Reactor Neutrino Flux Uncertainty Suppression on Multiple Detector Experiments

2.4 MB·English

by  Andi Cucoanes

Reactor Neutrino Flux Uncertainty Suppression on Multiple Detector Experiments
A. S. Cucoanes2∗, P. Novella,1 A. Cabrera1†, M. Fallot,2 A. Onillon,2 M. Obolensky,1 and F. Yermia2
1APC, Astro-Particule et Cosmologie, CNRS/IN2P3,
Universit´e Paris Diderot, 75205 Paris Cedex 13, France
2SUBATECH, CNRS/IN2P3, Universit´e de Nantes,
Ecole des Mines de Nantes, F-44307 Nantes, France
(Dated: January 16, 2015)
This publication provides a coherent treatment for the reactor neutrino flux uncertainties suppres-
sion, specially focussed on the latest θ measurement. The treatment starts with single detector
13
in single reactor site, most relevant for all reactor experiments beyond θ . We demonstrate there
13
is no trivial error cancellation, thus the flux systematic error can remain dominant even after the
adoption of multi-detector configurations. However, three mechanisms for flux error suppression
have been identified and calculated in the context of Double Chooz, Daya Bay and RENO sites.
Ouranalysiscomputestheerrorsuppressionfractionusingsimplifiedscenariostomaximiserelative
5
comparison among experiments. We have validated the only mechanism exploited so far by exper-
1
iments to improve the precision of the published θ . The other two newly identified mechanisms
0 13
2 could lead to total error flux cancellation under specific conditions and are expected to have major
implications on the global θ13 knowledge today. First, Double Chooz, in its final configuration,
n
is the only experiment benefiting from a negligible reactor flux error due to a ∼90% geometrical
a
suppression. Second,DayaBayandRENOcouldbenefitfromtheirpartialgeometricalcancellation,
J
yielding a potential ∼50% error suppression, thus significantly improving the global θ13 precision
2 today. Andthird,weillustratetherationalebehindfurthererrorsuppressionupontheexploitation

of the inter-reactor error correlations, so far neglected. So, our publication is a key step forward in
]
the context of high precision neutrino reactor experiments providing insight on the suppression of
x
e theirintrinsicfluxerroruncertainty,thusaffectingpastandcurrentexperimentalresults,aswellas
- the design of future experiments.
p
e
PACSnumbers:
h
[

INTRODUCTION mospheric mass hierarchy [10].
1
v Tomaximisethesensitivitytoθ ,reactorexperiments
13
6 wereforcedtoconceivedexperimentalsetupswhereflux,
5 Reactor neutrinos have been used for fundamental re-
detection and background systematics are controlled to
3 search since the discovery of neutrinos. The last decade
the unprecedented level of a few per-mille each contribu-
0 have witnessed a remarkable reduction of systematic
0 tion (<1% total). The statical resolution is boosted by
error in reactor neutrino experiments, by about one
. using multi-reactors sites. The unprecedented precision
1 of order of magnitude, imposed by the high precision
achievedisexperimentallyverychallenging,thereforethe
0
needed to measure θ by Double Chooz [1] (DC), Daya
5 13 redundancy θ13-experiments is critical, specially if their
Bay[2,3](DB)andRENO[4], amilestonefortheworld
1 uncertaintybudgetsarecomplementarytooffermaximal
strategyofneutrinoflavourresearch. RENOhasreleased
: cross-validation. Despite some complementary, the reac-
v
severalupdatesoftheθ analysisinconferencesheredis-
i 13 tor experimental setups are unavoidably similar and suf-
X regarded until publications follow. The reactor measure-
ferfromsimilarlimitations, hencevalidationbydifferent
r ments are consistent with all measurements sensitive to techniques would be important, although the precision
a θ [5,6]obtainedviaothertechniquesprovidingacoher-
13 needed is unattainable today. The fore-mentioned preci-
entθ perspective,asobtainedbyglobalfitanalyses[7–
13 sionimprovementwasobtainedviamulti-detectorexper-
9]. Since the reactor θ experiments precision is unri-
13 imental setups, whereby, at least, two detectors are used
valled, they are expected to dominate the world knowl-
for the reduction of the overall systematic budget since
edgeonθ ,likely,forafewdecadestogo. Hence,reactor
13 correlated systematics among detectors cancel out. This
systematic dominates much of the θ world knowledge,
13 way, while the absolute systematics are the same, the
as experiments reach their final sensitivities. The mea-
relative systematics are much lower. The absolute sys-
suredθ (anditsuncertainty)isexpectedtoplayacrit-
13 tematics are still dominant in any single-detector setup,
icalroletoconstrain,ormeasure,stillunknownneutrino
suchasDC(singledetector),butalsoallpastand,likely,
oscillation observables, such as CP-violation and the at-
most future reactor experiments.
The systematics reduced by the multi-detector config-
uration are: detection and flux systematics. Detection
∗Corresponding:[email protected] systematics benefit from dedicated detector design for
†Corresponding:[email protected] them to be effectively identical, typically so, only upon
2
far
R6
far
R5
near 2
near R6
R4
R3 R5
R1
R4
R2
near near 1 R3
R2
far
R1
R1 R2
FIG.1: The2DgeometryoftheexperimentalsetupsofDoubleChooz(left),RENO(middle)andDayaBay(right)experiments
is shown. The squares indicate the detectors and the circles indicate the reactor cores. The dotted lines depict the baselines
between detectors and reactors, while distances are summarised in Table II. Notation on Daya Bay: AD1, AD3 and AD4
correspond to the here called near-1, near-2 and far, respectively.
fullcalibration,thusimplyingthesame(ornegligiblydif- tion, since the evolution of the fission elements depends
ferent) responses and composition (cross-section, proton on the running configuration of each reactor. Therefore
number,etc). Fluxsystematicsbenefitfromthefactthat flux systematics are expected, with current knowledge,
the near detector(s) is located closer the reactor(s) such to be the dominant contribution for some of the reactor
that the flux modulation originating from neutrino os- θ experiments, such as DB, itself leading the world θ
13 13
cillations, θ in this case, is negligible (or very small) precision.
13
compared to the far detector(s) located further away. This publication develops a framework for the non-
The far is placed at the expected maximal oscillation trivial propagation of the reactor flux uncertainties and
deficit driven by the ∆m2 (atmospheric) constraint by their suppression in the context of multi-detector and
MINOS [5] and T2K [6]. The suppression of the flux multi-reactor experimental setups, providing mechanism
systematic is highly non-trivial, being the main subject for the improvement of the global θ precision. Most
13
of this publication. While having effectively identical our discussion stays generic on flux systematic propa-
detectors suffices to reduce detection systematics, from gation; i.e. no need for the specific experiment error
∼2.0% [2] to ∼0.2% [2], just having near and far detec- breakdown, allowing easy relative comparison across all
torswillnotnecessarilyprovideafullcancellationofflux experiments. Our study cases, however, inspired on the
systematics - unlike what was originally thought. There specific reactor-θ experiments configurations for maxi-
13
are three mechanisms leading to possible flux system- mal pertinence. The core of our calculations is analyti-
atic reduction to be elaborated in detail within the pa- cal,butacross-checkwasimplementedusingadedicated
per. First, multi-reactor uncorrelated uncertainties can Monte-Carlo-based analysis, is also presented. The dis-
benefit from having several identical reactors. Second, cussion starts from the simplest single detector config-
the near-far geometry of the experimental setup could uration, then evolving towards a general formalism ap-
enhance ability for the near become an effective perfect plicable to any multi-detector and multi-reactor setups.
monitor to the far; i.e. cancelling fully the flux system- The numbers linked to the specific experimental setups
atic error. And, third, the nature of the reactor uncer- are however only guidelines as they are obtained in sim-
tainties; i.e. whether correlated or uncorrelated among plified scenarios, again, to maximise comparability.
reactors;mightbeexploited,asmeasuredbymultiplede-
tectors. Reactor systematics suppress from single detec-
torscenario,typically∼3%[2]to<1%formulti-detector
FLUX INDUCED UNCERTAINTIES
setups. DCachievesanimpressive∼1.7%[1]forasingle-
detectorsetupusingtheBugey4dataforthemeancross-
Sofar,thereactorθ -experimentsappeartotreatthe
section per fission normalisation of the fission; i.e. as 13
reactorsystematicssimilarly. However,allcollaborations
an effective normalisation-only near. The most accurate
are lacking dedicated publications on the topic, so ques-
reactor flux anti-neutrino spectrum predictions [11, 12]
tions about the coherence across collaboration remain.
rely on the ILL uranium and plutonium isotopes input
Below, we will briefly summarise the reactor flux sys-
data [13, 14]. However, the final flux systematics are a
tematics information, as provided by the different col-
combinationofthereactorandspectralsystematicerrors
laborations. Most experiments rely on PWR reactors,
and depends on depends on each experiment configura-
hence the anti-neutrino flux is dominated by four iso-
3
ative to its multi-detector configuration where only the
TABLE I: Reactor-θ experiments breakdown of detector
13 uncorrelated errors are to be considered, as quoted by
flux 1σ systematic error [1, 2, 4]. The Daya Bay and RENO
Daya Bay and RENO. DC is expected revise these num-
errors quoted here are correspond to their uncorrelated con-
tributions, as they relied on multi-detector setups. Instead, bers in forthcoming the multi-detector DC (data taking
the Double Chooz errors includes also the correlated contri- started in 2014). However, our description handles er-
butions, as quoted from single-detector analysis. Therefore, ror suppression in relative, thus avoiding absolute error
the Double Chooz error is expected to be overestimated and estimationand/orprematurediscussion,leftforthepub-
will be revised by the collaboration in future publications.
lications by the experiments.
Thermalpower(P ),fissionfractions(α )andspentfuelare
th f DB and RENO, instead, quote systematics for their
considered.
multi-detector configurations; i.e. only the detector un-
P α Spent Fuel Total
th f correlated components. DB quotes, as thermal power
(%) (%) (%) (%)
and fission fraction systematics, 0.5% and 0.6% respec-
Double Chooz 0.5 0.9 included 1.0
tively [2]. An extra contribution from the spent fuel is
Daya Bay 0.5 0.6 0.3 0.8
RENO 0.5 0.7 unknown 0.9 estimatedtobe0.3%,increasingtheoverallsystematicto
0.8% [2]. DB presented a reduction of the total system-
atic uncertainty to 0.04% on its ratio between observed
over predicted rate upon optimisation, allowing modula-
topes: 235U,239Pu,241Puand238U(orderedbycontribu- tionofthedifferentnearcontributionsrelativetothefar.
tion relevance). This analysis, although not used for the measurement of
In general, flux systematics uncertainties are typically θ , raises unsettled debate about the physical meaning
13
broken down into three terms: thermal power, fission of such a modulation, a priori fixed by the geometrical
fractions and spent fuel, as summarised in Table I. The configurationofthesite. Forcompleteness,wehaverepli-
thermal power history is usually provided by the elec- cated such a result in this publication, however we disre-
tricity company exploiting the reactor. The associated garditfullyfordiscussionontheprecisionofθ . RENO,
13
error is related to the measurement method, the preci- ontheotherhand,quotes0.5%and0.7%[4],respectively,
sion of the installed sensors and the employed sensors forthethermalpowerandfissionfractions, leadingtoan
calibration [15]. Each experiment has to precisely study overall total reactor uncorrelated flux systematic uncer-
the way how the thermal power is measured in order to tainty of 0.9%, in consistent agreement with DB. RENO
estimate the corresponding error and the possible corre- quotes no error for the spent fuel, so it is neglected here-
lations among all the reactors involved, as typically, the after,howeverthisissomewhatunexpectedsinceitscon-
same measurement techniques are applied to all identi- tribution is expected to be non-negligible given the large
calreactorsinapowerplant. Thefractionalfissionrates number of reactors, like in the case of DB.
have to be computed through reactor simulations, while
their uncertainty estimation is not simple as they de-
pend on the reactor model and the approximations used THE ISO-FLUX CONDITION
for each setup [16]. After each cycle, the reactors are
stopped for refuelling, typically once per year in a few An important consideration for the reduction of reac-
weeks lasting operation. In this operation, part of the tor flux systematics is the geometry of the experimental
fuel assemblies are exchanged with new ones. The spent setup, illustrated in Fig. 1, which might lead to total er-
fuel are stored in the dedicated pools, typically located ror cancellation. The location of the detector strongly
next to the reactor sites with slightly different baselines. depends on the overburden topology of the site, critical
The spent fuel long lived fission products can generate for background suppression. In a multi-detector exper-
a small fraction of anti-neutrinos above the inverse beta imental site configuration, the flux uncertainties would
decay threshold [17]. Therefore, each experiment has to cancel entirely, regardless of the nature of the uncertain-
evaluatealsothecontributionfromspentfuelanditsun- ties,iftherelativecontributionmadebyeachgivenreac-
certainty. tortothetotaldetectedantineutrinofluxisthesamefor
DC, until now operating in single-detector mode (DC- all the detectors. This is a simple acceptance condition,
I), quotes its reactor uncertainties as fully correlated be- as it will be demonstrated later on. If met, the near be-
tween reactors, which is the most conservative approach comesaneffectiveperfectmonitorofthefar,thustheflux
forthisconfigurationasitwillbedemonstratedlateron. systematic uncertainty is null. This condition is, in fact,
Thevaluesforthethermalpowerandthefissionfraction triviallypossibleincaseofisotropicsources,likereactors
are 0.5% and 0.9%, respectively [1]. The spent fuel con- and, unfortunately, typically impractical in meson-decay
tribution was estimated within the dominant 1.7% flux neutrinoinducedbeams(i.e. decayinflight),wherelabo-
systematic error quoted for all DC-I publications. The rious dedicated efforts, including dedicated experiments,
DC-I errors quoted account for both correlated and un- are needed for an accurate spectral extrapolation from
correlatedcontributions,thustheyareoverestimatedrel- near to far, like in the case of MINOS and T2K. Com-
4
TABLE II: The distances in meters between detectors and reactors used for calculation, as illustrated in Fig 1. In the cases
of DC and RENO, we inferred from [1, 4], while for DB the information is clearly provided in [2].
Setup R R R R R R
1 2 3 4 5 6
Double Chooz
far 997.9 1114.6
near 351 466
Daya Bay
far 1920 1894 1533 1534 1551 1525
near-1 362 372 903 817 1354 1265
near-2 1332 1358 468 490 558 499
RENO
far 1556.5 1456.2 1395.9 1381.3 1413.8 1490.1
near 667.9 451.8 304.8 336.1 513.9 739.1
plex multi-reactor sites experiments might not meet iso- They characterise the systematic error associated to the
flux condition. However, the iso-flux condition might be emitted anti-neutrino flux contributing to the error of
fulfilledpartially,thusyieldingpartialsuppressionofsys- the measurement made by detectors. δ represents the
Ri
tematics. As the iso-flux condition is not fully fulfilled relative error of the emitted anti-neutrino flux per sin-
by any of the reactor-θ experiments, the estimation of gle reactor core Ri; i.e. the relative error of Φ . δ
13 Ri D
the reactor systematics is not straightforward. So, it is is the flux reactor uncertainty, as measured by the ex-
mandatorytoaccountforthedifferencesamongthereac- periment; i.e. the relative error of Φ . For an experi-
D
tor compositions fluxes and their corresponding system- mentalist considering a precise measurement of θ , the
13
atic errors. In fact, this will be investigated and quanti- most important quantity is δ , but the raw input is δ .
D R
tatively estimated in the following sections for each ex- Theratiobetweenδ overδ providesameasureforthe
D R
periment, using some simple approximations. effective error suppression fraction (SF) obtained upon
error propagation, which is the core parameter used for
error suppression characterisation. This factor reflects
REACTOR UNCERTAINTY SUPPRESSION the ability of each experiment to reduce the overall re-
actor uncertainty relative to the simplest case of one de-
Letusconsiderageneralexperimentalsetupwhereone tector with one reactor configuration (i.e. δ ), where no
R
or more detectors D measure antineutrino fluxes gener- cancellation is expected. The convention use is that the
ated by N reactors. For each detector, the incoming smaller the SF, the smaller the final reactor flux error
R
flux Φ represents a superposition of the fluxes emitted systematic as measured at the detector (δ ), so SF is
D D
by N reactors having the magnitudes Φi weighted by ranges within the interval [0,1]. The extreme values 0
R R
thecorrespondingsolidanglesΩi subtendedbetweenthe and 1 stand for total suppression (δ = 0, regardless of
D D
detector and each reactor. We express this as δ ) and no suppression (δ =δ ), respectively.
R D R
(cid:88)NR
Φ = Ω Φ (1)
D DRi Ri
Ri=R1
In this section, our formalism focuses on the propa-
whereindexRirunsoverallthereactors(R1,R2,...,N ) gation of the flux uncertainties, as characterised by the
R
persiteandtheindexDstandsforthedetector(s),where effective SF starting from Eq. 1 as applied to the spe-
thenear(s)andfar(s)willbeindicatedlateronbynand cificconfigurationofeachsetupandconsideringthesame
f, respectively. contribution for each individual reactor and ignoring the
Theaccuracyandtheprecisionofthegeometricalsolid nature of each error component. Hence, most of our de-
angles is carefully controlled by the experiments by ded- scriptionisdoneinrelative;i.e. nonecessitytointroduce
icated surveys. The relevant distances are summarised absolute error discussion; providing the ideal framework
in Table II. The associated uncertainty is generally so for highlighting the main features of each experiment
small relative to other terms, that we can neglect them in a comparable basis. As convention, we took the ac-
for our analysis. We can also neglect the effect of neu- tualexperimentalsetups(nextsub-sections)toillustrate
trinooscillation,asweconsidertheun-oscillatedintegral our discussion. However, most of the discussion remains
flux prediction. generic and figures will aim to be exemplify the general
Before going through the specific cases, let us define features. To simplify the description flow, some calcula-
three quantities to be used throughout the formalism. tions are only shown in the Appendix.
5
Single Detector: the DC-I Configuration
1.0 1.00
y
This case is best illustrated by the DC-I (far only etr
m
phase), where N stands for 2 reactors, R1 and R2. m 0.95
Propagating the eRrrors on Φ (Eq. 1) for the sole detec- sy 0.5
D A
tor involved, we obtain the corresponding SF expressed e 0.90
p
as Ty
y 0.0 0.85
nt
ai
SF2 =1− 2ΩR1ΦR1ΩR2ΦR2(1−k) (2) cert 0.80
(ΩR1ΦR1+ΩR2ΦR2)2 Un -0.5
0.75
wherekstandsforthecorrelationfactorbetweenthegen-
erated fluxes defined (see Eq. A34) as -1.0
-1.0 -0.5 0.0 0.5 1.0
Flux Asymmetry
1+ δRc−δRu
δc+δu
k = R R (3) FIG. 2: The suppression fraction (SF) for DC-I, coloured
(cid:115)
(cid:20) (cid:16)δc−δu(cid:17)2(cid:21) codedinthez-axis,isshownagainstthereactorfluxesasym-
2 1+ δRRc+δRRu mtaeintrtyy,cdoerfirenleadtioans (aΦsyRm2m−eΦtrRy1,)d/e(fiΦnRe2d+asΦ(Rδ1c)−, aδnud)/t(hδecu+nδcuer)-.
NoreductionofuncertaintiesisindicatedbyanSF=1.0(red).
whereδc andδu are, respectively, thecorrelatedandthe
R R Whenever one reactor dominates (i.e. the reactor flux asym-
uncorrelated components of the total reactor flux uncer-
metry tends to ±1.0) or the reactor uncertainties are mostly
tainty per individual core, defined as (δR)2 = (δRc)2 + correlated (i.e. (δc −δu)/(δc +δu) → +1.0), there is small
(δu)2. Then, we can rewrite Eq. 2 as or even no SF. The maximal suppression (violet) is obtained
R
wheneverthereactoruncertaintiesarefullyuncorrelatedand
both reactors are delivering similar fluxes. The expected SF
√
1+Ω2 +2kΩ minimum is 1/ 2 (∼0.7), only achievable for a slight offset
SF2 = R R (4)
(1+Ω )2 to R2 to compensate the fact that the far and R1 are closer
R
together, as shown in Fig. 1.
using a term proportional to the difference between the
reactorfluxesweightedbythecorrespondingsolidangles
• when either reactor is working, represented by the
Ω = ΩR2ΦR2 ∼= L2R1ΦR2 (5) condition (ΦR2−ΦR1)/(ΦR1+ΦR2)→±1.0
R Ω Φ L2 Φ
R1 R1 R2 R1
where L and L are the distances between the detec-
R1 R2 this is because those cases cannot be effectively distin-
tor and the reactors. The proportionality between the
guished from the case of single detector with a single re-
solid angles and the corresponding detector-reactor dis-
actor scenario. If two reactors uncertainties are totally
tances is given by the inverse-square law. The results
correlated, this is like if it was one effective reactor, re-
provided by Eq. 4 remain the same for a different choice
gardless of the geometry of the setup.
of the reference reactor in Eq. 5, more precisely consid-
ering the ratio (Ω Φ )/(Ω Φ ). Indeed, this case Conversely, some error reduction (SF <1) is expected
R1 R1 R2 R2
is equivalent to a transformation Ω → 1/Ω which elsewhere: whenever the source is consistent made up of
dR dR
leaves Eq. 4 unchanged. two independent (i.e. uncorrelated errors; i.e. k = 0)
The implications of Eq. 4 are best illustrated in Fig. 2 reactors working at about the same contribution. Error
where the SF is shown against the reactor flux asym- suppression is expected to be maximal (i.e. SF minimal)
metry between both Chooz reactors, defined as (ΦR2 − when the two reactors have totally unc√orrelated errors,
Φ )/(Φ + Φ ), and the reactor uncertainty type since those reactor errors scale with 1/ 2 (∼0.7). The
R1 R2 R2
asymmetry, definedas(δc−δu)/(δc+δu), characterising slight asymmetry in the reactor flux asymmetry (x-axis)
the fraction of reactor error (un)correlation generically. is a consequence of the distance far-R1 being slightly
TherearetwointerestinglimitcasesinFig.2wherethere closer than far-R2, as shown in Fig. 1. While rather
is no error reduction (i.e. SF = 1): trivial, the one detector case is important for many ex-
perimentsanditisremarkablyillustrativeinthecontext
• when the errors are maximally correlated, repre- of this discussion, since the fore-mentioned description
sented by the condition (δc−δu)/(δc+δu) → 1.0 inverts when considering a multi-detector sites, as de-
(or k →1 in Eq. 4) scribed next.
6
Multi-Detector: the DC-II Configuration representing SF general expression for the near-far setup
with two reactors.
Let us consider now the SF behaviour for the DC-II As before, ΩnR and ΩfR are independent from any
scenario with two detectors with two reactors. In this factor which multiplies both solid angles or both reac-
case, it is more reasonable to characterised the SF asso- tor rates, thus the weight parameter from Eq. 6 does
ciated to the near-far ratio, most relevant for oscillation not contribute to the expression from Eq. 10. This
analyses, as indicated by expression is also independent from the transformation
Ω → 1/Ω (with D = n,f) given by a different way
DR DR
ofdefiningthefractiontermsinEq.5. Bydefinition,the
Φ
Ratio(f/n)= f (6) iso-flux condition implies that the relative antineutrino
C×Φ
n fluxes from reactors is the same for both detectors. This
when C represents a constant such that the predicted can be mathematically represented as
un-oscillated flux at the far is obtained from the near
(i.e. Φpred = C×Φ ). This constant accounts for the
f n Ω =Ω (11)
differenceinthebaselinesbetweenthenearandfarsites, nR fR
duetotheisotropicemissionofneutrinosinreactor. The The implication of Eq. 10 are best illustrated in Fig. 3
nearandfararerespectivelyindicatedbythesubindexes where the evolution of the SF is shown. Fig. 3-Left plot
n and f. Let us reconsider Eq. 1 now generalised to shows SF evolution relative to both the reactor accep-
considerthecontributionofthetwodetectorstocompute tance asymmetry, defined as (Ω −Ω )/(Ω +Ω ),
R2 R1 R2 R1
the ratio Φf/Φn. As in the case of the single detector and the reactor uncertainty type asymmetry, defined as
case, we are interested in the expression for SF but now (δc−δu)/(δc+δu). The iso-flux condition is met when-
relative to the ratio between detectors, indicated by δnf, ever the reactor acceptance asymmetry is ∼0, leading to
to the relative error of the flux emitted by each reactor, a SF to fully cancel, regardless of the nature of reactor
thus uncertainties, demonstrating so the near is a geometrical
perfect monitor to the far. This is the same as impos-
(cid:18)δ (cid:19)2 (cid:18)δ (cid:19)2 (cid:18)δ (cid:19)2 cov(Φ ,Φ ) ing the condition in Eq. 11 in Eq. 10. The DC geometry
SF2 = nf = n + f −2 n f (7) is such that the iso-flux-ness is partially met along the
δ δ δ δ2Φ Φ
R R R R n f projection (Ω −Ω )/(Ω +Ω ) ≈ −0.28, hence a
R2 R1 R2 R1
whereδnandδf aretherelativeerrorsofthefluxesreach- majorreductionofSFisobtainedwhereSFcan,atmost,
ingtheindividualdetectors,respectivelyforthenearand be ∼0.12; i.e. ∼90% of the original uncertainty is thus
far. ThefirsttwotermsinEq.7arethesameastheones suppressed. Fig. 3-Right plot shows SF evolution rela-
obtained for single detector configuration (Eq. 4). The tive to both the reactor power flux asymmetry, defined
newtermcorrespondstothecovarianttermcomputedas as(Φ −Φ )/(Φ +Φ ),andthereactoruncertainty
R2 R1 R2 R1
typeasymmetry,asdefinedbefore. Theeffectofthereac-
torfluxasymmetrycanonlyexemplifiedinthecaseofDC
cov(Φ ,Φ ) = δ2[Ω Ω Φ2
n f R nR1 fR1 R1 with two reactors. Total cancellation of SF also is found
+ ΩnR2ΩfR2Φ2R2 iftheuncertaintiesarereactorcorrelatedmaximally(i.e.
+ kΦ Φ (Ω Ω k = 1) or either reactor is off. This is expected because
R1 R2 nR1 fR2
both such conditions are equivalent as having one effec-
+ Ω Ω )] (8)
nR2 fR1
tive reactor as source, thus perfectly monitored by the
where k has the same definition as in Eq. 3 for the error near, regardless of the experimental setup geometry or
correlation factor. We introduce ΩnR and ΩfR parame- the type ofthe uncertainty. As expected, thisconclusion
ters, as in Eq. 5, to obtain now is general and true independently of how many reactors
areconsidered. Thisimpliesthat,forexample,whenever
cov(Φ ,Φ ) 1+Ω Ω +k(Ω +Ω ) DC runs with 1 reactor and both detectors, the flux er-
n f = nR fR nR fR (9)
δ2Φ Φ (1+Ω )(1+Ω ) ror is zero for that data set. It is worth noting that the
R n f fR nR
pattern shown for DC-II, illustrated in Fig. 3-Right, is
such that the previous Eq. 7 now becomes exactly the opposite to the one exhibited by DC-I, illus-
tratedinFig.2,wherenosuppression(SF=1)isobtained
1+Ω2 +2kΩ in the case of total correlated reactor uncertainties. In
SF2 = nR nR
brief, the DC-II performance, SF ∼0.12, exemplifies the
(1+Ω )2
nR
best multi-detector error cancellation due its almost iso-
1+Ω2 +2kΩ
+ fR fR flux geometry where almost ∼90% of the SF is due ge-
(1+ΩfR)2 ometry. The remaining SF fraction (∼30%) is caused by
√
1+Ω Ω +k(Ω +Ω ) the 1/ N scaling of the remaining totally uncorrelated
− 2 nR fR nR fR (10) R
(1+ΩfR)(1+ΩnR) error, where NR =2.
7
y 1.0 y 1.0 0.12
etr 0.7 etr
m m
m m 0.10
sy 0.5 0.6 sy 0.5
A A
e 0.5 e 0.08
p p
y y
T T
y 0.0 0.4 y 0.0 0.06
nt nt
ai 0.3 ai
ert ert 0.04
c c
n -0.5 0.2 n -0.5
U U
0.02
0.1
-1.0 0.0 -1.0 0.00
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
Acceptance Asymmetry Flux Asymmetry
FIG. 3: The suppression fraction (SF) for DC-II is illustrated coloured coded in the z-axis. Left shows the evolution of SF
against both the reactor acceptance asymmetry (x-axis), defined as (Ω −Ω )/(Ω +Ω ), and the reactor uncertainty
R2 R1 R2 R1
type asymmetry (y-axis), defined as (δc−δu)/(δc+δu). The iso-flux condition is met whenever the acceptance asymmetry
is ∼0, leading to a SF →0 (violet), regardless of the nature of reactor uncertainties, thus demonstrating that the near is a
geometrical perfect monitor of the far. The DC geometry is such that the iso-flux-ness is partially met along the projection
(Ω −Ω )/(Ω +Ω ) ≈ −0.28, hence SF is, at most, ∼0.12. Right shows the evolution of SF against both the reactor
R2 R1 R2 R1
powerfluxasymmetry(x-axis),definedas(Φ −Φ )/(Φ +Φ )(i.e. thefluxdifferencebetweenreactorR1andR2),and
R2 R1 R2 R1
the reactor uncertainty type asymmetry (y-axis), as defined before. Note that the maximal value possible of SF is the same
∼0.12, as explained. Total cancellation (i.e. lowest SF) is found if the uncertainties are reactor correlated maximally or either
reactor is off. This is because both cases are mathematically identical as to having one effective reactor as source, perfectly
monitored by the near, regardless of the experimental setup geometry.
Multi-Detector: the RENO Configuration S(cid:48) = N(cid:88)R−1 Ω(i)Ω(j);
nf nR fR
i,j>1;i(cid:54)=j
Generalising Eq. 10 to account for a larger number of
reactors (NR =6), RENO is becomes our next case with and Ω(i) is a generalization from the Eq. 5, to obtain
two detector sites and N = 6, as illustrated in Fig. 1. dR
R
The new general expression is
Ω Φ L2 Φ
Ω(i) = dRi Ri ∼= dR1 Ri; i=2...N −1 (13)
dR Ω Φ L2 Φ R
SF2 = (cid:88) (cid:18)1+Sd2 +2k(Sd+Sd(cid:48))(cid:19) dR1 R1 dRi R1
(1+S )2 where reactor R1 has been arbitrary chosen as reference
d
d=n,f reactor. Nevertheless, choosing a different reactor leave
1+S +k(S +S +S(cid:48) )
− 2 nf n f nf (12) Eq. 12 unchanged.
(1+S )(1+S ) Theiso-fluxcondition,originallyinEq.11,isnowmore
n f
generally expressed as
where we have introduced the following notation
Ω(i) =Ω(i) (14)
nR fR
S =N(cid:88)R−1Ω(i); which, like in the case of DC-II, would cause SF to go
d dR
i>1 to zero, regardless of any other error dependence, upon
S =N(cid:88)R−1(Ω(i))2; ismidperoesditiaolnlo,wifs.the geometry of the experimental site con-
d2 dR
i>1 UponintroducingthegeometryofRENO,wecansum-
S(cid:48) = N(cid:88)R−1 Ω(i)Ω(j); marise our finding as follows,
d dR dR
• SF will obtained full cancellation if reactor errors
i,j>1;i(cid:54)=j
are fully correlated, as demonstrated and illus-
S =N(cid:88)R−1Ω(i)Ω(i); trated for DC-II (Fig. 3), regardless of the experi-
nf nR fR
mental site geometry.
i>1
8
• SF will be largest in case of the fully uncorrelated explicitcontributionoftwonearsitesinthedenominator.
reactor systematics errors when they deliver sim- Hence, the new expression becomes
ilar fluxes, also as obtained for DC-II. However,
√
an overall SF follows 1/ N reduction, hence, SF
R Φ
will benefit sites with many independent (i.e. un- Ratio(f/n)= f (15)
βΦ +γΦ
correlatederrors)identicalreactors. Inasiteswith n1 n2
different reactor types, the effective SF might be where β and γ are two constants, whose values are
deteriorated. fixed by the experimental geometry and the fluxes of the
runningreactors. Ifoptimisationofthesite,upondesign,
Mathematically, this can be seen when considering
wanted to be considered, the effective SF of DB would
k = 0 and N → ∞ into Eq. 12. The factors on
R
havedependedonthem. Fortheprediction,thenewflux
the denominator, (1+S )2 and (1+S )(1+S )
d n f
isthusconstructedfromthecombinationofthetwonear
will increase faster than the corresponding factors
sites measurements; i.e. Φ , Φ . However, the near
at numerator, S respectively S , hence leading n1 n2
d2 nf
sites do not only monitor their respective reactors, but
to cancellation.
instead,theyalsoseeafractionoftheotherreactors(not
• RENO has some SF reduction due to some degree supposed to be monitoring); i.e. near-2 also sees ∼6.5%
ofiso-flux-ness. TheamountofSFreductionislim- ofreactorsR1andR2andnear-1seeabout,∼17%ofthe
ited, as expected, since the geometry of the site is remaining reactors. This makes this ratio very delicate,
not optimal to ensure that the near and far have since some degree of double-counting is unavoidable by
a similar contribution across all reactors equally. the nears is unavoidable, while this is not present in the
Thenearsees,respectively,∼78%,∼16%and∼6% far. Thiswilltranslateintoalossoftheiso-flux-nesscon-
from the different reactors pairs (from the nearest dition. The same conclusions listed for RENO are valid
tothefarthest),whilethefarseesamoreevencon- for DB (previous section). The overall SF estimated for
tribution from all reactors. Thus, the RENO site this site is 0.18 benefiting for slightly better partial iso-
is not expected to meet the iso-flux condition fully. flux-ness relative to RENO, despite the visual appealing
The total SF measured was estimated to be ∼0.23 geometrical symmetry of the RENO site. In the case of
in the full power scenario. bothRENOandDB,mostoftheSF(∼60%)arisesfrom
√
the1/ N scalingduetotheirlargenumberofidentical
• There is a small difference between the nominal R
reactors.
power of two RENO reactors [4]. In our analysis,
we took this into account by weighting the corre-
spondent Ω(i) terms. Suppression Fraction Estimation via Monte-Carlo
DR
Unlike so far assumed, commercial reactors do not de-
Together with the analytical formalism presented, we
liver their flux constantly over time. Some unavoidable
haveanalysedthereactorfluxsystematicsusingaMonte-
variations expected are due to refuelling, a total reactor
Carlo based flux simulation method. Likewise, the full
stop once a year over a few weeks, and reactor burn-up
geometry and ingredients to each experimental sites are
effects, typically a few percent decrease in flux over the
simulated. Eachreactorisdescribedbyitsthermalpower
entire year. In addition, reactors might be run differ-
(P )andfractionalfissionrates(α withk =235U,238U,
ently every year, so each reactor cycle history is a priori th k
239Pu and 241Pu, the main core isotopes), while each de-
expected to be unique. When that happens in site with
tector is described by the number of free protons in the
many reactors, the effective SF can only deteriorate as
targetandbythedetectionefficiency. Forsimplicity,the
thetwoaboveconditionswillbevariedawayfromitsop-
samemeanα ’shavebeenusedforallexperiments,how-
timal spot. This effect will be further studied and quan- k
ever this is not totally correct, as different experiments
tified later on.
not only have different reactors but also the fuel com-
position, namely the enrichment of 235U, is expected to
Multi-Detector: the Daya Bay Configuration be somewhat different. Since the detector correlated er-
rors are canceled in multi-detector setups, we have prop-
agated only the detector uncorrelated errors.
Let us now consider the setup of the DB experiment
The antineutrino flux emitted by a given reactor is a
now. As illustrated in Fig. 1, this site differs from the
function of thermal power and the fission rates define
other on having 2 near sites to monitor the two sets of
reactors geometrically grouped of the power plant. The
near-1 monitors mainly reactors R1 and R2, while and
Φ=Φ(P ,α ). (16)
th k
the near-2 monitors all others.
Usingsimilarformalismtillnow,theonlymodification As the error propagation formalism does not depend on
here is to build the Ratio(f/n), used in Eq. 6, with the the specific form of the function Φ, we do not describe
9
it in detail. For a given binning of the antineutrino en-
ergy spectrum, the error of the flux is expressed as the
covariance matrix SF = δNfF, (23)
δ
f
MΦ =MPth +Mα, (17) where δf is the error on the flux at the far detector
ij ij ij NF
when computed as ratio(fn)×Φn. Such a SF is inde-
where i and j stand for the energy bins, and MPth ≡ pendent of the parameterisation of Φ and the specific
ij
(cid:104)δPith δPjth(cid:105) and Miαj ≡ (cid:104)δαi δαj(cid:105) are the uncertainty con- values of σPth and σα.
tributions from the thermal power and the fission rates, In this calculation, the SF accounts only for the error
respectively. As the error on P does not depend on suppression provided by the iso-flux-ness of the experi-
th
the antineutrino energy, MPth is a diagonal matrix with mental setup. In order to be compared with the analyti-
δi =δj =δ . cal estimation, developed in the previous sections based
PtFhoragPitvhenrePatchtorr,theerroronΦduetothethermal on the δf/δ ratio, one needs to incorporate the uncor-
R
power uncertainty δ is computed as related error reduction due to the number of reactors. If
Pth
detectorsfulfilledtheiso-fluxcondition,suchareduction
√
would be N . As this is not the case for any of the
R
δPrth =Φr(Ptrh+σPrth,αk)−Φr(Pth,αk), (18) experiments considered, an effective number of reactors
(Neff) has to be computed. According to our simula-
where σr is the 1-σ error of the thermal power. R
Mα isPcthomputedbypropagatingthecovariancematrix tion, the NReff values are 1.98, 5.95 and 5.78, for DC,
ij DB and RENO, respectively.
of the fission rates, Cα ≡ρ σ σ , being ρ the correla-
kl kl αk αl The SF obtained via Monte-Carlo, once corrected by
tion matrix and σ the 1-σ error of the fission rate
α the Neff, are in remarkably good agreement with the
R
ones from the analytic calculations: identical numbers
4 reproduced to the 3rd digit. This provides a mutual val-
(cid:88)
Miαj,r = DikCkαlDjTl, (19) idation of the procedures presented.
k,l
where D stands for the derivatives defined as
Reactor Time Variations: Refuelling Scenario
Φi(α +σ ,P )−Φi(α ,P ) Consideringtimevariationofthefluxdeliveredperre-
D ≡ r k k th r k th . (20)
ik σ actorwillmodifytheeffectiveSFbehaviourdescribedso
k
far, as the effective site reactor power configuration de-
Theα-relatederrorontheintegratedfluxfromareac- partsfromthefull-powerover-simplifiedscenarioconsid-
tor R can be expressed as ered so far. In this section, we shall consider the impact
to the effective SF for each experiments upon reactor
(cid:115) refuelling estimated analytically, which a more realistic
(cid:88)
δαR = EiMiαj,REj, (21) experimental scenario since this is, by far, the largest
ij expected reactor-to-reactor variation that can be mod-
elled similarly to all sites considered. The effects asso-
where E stands for the energy phase space of bin i.
i ciated to burn-up, fuel composition, variations caused
For an experimental setup with N reactors, the final
R by running operation constraints (unicity reactor cycle)
integrated error on the antineutrino flux is derived as
will all be neglected, as they are expected to be both
the quadratic sum of the contributions from individual
smallerinmagnitudeandsitedependent,thusoutofthe
reactors, provided by Eq. 18 and Eq. 21
scope of the generic description considered here. How-
ever,moreaccurateestimationsexpectedbyeachspecific
(cid:118) experiment should consider its impact carefully. There-
(cid:117)(cid:117)(cid:88)NR fore,forsimplicity,letusconsiderthefollowingrefuelling
δ =(cid:116) (δr )2+(δr)2. (22)
Φ Pth α scenario: for a total period of one year, each reactor is
r
stopped for two months for refuelling, while during this
Usingtheproceduredescribedabove,wecomputedthe period the other reactors are running. In this scenario,
centralvalues(Φn andΦf)anduncertainties(δn andδf) we consider also fully uncorrelated reactors errors, rep-
ofthefluxesatbothnear(n)andfar(f)detectorsofDC- resenting the worst case scenario. The impact on SF
II, DB and RENO experiments. We also computed the due to one reactor going down is, obviously, site depen-
far-to-near flux ratio, ratio(fn) = Φf/Φn, along with its dent. For DC-II, the overall SF is zero whenever one
error and, finaly, derived the error SF as reactor is monitored by two detectors, regardless of the
10
TABLEIII: SFvaluesfordifferentRENOreactorconfigura-
0.5
tions: all ON and each off. Notation as shown in Fig. 1. n
o
cti All reactors on
ON R1OFF R2OFF R3OFF R4OFF R5OFF R6OFF a
0.227 0.226 0.265 0.234 0.285 0.244 0.206 n Fr 0.4 Mean for 1y runtime
o
si Max. variation for 1y runtime
es 0.3
pr
geometry, as demonstrated before. For RENO and DB, up
S
contrarytoDC-II,weexpectaneffectivedeteriorationof 0.2
SFsinceboththeeffectiveiso-flux-nesscondition(power
symmetrygeometry)andthenumberofreactorsrunning 0.1
worsens upon refuelling.
Our results can be summarised as follows:
0.0
0 1 2 3 4 5 6
DC-II: the year average SF is 0.08 (refuelling scenario), Relative Weighting Parameter
to be compared with 0.12 (full power). Any one
reactor off periods will only benefit DC-II as it has
FIG. 4: The SF variation of DB for the near sites relative
nullfluxuncertainty,thereforethelongerthebetter weighting is illustrated. The minimum obtained is in agree-
for systematics, although that will imply a loss in ment with DB published results. The case for all reactors at
luminosity. full power is indicated by the dashed line, while the maxi-
mal variations obtained due to refuelling scenario (simplified
DB: the tear average SF is 0.20, against the 0.18 ob- casedescribedintext)isindicatedbytheshadedregion. The
tained at full power. physicalinterpretationofthisanalysisanditsapplicationfor
meaningfulreactorfluxerrorsuppressionisadebatewithno
RENO: theresultsareshowninTableIIIforeachreac- consensus within the field. Our result is here illustrated as
tor, however the year average SF was found to be validationofpublishedresults,butitwilldisregardedforfur-
ther discussion.
0.24, to be compared when considering all reactors
(0.23).
relative normalisation of each near are not fully in-
Finally, the refuelling scenario here described (one re-
dependent.
actorstoppedatthetime)isspeciallynaiveinthecaseof
RENOandDB,astheirdatashowssimultaneousstopof • the effect of the neutrino oscillations measured by
up 3 reactors for some time, hence the site power might the near detectors are not fully negligible.
swing up to ∼50%. This will have implications, typi-
If we ignored both physical conditions, we could con-
cally, towards the deterioration of the effective SF to be
sider the optimisation of the contribution of the near
computed for those sites, when integrating over all those
sites, as done by DB in [2]. As we have shown previ-
effects, including fuel burn-up. That level of accuracy is
ously, our formalism is transparent to any factor which
beyond the scope of this estimation.
multiplies all solid angles or all fluxes provided by re-
actors with the same factor for a given detector. Thus,
we consider only the relative difference of the weighting
Daya Bay Near Detectors Optimisation
parameters as
As indicated in Eq. 15, combining flux measurements
from different nears with different weights seems an ap- (cid:88)NR
pealingconsequenceforthecaseofDB.Thisrepresentsa Φ ≡Φprediction =Φ +ρΦ = Ω (ρ)Φ (24)
n f n1 n2 nRi Ri
fundamental difference with respect to the experimental i
configuration with a single near. Any scaling of a flux
where ρ=γ/β is the relative weight parameter and
measurement in Eq. 1 is equivalent to a change to the
baseline of the detectors. So, a difference between the
valuesofβ andγ,inEq.15,isequivalenttoachangebe- Ω (ρ)=Ω +ρΩ (25)
nRi n1Ri n2Ri
tweentherelativebaselinesofthenearsites. Theauthors
think that such transformation might become problem- is a linear combination of the omega terms as described
atic given two possible effects: by Eq. 13.
Since both Φ and Φ are given by Eq. 1, we can use
n f
• the near sites do not only monitor the closest reac- directly Eq. 12 in order to get the expression for the
tors, but there is a sizeable fraction of flux coming corresponding SF. Inserting the values of the geometri-
from the further reactors. This implies that the cal parameters, we show the variation of the SF relative