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					Long baseline phenomenology with an intermediate γ beta-beam


                                     Christopher Orme




                                       March 7th 2008




                               Based on arXiv:0802.0255
Davide Meloni, Olga Mena, Christopher Orme, Sergio Palomares-Ruiz and Silvia Pascoli.




Christopher Orme ()     Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   1 / 23
1   Introduction




2   3-ν phenomenology




3   Determining (θ13 , δ) and the resolution of degeneracies




4   Case study: A beta-beam for the CERN-Boulby baseline (1050 km)




         Christopher Orme ()     Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   2 / 23
2-ν oscillations



                               "solar" parameters                                 "atmospheric" parameters                              -2
                                                                              5                                                        10
                                                                                                                                                             90% CL (2 dof)
                        20                                                                             global                                                                  2007
                                                                                                                                                            GLOBAL
                                                                              4




                                                                                                                          ∆m31 [eV ]
                                                              ∆m31 [10 eV ]
       ∆m21 [ 10 eV ]




                                                                                                                          2
                                                                                      MINOS




                                                              2
      2




                        15     solar
                                                                              3                                                                                   SK+K2K+MINOS
                                           global
      -5




                                                              −3




                                                                                                                          2
                                                                                                  5
                        10
                                                                              2



                                                              2
      2




                                       5
                                                                                                                                                                         CH
                                              KamLAND                                                                                                                          OO
                         5                                                    1                                                             -3   SOL+KAML                        Z
                                                                                                        atmospheric                    10        +CHOOZ

                         0                                                    0                                                                       -2                  -1
                             0.2       0.4       0.6    0.8                       0      0.25    0.5        0.75      1                          10                     10
                                                                                                                                                                    2
                                             2
                                          sin θ12
                                                                                                   2
                                                                                                sin θ23                                                         sin θ13


                                                                                                                                                           Schwetz, arXiv:0710.5027

    Present oscillation data and solar neutrino data points to 2 2-ν mixing schemes

    There are two mass squared splitings of different orders.

    Only a bound on θ13 presently exists.




                 Christopher Orme ()                    Long baseline phenomenology with an intermediate γ beta-beam                                                          March 7th 2008   3 / 23
3-ν mixing


   The neutrino mixing matrix (PMNS) can be parameterised by three mixing angles and a
   complex phase

                                                                                                             
                             1   0           0             c          0      s13 e−i δ              c   s12   0
                                                13                                      12                 
                       U = 0             −s23   0                  1           0       −s12              0                       (1)
                                                                                                           
                                 c23                                                                    c12
                                                                                                           
                                                                 iδ
                             0   s23        c23          s13 e        0         c13          0          0     1



   If θ13 = 0 then the solar and atmospheric regimes decouple into 2 2-ν oscillation schemes.
   The CP phase is not physical.

   If θ13 = 0 then the two regimes are weakly coupled with θ13 and δ manifesting themselves
   through sub-dominant oscillation processes.

   Experimentally, θ13 is known to be small or zero. Need high luminosity experiments to
   observe it.

   Degeneracies (later) and systematics will present a major stumbling block.

       Christopher Orme ()           Long baseline phenomenology with an intermediate γ beta-beam                     March 7th 2008   4 / 23
Present knowledge and unknowns


                               Parameter                Best fit                      3σ

                              ∆m21 (eV2 )
                                2
                                                     7.9 × 10−5             7.1-8.9 ×10−5
                                 sin θ12
                                    2
                                                          0.30                 0.24-0.40
                                                                  −3
                              |∆m31 | (eV2 )
                                 2
                                                     2.5 × 10               1.9-3.2 ×10−3
                                 sin θ23
                                    2
                                                           0.5                 0.34-0.68
                                 sin θ13
                                    2
                                                      unknown                    < 0.041
                                    δ                 unknown                  no bound

   Task of current and next-generation experiments is improve accuracy of the knowns and pin
   down the unknowns.

   The ultimate goal in neutrino oscillation physics is to measure or, at least get a bound on,
   θ13 , δ and sign(∆m31 ).
                      2


   Experiments running or about to will improve accuracy and make the first attempt to
   determine the unknowns. Superbeams, Beta-beams and Neutrino Factories will be the
   experiments of choice for the long term future.
       Christopher Orme ()         Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   5 / 23
We are in the era of precision not discovery




       Christopher Orme ()   Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   6 / 23
3-ν oscillations in matter




    To find neutrino oscillation probabilities one needs to solve the Schrondinger equation for
    neutrinos in the flavour basis
                                                            d
                                                        i        |ν(t ) = H |ν(t )                                                          (2)
                                                            dt
    where                                                                                                         
                                            0                0            0                   A        0         0
                                       1                                   † 
                                                                                                                 
                               H        U 0             2
                                                       ∆m21            0  U + 0                      0         0                         (3)
                                                                                                                  
                                      2E                                                                        
                                                                        2
                                            0                0        ∆m31       0                     0         0



    Here, U is the usual PMNS mixing matrix in vacuum and A is given by
                                   √                                                           1
                                                                                                   Z       L
                              A=    2GF ¯e (L)
                                        n                          with               ¯e =
                                                                                      n                        ne (L )dL
                                                                                               L       0




        Christopher Orme ()            Long baseline phenomenology with an intermediate γ beta-beam                        March 7th 2008   7 / 23
Probabilities


For θ13 = 0.05 and L = 1050 km:

                              0.07
                                                                                          νe → νµ, δ=0
                                                                                         νe → νµ, δ=90
                              0.06                                                      νe → νµ, δ=-90


                              0.05
                Probability




                              0.04

                              0.03


                              0.02


                              0.01


                                0
                                     0.5            1             1.5      2                      2.5     3
                                                                   Energy GeV




        Christopher Orme ()                Long baseline phenomenology with an intermediate γ beta-beam       March 7th 2008   8 / 23
Analytical approach - solar oscillations as a perturbation



    It is desirable to have an analytical form for the probabilities, however, the complete
    diagonisation gives a result lacking in physical intuition. Instead, treat the solar effects as a
    perturbation of the 2-ν atmospheric oscillation by noting the following
                                                               2
                                                             ∆m21            1
                                                               2
                                                             ∆m31           30
                            2
    Then to first order in ∆m21 we may write
                                                                                                             
                                  ∆13 ±A−B
                                      2           0             0                    0                 0    0
                                                                          ¯† ¯†                              †¯
                                                                                                           
                      ¯ 
                   H =U             0            0             0         U + U U 0                 ∆12   0 U U                   (4)
                                                                                                           
                                                          ∆13 ±A+B
                                     0            0              2
                                                                                     0                 0    0



    Eigenvalues and eigenvectors are then found to diagonalise the above.

    With the mixing matrix to first order in ∆12 known, one can then get the probabilities.


        Christopher Orme ()            Long baseline phenomenology with an intermediate γ beta-beam                 March 7th 2008   9 / 23
The oscillation probability formula



                                                         2
                                                       ∆mij
    Introducing the abbreviation ∆ij ≡                  2E
                                                              , the oscillation probability can be expressed as

                              Peµ = Patm + Pint + Psol


    where
                                                                               2
                                                                ∆13                         (A     ∆13 )L
                              Patm =    s23 sin2 2θ13
                                         2
                                                                                    sin2
                                                               A ∆13                               2
                                           ∆12       ∆13      AL     (A                    ∆13 )L           ∆13 L
                              Pint   = J                  sin    sin                              cos δ
                                             A A      ∆13     2                            2                 2
                                                                         2
                                                               ∆12                  AL
                              Psol   = c23 sin2 2θ12
                                        2
                                                                             sin2
                                                                 A                   2

    The result is accurate for the baseline and energies considered here. Fit is still good for
    longer baselines and higher energies.




        Christopher Orme ()                Long baseline phenomenology with an intermediate γ beta-beam             March 7th 2008   10 / 23
The problem of degeneracies



   Determination of the unknown parameters (θ13 , δ, sign(∆m31 ), θ23 ) suffers from the problem of
                                                            2


   degeneracies.

   In general, we can have (θ13 , δ ) such that

                                 Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , ∆m31 , θ23 )
                                  +              2             +               2


                                 Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , ∆m31 , θ23 )
                                  −              2             −               2



   Known as the intrinsic degeneracy

   The sign of ∆m31 and octant of θ23 are also unknown so one can also have
                 2



                                Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , −∆m31 , θ23 )
                                                2                              2



   and
                                                                                               π
                               Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , ∆m31 ,
                                               2                             2
                                                                                                 − θ23 )
                                                                                               2



         Christopher Orme ()        Long baseline phenomenology with an intermediate γ beta-beam           March 7th 2008   11 / 23
Degenerate solutions graphically


                                              L = 130 km, E = 250 MeV
                 4.0
                               positive sign solution
                               negative sign solution
                 3.5
                               mixed sign solution

                 3.0
                          no solution
                                                                                                            Suppose that the appearence
                 2.5                                                                                        probability has been measured for both
  CP[P(ν)] (%)




                 2.0                                                                                        a neutrino and anti-neutrino run

                 1.5                                                                                        Set θ23 = π/4

                 1.0                                                         no solution                    There are 4 solutions that can fit the
                                                              CP+, sin 2θ13 = 0.055 (upper)
                                                                       2


                 0.5                                          CP+, sin 2θ13 = 0.05 (lower)
                                                                      2                                     data - 2 for each mass hierarchy.
                                                              CP−, sin 2θ13 = 0.0586 (upper)
                                                                      2


                                                              CP−, sin 2θ13 = 0.0472 (lower)
                                                                      2


                 0.0
                    0.0      0.5        1.0     1.5     2.0      2.5       3.0    3.5      4.0
                                                      P(ν) (%)



Minakata, Nunokawa and Parke, Phys.Rev. D66 (2002) 093012, arXiv:hep-ph/0208163



                       Christopher Orme ()                        Long baseline phenomenology with an intermediate γ beta-beam        March 7th 2008   12 / 23
Techniques for the removal of degeneracies



   The location of the intrinsic degeneracy in parameter space is energy dependent. All next
   generation or later experiments should therefore have good energy resolution to eliminate
   these clone solutions. Count rates, systematics (and correlations) are then the only barriers
   to (θ13 , δ).



                                                                                 2
   In the presence of matter, the oscillation probability is dependent on sign(∆m31 ). The effect
   mimics that of δ so one needs a large matter effect or favourable δ to resolve the degeneracy.
   Baselines L ≥ 700 km are necessary to have any possibility of measurement.



   The θ23 octant is very hard to get a handle on - an effect relating to solar oscillations. For a
   LBL optimised for atmospheric oscillations, this is a subdominant process requiring small θ13
   or low E to be seen.



        Christopher Orme ()    Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   13 / 23
The Durham-Roma approach




   The crucial difference to most studies is that we propose the use of 2 maxima as opposed to
   the usual neutrino and anti-neutrino runs (5+5). To see this, write the consider an experiment
   with flux Φνe (νe ) (Eν ), the number of events in the ith bin is being given by
                                          Z    Ei +∆E
                     Ni (¯13 , ¯ = NT t
                         θ δ)                           ε(Eν )σνµ (νµ ) (Eν )Peµ (Eν , ¯13 , ¯ νe (νe ) (Eν )dEν
                                                                              ±
                                                                                       θ δ)Φ
                                              Ei

   COnsider further infinite energy resolution so that

                                                  ±                                           ±
                                   N1 (E1 ) = c1 Peµ              and          N2 (E2 ) = c2 Peµ


   This type of analysis allows one to get at the features of the probability instrumental to the
   existence of the degeneracies.




       Christopher Orme ()           Long baseline phenomenology with an intermediate γ beta-beam            March 7th 2008   14 / 23
The Durham-Roma approach Contd.



For the intrinsic degeneracy, the clone solution (θ13 , δ) is found by solving

                           N1 (¯13 , ¯ sign(∆m31 ), ¯23 )
                               θ δ,           2
                                                    θ                                          θ
                                                                   = N1 (θ13 , δ, sign(∆m31 ), ¯23 )
                                                                                         2


                           N2 (¯13 , ¯ sign(∆m31 ), ¯23 )
                               θ δ,           2
                                                    θ                                          θ
                                                                   = N2 (θ13 , δ, sign(∆m31 ), ¯23 )
                                                                                         2



By taking E1 as first oscillation maximum and E2 and solving these equations, one finds no intrinsic
clone solutions

Similarly, for the sign degeneracy we can show

                                                                                A
                                       sin2 2θ13                     θ
                                                               sin2 2¯13 1 + 4
                                                                               ∆13

                                            sin δ              sin ¯
                                                                   δ

The octant degeneracy persists for sin2 2θ13 > 10−3 .




         Christopher Orme ()            Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   15 / 23
The beta-beam


   Proposal to accelerate then store β emitting ions to source a pure neutrino beam




                                                                           Flux well-known theoretically

                                                                           No contamination of beam as in
                                                                           Superbeam

                                                                           No need for magnetised detectors



   18
        Ne, 6 He, 8 B and 8 Li are favoured nuclides. Need ∼ 1018 ions per year to compare with
   physics reach of Superbeams and Neutrino Factories.

   Beam collimated with opening angle 1/γ with rest frame energies E0 being boosted to 2γE0
   on-axis

          Christopher Orme ()    Long baseline phenomenology with an intermediate γ beta-beam        March 7th 2008   16 / 23
A beta-beam for the CERN-Boulby baseline (1050 km)


   1st atmopsheric maximum at Boulby is at E ∼ 2 GeV (from ∆m31 L/4E = π/2).
                                                             2


   18
        Ne boosted to γ = 450 covers this and beyond.

   Assuming the detector has a low energy threshold, second oscillation maximum is avaliable
   as well.




          Christopher Orme ()   Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   17 / 23
Details of simulation



    Two exposures are considered
         1 × 1021 ions per kton-year        (low statistics)
         5 × 1021 ions per kton-year        (high statistics)

    For predicted number of events Ni and simulated number of event ni , we calculate a χ2 of the
    following form
                                                χ2 = ∑(ni − Ni )Ci−1 (nj − Nj )
                                                                  ,j
                                                         i ,j

    with the covariance matrix

                               Ci−1 = δij (δni )2
                                 ,j                             with        δni =        ni + (fsys · ni )2


    We set fsys = 0.02 and include a background of 0.03 atmospheric events per kton-year
    (assuming 10−3 duty cycle)

    We bin with 200 MeV in the range [0.4, 2.0], with a final bin [2.00, 3.06]



        Christopher Orme ()            Long baseline phenomenology with an intermediate γ beta-beam           March 7th 2008   18 / 23
Fits to simulated data


θ13 = 1 deg                                                 θ13 = 3 deg




The black contours are the 1σ, 2σ and 3σ fits. The red contours are a clone solution.



        Christopher Orme ()    Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   19 / 23
CP-violation


                                 CP discovery potential

    180
                                                                                                  Blue line - high statistics,
                                                                                                  red line - low statistics
     90

     δ                                                                                            A 1050 km beta-beam will have
                                                                                                  good sensitivity for
      0
                                                                                                  sin2 2θ13 ∼ few × 10−4

    −90                                                                                           For low statistics, still better than
                                                                                                  superbeam. Sensitivity for
   −180 −5                  −4                 −3              −2              −1
                                                                                                  sin2 2θ13 ∼ few × 10−3
      10               10                 10              10              10
                                          2
                                       sin (2θ13)

   Test the hypothesis δ = 0. Apply a χ2 test over the (θ13 , δ) plane

   Contours are for 99 % confidence and 2 d.o.f.


          Christopher Orme ()                       Long baseline phenomenology with an intermediate γ beta-beam              March 7th 2008   20 / 23
Mass hierarchy




                                                                  Blue line - high statistics,
                                                                  red line - low statistics

                                                                  Can determine the mass hierarchy for all δ
                                                                  when sin2 2θ13 < 0.03 (low statistics)
                                                                  sin2 2θ13 < 0.01 (high statistics)

                                                                  Need longer baseline for a dominant matter
                                                                  effect and resolution of degeneracy




   Calculate χ2 assuming NH and ruling out IH and vice versa.

   Contours are for 99 % confidence and 2 d.o.f.


      Christopher Orme ()    Long baseline phenomenology with an intermediate γ beta-beam        March 7th 2008   21 / 23
Octant degeneracy


                                octant sensitivity
    180



     90                                                                                      High statistics only
     δ                                                                                       Sensitivity is best for small θ13 as
      0                                                                                      this is the region where the solar
                                                                                             term dominates
    −90                                                                                      If θ13 is close to the present limit
                                                                                             then sensitivity is poor.
   −180 −3                                −2                              −1
      10                             10                              10
                                     2
                                 sin (2θ13)

   For θ13 close to the present limit, need info from low energies in this case (region of solar
   oscillations). Limited by low cross-sections and flux.



          Christopher Orme ()                  Long baseline phenomenology with an intermediate γ beta-beam              March 7th 2008   22 / 23
Summary and conclusions




   The physics that links solar and atmospheric neutrino mixing is still undermined. This regime
   is to be explored by looking for sub-dominant processes in near and long-term future neutrino
   oscillation experiments.

   It has been demonstrated that the problem of degenerate solutions can be overcome with a
   single neutrino helicity when detectors have a good energy resolution and low energy
   thresholds. The information provided by second oscillation maximum compensates for not
   running in anti-neutrinos.

   The beta-beam is an experimental option that can be used on the intermediate baselines of
   Europe, for example the CERN-Boulby baseline. The physics reach is better than for a
   Superbeam, providing good sensitivities to the unknown parameters if sin2 2θ13 ∼ few × 10−3 .




       Christopher Orme ()      Long baseline phenomenology with an intermediate γ beta-beam   March 7th 2008   23 / 23

				
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