What is $\Delta m^2_{ee}$ ?

0.43 MB·

by  Stephen Parke

FERMILAB-PUB-16-020-T
What is ∆m2 ?
ee
Stephen Parke∗
Theoretical Physics Department,
Fermi National Accelerator Laboratory,
P. O. Box 500, Batavia, IL 60510, USA
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(Dated: January 26, 2016)
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Abstract
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The current short baseline reactor experiments, Daya Bay and RENO (Double Chooz) have
n
a measured(orarecapableofmeasuring)aneffective∆m2associatedwiththeatmosphericoscillation
J
scaleof0.5km/MeVinelectronanti-neutrinodisappearance. Inthispaper,Icompareandcontrast
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2 the different definitions of such an effective ∆m2 and argue that the simple, L/E independent,
definition given by ∆m2 ≡ cos2θ ∆m2 +sin2θ ∆m2 , i.e. “the ν weighted average of ∆m2
ee 12 31 12 32 e 31
]
h and ∆m2 ,” is superior to all other definitions and is useful for both short baseline experiments
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mentioned above and for the future medium baseline experiments JUNO and RENO 50.
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e PACS numbers: 14.60.Lm, 14.60.Pq
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∗Electronic address: [email protected]
Typeset by REVTEX 1
I. INTRODUCTION
The short baseline reactor experiments, Daya Bay [1], RENO [2], and Double Chooz [3] ,
have been very successful in determining the electron neutrino flavor content of the neutrino
mass eigenstate with the smallest amount of ν , the state usually labelled ν . The parameter
e 3
which controls the size of this flavor content is the mixing angle θ , in the standard PDG
13
convention1,andthecurrentmeasurementsindicatethatsin22θ ≈ 0.09withgoodprecision
13
(∼ 5%).
The mass of the ν eigenstate, has a mass squared splitting from the other two mass
3
eigenstates, ν and ν , of approximately ±2.4 × 10−3 eV2 given by ∆m2 ≡ m2 − m2 and
1 2 31 3 1
∆m2 ≡ m2 −m2, the sign determines the atmospheric mass ordering. The mass squared
32 3 2
difference between, ν and ν , ∆m2 ≡ m2−m2 ≈ +7.5×10−5 eV2 is about 30 times smaller
2 1 21 2 1
than both ∆m2 and ∆m2 , hence ∆m2 ≈ ∆m2 . However, the difference between ∆m2
31 32 31 32 31
and ∆m2 is ∼3%.
32
Recently, two of these reactor experiments, Daya Bay, see [4] - [6] and RENO [7], have
extended their analysis of their data, from just fitting sin22θ , to a two parameter fit of
13
both sin22θ and an effective ∆m2. The measurement uncertainty on this effective ∆m2
13
is approaching the difference between ∆m2 and ∆m2 . So it is now a pertinent question
31 32
“What is the physical meaning of this effective ∆m2?” Clearly, the effective ∆m2 measured
by these experiments is some combination of ∆m2 and ∆m2 . Answering the question
31 32
“What is the combination of ∆m2 and ∆m2 is measured in such a short baseline reactor
31 32
experiment?” is the primary purpose of this paper,
Theoutlineofthispaperisasfollows: inSectionIItheν¯ survivalprobabilityiscalculated
e
in terms of an effective ∆m2 which naturally arises in this calculation, then this definition is
applied to the short baseline reactor experiments, L/E < 1 km/GeV. In Section III, I review
other possible definitions of an effective ∆m2, including the two invented by the Daya Bay
collaboration. These new effective ∆m2’s are either essentially equal to the effective ∆m2
of section II or are L/E dependent. This is followed by a conclusion and two appendices.
II. ν¯ SURVIVAL PROBABILITY IN VACUUM:
e
The exact ν¯ survival probability in vacuum, see Fig. 1, is given by2
e
P (ν¯ → ν¯ ) = 1−4|U |2|U |2sin2∆
x e e e2 e1 21
−4|U |2|U |2sin2∆ −4|U |2|U |2sin2∆
e3 e1 31 e3 e2 32
= 1−cos4θ sin22θ sin2∆
13 12 21
− sin22θ (cos2θ sin2∆ +sin2θ sin2∆ ), (1)
13 12 31 12 32
using ∆ ≡ ∆m2 L/4E.
ij ij
1 A more informative notation for mixing angles (θ , θ , θ ) is (θ , θ , θ ), respectively, such that
12 13 23 e2 e3 µ3
U =cosθ sinθ , U =sinθ e−iδ and U =cosθ sinθ .
e2 e3 e2 e3 e3 µ3 e3 µ3
2 ThestandardPDGconventionswiththekinematicalphasegivenby∆ ≡∆m2 L/4E or1.267∆m2 L/E
ij ij ij
depending on whether one is using natural or (eV2, km, MeV) units. Also, matter effects shift the ∆m2
by (1+O(E/10GeV)), where E <10 MeV, so are negligible for typical reactor neutrinos experiments.
2
FIG. 1: The vacuum survival probability for ν¯ as a function of L/E. Blue is for the normal
e
mass ordering (NO) and red is the inverted mass ordering (IO) with ∆m2 and ∆m2 chosen in
31 32
such a fashion that the two survival probabilities are identical at small L/E, ie. ∆m2 (IO) =
31
−∆m2 (NO) + 2sin2θ ∆m2 . Near the solar oscillation minimum, L/E ∼ 15 km/MeV, the
31 12 21
phase of the θ oscillations advances (retards) for the normal (inverted) mass ordering and the
13
two oscillation probabilities are distinguishable, in principle. Also near the solar minimum, the
amplitude of the θ oscillations is significantly reduced compared to smaller values of L/E. The
13
short baseline experiments, Daya Bay, RENO and Double Chooz, probe L/E < 0.8 km/MeV and
the medium baseline, JUNO and RENO 50, probe 6 < L/E < 25 km/MeV, as indicated.
It was shown in [8], that to an excellent accuracy
cos2θ sin2∆ +sin2θ sin2∆ ≈ sin2∆
12 31 12 32 ee
where ∆m2 ≡ cos2θ ∆m2 +sin2θ ∆m2
ee 12 31 12 32
for L/E < 0.8 km/MeV. A variant of this derivation is given in the Appendix V.
However, in this article we will use an exact formulation given in [9] which follows
Helmholtz, [10], in combining the two oscillation frequencies, proportional to ∆ and ∆
31 32
into one frequency plus a phase. The exact survival probability is given by (see Appendix
VI)
P (ν¯ → ν¯ ) = 1−cos4θ sin22θ sin2∆
x e e 13 12 21
(cid:18) (cid:19)
1 (cid:112)
− sin22θ 1− 1−sin22θ sin2∆ cosΩ (2)
13 12 21
2
with Ω = (∆ +∆ )+arctan(cos2θ tan∆ ).
31 32 12 21
Ω consists of two parts: one that is even under the interchange of ∆ and ∆ and is linear
31 32
in L/E, (∆ +∆ ), and the other which is odd under this interchange and contains both
31 32
linear and higher (odd) powers in L/E, arctan(cos2θ tan∆ ), remember ∆ = ∆ −∆ .
12 21 21 31 32
3
The key point is the separation of the kinematic phase, Ω, into an effective 2∆ (linear in
L/E) and a phase, φ. For short baseline experiments, it is natural to expand Ω in a power
series in L/E and identify the coefficient of the linear term in L/2E as the effective ∆m2
and include all the higher order terms in the phase3. Then,
Ω = 2∆ +φ (3)
ee
∂ Ω (cid:12)
where ∆m2 ≡ (cid:12) = cos2θ ∆m2 +sin2θ ∆m2 (4)
ee ∂(L/2E) (cid:12)EL→0 12 31 12 32
and φ ≡ Ω−2∆ = arctan(cos2θ tan∆ )−∆ cos2θ . (5)
ee 12 21 21 12
With this separation, 2∆ varies at the atmospheric scale, 0.5 km/MeV, whereas φ varies
ee
at the solar oscillation scale, 15 km/MeV, and
∂ φ ∂2 φ L
φ = 0, = 0 and = 0 at = 0,
∂(L/2E) ∂(L/2E)2 E
therefore, in a power series in L/E, φ starts at (∆m2 L/E)3 (see eqn. 11).
21
Since Ω only appears as cosΩ, it is useful to redefine Ω = 2|∆ |±φ , so that the sign
ee
associated with the mass ordering appears only in front of φ. If and only if this sign is
determined, is the mass ordering determined in ν disappearance experiments.
e
There are three things worth noting about writing the exact ν survival probability as in
e
eqn. 2, with Ω given by eqn. 3:
• The effective atmospheric ∆m2 associated with θ oscillation is a simple combination
13
of the fundamental parameters, see eqn. 4 above or in ref. [8] as they are identical,
∆m2 = cos2θ ∆m2 +sin2θ ∆m2
ee 12 31 12 32
= ∆m2 −sin2θ ∆m2 = ∆m2 +cos2θ ∆m2
31 12 21 32 12 21
= m2 −(cos2θ m2 +sin2θ m2).
3 12 1 12 2
Thus ∆m2 is simple the “ν average of ∆m2 and ∆m2 ,” since the ν ratio of ν to ν
ee e 31 32 e 1 2
is cos2θ to sin2θ , and determines the L/E scale associated with the θ oscillations.
12 12 13
• The modulation of the amplitude associated with the θ oscillation, is manifest in the
13
square root multiplying the cosΩ oscillating term, where
(cid:26)
(cid:112) 1 at ∆ = nπ
1−sin22θ sin2∆ = 21 (6)
12 21 cos2θ ≈ 0.4 at ∆ = (2n+1)π/2
12 21
for n = 0,1,2,···. Thus, at solar oscillation minima, when ∆ = 0,π,2π,..., the
21
oscillation amplitude is just sin22θ , whereas at solar oscillation maxima, when
13
∆ = π/2,3π/2,..., the oscillation amplitude is cos2θ sin22θ i.e. reduced by ap-
21 12 13
proximately 60%.
3 Appendix A of [9] contains a discussion of an effective ∆m2 as a function of L/E for arbitrary L/E. At
L/E =0 this definition is identical to ∆m2 .
ee
4
FIG. 2: The L/E dependence of the two components that make up the kinematic phase Ω =
2|∆ | ± φ associated with the θ oscillation, eqn. 2. φ is the black staircase function which
ee 13
increases by 2πsin2θ for every increase in ∆ by π, see eqn. 7. The blue straight line is
12 21
2|∆ |/80, which is always greater than or equal to φ. The green curve is the ∆3 approximation
ee 21
to φ given in eqn. 11, which is an excellent approximation for L/E < 8 km/MeV.
• The phase, φ, causes an advancement (retardation) of the θ oscillation for the normal
13
(inverted) mass ordering of the neutrino mass eigenstates. φ is a “rounded” staircase
function4, whichiszeroandhaszerofirstandsecondderivativesatL/E = 0(∆ = 0),
21
but then between L/E ∼ 10−20 km/MeV (∆ ∼ π−2π) rapidly jumps by 2πsin2θ ,
21 3 3 12
and this pattern is repeated for every increase of L/E ∼ 30 km/MeV (∆ by π), i.e.
21
φ(∆ ±π) = φ(∆ )±2πsin2θ , (7)
21 21 12
see Fig. 2. Also shown on the same plot is 2|∆ | divided by 80. This number 80 was
ee
chosen so that 2|∆ | fits on the same plot and to demonstrate that 2|∆ | ≥ 80 φ
ee ee
so that the shift in phase caused by φ is never bigger than a 1.25% effect. Also for
L/E < 5 km/MeV, the shift in phase is much smaller than this, see next section.
A. Short Baseline Experiments (0 < L/E < 1 km/MeV)
For reactor experiments with baselines less than 2 km, the exact expression eqn. 2
contains elements which require measurement uncertainties on the oscillation probability to
better than one part in 104. This is way beyond the capability of the current or envisaged
experiments. Thisoccursbecauseforexperimentsatthesebaselines, thefollowingconditions
4 Inthelimit,sin2θ → 1,onerecoversthewellknownresultthatthisroundedstaircasefunctionbecomes
12 2
a true staircase or step function.
5
on the kinematic phases are satisfied,
0 < |∆ | ≈ |∆ | < π ⇒ 0 < ∆ < 0.1, (8)
31 32 21
and some elements of eqn. 2 dependent on higher powers of ∆ . These elements are
21
• The modulation of the θ oscillation amplitude which when expanded in powers of
13
∆ is given by
21
(cid:113)
(1−sin22θ sin2∆ ) = 1−2sin2θ cos2θ ∆2 +O(∆4 ) (9)
12 21 12 12 21 21
= 1+O(< 10−3) (10)
Remember, this amplitude modulation factor is multiplied by 1 sin22θ ∼ 0.05. Re-
2 13
ducing the effect of the amplitude modulation to less than one part in 104.
• The advancement or retardation of the kinematic phase, Ω, caused by φ whose sign
depends on the mass ordering. For small values of ∆ the advancing/retarding phase
21
can be written as
1
φ = cos2θ sin22θ ∆3 +O(∆5 ) (11)
3 12 12 21 21
then using this approximation in the kinematic phase Ω, we have
cos(2|∆ |±φ) = cos(2|∆ |)cosφ∓sin(2|∆ |)sinφ
ee ee ee
1
= cos(2|∆ |)∓ cos2θ sin22θ ∆3 sin(2|∆ |)+O(∆5 )
ee 3 12 12 21 ee 21
(12)
= cos(2|∆ |)+O(< 10−4)
ee
Again remember, that we have a further reduction by 1 sin22θ ∼ 0.05. Making
2 13
the phase advancement or retardation significantly smaller than even the amplitude
modulation for these experiments.
Using this information in the ν survival probability, we can replace eqn. 2 by
e
P (ν¯ → ν¯ ) = 1−cos4θ sin22θ sin2∆ −sin22θ sin2|∆ |. (13)
short e e 13 12 21 13 ee
which is accurate to better than one part in 104. In Fig. 3 the fractional difference between
eqn. 2 and 13 is shown for an experiment with a baseline of 1.6 km. Since the measurement
uncertainty on the ν survival probability is much greater (> 0.01%) than the difference
e
between the exact, eqn. 2, and the approximate, eqn. 13, survival probabilities, use of
either will result in the same measured values of the parameters sin22θ and |∆m2 | i.e.
13 ee
the measurement uncertainties will dominate.
If new, extremely precise, short baseline experiments ever need a more accurate survival
probability, one could easily add the first correction of the amplitude modulation, giving
P (ν¯ → ν¯ ) = 1−cos4θ sin22θ sin2∆
xshort e e 13 12 21
−sin22θ [ sin2|∆ |+sin2θ cos2θ ∆2 cos(2|∆ |) ] (14)
13 ee 12 12 21 ee
and this would improve the accuracy of the approximation to better than one part in 105.
An alternative way to derive these approximate survival probability, eqn 13 & 14, is given
in the Appendix V.
6
FIG. 3: The fractional difference between the exact survival probability, eqn. 1, and a sequence
of approximate survival probabilities, where cos2θ sin2∆ + sin2θ sin2∆ is replaced with
12 31 12 32
sin2(∆m2 L/4E) with ∆m2 ≡ (1−r)∆m2 +r∆m2 . Clearly, r = sin2θ minimizes the absolute
rr rr 31 32 12
value of the fractional difference between the exact and approximate survival probabilities. Thus,
the approximation of replacing cos2θ sin2∆ +sin2θ sin2∆ with sin2∆ gives an approxi-
12 31 12 32 ee
mate survival probability that is better than one part in 104 over the L/E range of the Daya Bay,
RENO and Double Chooz experiments.
III. OTHER POSSIBLE DEFINITIONS OF AN EFFECTIVE ∆m2
A. A New Definition of the Effective ∆m2
Another possible way to define an effective ∆m2, here I will use the symbol ∆m2 , is as
XX
follows
(cid:113)
∆m2 ≡ cos2θ (∆m2 )2 +sin2θ (∆m2 )2 . (15)
XX 12 31 12 32
Clearly this definition is independent of L/E and it guarantees that, in the limit L/E → 0,
that
sin2∆ = cos2θ sin2∆ +sin2θ sin2∆ . (16)
XX 12 31 12 32
One can then show, that
(cid:20) (cid:18)∆m2 (cid:19)2 (cid:21)
|∆m2 | = |∆m2 | (1+O 21 ) (17)
XX ee ∆m2
ee
So |∆m2 | is essentially equal to |∆m2 | up to correction on the order of 104, including the
XX ee
effects of the solar mixing angle5.
5 The following, useful identity is easy to prove by writing ∆m2 =∆m2 −∆m2 :
21 31 32
(cos2θ ∆m2 +sin2θ ∆m2 )2 =[cos2θ (∆m2 )2+sin2θ (∆m2 )2]−cos2θ sin2θ (∆m2 )2.
12 31 12 32 12 31 12 32 12 12 21
7
A variant of this definition of an effective ∆m2 (here I will used the subscripts “xx”), is
defined in terms of the position of the first extremum of (cos2θ sin2∆ +sin2θ sin2∆ )
12 31 12 32
in L/E. If this extremum occurs at (L/E)| , then define
1
2π
∆m2 ≡ , (18)
xx (L/E)|
1
so that, at this extremum, ∆m2xxL = π. With this definition it is again easy to show that,
4E 2
(cid:20) (cid:18)∆m2 (cid:19)2 (cid:21)
|∆m2 | = |∆m2 | (1+O 21 ). (19)
xx ee ∆m2
ee
Again, essentially equal to ∆m2 .
ee
(cid:16) (cid:17)2
In both |∆m2 | and |∆m2 |, the corrections of order ∆m221 , come from the amplitude
XX xx ∆m2
ee
modulation of the θ oscillation and the coefficients are 1 sin2θ cos2θ and sin2θ cos2θ
13 2 12 12 12 12
respectively. Note, these corrections are mass ordering independent.
B. Daya Bay’s Original Definition of the Effective ∆m2
In ref. [4] & [5], the Daya Bay experiment used the following definition for an effective
∆m2, here I will use the symbol ∆m2 ,
YY
sin2∆ ≡ cos2θ sin2∆ +sin2θ sin2∆ . (20)
YY 12 31 12 32
which implies that
(cid:18)4E(cid:19) (cid:20)(cid:113) (cid:21)
∆m2 ≡ arcsin (cos2θ sin2∆ +sin2θ sin2∆ ) . (21)
YY L 12 31 12 32
For L/E < 0.3 km/MeV, so that sin2∆ = ∆2 is a good approximation, ∆m2 is approx-
3i 3i YY
imately independent of L/E. However, for larger values of L/E, ∆m2 is L/E dependent,
YY
exactly in the L/E region, 0.3 < L/E < 0.7 km/MeV, where the bulk of the experimental
data from the far detectors of the Daya Bay experiment is obtained. In the center of this
L/E region, L/E ≈ 0.5 km/MeV, is the position of the oscillation minimum.
Furthermore, the definition given by Eqn. 20, is discontinuous at oscillation minimum
(OM). This occurs because as you increase L/E, the L.H.S. eqn. 20 can go to 1, whereas
the R.H.S. never reaches 1. So to satisfy Eqn. 20, as you increase L/E, your effective ∆m2
must be discontinuous at OM and the size of this discontinuity is given by6
δ ∆m2 | = sin2θ ∆m2 (22)
EE OM 12 21
which is of order of 3%. In Fig. 4, the various ∆m2’s are plotted as a function of L/E.
6 The following identity is useful to understand this point, sin2(π ±(cid:15))≈1−(cid:15)2 where here (cid:15)=s c ∆ .
2 12 12 21
Similarly at oscillation maximum, sin2(π±(cid:15))≈(cid:15)2.
8
FIG. 4: Daya Bay’s original definition, see [4] and [5], for an effective ∆m2, ∆m2 , is given by
YY
the solid red line. Notice the sizeable L/E dependence near oscillation minimum and maximum
(vertical black dotted lines). At all oscillation extrema, this definition is discontinuous and the size
of the discontinuity is sin2θ ∆m2 ∼ 3%. The first discontinuity occurs in the middle of the
12 21
experimental data of the Daya Bay, RENO and Double Chooz experiments. The L/E independent
lines: ∆m2 ≡ cos2θ ∆m2 +sin2θ ∆m2 is the blue dashed, ∆m2 and ∆m2 are the labelled
ee 12 31 12 32 31 32
black lines. This figure is for normal mass ordering with sin2θ = 0.30 and ∆m2 = 2.453×10−3
12 ee
eV2.
The relationship between Daya Bay’s ∆m2 and that of the previous section is as follows
YY
(cid:118)
(cid:117)(cid:117)(cid:32) (cid:18)∆m2 (cid:19)2(cid:33)
∆m2 | = ∆m2 (cid:116) 1+sin2θ cos2θ 21 . (23)
YY L/E→0 ee 12 12 ∆m2
ee
Therefore they are identical up to corrections of O(10−4) as L/E → 0.
Given that ∆m2 is L/E dependent one should take the average of ∆m2 over the L/E
YY YY
range of the experiment
(cid:82)(L/E)maxd(L/E) ∆m2
(cid:104)∆m2 (cid:105) = (L/E)min YY . (24)
YY [(L/E) −(L/E) ]
max min
For the current experiments this range is from [0,0.8] km/MeV and then from Fig, 4 it is
clear that
(cid:104)∆m2 (cid:105) ≈ ∆m2 , (25)
YY ee
if the discontinuity at OM is averaged over in a symmetric way. In practice, of course, one
needs to weight the average over the L/E range by the experimental L/E sensitivity. This
is something that can only be performed by the experiment. This was not performed in ref.
[4] or [5].
9
FIG. 5: Daya Bay’s new definition, see [6], of an effective ∆m2, ∆m2 , for ν¯ disappearance
ZZ e
compared ∆m2 ≡ cos2θ ∆m2 + sin2θ ∆m2 . The L/E range appropriate for JUNO and
ee 12 31 12 32
RENO-50 is 6 to 25 km/MeV, exactly the range in which ∆m2 changes by ±1%. Yet, the
ZZ
expected accuracy of these two experiments is better than 0.5%. The sign of the variation of
∆m2 is mass ordering dependent. The blue and red dashed lines are ∆m2 for NO and IO
ZZ 31
respectively.
C. Daya Bay’s New Definition of the Effective ∆m2
After the issue with ∆m2 was pointed out to the Daya Bay collaboration [11], the Daya
YY
Bay collaboration defined a new effective ∆m2 in the supplemental material of ref. [6]. Here
I will use the symbol ∆m2 for this new definition which is defined in terms of the kinematic
ZZ
phase, Ω, given eqn. 3, as
2E
∆m2 ≡ Ω, (26)
ZZ L
2E
= |∆m2 |± φ.
ee L
Unfortunately, since φ is not a linear function in L/E, ∆m2 is also L/E dependent. In
(cid:12) ZZ
contrast remember, from eqn. 4, ∆m2 ≡ ∂ Ω (cid:12) .
ee ∂(L/2E) (cid:12)L→0
E
For short baseline experiments, such as Daya Bay, RENO and Double Chooz, this de-
pendence is small, and can be calculated analytically from eqn. (11),
(cid:20) 1 (cid:18)∆m2 (cid:19) (cid:18)(cid:18)∆m2 (cid:19) (cid:19)(cid:21)
∆m2 = |∆m2 | 1± cos2θ sin22θ 21 ∆2 +O 21 ∆4
ZZ ee 6 12 12 ∆m2 21 ∆m2 21
ee ee
(cid:34) (cid:35)
(cid:18) L/E (cid:19)2
≈ |∆m2 | 1±6×10−6 . (27)
ee 0.5 km/MeV
10