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112-533: The Hyper-Kamiokande experiment facility will be located in two places: Neutrino oscillations are a quantum mechanical phenomenon in which neutrinos change their flavour (neutrino flavours states: ν e , ν μ , ν τ ) while moving, caused by the fact that the neutrino flavour states are a mixture of the neutrino mass states (ν 1 , ν 2 , ν 3 mass states with masses m 1 , m 2 , m 3 , respectively). The oscillation probabilities depend on

224-506: A − b b a ] / | c | {\displaystyle {\boldsymbol {Q}}={\boldsymbol {P}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]/|c|} and identify with the complex signum of c {\displaystyle c} , sgn ⁡ c = c / | c | {\displaystyle \operatorname {sgn} c=c/|c|} . In this sense, polar decomposition generalizes to matrices

336-518: A {\displaystyle a} as n {\displaystyle n} becomes sufficiently large. In the notation of mathematical limits , continuity of f {\displaystyle f} at a {\displaystyle a} requires that f ( a n ) → f ( a ) {\displaystyle f(a_{n})\to f(a)} as n → ∞ {\displaystyle n\to \infty } for any sequence (

448-390: A n {\displaystyle a_{n}} to be the sequence 1 , 1 2 , 1 3 , 1 4 , … , {\displaystyle 1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\dots ,} which tends towards zero as n {\displaystyle n} increases towards infinity. In this case,

560-436: A n → a {\displaystyle a_{n}\to a} as required, but sgn ⁡ ( a ) = 0 {\displaystyle \operatorname {sgn}(a)=0} and sgn ⁡ ( a n ) = + 1 {\displaystyle \operatorname {sgn}(a_{n})=+1} for each n , {\displaystyle n,} so that sgn ⁡ (

672-468: A n ) n = 1 ∞ {\displaystyle \left(a_{n}\right)_{n=1}^{\infty }} for which a n → a . {\displaystyle a_{n}\to a.} The arrow symbol can be read to mean approaches , or tends to , and it applies to the sequence as a whole. This criterion fails for the sign function at a = 0 {\displaystyle a=0} . For example, we can choose

784-416: A n ) → 1 ≠ sgn ⁡ ( a ) {\displaystyle \operatorname {sgn}(a_{n})\to 1\neq \operatorname {sgn}(a)} . This counterexample confirms more formally the discontinuity of sgn ⁡ x {\displaystyle \operatorname {sgn} x} at zero that is visible in the plot. Despite the sign function having a very simple form,

896-680: A r c t a n ( n x ) = lim n → ∞ 2 π tan − 1 ⁡ ( n x ) . {\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }{\frac {2}{\pi }}{\rm {arctan}}(nx)\,=\lim _{n\to \infty }{\frac {2}{\pi }}\tan ^{-1}(nx)\,.} as well as, sgn ⁡ x = lim n → ∞ tanh ⁡ ( n x ) . {\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }\tanh(nx)\,.} Here, tanh ⁡ ( x ) {\displaystyle \tanh(x)}

1008-433: A "charged-lepton-centric" superposition such as  | ν e ⟩ , which is an eigenstate for a "flavor" that is fixed by the electron's mass eigenstate, and not in one of the neutrino's own mass eigenstates. Because the neutrino is in a coherent superposition that is not a mass eigenstate, the mixture that makes up that superposition oscillates significantly as it travels. No analogous mechanism exists in

1120-607: A classical derivative. Although it is not differentiable at x = 0 {\displaystyle x=0} in the ordinary sense, under the generalized notion of differentiation in distribution theory , the derivative of the signum function is two times the Dirac delta function . This can be demonstrated using the identity sgn ⁡ x = 2 H ( x ) − 1 , {\displaystyle \operatorname {sgn} x=2H(x)-1\,,} where H ( x ) {\displaystyle H(x)}

1232-434: A constant function within the positive open region x > 0 , {\displaystyle x>0,} where the corresponding constant is +1. Although these are two different constant functions, their derivative is equal to zero in each case. It is not possible to define a classical derivative at x = 0 {\displaystyle x=0} , because there is a discontinuity there. Nevertheless,

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1344-542: A few MeV. The baselines of these experiments have ranged from tens of meters to over 100 km (parameter θ 12 ). Mikaelyan and Sinev proposed to use two identical detectors to cancel systematic uncertainties in reactor experiment to measure the parameter θ 13 . In December 2011, the Double Chooz experiment found that θ 13 ≠ 0 . Then, in 2012, the Daya Bay experiment found that θ 13 ≠ 0  with

1456-556: A few TeV, and with a baseline of the diameter of the Earth ; the first experimental evidence for atmospheric neutrino oscillations was announced in 1998. Many experiments have searched for oscillation of electron anti -neutrinos produced in nuclear reactors . No oscillations were found until a detector was installed at a distance 1–2 km. Such oscillations give the value of the parameter θ 13 . Neutrinos produced in nuclear reactors have energies similar to solar neutrinos, of around

1568-486: A few tens of seconds. For Betelgeuse at the distance 0.2 kpc, this rate could reach up to 10 interactions per second and such a high event rate was taken into account in the detector electronics and data acquisition (DAQ) system design, meaning that no data would be lost. Time profiles of the number of events registered in HK and their mean energy would enable testing models of the explosion. Neutrino directional information in

1680-477: A low flux (few tens/cm/sec.), they have not yet been discovered. With ten years of data taking, HK is expected to detect about 40 SRN events in the energy range 16–30 MeV. For the solar ν e 's, the HK experiment goals are: Geoneutrinos are produced in decays of radionuclides inside the Earth. Hyper-Kamiokande geoneutrino studies will help assess the Earth's core chemical composition, which

1792-556: A matrix product: The 2×2 matrix is real symmetric and so (by the spectral theorem ) it is orthogonally diagonalizable . That is, there is an angle θ such that if we define then where λ 1 and λ 2 are the eigenvalues of the matrix. The variables x 1 and x 2 describe normal modes which oscillate with frequencies of λ 1 {\displaystyle {\sqrt {\lambda _{1}\,}}} and λ 2 {\displaystyle {\sqrt {\lambda _{2}\,}}} . When

1904-407: A multitude of experiments in several different contexts. Most notably, the existence of neutrino oscillation resolved the long-standing solar neutrino problem . Neutrino oscillation is of great theoretical and experimental interest, as the precise properties of the process can shed light on several properties of the neutrino. In particular, it implies that the neutrino has a non-zero mass outside

2016-450: A neutrino changing its flavor is Or, using SI units and the convention introduced above This formula is often appropriate for discussing the transition ν μ ↔ ν τ in atmospheric mixing, since the electron neutrino plays almost no role in this case. It is also appropriate for the solar case of ν e ↔ ν x , where ν x is a mix (superposition) of ν μ and ν τ . These approximations are possible because

2128-924: A number falls into by mapping it to one of the values −1 , +1 or 0, which can then be used in mathematical expressions or further calculations. For example: sgn ⁡ ( 2 ) = + 1 , sgn ⁡ ( π ) = + 1 , sgn ⁡ ( − 8 ) = − 1 , sgn ⁡ ( − 1 2 ) = − 1 , sgn ⁡ ( 0 ) = 0 . {\displaystyle {\begin{array}{lcr}\operatorname {sgn}(2)&=&+1\,,\\\operatorname {sgn}(\pi )&=&+1\,,\\\operatorname {sgn}(-8)&=&-1\,,\\\operatorname {sgn}(-{\frac {1}{2}})&=&-1\,,\\\operatorname {sgn}(0)&=&0\,.\end{array}}} Any real number can be expressed as

2240-497: A product Q P {\displaystyle {\boldsymbol {Q}}{\boldsymbol {P}}} where Q {\displaystyle {\boldsymbol {Q}}} is a unitary matrix and P {\displaystyle {\boldsymbol {P}}} is a self-adjoint, or Hermitian, positive definite matrix, both in K n × n {\displaystyle \mathbb {K} ^{n\times n}} . If A {\displaystyle {\boldsymbol {A}}}

2352-415: A range of energies, and oscillation is measured at a fixed distance for neutrinos of varying energy. The limiting factor in measurements is the accuracy with which the energy of each observed neutrino can be measured. Because current detectors have energy uncertainties of a few percent, it is satisfactory to know the distance to within 1%. The first experiment that detected the effects of neutrino oscillation

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2464-530: A significance of 5.2 σ  ; These results have since been confirmed by RENO . Neutrino beams produced at a particle accelerator offer the greatest control over the neutrinos being studied. Many experiments have taken place that study the same oscillations as in atmospheric neutrino oscillation using neutrinos with a few GeV of energy and several-hundred-km baselines. The MINOS , K2K , and Super-K experiments have all independently observed muon neutrino disappearance over such long baselines. Data from

2576-496: A vertical direction by a crane system, providing measurements of neutrino interactions at different off-axis angles (angles to the neutrino beam centre), spanning from 1° at the bottom to 4° at the top, and thus for different neutrino energy spectra. Combining the results from different off-axis angles, it is possible to extract the results for nearly monoenergetic neutrino spectrum without relying on theoretical models of neutrino interactions to reconstruct neutrino energy. Usage of

2688-457: A water-proof vessel. The OD will be instrumented with at least 3,600 8 centimetres (3.1 in) diameter PMTs coupled with 0.6×30×30 cm wavelength shifting (WLS) plates (plates will collect incident photons and transport them to their coupled PMT) and will serve as a veto to distinguish interactions occurring inside from particles entering from the outside of the detector (mainly cosmic-ray muons). HK detector construction began in 2020 and

2800-537: A wide range of neutrino energies and with many different detector technologies. The 2015 Nobel Prize in Physics was shared by Takaaki Kajita and Arthur B. McDonald for their early pioneering observations of these oscillations. Neutrino oscillation is a function of the ratio ⁠ L  / E ⁠   , where L is the distance traveled and E is the neutrino's energy. (Details in § Propagation and interference below.) All available neutrino sources produce

2912-509: Is sgn ⁡ x ≈ x x 2 + ε 2 . {\displaystyle \operatorname {sgn} x\approx {\frac {x}{\sqrt {x^{2}+\varepsilon ^{2}}}}\,.} which gets sharper as ε → 0 {\displaystyle \varepsilon \to 0} ; note that this is the derivative of x 2 + ε 2 {\displaystyle {\sqrt {x^{2}+\varepsilon ^{2}}}} . This

3024-475: Is differentiable everywhere except when x = 0. {\displaystyle x=0.} Its derivative is zero when x {\displaystyle x} is non-zero: d ( sgn ⁡ x ) d x = 0 for  x ≠ 0 . {\displaystyle {\frac {{\text{d}}\,(\operatorname {sgn} x)}{{\text{d}}x}}=0\qquad {\text{for }}x\neq 0\,.} This follows from

3136-403: Is a combination of both normal modes. Over time, these normal modes drift out of phase, and this is seen as a transfer of motion from the first pendulum to the second. The description of the system in terms of the two pendulums is analogous to the flavor basis of neutrinos. These are the parameters that are most easily produced and detected (in the case of neutrinos, by weak interactions involving

3248-431: Is accepted to be equal to 1, the signum can also be written for all real numbers as sgn ⁡ x = 0 ( − x + | x | ) − 0 ( x + | x | ) . {\displaystyle \operatorname {sgn} x=0^{\left(-x+\left\vert x\right\vert \right)}-0^{\left(x+\left\vert x\right\vert \right)}\,.} Although

3360-497: Is connected with the generation of the geomagnetic field . The decay of a free proton into lighter subatomic particles has never been observed, but it is predicted by some grand unified theories (GUT) and results from baryon number (B) violation. B violation is one of the conditions needed to explain the predominance of matter over antimatter in the universe . The main channels studied by HK are p → e + π which

3472-435: Is either positive or negative. These observations are confirmed by any of the various equivalent formal definitions of continuity in mathematical analysis . A function f ( x ) {\displaystyle f(x)} , such as sgn ⁡ ( x ) , {\displaystyle \operatorname {sgn}(x),} is continuous at a point x = a {\displaystyle x=a} if

Hyper-Kamiokande - Misplaced Pages Continue

3584-521: Is expected to confirm at the 5σ confidence level or better if CP symmetry is violated in the neutrino oscillations for 57% of possible δ CP values. CP violation is one of the conditions necessary to produce the excess of matter over antimatter at the early universe, which forms now our matter-built universe. Accelerator neutrinos will be used also to enhance the precision of the other oscillation parameters, |∆m 32 |, θ 23 and θ 13 , as well as for neutrino interaction studies. In order to determine

3696-413: Is expressed as The above formula is correct for any number of neutrino generations. Writing it explicitly in terms of mixing angles is extremely cumbersome if there are more than two neutrinos that participate in mixing. Fortunately, there are several meaningful cases in which only two neutrinos participate significantly. In this case, it is sufficient to consider the mixing matrix Then the probability of

3808-630: Is expressed as a unitary transformation relating the flavor and mass eigenbasis and can be written as where The symbol U α i {\displaystyle U_{\alpha i}} represents the Pontecorvo–Maki–Nakagawa–Sakata matrix (also called the PMNS matrix , lepton mixing matrix , or sometimes simply the MNS matrix ). It is the analogue of the CKM matrix describing

3920-525: Is favoured by many GUT models and p → ν + K predicted by theories including supersymmetry . After ten years of data taking, (in case no decay will be observed) HK is expected to increase the lower limit of the proton mean lifetime from 1.6  ·  10 to 6.3  ·  10 years for its most sensitive decay channel ( p → e + π ) and from 0.7  ·  10 to 2.0  ·  10 years for

4032-401: Is infeasible on multiple levels. The idea of neutrino oscillation was first put forward in 1957 by Bruno Pontecorvo , who proposed that neutrino–antineutrino transitions may occur in analogy with neutral kaon mixing . Although such matter–antimatter oscillation had not been observed, this idea formed the conceptual foundation for the quantitative theory of neutrino flavor oscillation, which

4144-619: Is inspired from the fact that the above is exactly equal for all nonzero x {\displaystyle x} if ε = 0 {\displaystyle \varepsilon =0} , and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of x 2 + y 2 {\displaystyle {\sqrt {x^{2}+y^{2}}}} ). See Heaviside step function § Analytic approximations . The signum function sgn ⁡ x {\displaystyle \operatorname {sgn} x}

4256-452: Is invertible then such a decomposition is unique and Q {\displaystyle {\boldsymbol {Q}}} plays the role of A {\displaystyle {\boldsymbol {A}}} 's signum. A dual construction is given by the decomposition A = S R {\displaystyle {\boldsymbol {A}}={\boldsymbol {S}}{\boldsymbol {R}}} where R {\displaystyle {\boldsymbol {R}}}

4368-449: Is non-zero only if neutrino oscillation violates CP symmetry ; this has not yet been observed experimentally. If experiment shows this 3×3 matrix to be not unitary , a sterile neutrino or some other new physics is required. Since | ν j ⟩ {\displaystyle \left|\,\nu _{j}\,\right\rangle } are mass eigenstates, their propagation can be described by plane wave solutions of

4480-466: Is not in a mass eigenstate; however, the charged lepton would lose coherence, if it had any, over interatomic distances (0.1  nm ) and would thus quickly cease any meaningful oscillation. More importantly, no mechanism in the Standard Model is capable of pinning down a charged lepton into a coherent state that is not a mass eigenstate, in the first place; instead, while the charged lepton from

4592-885: Is often represented as sgn ⁡ x {\displaystyle \operatorname {sgn} x} or sgn ⁡ ( x ) {\displaystyle \operatorname {sgn}(x)} . The signum function of a real number x {\displaystyle x} is a piecewise function which is defined as follows: sgn ⁡ x := { − 1 if  x < 0 , 0 if  x = 0 , 1 if  x > 0. {\displaystyle \operatorname {sgn} x:={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}} The law of trichotomy states that every real number must be positive, negative or zero. The signum function denotes which unique category

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4704-460: Is responsible for oscillation is often written as (with c and ℏ {\displaystyle \hbar } restored) where 1.27 is unitless . In this form, it is convenient to plug in the oscillation parameters since: If there is no CP-violation ( δ is zero), then the second sum is zero. Otherwise, the CP asymmetry can be given as In terms of Jarlskog invariant the CP asymmetry

4816-648: Is the Heaviside step function using the standard H ( 0 ) = 1 2 {\displaystyle H(0)={\frac {1}{2}}} formalism. Using this identity, it is easy to derive the distributional derivative: d sgn ⁡ x d x = 2 d H ( x ) d x = 2 δ ( x ) . {\displaystyle {\frac {{\text{d}}\operatorname {sgn} x}{{\text{d}}x}}=2{\frac {{\text{d}}H(x)}{{\text{d}}x}}=2\delta (x)\,.} The Fourier transform of

4928-569: Is the Hyperbolic tangent and the superscript of -1, above it, is shorthand notation for the inverse function of the Trigonometric function , tangent. For k > 1 {\displaystyle k>1} , a smooth approximation of the sign function is sgn ⁡ x ≈ tanh ⁡ k x . {\displaystyle \operatorname {sgn} x\approx \tanh kx\,.} Another approximation

5040-406: Is the complex argument function . For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines, for z = 0 {\displaystyle z=0} : sgn ⁡ ( 0 + 0 i ) = 0 {\displaystyle \operatorname {sgn}(0+0i)=0} Another generalization of

5152-411: Is the point on the unit circle of the complex plane that is nearest to z {\displaystyle z} . Then, for z ≠ 0 {\displaystyle z\neq 0} , sgn ⁡ z = e i arg ⁡ z , {\displaystyle \operatorname {sgn} z=e^{i\arg z}\,,} where arg {\displaystyle \arg }

5264-470: Is the standard gravity , L is the length of the pendulum, m is the mass of the pendulum, and x is the horizontal displacement of the pendulum. As an isolated system the pendulum is a harmonic oscillator with a frequency of g / L {\displaystyle {\sqrt {g/L\;}}\,} . The potential energy of a spring is 1 2 k x 2 {\displaystyle {\tfrac {1}{2}}kx^{2}} where k

5376-484: Is the constant value +1 , which equals the value of sgn ⁡ x {\displaystyle \operatorname {sgn} x} there. Because the absolute value is a convex function , there is at least one subderivative at every point, including at the origin. Everywhere except zero, the resulting subdifferential consists of a single value, equal to the value of the sign function. In contrast, there are many subderivatives at zero, with just one of them taking

5488-576: Is the loss of commutativity . In particular, the generalized signum anticommutes with the Dirac delta function ε ( x ) δ ( x ) + δ ( x ) ε ( x ) = 0 ; {\displaystyle \varepsilon (x)\delta (x)+\delta (x)\varepsilon (x)=0\,;} in addition, ε ( x ) {\displaystyle \varepsilon (x)} cannot be evaluated at x = 0 {\displaystyle x=0} ; and

5600-502: Is the probability that a neutrino originally of flavour α will be observed later as having flavour β. Comparison of the appearance probabilities for neutrinos and antineutrinos (P ν μ → ν e versus P ν μ → ν e ) allows measurement of the δ CP phase. δ CP ranges from −π to +π (from −180° to +180° ), and 0 and ±π correspond to CP symmetry conservation. After 10 years of data taking, HK

5712-551: Is the real part of z {\displaystyle z} and Im ( z ) {\displaystyle {\text{Im}}(z)} is the imaginary part of z {\displaystyle z} . We then have (for z ≠ 0 {\displaystyle z\neq 0} ): csgn ⁡ z = z z 2 = z 2 z . {\displaystyle \operatorname {csgn} z={\frac {z}{\sqrt {z^{2}}}}={\frac {\sqrt {z^{2}}}{z}}.} Thanks to

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5824-421: Is the spring constant and x is the displacement. With a mass attached it oscillates with a period of k / m {\displaystyle {\sqrt {k/m\;}}\,} . With two pendulums (labeled a and b ) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is This is a quadratic form in x a and x b , which can also be written as

5936-459: Is then a similar jump to sgn ⁡ ( x ) = + 1 {\displaystyle \operatorname {sgn}(x)=+1} when x {\displaystyle x} is positive. Either jump demonstrates visually that the sign function sgn ⁡ x {\displaystyle \operatorname {sgn} x} is discontinuous at zero, even though it is continuous at any point where x {\displaystyle x}

6048-484: Is unitary, but generally different than Q {\displaystyle {\boldsymbol {Q}}} . This leads to each invertible matrix having a unique left-signum Q {\displaystyle {\boldsymbol {Q}}} and right-signum R {\displaystyle {\boldsymbol {R}}} . In the special case where K = R ,   n = 2 , {\displaystyle \mathbb {K} =\mathbb {R} ,\ n=2,} and

6160-409: The p → ν + K channel. Dark matter is a hypothetical, non-luminous form of matter proposed to explain numerous astronomical observations suggesting the existence of additional invisible mass in galaxies. If the dark matter particles interact weakly , they may produce neutrinos through annihilation or decay. Those neutrinos could be visible in

6272-504: The Cauchy principal value . The signum function can be generalized to complex numbers as: sgn ⁡ z = z | z | {\displaystyle \operatorname {sgn} z={\frac {z}{|z|}}} for any complex number z {\displaystyle z} except z = 0 {\displaystyle z=0} . The signum of a given complex number z {\displaystyle z}

6384-710: The Einstein-Cartan torsion , which requires a modification to the Standard Model of particle physics . The experimental discovery of neutrino oscillation, and thus neutrino mass, by the Super-Kamiokande Observatory and the Sudbury Neutrino Observatories was recognized with the 2015 Nobel Prize for Physics . A great deal of evidence for neutrino oscillation has been collected from many sources, over

6496-838: The Iverson bracket notation: sgn ⁡ x = − [ x < 0 ] + [ x > 0 ] . {\displaystyle \operatorname {sgn} x=-[x<0]+[x>0]\,.} The signum can also be written using the floor and the absolute value functions: sgn ⁡ x = ⌊ x | x | + 1 ⌋ − ⌊ − x | x | + 1 ⌋ . {\displaystyle \operatorname {sgn} x={\Biggl \lfloor }{\frac {x}{|x|+1}}{\Biggr \rfloor }-{\Biggl \lfloor }{\frac {-x}{|x|+1}}{\Biggr \rfloor }\,.} If 0 0 {\displaystyle 0^{0}}

6608-625: The LSND experiment appear to be in conflict with the oscillation parameters measured in other experiments. Results from the MiniBooNE appeared in Spring ;2007 and contradicted the results from LSND, although they could support the existence of a fourth neutrino type, the sterile neutrino . In 2010, the INFN and CERN announced the observation of a tauon particle in a muon neutrino beam in

6720-499: The Lorentz factor , γ , is greater than 10 in all cases. Using also t ≈ L , where L is the distance traveled and also dropping the phase factors, the wavefunction becomes Eigenstates with different masses propagate with different frequencies. The heavier ones oscillate faster compared to the lighter ones. Since the mass eigenstates are combinations of flavor eigenstates, this difference in frequencies causes interference between

6832-588: The OPERA detector located at Gran Sasso , 730 km away from the source in Geneva . T2K , using a neutrino beam directed through 295 km of earth and the Super-Kamiokande detector, measured a non-zero value for the parameter θ 13 in a neutrino beam. NOνA , using the same beam as MINOS with a baseline of 810 km, is sensitive to the same. Neutrino oscillation arises from mixing between

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6944-492: The Polar decomposition theorem, a matrix A ∈ K n × n {\displaystyle {\boldsymbol {A}}\in \mathbb {K} ^{n\times n}} ( n ∈ N {\displaystyle n\in \mathbb {N} } and K ∈ { R , C } {\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}} ) can be decomposed as

7056-518: The W boson ). The description in terms of normal modes is analogous to the mass basis of neutrinos. These modes do not interact with each other when the system is free of outside influence. When the pendulums are not identical the analysis is slightly more complicated. In the small-angle approximation, the potential energy of a single pendulum system is 1 2 m g L x 2 {\displaystyle {\tfrac {1}{2}}{\tfrac {mg}{L}}x^{2}} , where g

7168-420: The neutrino mass ordering (whether the ν 3 mass eigenstate is lighter or heavier than both ν 1 and ν 2 ), or equivalently the unknown sign of the ∆m 32 parameter, neutrino oscillations must be observed in matter. With HK beam neutrinos (295 km, 600 MeV), the matter effect is small. In addition to beam neutrinos, the HK experiment studies atmospheric neutrinos , created by cosmic rays colliding with

7280-454: The (invertible) matrix A = [ a − b b a ] {\displaystyle {\boldsymbol {A}}=\left[{\begin{array}{rr}a&-b\\b&a\end{array}}\right]} , which identifies with the (nonzero) complex number a + i b = c {\displaystyle a+\mathrm {i} b=c} , then the signum matrices satisfy Q = P = [

7392-459: The 3×3 form, it is given by where c ij ≡ cos θ ij , and s ij ≡ sin θ ij . The phase factors α 1 and α 2 are physically meaningful only if neutrinos are Majorana particles —i.e. if the neutrino is identical to its antineutrino (whether or not they are is unknown)—and do not enter into oscillation phenomena regardless. If neutrinoless double beta decay occurs, these factors influence its rate. The phase factor δ

7504-452: The Earth's atmosphere, producing neutrinos and other byproducts. These neutrinos are produced at all points on the globe, meaning that HK has access to neutrinos that have travelled through a wide range of distances through matter (from a few hundred metres to the Earth's diameter ). These samples of neutrinos can be used to determine the neutrino mass ordering. Ultimately, a combined beam neutrino and atmospheric neutrino analysis will provide

7616-559: The HK detector as an excess of neutrinos from the direction of large gravitational potentials such as the galactic centre , the Sun or the Earth , over an isotropic atmospheric neutrino background. The Hyper-Kamiokande experiment consists of an accelerator neutrino beamline, a set of near detectors, the intermediate detector and the far detector (also called Hyper-Kamiokande). The far detector by itself will be used for proton decay searches and studies of neutrinos from natural sources. All

7728-410: The HK far detector can provide an early warning for the electromagnetic supernova observation, and can be used in other multi-messenger observations. Neutrinos cumulatively produced by supernova explosions throughout the history of the universe are called supernova relic neutrinos (SRN) or diffuse supernova neutrino background (DSNB) and they carry information about star formation history. Because of

7840-614: The Hyper-Kamiokande experiment: Neutrino oscillations Neutrino oscillation is a quantum mechanical phenomenon in which a neutrino created with a specific lepton family number ("lepton flavor": electron , muon , or tau ) can later be measured to have a different lepton family number. The probability of measuring a particular flavor for a neutrino varies between three known states, as it propagates through space. First predicted by Bruno Pontecorvo in 1957, neutrino oscillation has since been observed by

7952-893: The Inner Detector (ID) and the Outer Detector (OD) by a 60 cm-wide inactive cylindrical structure, with its outer edge positioned 1 meter away from vertical and 2 meters away from horizontal tank walls. The structure will optically separate ID from OD and will hold PhotoMultiplier Tubes (PMTs) looking both inwards to the ID and outwards to the OD. In the ID, there will be at least 20,000 50 centimetres (20 in) diameter PhotoMultiplier Tubes (PMT) of R12860 type by Hamamatsu Photonics and approximately 800 multi-PMT modules (mPMTs). Each mPMT module consists of nineteen 8 centimetres (3.1 in) diameter photomultiplier tubes encapsulated in

8064-407: The Standard Model that would make charged leptons detectably oscillate. In the four decays mentioned above, where the charged lepton is emitted in a unique mass eigenstate, the charged lepton will not oscillate, as single mass eigenstates propagate without oscillation. The case of (real) W boson decay is more complicated: W boson decay is sufficiently energetic to generate a charged lepton that

8176-503: The W ;boson decay is not initially in a mass eigenstate, neither is it in any "neutrino-centric" eigenstate, nor in any other coherent state. It cannot meaningfully be said that such a featureless charged lepton oscillates or that it does not oscillate, as any "oscillation" transformation would just leave it the same generic state that it was before the oscillation. Therefore, detection of a charged lepton oscillation from W boson decay

8288-477: The above elements will serve for the accelerator neutrino oscillation studies. Before launching the HK experiment, the T2K experiment will finish data taking and HK will take over its neutrino beamline and set of near detectors, while the intermediate and the far detectors have to be constructed anew. The Intermediate Water Cherenkov Detector (IWCD) will be located at a distance of around 750 metres (2,460 ft) from

8400-407: The analogous mixing of quarks . If this matrix were the identity matrix , then the flavor eigenstates would be the same as the mass eigenstates. However, experiment shows that it is not. When the standard three-neutrino theory is considered, the matrix is 3×3. If only two neutrinos are considered, a 2×2 matrix is used. If one or more sterile neutrinos are added (see later), it is 4×4 or larger. In

8512-414: The charged leptons (electrons, muons, and tau leptons) have never been observed to oscillate. In nuclear beta decay, muon decay, pion decay, and kaon decay, when a neutrino and a charged lepton are emitted, the charged lepton is emitted in incoherent mass eigenstates such as  | e ⟩ , because of its large mass. Weak-force couplings compel the simultaneously emitted neutrino to be in

8624-618: The corresponding flavor components of each mass eigenstate. Constructive interference causes it to be possible to observe a neutrino created with a given flavor to change its flavor during its propagation. The probability that a neutrino originally of flavor α will later be observed as having flavor β is This is more conveniently written as where Δ j k m 2   ≡ m j 2 − m k 2   . {\displaystyle \Delta _{jk}m^{2}\ \equiv m_{j}^{2}-m_{k}^{2}~.} The phase that

8736-464: The definition of the absolute value | x | {\displaystyle |x|} on the separate regions x < 0 {\displaystyle x<0} and x < 0. {\displaystyle x<0.} For example, the absolute value function is identical to x {\displaystyle x} in the region x > 0 , {\displaystyle x>0,} whose derivative

8848-585: The detector is the possibility to search for sterile oscillation patterns for different off-axis angles and to obtain a cleaner sample of electron neutrino interactions, whose fraction is larger for larger off-axis angles. The Hyper-Kamiokande detector will be built 650 metres (2,130 ft) under the peak of Nijuugo Mountain in the Tochibora mine, 8 kilometres (5.0 mi) south from the Super-Kamiokande (SK) detector. Both detectors will be at

8960-419: The differentiability of any constant function , for which the derivative is always zero on its domain of definition. The signum sgn ⁡ x {\displaystyle \operatorname {sgn} x} acts as a constant function when it is restricted to the negative open region x < 0 , {\displaystyle x<0,} where it equals -1 . It can similarly be regarded as

9072-409: The flavor and mass eigenstates of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different superposition of the three (propagating) neutrino states of definite mass. Neutrinos are emitted and absorbed in weak processes in flavor eigenstates but travel as mass eigenstates . As a neutrino superposition propagates through space,

9184-526: The form where In the ultrarelativistic limit , | p → j | = p j ≫ m j   , {\displaystyle \left|{\vec {p}}_{j}\right|=p_{j}\gg m_{j}~,} we can approximate the energy as where E is the energy of the wavepacket (particle) to be detected. This limit applies to all practical (currently observed) neutrinos, since their masses are less than 1 eV and their energies are at least 1 MeV, so

9296-460: The graph of the sign function with a vertical line through the origin, making it continuous as a two dimensional curve. In integration theory, the signum function is a weak derivative of the absolute value function. Weak derivatives are equivalent if they are equal almost everywhere , making them impervious to isolated anomalies at a single point. This includes the change in gradient of the absolute value function at zero, which prohibits there being

9408-432: The limits sgn ⁡ x = lim n → ∞ 1 − 2 − n x 1 + 2 − n x . {\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }{\frac {1-2^{-nx}}{1+2^{-nx}}}\,.} and sgn ⁡ x = lim n → ∞ 2 π

9520-411: The mixing angle θ 13 is very small and because two of the mass states are very close in mass compared to the third. The basic physics behind neutrino oscillation can be found in any system of coupled harmonic oscillators . A simple example is a system of two pendulums connected by a weak spring (a spring with a small spring constant ). The first pendulum is set in motion by the experimenter while

9632-477: The most sensitivity to the oscillation parameters δ CP , |∆m 32 |, sgn ∆m 32 , θ 23 and θ 13 . Core-collapse supernova explosions produce great quantities of neutrinos . For a supernova in the Andromeda Galaxy , 10 to 16 neutrino events are expected in the HK far detector. For a galactic supernova at a distance of 10 kpc about 50,000 to 94,000 neutrino interactions are expected during

9744-901: The near and intermediate detectors. For the HK/T2K neutrino beam peak energy (600 MeV) and the J-PARC – HK/SK detector distance (295 km), this corresponds to the first oscillation maximum, for oscillations driven by ∆m 32 . The J-PARC neutrino beam will run in both neutrino- and antineutrino-enhanced modes separately, meaning that neutrino measurements in each beam mode will provide information about muon (anti)neutrino survival probability P ν μ → ν μ , P ν μ → ν μ , and electron (anti)neutrino appearance probability P ν μ → ν e , P ν μ → ν e , where P ν α → P ν β

9856-406: The neutrino production place. It will be a cylinder filled with water of 10 metres (33 ft) diameter and 50 metres (160 ft) height with a 10 metres (33 ft) tall structure instrumented with around 400 multi-PMT modules (mPMTs), each consisting of nineteen 8 centimetres (3.1 in) diameter PhotoMultiplier Tubes (PMTs) encapsulated in a water-proof vessel. The structure will be moved in

9968-405: The point x = 0 {\displaystyle x=0} , unlike sgn {\displaystyle \operatorname {sgn} } , for which ( sgn ⁡ 0 ) 2 = 0 {\displaystyle (\operatorname {sgn} 0)^{2}=0} . This generalized signum allows construction of the algebra of generalized functions , but the price of such generalization

10080-1237: The product of its absolute value and its sign function: x = | x | sgn ⁡ x . {\displaystyle x=|x|\operatorname {sgn} x\,.} It follows that whenever x {\displaystyle x} is not equal to 0 we have sgn ⁡ x = x | x | = | x | x . {\displaystyle \operatorname {sgn} x={\frac {x}{|x|}}={\frac {|x|}{x}}\,.} Similarly, for any real number x {\displaystyle x} , | x | = x sgn ⁡ x . {\displaystyle |x|=x\operatorname {sgn} x\,.} We can also be certain that: sgn ⁡ ( x y ) = ( sgn ⁡ x ) ( sgn ⁡ y ) , {\displaystyle \operatorname {sgn}(xy)=(\operatorname {sgn} x)(\operatorname {sgn} y)\,,} and so sgn ⁡ ( x n ) = ( sgn ⁡ x ) n . {\displaystyle \operatorname {sgn}(x^{n})=(\operatorname {sgn} x)^{n}\,.} The signum can also be written using

10192-500: The quantum mechanical phases of the three neutrino mass states advance at slightly different rates, due to the slight differences in their respective masses. This results in a changing superposition mixture of mass eigenstates as the neutrino travels; but a different mixture of mass eigenstates corresponds to a different mixture of flavor states. For example, a neutrino born as an electron neutrino will be some mixture of electron, mu , and tau neutrino after traveling some distance. Since

10304-583: The quantum mechanical phase advances in a periodic fashion, after some distance the state will nearly return to the original mixture, and the neutrino will be again mostly electron neutrino. The electron flavor content of the neutrino will then continue to oscillate – as long as the quantum mechanical state maintains coherence . Since mass differences between neutrino flavors are small in comparison with long coherence lengths for neutrino oscillations, this microscopic quantum effect becomes observable over macroscopic distances. In contrast, due to their larger masses,

10416-501: The same off-axis angle (2.5°) to the neutrino beam centre and at the same distance (295 kilometres (183 mi)) from the beam production place in J-PARC . HK will be a water Cherenkov detector, 5 times larger (258 kton of water) than the SK detector. It will be a cylindrical tank of 68 metres (223 ft) diameter and 71 metres (233 ft) height. The tank volume will be divided into

10528-407: The same type of detector as the far detector with almost the same angular and momentum acceptance allows comparison of results from these two detectors without relying on detector response simulations. These two facts, independence from the neutrino interaction and detector response models, will enable HK to minimise systematic error in the oscillation analysis. Additional advantages of such a design of

10640-509: The second begins at rest. Over time, the second pendulum begins to swing under the influence of the spring, while the first pendulum's amplitude decreases as it loses energy to the second. Eventually all of the system's energy is transferred to the second pendulum and the first is at rest. The process then reverses. The energy oscillates between the two pendulums repeatedly until it is lost to friction . The behavior of this system can be understood by looking at its normal modes of oscillation. If

10752-803: The sign function for real and complex expressions is csgn {\displaystyle {\text{csgn}}} , which is defined as: csgn ⁡ z = { 1 if  R e ( z ) > 0 , − 1 if  R e ( z ) < 0 , sgn ⁡ I m ( z ) if  R e ( z ) = 0 {\displaystyle \operatorname {csgn} z={\begin{cases}1&{\text{if }}\mathrm {Re} (z)>0,\\-1&{\text{if }}\mathrm {Re} (z)<0,\\\operatorname {sgn} \mathrm {Im} (z)&{\text{if }}\mathrm {Re} (z)=0\end{cases}}} where Re ( z ) {\displaystyle {\text{Re}}(z)}

10864-529: The sign function takes the value −1 when x {\displaystyle x} is negative, the ringed point (0, −1) in the plot of sgn ⁡ x {\displaystyle \operatorname {sgn} x} indicates that this is not the case when x = 0 {\displaystyle x=0} . Instead, the value jumps abruptly to the solid point at (0, 0) where sgn ⁡ ( 0 ) = 0 {\displaystyle \operatorname {sgn}(0)=0} . There

10976-517: The signum function has a definite integral between any pair of finite values a and b , even when the interval of integration includes zero. The resulting integral for a and b is then equal to the difference between their absolute values: ∫ a b ( sgn ⁡ x ) d x = | b | − | a | . {\displaystyle \int _{a}^{b}(\operatorname {sgn} x)\,{\text{d}}x=|b|-|a|\,.} Conversely,

11088-421: The signum function is ∫ − ∞ ∞ ( sgn ⁡ x ) e − i k x d x = P V 2 i k , {\displaystyle \int _{-\infty }^{\infty }(\operatorname {sgn} x)e^{-ikx}{\text{d}}x=PV{\frac {2}{ik}},} where P V {\displaystyle PV} means taking

11200-463: The signum function is the derivative of the absolute value function, except where there is an abrupt change in gradient before and after zero: d | x | d x = sgn ⁡ x for  x ≠ 0 . {\displaystyle {\frac {{\text{d}}|x|}{{\text{d}}x}}=\operatorname {sgn} x\qquad {\text{for }}x\neq 0\,.} We can understand this as before by considering

11312-437: The signum-modulus decomposition of complex numbers. At real values of x {\displaystyle x} , it is possible to define a generalized function –version of the signum function, ε ( x ) {\displaystyle \varepsilon (x)} such that ε ( x ) 2 = 1 {\displaystyle \varepsilon (x)^{2}=1} everywhere, including at

11424-532: The six theoretical parameters: and two parameters which are chosen for a particular experiment: Continuing studies done by the T2K experiment , the HK far detector will measure the energy spectra of electron and muon neutrinos in the beam (produced at J-PARC as an almost pure muon neutrino beam) and compare it with the expectation in case of no oscillations, which is initially calculated based on neutrino flux and interaction models and improved by measurements performed by

11536-588: The source of the deficit until the Sudbury Neutrino Observatory provided clear evidence of neutrino flavor change in 2001. Solar neutrinos have energies below 20  MeV . At energies above 5 MeV, solar neutrino oscillation actually takes place in the Sun through a resonance known as the MSW effect , a different process from the vacuum oscillation described later in this article. Following

11648-553: The start of data collection is expected in 2027. Studies have also been undertaken on the feasibility and physics benefits of building a second, identical water-Cherenkov tank in South Korea around 1100 km from J-PARC, which would be operational 6 years after the first tank. A history of large water Cherenkov detectors in Japan, and long-baseline neutrino oscillation experiments associated with them, excluding HK: A history of

11760-413: The step change at zero causes difficulties for traditional calculus techniques, which are quite stringent in their requirements. Continuity is a frequent constraint. One solution can be to approximate the sign function by a smooth continuous function; others might involve less stringent approaches that build on classical methods to accommodate larger classes of function. The signum function coincides with

11872-484: The theories that were proposed in the 1970s suggesting unification of electromagnetic, weak, and strong forces, a few experiments on proton decay followed in the 1980s. Large detectors such as IMB , MACRO , and Kamiokande II have observed a deficit in the ratio of the flux of muon to electron flavor atmospheric neutrinos (see muon decay ). The Super-Kamiokande experiment provided a very precise measurement of neutrino oscillation in an energy range of hundreds of MeV to

11984-414: The two pendulums are identical ( L a = L b ), θ is 45°. Sign function In mathematics , the sign function or signum function (from signum , Latin for "sign") is a function that has the value −1 , +1 or 0 according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function

12096-413: The two pendulums are identical then one normal mode consists of both pendulums swinging in the same direction with a constant distance between them, while the other consists of the pendulums swinging in opposite (mirror image) directions. These normal modes have (slightly) different frequencies because the second involves the (weak) spring while the first does not. The initial state of the two-pendulum system

12208-485: The value f ( a ) {\displaystyle f(a)} can be approximated arbitrarily closely by the sequence of values f ( a 1 ) , f ( a 2 ) , f ( a 3 ) , … , {\displaystyle f(a_{1}),f(a_{2}),f(a_{3}),\dots ,} where the a n {\displaystyle a_{n}} make up any infinite sequence which becomes arbitrarily close to

12320-437: The value sgn ⁡ ( 0 ) = 0 {\displaystyle \operatorname {sgn}(0)=0} . A subderivative value 0 occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval [ − 1 , 1 ] {\displaystyle [-1,1]} , which might be thought of informally as "filling in"

12432-484: Was Ray Davis' Homestake experiment in the late 1960s, in which he observed a deficit in the flux of solar neutrinos with respect to the prediction of the Standard Solar Model , using a chlorine -based detector. This gave rise to the solar neutrino problem . Many subsequent radiochemical and water Cherenkov detectors confirmed the deficit, but neutrino oscillation was not conclusively identified as

12544-475: Was first developed by Maki, Nakagawa, and Sakata in 1962 and further elaborated by Pontecorvo in 1967. One year later the solar neutrino deficit was first observed, and that was followed by the famous article by Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge". The concept of neutrino mixing is a natural outcome of gauge theories with massive neutrinos, and its structure can be characterized in general. In its simplest form it

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