εfP[f¯γ Pf][ν¯ γµLν ] (1)
LNSI − F αβ µ β α
where P (L,R), L = (1−γ5), R = (1+γ5), f = e,u,d and εfP is the deviation from SM interactions and
∈ 2 2 αβ
can be called as the non-standard interactions(NSIs). A bound can be set to these NSI parameters [3, 4]
which are dependent on specific models [5, 6] or are also model independent [7]. These NSI parameters can
be reduced to the effective parameters and can be written as:
n
ε = εfP f (2)
αβ αβ n
Xf,P e
where n is the fermion number density and n is the electron number density. These NSIs play significant
f e
role in the context of neutrino oscillation experiments and modify the interaction of neutrinos with matter
and thus change the oscillationprobabilityof different flavorof neutrinos. The NSIs could be presentat the
sourceof neutrinos,during the propagationofneutrinos andalso during detection ofneutrinos [8]. The NSI
effects are expected to be smaller at the source and detector due to their stringent constraints [7]. We shall
consider the NSI effect during the propagation of neutrinos only and in section IV in presenting results of
3
ournumericalanalysisweshallconsiderthemodelindependentallowedrangeofrealvaluesofdifferentNSIs
as mentioned in reference [7] for earth like matter.
In vacuum, flavor eigenstates ν may be related to mass eigenstates of neutrinos ν as
α i
ν >= U ν >; i=1,2,3, (3)
α αi i
| |
Xi
where U is PMNS matrix [9] which depends on three mixing angles θ , θ and θ and one CP violating
12 23 13
phase δ. Although two more Majorana phases could be present in U but are not relevant for neurino
oscillation experiments. The Hamiltonian due to standard (H ) and non-standardinteractions (H ) of
SM NSI
neutrinos interacting with matter during propagation can be written in the flavor basis as:
H =H +H (4)
SM NSI
where
0 0 0 A 0 0
∆m2
HSM = 2E31 U0 α 0U†+0 0 0, (5)
0 0 1 0 0 0
    
ε ε ε
ee eµ eτ
HNSI =Aε∗eµ εµµ εµτ (6)
ε∗ ε∗ ε
 eτ µτ ττ 
In equations (5) and (6)
2E√2G n ∆m2
A= F e; α= 21; ∆m2 =m2 m2 (7)
∆m2 ∆m2 ij i − j
31 31
wherem isthemassofthei-thneutrinoandAisconsideredduetotheinteractionofneutrinoswithmatter
i
in SM, G is the Fermi constant and n is the electron number density of matter . ε , ε , ε , ε , ε
F e ee eµ eτ µµ µτ
and ε are considered due to the non-standard interaction (NSIs) of neutrinos with matter. In equation
ττ
(6), (∗ ) denotes complex conjugation. In general, the NSIs - ε , ε and ε could be complex. However,
eµ eτ µτ
in our numerical analysis we have considered those to be real.
ForunderstandingqualitativelywhichbaselineissuitablefordiscoveryreachofCP violationandhierarchy
in presence of standard and non-standrad interactions we present below the oscillation probability P
νµ→νe
forbothshortandlongbaseline. ν ν oscillationchannelisparticularlysensitivetoCP violation. Inour
µ e
→
numerical analysis however, we have considered other channels also like ν ν . To get these expressions
µ µ
→
of probability we have followed the perturbation method adopted in references [8, 10–12]. At first, we shall
present the oscillation probability upto order α2 as the CP violating phase δ appears at this order for only
SM interactions as well as for small NSI of the order of α. For short baseline of 130 Km keeping the matter
effect parameter A at the (α) and considering NSI parameters ε of the order of α one obtains
αβ
O
4
L2α2∆m4 cos[θ ]2sin[2θ ]2 L∆m2 L∆m2
P = 31 23 12 +sin 31 2AL∆m2 cos 31
νµ→νe 16E2 (cid:20) 4E (cid:21)(cid:18)− 31 (cid:20) 4E (cid:21)
L∆m2 sin[θ ]2sin[θ ]2
+ 2E(1+4A+cos[2θ ])sin 31 13 23
13 (cid:20) 4E (cid:21)(cid:19) E
Lα∆m2 sin[θ ]sin[θ ] L∆m2 L∆m2
+ 31 13 23 cos δ+ 31 cos[θ ]sin 31 sin[2θ ]
E (cid:18) (cid:20) 4E (cid:21) 23 (cid:20) 4E (cid:21) 12
L∆m2
sin 31 sin[θ ]2sin[θ ]sin[θ ] (8)
− (cid:20) 2E (cid:21) 12 13 23 (cid:19)
For 2300Km the matter effect is large. Keeping A in the leading order of perturbationand considering NSI
parameters ε of the order of α one obtains
αβ
α2cos2[θ ]sin2 AL∆m231 sin2[2θ ]
23 4E 12
P = h i
νµ→νe A2
a 8Esin2 (−1+A)L∆m231 ( 1+A)L∆m2 sin (−1+A)L∆m231 sin2[θ ]sin2[θ ]
+ 6(cid:16) h 4E i− − 31 h 2E i(cid:17) 13 23
( 1+A)3E
−
a 8Esin2 (−1+A)L∆m231 +( 1+A)L∆m2 sin (−1+A)L∆m231 sin2[θ ]sin2[θ ]
+ 1(cid:16)− h 4E i − 31 h 2E i(cid:17) 13 23
( 1+A)3E
−
sin (−1+A)L∆m231 sin2[θ ]sin2[θ ]
4E 13 23
+ h i (2E(1+( 6+A)A
( 1+A)4E −
−
( 1+A)L∆m2
+ (1+A)2cos[2θ ] sin − 31
13 (cid:20) 4E (cid:21)
(cid:1)
( 1+A)L∆m2
+ 4( 1+A)AL∆m2 cos − 31 sin2[θ ]
− 31 (cid:20) 4E (cid:21) 13 (cid:19)
4a cos[θ ] AL∆m2 AL∆m2
+ 2 23 sin 31 ( 1+A)cos[θ ]sin 31 (a +αcos[φ ]sin[2θ ])
( 1+A)A2 (cid:20) 4E (cid:21)(cid:18) − 23 (cid:20) 4E (cid:21) 2 a2 12
−
L∆m2 ( 1+A)L∆m2
+ 2Acos δ+ 31 +φ sin − 31 sin[θ ]sin[θ ]
(cid:20) 4E a2(cid:21) (cid:20) 4E (cid:21) 13 23 (cid:19)
αsin[θ ]sin[θ ]sin[θ ] L∆m2
+ 12 13 23 8( 1+A)2Ecos δ+ 31 cos[θ ]cos[θ ]
( 1+A)3AE (cid:18) − (cid:20) 4E (cid:21) 12 23 ×
−
( 1+A)L∆m2 AL∆m2 ( 1+A)L∆m2
sin − 31 sin 31 +A 8AEsin2 − 31
(cid:20) 4E (cid:21) (cid:20) 4E (cid:21) (cid:18)− (cid:20) 4E (cid:21)
( 1+A)L∆m2
+ ( 1+A)L∆m2 sin − 31 sin[θ ]sin[θ ]sin[θ ]
− 31 (cid:20) 2E (cid:21)(cid:19) 12 13 23 (cid:19)
4a a sin[2θ ] L∆m2 (1 A)L∆m2 AL∆m2
2 3 23 cos 31 +E φ sin − 31 sin 31
− ( 1+A)A (cid:20) 4E − a3(cid:21) (cid:20) 4E (cid:21) (cid:20) 4E (cid:21)
−
+ 4a3sinh(−1+A4)EL∆m231isin[θ23] ( 1+A)αcos[θ ]cos L∆m231 φ
( 1+A)2A (cid:18) − 23 (cid:20) 4E − a3(cid:21)×
−
5
AL∆m2 ( 1+A)L∆m2
sin 31 sin[2θ ]+Asin − 31 sin[θ ](a +2cos[δ+φ ]sin[θ ])
(cid:20) 4E (cid:21) 12 (cid:20) 4E (cid:21) 23 3 a3 13 (cid:19)
+ a5sinh(−1+A4)EL∆m231isin2[θ13]sin[2θ ] 4Acos AL∆m231 cos[φ ]sin L∆m231
( 1+A)2A 23 (cid:18)− (cid:20) 4E (cid:21) a5 (cid:20) 4E (cid:21)
−
AL∆m2 L∆m2 L∆m2
+ 4sin 31 cos 31 cos[φ ] ( 1+A)sin 31 sin[φ ] (9)
(cid:20) 4E (cid:21)(cid:18) (cid:20) 4E (cid:21) a5 − − (cid:20) 4E (cid:21) a5 (cid:19)(cid:19)
where
a =Aε
1 ee
A ε 2+ ε 2+(ε 2 ε 2)cos2θ 2ε ε cos[φ φ ]sin2θ
eµ eτ eµ eτ 23 eµ eτ eµ eτ 23
a = | | | | | | −| | − | || | −
2 p √2
A ε 2+ ε 2+( ε 2+ ε 2)cos2θ +2ε ε cos[φ φ ]sin2θ
eµ eτ eµ eτ 23 eµ eτ eµ eτ 23
a = | | | | −| | | | | || | −
3 p √2
1
a =A ε 2cos22θ cos2φ +(ε ε )2cos2θ sin2θ + ε ((ε ε )cosφ sin4θ
5 µτ 23 µτ µµ ττ 23 23 µτ µµ ττ µτ 23
(cid:18)| | | |−| 2| | | |−| |
1/2
+ 2ε sin2φ
µτ µτ
| | (cid:19)
(cid:1)
a =A ε cos2θ + ε sin2θ + ε cosφ sin2θ
6 ττ 23 µµ 23 µτ µτ 23
| | | | | |
(cid:0) ε cosθ sinφ ε sinθ sinφ (cid:1)
φ =tan−1 | eµ| 23 eµ−| eτ| 23 eτ
a2 (cid:20) ε cosθ cosφ ] ε cosφ ]sinθ (cid:21)
eµ 23 eµ eτ eτ 23
| | −| |
ε sinθ sinφ + ε cosθ sinφ
φ =tan−1 | eµ| 23 eµ | eτ| 23 eτ
a3 (cid:20) ε cosθ cosφ + ε cosφ sinθ (cid:21)
eτ 23 eτ eµ eµ 23
| | | |
ε sin[φ ]
φ =tan−1 | µτ| µτ (10)
a5 (cid:20) ε cos2θ cosφ +(ε ε )cosθ sinθ (cid:21)
µτ 23 µτ µµ ττ 23 23
| | | |−| |
For CP violationthere is difference of probabilityin the neutrino oscillationandprobability of antineutrino
oscillation. One can relate the oscillation probabilities for antineutrinos to those probabilities given for
neutrinos above by the following relation:
Pα¯β¯ =Pαβ(δCP →−δCP, A→−A). (11)
In addition, we also have to replace ε with their complex conjugates, in order to deduce the oscillation
αβ
probability for the antineutrino, if one considers non-standard interaction during propagation.
To estimate the order of magnitude of different terms in the above two oscillation probabilities we shall
consider A α for short baseline and A 0.5 for longer baseline and following Daya Bay result we
∼ ∼
shall consider sinθ √α. For only SM interactions, (i.e ε 0) in above expressions of oscillation
13 αβ
∼ →
probabilities one finds that for both short and long baseline the δ dependence occurs at order of α3/2.
This order of dependence with δ remains same in the difference of neutrino oscillation probabilities and
antineutrino oscillation probabilities (represented by ∆P later). However, the δ independent part in ∆P
(which could mimic CP violation) for short baseline is at order α2 but for long baseline this is at order α.
This happens due to matter effect through A for SM as can be seen fromabove expressions. For this reason
the discovery reach of CP violation is better in short baseline than that in long baseline. However, when
6
NSIsarealsotakenintoaccountonecanseethatforlongerbaselinefurtherδ dependencein∆P couldoccur
at the order of α3/2 through a and a containing terms in (9) for NSIs of the order of α. We have checked
2 3
that for slightly higher NSIs of the order of √α using perturbation method the same δ dependent terms in
∆P appears with a and a in the oscillation probability for long baseline as given in (9) and this slightly
2 3
higher NSI makes these terms at the order of α which could compete with the δ independent part (which
could mimic CP violation) in ∆P for long baseline as that is also at the order of α. This improvement of δ
dependent part over independent part for long baseline does not happen for short baseline. First of all for
NSIoftheorderofαoscillationprobabilityin(8)forshortbaselineisindependentofNSIs. Evenforslightly
higher NSIs of order √α we have checked although NSIs like a and a enter into the only δ dependent
2 3
part of ∆P for short baseline but that is at the order of α2. So not much improvement in the discovery
reach of CP violation is expected for short baseline for NSIs which are present in a and a . So presence of
2 3
slightlyhigherNSIsoforder√αpresentina anda improvesthediscoveryreachofCP violationforlonger
2 3
baseline in comparisonto the shortbaseline. As a anda contains NSIs like ε andε itis expected that
2 3 eµ eτ
in presence of these NSIs the long baseline could provide a better discovery reach for CP violation.
For hierarchy discovery the difference in the oscillation probabilities due to two different hierachies mat-
ters. If we change hierarchy then in above oscillation probabilities the following transformations are to be
considered:
∆m2 ∆m2 ; A A; α α
31 →− 31 →− →−
For SM only the above differences of oscillation probabilities between two different hierarchies (∆P ) for
H
longerbaselineisoftheorderαduetomatterinteractionthroughA. However,forshorterbaselinethe∆P
H
is of the order α3/2. So longer baseline is more suitable for discovery of hierarchy. This remains unaltered
even after including NSIs.
III. ANALYSIS AND EXPERIMENTAL SETUP
Inthisworkforthenumericalsimulationweconsidertwoset-ups: (a)ASuperbeamsetupwhichoriginates
inCERNandreachesa100ktLiquidArgondetectorplacedatadistanceof2300KmatPyha¨salmi(Finland)
(b) A Superbeam setup originating in CERN and reaching a 500 Kt Water Cherenkov detector placed at a
distance of 130 Km at Fr´ejus (France).
We consideraflux withmeanneutrino energyaround5 GeVand3 1021 protonsontargetper year. We
×
considerthesamefluxasin[13]. Forset-up(a)weconsiderapowerof0.8MWperyear. Indoingtheanalysis
we consider a signal efficiency of 90% in both the appearance and disappearance channels. As backgrounds
we consider a 0.5% neutral current events, 1% of ν misidentified and the complete intrinsic contamination
µ
of the beam. The background rejection efficiencies were assumed to be constant over the energy window of
0.5 to 10 GeV. The calibration error has been considered to be 2% for ν disappearance channel and 0.01%
µ
for ν appearance channel. The Gaussian energy resolution is considered to be 150 MeV for electrons and
e
positrons and 0.2√E for muons. The correlation between the visible energy of background NC events and
the neutrino energy is implemented by migration matrices which has been provided by L. Whitehead [14].
In the case of set-up(b) we have considered a beam intensity of 4 MW and a systemetics on signal and
backgroundas2%basedonref[15]. Aneutrinoenergywindowof0.5to10GeVhasbeenconsideredforboth
7
the set-ups and both ν and ν¯ beams have been used simultaneously for a time period of 5 yrs for neutrino
and 5 yrs for antineutrino in case of set up (a) 2300 km and 2 yrs for neutrino and 8 yrs for antineutrino in
case of set up (b) 130 km.
Followingref[13]weconsiderthetruevaluesoftheneutrinooscillationparametersas ∆m2 =2.45 10−3
| 31| ×
eV2,∆m2 =7.64 10−5eV2,θ =9◦,θ =34.2◦andθ =45◦. Alsoincalculatingthepriorsweconsider
21 × 13 12 23
an error of 3% on θ , 0.005 on sin22θ , 8% on θ , 4% on ∆m2 and 2.5% on ∆m2 . Also we consider
12 13 23 | 31| 21
an error of 2% on matter density and a systematic uncertainties of 5 and 10% on signal and background
channels. In doing the whole analysis we have used GLoBES software [2]. In this work we have done a
comparative study of the two set-ups (a) and (b) in finding the discovery reachof the CP violating phase δ
depending on different NSI parameters and the hierarchies. For the longer baseline we have also discussed
about discovery reach of hierarchy and NSI.
IV. RESULTS
A. Discovery of CP violation
103 130 km 103 130 km
2300 km 2300 km
2χ102 2χ102
∆ ∆
5 σ 5 σ
101 3 σ 101 3 σ
100 100
-180 -90 0 90 180 -180 -90 0 90 180
δ(true) δ(true)
FIG. 1: CP violating phase δ discovery for two different baselines 130 Km and 2300 Km. The right hand panel is
for IH and theleft hand panel is for NH .
Infigure1wehaveshownthediscoveryreachofCP violationforSMinteractionsofneutrinoswithmatter.
For130KmbaselinetheCP violationcouldbediscoveredoverabout67%ofthepossibleδ valuesfornormal
hierarchy and this is about 68% for inverted hierarchy whereas for 2300 Km baseline these values are about
46%and47%respectively. Thediscoveryreachforlongerbaselineof2300KmwasshownearlierbyColoma
et al[13]. SowithonlySMtheshortbaselinelike130KmseemstobebetterforgooddiscoveryreachofCP
violation. However,this could change depending on the type of NSIs takeninto accountas discussedbelow.
8
360 360

270 270

e) e)
u u
δ (tr180 2310300 kkmm δ (tr180 2133000 kkmm

90 90

0 0
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
εee (true) εee (true)
360 360

270 270

e) e)
u u
δ (tr180 δ (tr180

130 km 130 km
90 2300 km 90 2300 km

0 0
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
εe µ (true) εe µ (true)
360 360

270 270

e) e)
u u
δ (tr180 δ (tr180

130 km 130 km
90 2300 km 90 2300 km

0 0
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
εe τ (true) εe τ (true)
FIG. 2: CPviolating phaseδ discoveryfortwodifferentbaselines 130Kmand2300 Kmat5σ considering NSIsεee,
εeµ and εeτ. The right hand panel is for IH and theleft hand panel is for NH .
In figure 2 we have compared the discovery reach of CP violation for 130 Km baseline and 2300 Km
baseline in presence of NSIs like ε , ε and ε . For ε in the range of about - 2.5 to 0.5 the 2300 Km
ee eµ eτ ee
baseline is found to be better for the discoveryreach which could be possible for about 49 % of the possible
9
360 360

270 270

e) e)
u u
δ (tr180 δ (tr180

130 km 130 km
90 2300 km 90 2300 km

0 0
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
εµ µ (true) εµ µ (true)
360 360

270 270

e) e)
u u
δ (tr180 δ (tr180

90 90
130 km 130 km
2300 km 2300 km

0 0
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
εµ τ (true) εµ τ (true)
360 360

270 270

e) e)
u u
δ (tr180 2133000 kkmm δ (tr180 2133000 kkmm

90 90

0 0
-10 -5 0 5 10 -10 -5 0 5 10
ετ τ (true) ετ τ (true)
FIG. 3: CPviolatingphaseδ discoveryfortwodifferentbaselines130Kmand2300Kmat5σ consideringNSIsεµµ,
εµτ and εττ. The right hand panel is for IH and theleft hand panel is for NH .
δ values whereas 130 Km is not so good in this range of NSI ε . For somewhat larger NSI around 4 or -4
ee
the short baseline of 130 Km is found to be better for discovery reach which could be possible for about
36% of the possible δ values. It is interesting to note that for NSI - ε - the discovery reach is consistently
eµ
better for 2300Km than that for 130Km and couldbe discoveredover90%of the allowedδ values. For ε
eτ
10
in 2300 Km CP violation could be discoveredover about 70% of the δ values over the entire allowed region
of ε . 130 Km baseline is found to be better for very high ε values around 3 and -3 only.
eτ eτ
In figure 3 we have compared the discovery reach of CP violation for 130 Km baseline and 2300 Km
baseline in presence of NSIs like ε ,ε and ε . For particularly ε and ε , 130 Km baseline is found
µµ µτ eτ µµ µτ
to be better for CP discovery which could be possible over 70 % of the δ values for both the hierarchies
throughout the entire allowed region of these NSIs. This is about 43 % for 2300 Km baseline. For ε in
ττ
2300Kmbaselinethe discoveryreachofCP violationcouldbe possibleover50%ofδ valuesforsmallNSIs.
For 130 Km for large NSIs only good discovery reach of about 61% could be possible. So it is found that
except ε and ε for other NSIs for their small values the discovery reach for CP violation is better for
µµ µτ
longer baseline than the shorter baseline.
B. Discovery of hierarchy
103 103
130 km 130 km
2χ102 2300 km 2χ102 2300 km
∆ ∆
5 σ 5 σ
101 3 σ 101 3 σ
100 100
-180 -90 0 90 180 -180 -90 0 90 180
δ(true) δ(true)
FIG. 4: Discoveryreachofhierarchyfortwodifferentbaselines130Kmand2300KmforSMinteractionsonly. The
right hand panel is for IH and theleft hand panel is for NH .
In figure 4 it is seen that with only SM interactions the discovery reach of hierarchy could be possible at
above 5σ confidence level for 2300 Km baseline. It was shown earlier by Coloma et al [13]. The 130 Km
baseline is found to be not so suitable for hierarchy as the discovery could be possible at only below 3σ.