JHEP05(2021)139 in the Springer PDG May 5, 2021 can be mis- April 9, 2021 δ May 17, 2021 : δ : : November 20, 2020 : Revised Accepted Published Received . We demonstrate how the δ Published for SISSA by https://doi.org/10.1007/JHEP05(2021)139 2 a,b, . . [email protected] , . 3 and Rebekah Pestes 1 2006.09384 governs the scale of these results. We then demonstrate how a, | The Authors. 3 e c CP violation, Neutrino Physics
U CP violation in the lepton mass matrix will be probed with good precision in | , [email protected] https://orcid.org/0000-0002-5209-872X https://orcid.org/0000-0002-9634-1664 1 2 Upton, NY 11973, U.S.A. Center for Neutrino Physics,Blacksburg, Department VA of 24061, Physics, U.S.A. Virginia Tech, E-mail: High Energy Theory Group, Physics Department, Brookhaven National Laboratory, b a Open Access Article funded by SCOAP considered, the standard PDG parameterization has a number ofKeywords: convenient features. ArXiv ePrint: given the current data, differentworking parameterizations with can parameterization lead dependentsmallness to variables, of different such conclusions as when leading and argue that theviolation Jarlskog in the is lepton the sector. cleanest We means also of confirm presenting that, the among amount the of different parameterizations CP Abstract: upcoming experiments. The amounttified of in numerous CP ways violation andstandard present is PDG in typically definition oscillations parameterized can of byzations be the the of complex quan- lepton the phase mixing lepton matrix. mixing matrix There as are well. additional Through parameteri- various examples, we explore how, Peter B. Denton The impact of differentinterpretation parameterizations of on CP the violation in neutrino oscillations JHEP05(2021)139 ) via (1.1) 3 , 2 , ) submatrix = 1 τ i ]. For leptons, 4 ]: 7 for i i ν | ) for concreteness. Any , however, is convention δ 1.1 . ] 2 µ row in eq. ( U 1 τ ∗ µ U 2 ∗ e ) is analogous to the KM phase in the CKM ) is arbitrary; we chose the (3, δ U ) and mass eigenstates ( 1 1.1 e – 1 – 2 U ] provides a small amount of CPV [ 3 e, µ, τ , Im[ 2 = ≡ α J for i ], one of which ( α , like the CKM matrix for quarks, describes the mismatch of 6 ν , | U 5 16 submatrix in eq. ( [ ] is a frequently used motivation for searching for sources of CPV, 1 2 U × 18 17 4 2 1 . While all three phases contribute to leptonic CPV, the complex phase i 15 i 12 ν | ∗ αi U i P comparison = There are many different ways of parameterizing the lepton mixing matrix, all of which δ i contributes to neutrino oscillation while the two Majorana phases, though important α ν The choice of the resulting from deleting therow third and column any and column the can be removed. must have at least one complex phase. Depending on the details of the parameterization, for neutrinoless double betadependent decay, and do thus not. not a Thedescribe fundamental quantity. complex the The amount phase most of useful CPV fundamental in quantity a to mass matrix is the Jarlskog invariant [ matrix while the othertribution to two their are mass. physicallepton flavor if eigenstates and ( only| if neutrinos haveδ a Majorana con- portant part of the ongoingalso goal for of understanding measuring when theKobayashi-Maskawa (CKM) CP parameters matrix is of [ and the Standard isthree Model, not phases but relating violated. to In howlepton neutrinos the mixing and quark matrix anti-neutrinos sector, behave the differently appear Cabibo- in the 1 Introduction While the fact that CPmetry violation in (CPV) the is universe needed [ identifying to sources explain the of matter-antimatter CPV asym- is interesting in its own right. Not only does it play an im- B Quark mixing C Flavor models 3 4 Discussion 5 Conclusion A Unitarity violation Contents 1 Introduction 2 Mixing matrix parameterizations JHEP05(2021)139 3 e U , we 2 unitary or larger 3 ) and six 4 23 × . Rephasing of the choice × δ , θ 3 4 13 , θ (or . 12 3 5 θ × 3 independent and the key role that 0 δ , should be unitary up to any corrections U – 2 – ). Thus, the matrix can be parameterized as three A ]. Some of these degeneracies can be differentiated 32 – 8 complex matrix has 18 free parameters; for neutrino oscillations five of the 3 × 3 can all lead to confusion related to the extraction of the standard 1 . Finally, we will discuss our results in the context of T2K’s measurements 3 and we will draw several interesting conclusions in section 4 In this paper, we will demonstrate exactly how the complex phase of the lepton mixing In addition, new physics scenarios, if they exist in reality, can fundamentally complicate A general non-unitary 1 invariance also allows us tochanging constrain the each quadrant of mixing a angle mixing to angle a is quadrant equivalent to of specific ourcomplex combinations choice, of phases since rephasing can bereal removed (or via any rephasing. other rephasing While choice) one this has only a leads choice to in a which constant row shift and in column the to complex make phases. complex phases. However, allare rows and Dirac, columns or can ifoscillations. be we individually Rephasing is restrict rephased the if ourselves factphase. neutrinos that to any relativistic column While or neutrinos, this row can appearsmatrix, as be one to is multiplied of by remove these the an six all arbitrary case rephasingsphase six is for remaining phases dependent for neutrino on present describing the in neutrino other five, oscillations an which and arbitrary leaves is one usually complex labeled as We anticipate that the leptonfrom mixing sterile matrix, neutrinos, whichunitary we violation ignore schemes, (for see appendix arotation discussion of matrices, parameterizations containing and a general total of three Euler mixing angles ( how these different parameterizationsplays affect in section their respective in section 2 Mixing matrix parameterizations matrix is not thewhen optimal it parameter can for be understandingthe misleading. CPV mixing We in matrix will neutrino show withthe how oscillations one complex different, and complex phase perfectly is phase valid, used canwill representations as define lead of the the to primary parameterizations indicator rather we of are different CPV going conclusions in to when consider oscillations. in In this section paper. Next, we will show of parameterization. The optimal choicematrix of in parameterization of the the presencenew of physics scenarios additional is neutrinos) beyondthree-flavor mixing the oscillation matrix scope picture in of with this the no article new presence and interactions. of we will various focus on the standard violating phases), sterile neutrinosor (also with unitary or violation withoutCP additional violating CP violating three-flavor phases), phaseby [ combining measurements from different experiments.those discussed These in scenarios this are paper differentabout in from CP that violation in would the suffer presence from various of partial new degeneracies physics, extracting information results and goals of experimentsnot in a terms fundamental description of of the our complex understanding phase of of CPV. onethe parameterization extraction of is CP violationexample, independent the of addition how of the non-standard mixing neutrino matrix interactions is (with parameterized. or without For additional CP the constraint on the complex phase can be quite different. This means that interpreting JHEP05(2021)139 – 36 , (2.2) (2.1) is on δ 0 0 parameter- 12 12 ]. As these , c s 132 ] without the 52 U 4 – 12 iδ 12 0 0 1 s are all different. 23 23 − c 45 c s − e k ], or five rotations 13 13 13 c c 41 s ≡ iδ 3 , and iδ e j e U 23 , 23 c i s 13 13 s s 13 12 ] is advantageous over the PDG c 12 ), so we will only consider having s s δ 12 44 − , and s , − in which 23 43 3 23 → − s ), but changing which rotation iδ , for neutrino oscillations, the magnitudes c δ 12 2 − 2 12 c , 13 U e c c − 13 s = 1 iδ iδ 0 0 e ] has the same complex phase as in the – 3 – e 23 42 iδ 23 s e c (see table i, j, k . Thus, the rotation matrices are defined as follows: 13 13 0 1 0 . In this paper, we are only considering parameteri- 13 ] including those using the same rotation axis twice 13 δ s c ) s 13 s 13 θ ij for 12 − c confirming that they are qualitatively the same as those 34 θ 12 c , k c 12 used by the Particle Data Group (PDG) [ iji − U c 33 ). The three rotation matrices can be in any order, and the − j sin( ≡ ) U U 23 U ◦ 2 23 ], the exponential of a complex matrix [ i If one wanted to include the Majorana phases in the param- c ≡ s U U 90 12 40 2 , ij 12 s ]. s s ≡ [0 − , 42 ∈ ijk and U 23 = 23 23 ) c s , θ ij can be put into any of the rotation matrices. , the symmetric parameterization [ . In addition, we discuss the CKM matrix in different parameterizations θ 123 13 23 δ U by a constant. We choose to constrain all of the mixing angles to the first U 23 s . ijk c , θ δ − cos( U = B 12 1 00 0 0 θ ≡ ], three of the Gell-Mann matrices parameterizing the generators of SU(3) [ ij PDG 35 c ≡ U 1 )[ The parameterization of We also note that there are numerous other models of neutrino mixing designed to When we make various parameterizations of The primary parameterization presented in [ U 2 iji ], four complex phases [ in the rotation matrix containing U ization below. impact of parameterizationparameterization in to neutrino be carefully flavor examined models, as these but models we areMajorana developed. encourage phases is the choice of predict certain relationships amongmodels the have oscillation fewer parameters, than seethus four e.g. any [ parameters, features they relatedother don’t parameters to span introduced the the by complexlarge the entire phase body model relevant would of space choices. depend literature and For on on this this correlations reason, topic, with along we with the do the not very present a comprehensive review of the the context neutrino oscillations.three In rotations this article, and we one focus complexrepeated on phase, rotations parameterizations of but the containing we form alsoof show the numerical form results forin the appendix case of ( 39 and a complex phaseeterization [ for method of including the Majorana phases, and it simplifies to the PDG parameterization in where zations of the form Other parameterizations exist [ the values of thedoes mixing not angles (other and than trivialδ redefinitions, such as quadrant ( complex phase of the matrix elementschange; this remain is the the same, crux of as our does paper. the Changing Jarlskog, the order but of the the four rotation matrices parameters changes and shifting JHEP05(2021)139 , ) ◦ 7 123 . (2.3) U terms ], and iδ 56 row and = 49 . Within – ± δ e th . (3.1) 23 54 i , we use the − θ U ) = ]. 3 150 754 640 . . . e while the complex 59 , and ◦ U ) 0 ) 0 61 . PDG arg( δ and then the second and PDG PDG δ δ = 8 iδ e 13 . θ remains largely unconstrained. ) , to be completely unconstrained δ ◦ 8 550 0 068 cos( 068 cos( . . . . PDG . (3.2) 0 δ ) PDG will be called δ − = 33 sin( is set such that . Recently, T2K has placed some con- PDG 0 δ 12 δ 293 405 + 0 δ 23 . . PDG θ s has simple elements along the 0 0 U 23 ]( p p c ijk 60 ) ) 13 – 4 – U s 0334 sin( ), 2 13 . PDG PDG c , its data is currently not constraining [ δ δ 2.1 12 = 0 with good precision in future experiments [ s between the various parameterizations of PDG δ δ J 12 δ c are the least precise), but = 23 822 0 068 cos( 068 cos( . . . θ 0 J 0 for now, we will consider ] for a related discussion for the quark mixing matrix. − 3 ]; 53 138 + 0 186 , discuss the interplay between the T2K measurements and different . . 58 0 0 , 4 so that the first row and third column were all real. p p 57 [ iδ − e = | PDG δ U | , see also ref. [ , which gives us the following for the magnitudes of the elements in the neutrino column. 1 comparison th PDG In the PDG parameterization, experiments have measured all of the mixing angles In the PDG convention, the complex phase We note some important features of this parameterization. The most important of δ k While NOvA also has sensitivity to U 3 The preferred value of thephase Jarlskog is for this parameterization as a function of the complex mixing matrix: 3 In order to compare thebest values of fit values forin the mixing angles from [ phase in other parameterizationsstraints will on be called and then, in section parameterizations of the mixing matrix. (though the measurements of Plans have been made tothis measure paper aims at discussingIn the the meaning rest of of the this precision paper, claimed the by complex these phase measurements. in the the PDG convention, thisphysics. phase For example, can one could bethird multiply rows shifted the by third with column no by effect on any of the oscillation which is the difference betweena “simple” simple element elements to and be “complicated”while one elements. that complicated is We functions only define can a alsowe product be of see the trigonometric that sum functions or the and general, difference first of using row such the and terms. definitions third Thus in column in eq. are ( composed entirely of simple elements. In The mixing matrix and Jarlskogin in table the other five parameterizations considered are shown and the Jarlskog in this parameterization is
JHEP05(2021)139
ij ij ij ij PDG PDG ij
≡ s θ ≡ c θ δ U θ sin( , ) cos( , , ( . in parameters denote ) )
ij ij ij
ij ij
s θ ≡ c θ δ θ ≡
0 0
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al 1 Table U aaeeiain ftenurn iigmatrix mixing neutrino the of Parameterizations . ne osdrto nti ae n hi orsodn alkgivrat The invariant. Jarlskog corresponding their and paper this in consideration under
23 13 23 13 13
s c − e s − c c
0 0 0 0 0
iδ
0
13
12 13 12 23 13 12 23 12 23 13 12 23
23 23 13 12 12 321
s c e s s s c c c s − e c s s − U c s c δ s c s 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+
) sin(
iδ − iδ − 0 2
0 0
23 13 12 23 12 13 12 23 13 12 23 12
s s c − c s c c e c s c s s e
0 0 0 0 0 0 0 0 0 0 0 0
+
iδ − iδ −
0 0
23 13 23 23 13
c c s − e c s −
0 0 0 0 0
iδ
0
23
23 13 12 23 12 23 13 12 13 12 13 12
23 13 13 12 12 312
c e s s c − c s − e s s − s c c c U s c s c s c δ 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
) sin(
iδ iδ − 2 0
0 0
13 12 23 13 12 23 12 23 13 12 13 12
c e s c s c s c s e s s s − c
0 0 0 0 0 0 0 0 0 0 0 0
+
iδ − iδ
0 0
23 13 12 23 13 13 12 23 13 12 23 13
c − s − c c e c s s − s c − e s e s s
0 0 0 0 0 0 0 0 0 0 0 0
iδ iδ iδ
0 0 0
12
23 12 12 23 12
23 23 13 13 12 231
c c s − s c U δ s c s c s c 0 0 0 0 0 0 0 0 0 0 0
) sin( 0 2
23 13 12 23 13 23 13 12 13 12 23 13
e c s s c s e s s − c c s c c
0 0 0 0 0 0 0 0 0 0 0 0
+
iδ − iδ −
0 0 – 5 –
13 12 23 13 12 23 13 13 12 23 13 12
c c e s s − s c c − e s c − s c s
0 0 0 0 0 0 0 0 0 0 0 0
iδ iδ
0 0
23
23 12 23 23 12
23 13 13 12 12 213
c s − s c c U δ s c s c s c 0 0 0 0 0 0 0 0 0 0 0
) sin( 2 0
12 23 13 12 13 12 23 13 23 13 12 13
s e s s s c c e c s e s s c − c
0 0 0 0 0 0 0 0 0 0 0 0
+
iδ − − iδ − iδ
0 0 0
23 13 23 13 12 23 13 12 23 12 23 12
− e c s − s c s e s s s c c s c
0 0 0 0 0 0 0 0 0 0 0 0
+
iδ iδ −
0 0
12
23 13 23 13 12 23 13 12 23 13 23 12
23 23 13 13 12 132
e s s − c c s − e c s s − s c c c U s c s c s c δ 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
sin( )
iδ iδ − 2 0
0 0
13 12 13 12 12
c c e s c s
0 0 0 0 0
iδ −
0
23 13 12 23 12 23 13 12 23 12 23 13
e c s c − s s c c e c s s − s c −
iδ iδ
13
23 12 23 13 23 13 12 23 12 23 13 12 PDG 123 12 12 23 23 13
c s − s c e s s s − c c e s s c − U c s c δ s c s
) sin( iδ iδ 2
13 13 12 13 12
e s c s c c
iδ −
ijk U J Matrix
JHEP05(2021)139
δ
eedrvdfo h Jarlskog. the from derived were ) sin( for formulas 0
δ U U
sml”eeet sedfiiini h et of text) the in definition (see elements “simple” r eie rmtemgiueo cmlctd lmn of element “complicated” a of magnitude the from derived are ) cos( for formulas The . The . 0
ij ij
θ − θ ,
ahmxn nl si h interval the in is angle mixing each h omlsfrtemxn nlsaedrvdfo h antdso the of magnitudes the from derived are angles mixing the for formulas The . sin 1 = cos , ) 90 [0
0 0 ◦ 2 q
al 2 Table U U omlsfrtefu aaeesi ie aaeeiainof parameterization given a in parameters four the for Formulas . ntrso h bouevleo lmnsof elements of value absolute the of terms in n h alkg Since Jarlskog. the and
1 τ 1 τ
| U −| 1 | U −| 1
3 τ 2 τ 1 τ 1 µ 1 e 3 τ 2 τ 1 τ 1 µ 1 e
|| U | 2 | U || U || U || U || U | | U || U || U || U
2 2
1 τ 321
| U | U
√ √
2 τ 1 µ | U | | U |
1 τ 3 τ 1 τ 1 e 2 τ 1 µ 1 τ 3 e
U | 1 J | U | | U | | U −| | U | | U −| | U −| 1 | | U −| ) ( ) (
2 2 2 2 2 2 2 2 2
2 τ 2 τ
| U −| 1 | U −| 1
3 τ 2 τ 1 τ 2 µ 2 e 3 τ 2 τ 1 τ 2 µ 2 e
| U || U || U || U || U | 2 | U || U || U || U || U |
2 2
2 τ 312
U | U |
√ √
1 τ 2 e | U | | U |
1 τ 2 µ 3 τ 2 e 2 τ 1 µ 2 τ 2 τ
U −| | U −| 1 | U | | U −| 1 J | U | | U | | U −| | U | | ( ) ( )
2 2 2 2 2 2 2 2 2
1 µ 1 µ
| U −| 1 | U −| 1
1 µ 1 e 1 τ 3 µ 2 µ 1 µ 1 e 1 τ 3 µ 2 µ
| U || U || U || U || U | 2 | U || U || U || U || U |
2 2 1 µ 231
| U | U
√ √
1 τ 3 µ | U | | U |
3 µ 1 µ 1 τ 2 µ 1 µ 3 e 1 µ 1 e
| U −| 1 J | U | | U | | U −| | U | | U −| | U −| 1 | U | – 6 – ) ( ) (
2 2 2 2 2 2 2 2 2
3 µ 3 µ
| U −| 1 | U −| 1
3 µ 2 µ 1 µ 3 e 3 τ 3 µ 2 µ 1 µ 3 e 3 τ
| U || U || U || U || U | | U || U || U || U || U | 2
2 2 213 3 µ
U | U |
√ √
3 e 1 µ | U | | U |
3 τ 2 µ 3 µ 1 e 3 µ 3 µ 1 µ 3 e
−| | U | | U −| | U −| 1 | U | | U −| 1 J | U | | U | | U ) ( ) (
2 2 2 2 2 2 2 2 2
2 e 2 e
U −| 1 | U −| 1 |
2 µ 3 e 2 e 1 e 2 µ 3 e 2 e 1 e 2 τ 2 τ
|| U || U || U || U | 2 || U || U || U || U | | U | U
2 2
132 2 e
U | U |
√ √
2 τ 3 e | U | | U |
2 τ 3 e 2 e 1 µ 2 e 2 µ 2 e 1 e
−| | U | | U −| | U −| 1 | U | | U −| 1 J | U | | U | | U ) ( ) (
2 2 2 2 2 2 2 2 2
3 e 3 e
U −| 1 | U −| 1 |
3 µ 3 e 2 e 1 e 3 µ 3 e 2 e 1 e 3 τ 3 τ
|| U || U || U || U | 2 || U || U || U || U | | U | U
2 2
123 3 e
U | U |
√ √
2 e 3 µ | U | | U |
3 τ 2 e 3 e 1 µ 3 e 3 µ 3 e 1 e
−| | U | | U −| | U −| 1 | U | | U −| 1 J | U | | U | | U
) ( ) (
2 2 2 2 2 2 2 2 2
23 13 12
ijk
U θ θ θ δ δ
sin( ) sin( ) sin( ) cos( ) ) sin( 0 0 0 0 0 JHEP05(2021)139 . . . ) ) 1 0 1.0 1 δ (3.3) ), we PDG δ ∼ − sin( allowed 3.2 ) 0.5 0 σ sin( δ ). In each ) in the PDG ) in figure 3 ) PDG cos( plot. That is, δ 0.0 ) and ( 13 ) in six different 0 ) θ is bounded and cos(δ in terms of PDG δ δ 0 3.1 cos( 0 is simple, whereas δ δ | -0.5 cos( 3 e 132 213 231 312 321 , and and (see table U U U U U . One can see this by U | | 0 23 ) 3 is complicated. Thus, in θ δ , e | , -1.0 ]. U 1.0 0.5 0.0 3
1
PDG -0.5 -1.0 |
e
0
13 ) ( ' cos δ ) and that of other parameteri- sin( 60 U θ U | , 1.0 in , 0 12 PDG δ θ 1 ( 13 does not uniquely determine − s in those parameterizations for which are dependent on the mixing angles. ) 0.5 | PDG 2 ijk ◦ 3 in other parameterizations. Varying the (as well as U e d ) plot but not in the PDG 0 1 2 U δ | δ PDG is bounded, ) to find on the allowed range of 0.0 δ ≈ 0 PDG | sin(δ δ = 180 ] δ 13 sin( 3 e – 7 – s 0 0 sin( δ δ U vs | comparatively large, we must have -0.5 0 is unbounded in figure δ or 132 213 231 312 321 , but ). The shaded regions on the left plot show the range of is restricted to a small domain just by our constraint 0 0 U U U U U B ) comes from the Jarlskog, so it depends on [cos 0 δ δ 0 δ ) δ 0 range of mixing angles from [ , -1.0 δ = 0 PDG 1.0 0.5 0.0 and
-0.5 -1.0 δ max
σ ) ( 0 ' sin δ cos( 321 δ A sin( U 360° with in terms of the magnitudes of the elements of the matrix and the , and comes from the norm of the elements of the matrix, and thus only ) 270° 0 , while ) U ) 0 δ 312 δ U , PDG cos( in each parameterization, which is PDG δ cos( 180° δ | 231 through the mixing angles. B 3 U e ) + cos( U | uniquely determines A 90° PDG , we show how to calculate the four parameters ( ) . Comparison of the CP-violating phase of also shows that the effect of varying the mixing angles within their 3 δ p 2 1 321 132 213 231 312 U U U U U = PDG cos( , namely An interesting feature of this plot is that for some of the alternative parameterizations We note that there are loops in the | δ 0° 0° 3 90°
U 270° 180° 360°
e
' δ U | We can approximate the effectparameters by of leveraging small the smallness of parameterization or, alternatively, thelooking relative at smallness of of the parameterizations forin which each of theorder parameterizations to for get which the comparatively small value of of of the threewhether PDG it mixing is angles. boundedBased on about The current parameterizations measurements, for the which restriction is driven by the size of cos( This is because depends on and have plotted the relationship between Figure values has a marginalmixing impact angles on significantly the outside values ofthis of these figure. ranges, however, can have a dramatic effect on variations of the curves over the 3 In table parameterizations of Jarlksog. Using the expressions in this table combined with equations ( Figure 1 zations of the neutrino mixing matrix ( JHEP05(2021)139 4 in 231 ) 0 PDG U . On (3.4) (3.5) δ δ 213 , cos( U (and thus is approx- 39 ) . This is all in 0 . 312 δ 2 12 s U should be quite = 0 ; these correlations PDG ). This provides sin( is unconstrained) δ 23 321 PDG 2 23 c 213 U δ s U 23 ranges. We note that and is accurate at the 2 12 PDG s in the PDG definition c δ 1 σ 12 and c . We can also see how the − 13 θ 12 1 , and 213 s 213 U U 132 13 s U is the smallest for ), we can easily see how the three , , ≈ and ) ) 3.4 . We then see that 123 | 312 U 3 132 instead of the factor of PDG e ( PDG . (3.6) U δ δ U is only independent of 3 ) | 2 12 0 e c δ 0 U 0 23 sin( sin( ). J , d θ . on the tightly constrained nature of | sin( 1 ijk 3 57 54 3.5 in both . . d e , which are quite similar. Thus, swapping – 8 – ). From eq. ( ≡ U 2 23 )–( | ≈ while 0 r s = 0 = 0 J 2% PDG in some parameterizations is restricted in its allowed 0 3.3 δ δ 132 ↔ ) 23 23 < 0 23 2 2 23 c c U δ c s 2 23 23 23 sin c 2 12 12 2 s s in s s . In fact, sin( as a function of 12 12 c c ) − − 0 ijk ), the reduced Jarlskog is notably much larger than it is in the 12 12 δ 1 PDG 1 d and eqs. ( s s δ ± 321 1 13 13 U sin( s s ≈ ≈ ) is accurate to is independent of , and 3.3 321 231 0 13 elements. d d are allowed to vary independently within their 3 θ 312 3 e U parameterization, there is some dependence on the constraints from U , is quite a bit less precise in other parameterizations, in particular those with for explicit expressions in each of the parameterizations considered.) This PDG 123 . The reduced Jarlskog is defined by, 1 0 231 13 U is the same (up to higher order corrections) as changing the octant. Meanwhile, U θ U 3 and ( , along with bands representing the region covered by these curves when the mixing level (eq. ( 321 3 for various different parameterizations as a function of 2 e U To further understand the impact of We also show the reduced Jarlskog in different parameterizations as a function of We show the values of the three mixing angles in the different parameterizations in 0 is only independent of In the U δ 4 is the most constrained), since it has a factor 10% 0 0 12 other parameterizations, we plottedof the allowed range (assuming are ignored here for simplicity. parameterization. We can seefor that in the parameterizationsparameterizations with complicated with expressions simple expressionsanother for means of showing why range, as shown in figure in figure (See table quantity represents the contribution to the Jarlskog from the mixing angles in a given the other hand, precision on the differentNotably oscillation parameters changescomplicated in different parameterizations. consistent with the top right panel in figure figure angles in θ different parameterizations behave. For example, wesimilar see since that they differand only by we see that the slopeδ of imately contained within provides a simple approximation for the. expressions shown in figure for the three parameterizations with complicated with
JHEP05(2021)139
3 e PDG
δ | U | ≈ prxmto for approximation ) cos( at variables parameterization PDG the of terms in . 0
PDG PDG δ PDG ij
δ θ U δ ≡ c , ( variables unprimed the and parameterization alternative the eas hwan show also We . in parameters the denote ) ) cos( ,
ij ij ij
3 e ij ij
U 3 Table δ θ ≡ c θ ≡ s θ U
0 0
nvrosprmtrztosof parameterizations various in . eoeteprmtr in parameters the denote ) , cos , sin , ( variables primed The .
0 0 0 0
23 12
δ
s s − 1
c 068 . 814+0 . 0
23 13 12 23 12 321
2 2
e c s c s s U | / δ |
0 0 0 0 0
+ 2) cos(
12 12 23 23 iδ − 0
c s c s 2
δ δ δ 0
δ δ
1+0 ) δ cos( 108 . +0 c 005 . +0 c 018 . 108+0 . 0 c 03 . 0 − c 16 . 0 − c 29 .
0
3 2 2
√ q
23 12
δ
s c − 1
c 068 . 0 − 595 . 0
23 13 12 13 12 312
2 2
s c s e s c U | / δ |
0 0 0 0 0
+ 2) cos(
23 23 12 12 iδ − 0
c s c s 2
δ δ δ 0
δ δ
c 07 . +0 c 08 . 0 − c 04 . 2 − 74 . 4 ) δ cos( 048 . +0 c 005 . +0 c 024 . 0 − 105 . 0
0 3 2 2
√ q – 9 –
23 12
δ
c s − 1
c 068 . 0 − 861 . 0
23 13 23 13 12 231
2 2
c s U s c s e | | / δ
0 0 0 0 0
+ cos( 2)
23 23 12 12 iδ − 0
c s c s 2
δ δ δ 0
δ δ
c 07 . +0 c 34 . 0 − c 17 . 0 − 62 . 1 ) δ cos( 085 . +0 c 005 . +0 c 007 . 0 − 108 . 0
0 2 3 2
√ q
23 13 213 13
U c s . s 0 0 150 0
13 12 132 13
U s c . s 0 0 150 0
23 13 12
ijk δ PDG 3 e 3 e 23 13 12 3 e 23 13 12
U U δ , δ U | δ , θ , θ , θ θ , . θ , . θ , θ , ≈ c , δ U | ≈ | . | θ , θ ,
( ) ( 61 8 = 8 33 = 0 ( ) 7 49 = ) 0 0 0 0 0 ◦ ◦ 0 ◦ JHEP05(2021)139 . . is 360° 23 0 θ δ PDG δ ] while , which ] and the 60 270° 13 θ 60 ranges) in the PDG 180° δ σ . The shaded regions range of mixing angles and that the allowed 90° is unconstrained until σ parameterization, 1 0 132 213 231 312 321 PDG U U U U U δ 360° − δ , 132 0° 0°
90° 75° 60° 45° 30° 15°
U
23 ' 213 θ when allowing the mixing angles U for smaller values of 360° 13 PDG 270° θ U range of mixing angles from [ . For ◦ 270° σ for r J PDG PDG 180° δ 180° δ for each parameterization as a function of using the best fit mixing angles from [ up to 45 ) 0 – 10 – δ J )). 13 PDG θ 90° is constrained to be near sin( U 3.4 ) 0 is constrained to be near 1. In the three remaining 132 213 231 312 321 ≡ U U U U U 90° δ for 0 ) r 0 J r 0° δ 0° 45° 30° 15° J
cos(
13 ' θ 132 213 231 312 321 U U U U U cos( (see eq. ( 360° 0° 13 s
0.00 0.08 0.06 0.04 0.02
r ' J increases roughly linearly with ∝ 270° 13 ranges. The sharp features are due to switching the preferred octant of θ ijk σ d PDG 180° δ past which point , with all the other mixing angles fixed. In the 90° . Mixing angles in each parameterization as a function of ◦ . The “reduced Jarlskog” 4 ]. The dashed gray line is 132 213 231 312 321 35 U U U U U 60 > 0° 0°
15° 75° 60° 45° 30° 90°
12 ' θ 13 always unconstrained for anyθ value of parameterizations, we see that region increases as is consistent with from [ dotted gray lines are theto maximum vary and over minimum their of 3 in figure Figure 3 The shaded regions show the range of variations of the curves over the 3 the dashed (dotted)standard gray parameterization. lines are best fit values (maxima and minima of 3 Figure 2 show the range of variations of the curves over the 3 JHEP05(2021)139 , , ◦ ). 5 321 15 231 5 PDG U have δ U 45° in the 13 PDG θ δ , and (with fixed 0 . In figure 30° δ 312 to within 3 (see figure as in the PDG e U ∆ 13 U 321 θ , as an important U 213 PDG ] as a useful target. ◦ 45° PDG U 231 δ 15° δ 62 U [ elements. ◦ and 3 = 15 e U 30° 132 0° = 10 0°
90° U
360° 270° 180° PDG
' 13 of Range 213 δ δ θ U ∆ PDG 45° δ ) becomes more precise in 0 15° δ ∆ sin 30° 0° 0°
90°
and plot the corresponding 360° 270° 180°
'
ag of Range indicates that the precision on the mixing 13 312 δ θ ◦ U , depending on the parameterization and the 5 0 , is comparable in 45° δ 15° – 11 – 23 = 15 (and thus s 0 23 δ . This is due to the fact that c PDG 30° in different parameterizations as a function of 13 3 δ 0° s 0° 0 90° aren’t too different from the PDG parameterization (as
180° 360° 270° ∆
δ
'
13 of Range 2 13 132 θ δ U c precision on 213 12 45° s ◦ U 15° 1 12 c ∼ and ≡ elements while the others all have simple 30° r 0° 0° J 3
132 90°
e 180° 360° 270°
' ag of Range 13 231 δ U θ U U , the mixing angles must become less precise in those parameterizations to 15° . In addition, since the Jarlskog (and thus the precision on the Jarlskog) will lead to extremely different precision in different parameterizations, in 321 ◦ U PDG . The allowed range of δ ] and a more recent theory overview emphasized 0° 0° 90° of 15
270° 180° 360°
61
' of Range δ , and Next generation long-baseline accelerator experiments are aiming to not only detect 312 PDG angles, in particular parameterization. Meanwhile, since U compensate, as confirmed inhave complicated figure particular in the three parameterizationswe with see complicated expressions that for while seen before), in theprecision other could three result parameterizations, in avalue measurement of of is the same in each parameterization, figure goal [ We take as a benchmarkmixing number angles) in theWe find other that parameterizations in forδ different various parameterizations values of of the lepton mixing matrix, a precision on CPV, but measure itbeen with presented some based precision. on Various theoreticalflavor targets concerns symmetries. on motivated by the discrinimating The precision among on 2013 different Snowmass report stressed Figure 4 PDG parameterization. We haveis fixed unconstrained. the other mixing angles to their best fit values while JHEP05(2021)139 are , and iki parame- U 312 is largely ijk U . That is, . U , 6 . and 1 0 . In reality, of PDG 213 given the other , and there is a δ iji U 1 . U − , this implies that | PDG . ) ◦ δ ◦ 0 δ 15 360 of sin( . | 360° by itself is not maximal in =15° PDG ◦ δ PDG PDG (and even quite a bit beyond Δδ 90 δ ◦ ± . 50 for six parameterizations involving = ◦ 270° . 170 PDG 23 PDG δ . We make two observations. First, the δ θ iji . PDG U 180° δ ◦ 40 – 12 – , and correspondingly, 8 . in parameterizations of the forms 0 , given an uncertainty on 0 − 90° δ are very similar in these parameterizations as they are , as shown by the shaded regions in figure ) 0 PDG . of the form range of δ δ as a function of . 231 312 321 132 213 5 0 U U U U U σ PDG δ , in some parameterizations, in particular δ sin( 1 0 0 all different.
5°
and 25° 20° 15° 10° 30°
' Δδ k 0 δ (and, given T2K’s preference for , and ). This highlights the fact that j ) as a function of ◦ ◦ 0 , we also show δ i is restricted to be is restricted in 3 of the parameterizations to a region near 6 210 210 parameterizations while it can be associated with any of the rotations in the ) ) style parameterizations. In fact, they look nearly identical, but they have slight 0 0 . . δ δ iji . Uncertainty of the CP-violating phase of alternate parameterizations of the neutrino 0 0 U ijk δ δ U cos( cos( . . , In much of the previous discussion and figures, we have assumed that we have measured In figure Note that the complex phase must be associated with either the first or the third rotation in the repeated ◦ ◦ 5 321 that even over thethis), allowed 3 corresponding constraint on rotations terizations. measured oscillation parameters the three oscillation parameterscourse, perfectly and each have of no theunconstrained information four at on the parameters moment are except via somewhat T2K constrained, appearance data. although We have verified, however, U if the lepton mixing150 matrix was parameterized170 differently, we wouldany already fundamental know way; that it does lead to the largest amount of CPV allowed differences. Second, weequivalent note for that 4 Discussion As we have seen in figure a single repeated rotationrelationships axis between in the Figure 5 mixing matrix ( JHEP05(2021)139 6 2 21 2 31 are m m ∆ ∆ 2 21 23 m s ∆ 23 c 12 s 12 , and c 2 31 13 m s ◦ ]. T2K has measured ∆ 2 13 c given other constraints, , 58 ], as can the corrections 360 ◦ 12 66 s 12 c ◦ , = 270 from T2K, this results in a 14% 13 s are equal. 270 13 PDG 2 13 θ δ c iki U ◦ and except for parameterizations with a repeated PDG from reactors was used since it is extremely 180 δ iji 1 13 ]. In this analysis, however, they assume priors U θ – 13 – 65 – ◦ 63 90 212 232 313 323 121 131 . In addition, they seem to see slightly more neutrino U U U U U U σ allowed regions for σ ◦ 0 ◦ ◦ ◦ ◦ ◦ 0
90
180 360 270 0 δ at close to 3 ]. in the parameterizations , which (very weakly) suggests a larger value than that from precise 0 0 δ 68 13 , θ 67 J < ’s [ 2 ij . The same as the left panel of figure m ∆ We now take this slight tension seriously and consider the implications if it persists with The strongest constraint on CPV currently comes from T2K [ While it is interesting to note that there was a slight tension in the solar mass splitting data, it goes in 6 We note that the matter effect onlyto slightly the modifies Jarlskog the situation can at be T2K,to as determined the the in corrections a straightforward fashion [ the wrong direction to explain the T2K tension. the other oscillation parameters, since theup measurement depends to on matter effects.excess Based in on this the quantity.1.4%, best The 1.1%, fit 1 1.3%, value and for 2.8%.in the This T2K suggests data with that the the rest best of place the oscillation to data understand is any in tension terms of the solar mass splitting. on solar parameters. If, instead,well measured, a one prior would on findaccommodated that this in slight terms tension of could a be constraint more on somewhat solar more easily parameters. additional data. If it is confirmed at high precision, then we would look to the constraint on implying that events and fewer anti-neutrino events thanalthough allowed by this tensiona is constraint at on lowmedium significance. baseline reactor experiments Nonetheless, [ T2K has interpreted this as Figure 6 rotation. Note that a high amount of neutrino appearance and a low amount of anti-neutrino appearance, JHEP05(2021)139 ] ◦ ◦ PDG 358 δ = 18 , 0 ) = +1 ◦ δ However, , which is in a given ∆ ]. PDG [165 δ 7 69 . δ 22% . In addition, ) as opposed to in the different sin( PDG 132 2 31 0 δ roughly commute is small compared δ U m 3 , the allowed range . This makes sense ◦ are the two smallest 3 ∆ 6 U . , and U 312 12 ◦ θ 3 U . = 8 and quantity. In particular, it and 2 13 and δ 2 θ U = 35 uncertainty on the oscillation parameterizations. Note that . These maximum (minimum) U 13 σ θ 2 321 or other possible choices [ √ 096] . The Jarlskog is always within the U ) . / δ 0 23 1 as measuring parameters in the Stan- , θ δ sin( and , as T2K measures 096 information. This represents 35% of the . 2 21 0 ) 231 , and m − U [ = atan 12 space, compared with 53.6% of the total is that it is constrained to within ∆ θ PDG 0 13 , δ including the 3 δ – 14 – θ ] 13 , ◦ θ PDG ◦ sin( δ in general and as a proxy for understanding CP vio- does not affect CP violation, it is still a fundamental . A subleading contribution comes from the fact that 208 ◦ , δ δ 3 ◦ = 45 . 35 cos 23 θ [173 . Thus, we can conclude that , which results in a reduction in the allowed space to 93%. in the usual parameterization). While we have extensively = ◦ parameterization, the uncertainty is already down to 321 ’s (and really just 45 12 2 U θ 312 PDG m U , which numerically is δ (the PDG parameterization) is quite similar to ] ∆ 3 1 √ 123 6 ). From nu-fit v4.0, it is found that the allowed range is reduced to , U 1 3 without including any − 1 √ 6 under the assumption of anarchy, instead of . While the sign of occur when δ − δ [ that 033] , compared to . ) = , which also requires knowing J ◦ ) 0 1 ∈ , cos CL, accounting for the uncertainty in the oscillation parameters. , which is about half the allowed space using a uniform prior on J PDG is quite similar to σ PDG σ δ Before concluding this section, we make one comment on the usefulness of We also note that certain parameterizations are very similar. In particular, we see Thus, we propose that a more useful metric for quantifying CP violation is amount In addition, T2K’s constraint on 033 = 34 δ . The Haar measure applied to lepton mass mixing indicates that the correct prior on the complex phase 7 0 231 12 sin( − parameter in the Standard Modeldo and not is advocate physical, entirely and ignoringdard thus measurements it Model of must is be the measured.and primary theorists Thus goal we to of focus particle on physics, the but Jarlskog we when do discussingis encourage CP uniform experimentalists in violation. lation, as well as thein benefit the of leptonic using mass thecomes matrix, Jarlskog from to there the quantify is goal thewith still amount arbitrary of some precision, of physics as value CP to well in violation assign measure the three of rotation everything. angles, leaves That one sign is, undetermined: measuring the the Jarlskog angles, so while rotationssize are of not the commutative rotations. in general, the correctionparameterization scales (or with the demonstrated the shortcomings of in figure U within the contextparameterizations of doesn’t the change discussion muchsince of when the CPV. associated commuting angles in That the is, PDG parameterization, the effect of to the value for maximalθ CPV, Thus given T2K’s measurement, thea allowed parameterization space independent of statement. the Jarlskog is about range values of ( [ total allowed space, and it mostly comes from the fact that this means that in the at 3 of allowed Jarlskog space.knowing the ratio Extracting of thesin( Jarlskog only requires (up to matter effects) at 3 we find that ifon the the complex lepton phase mixingparameters. is matrix This only is was only parameterized 7.5%space. of as the Similarly, total small regions exist for the JHEP05(2021)139 ) 0 δ gen- ) is the cos( N may be is proof 0 ), so the δ SU( PDG δ 5 6= 0 ) PDG δ 323 321 and CPV in neutrino U U sin( . 323 PDG ), and long-baseline acceler- δ U 313 312 3 e U U U , or 313 232 231 U U U , , without even considering any constraint 131 212 213 U 1 , is related to the amount of CP violation U U , ), we find that the preferred parameterizations 121 PDG 3 – 15 – U δ µ 131 132 ]. , U U U 70 213 U , 121 123 U U 132 U ), medium-baseline reactor ( While this is of course a matter of taste, we here list some 2 | | | | | | , is measured is not truly fundamental (see figure e 3 3 3 2 3 2 e e e e µ µ U 8 123 U U U U U U | | | | | | U PDG δ implies no CPV in neutrino oscillations and (i.e. parameterizations in which the element concerned is simple) are: is small, it is favorable to have a simple element there; this picks out , which requires a fairly strong cancellation, which only happens for . The final parameter, | in certain other parameterizations without any information on 2 | 3 23 3 0 e e ) = 0 θ δ U U | | PDG coming from T2K or other experiments. This also highlights the fact that the δ . , and 1 ator/atmospheric disappearance ( from table Since parameterizations It is useful to be ableof to a write single approximate mixing two-flavor oscillations angleBy as for a looking the simple at function various solar experimental ( probes that are available to us. 13 sin( − θ PDG We use this time to revisit the topic of optimality with regards to the choice of pa- There are other potentially interesting criteria such as computational efficiency, for which δ , • • 8 12 erators may be more efficient than rotations [ rameterization for the leptonic mixingthe matrix in presently the known context of data. desirable neutrino features oscillations, giving of a matrix: precision with which precision on the Jarlskog should beconstraint considered on instead. The causesmallness of of the surprisingly strong near of CPV (givenoscillations that are not all exactly three equivalent concepts.parameterizations mixing One with clear angles one way to complex are see phase,already this non-zero), is severely the that constrained, allowed in as range different shown ofon in the figure new phase Three of the fourreasonably degrees well. of In freedomθ the in standard the PDG lepton(CPV) in parameterization, mixing the matrix these lepton havethat parameters sector been are affecting measured labeled neutrino oscillations. While it is certainly the case 5 Conclusion JHEP05(2021)139 13 θ ] or 81 mixing , (the . As this 3 3 2 80 , × U U 3 33 , or 2 U ). , , which is the same 1 2.2 U 123 U ]. For consistency, however, we , then in a different parameteri- 71 , PDG δ 67 . Nonetheless, one could investigate whether 1 for an example. – 16 – C rotation, should be used; see eq. ( rotation) instead of the more conventional ]. Since these are typically written in the PDG parame- 77 13 23 θ – θ 72 , (the 45 ]. In the various different UV schemes the accessible 1 79 U . ], we here investigate the impact of new physics on the above discussion. PDG 78 δ Finally, as neutrino oscillation measurements improve and the values of the lepton In addition, one could examine the impact the choice of parameterization has on a One popular general new physicstarity scenario violation (UV) that [ directly modifiesmatrix the is not mixing unitary, matrix likely is becauseexpectation it uni- if the neutrinos submatrix get of their adue masses larger to from unitary any matrix. one number of This of the is other seesaw the mechanisms new [ physics scenarios. While not confirmed, there are several A Unitarity violation The discussion in the main textbut has as focused testing new on physics the scenarios standardtrino in three program neutrino flavor [ oscillations oscillation is picture, an important goal of the neu- force Development for Teachers and Scientists, Office(SCGSR) of program. Science Graduate Student The Research SCGSRScience program and is Education administered (ORISE) bytract for the number the DE-SC0014664. Oak DOE. Ridge All opinions Institutenot ORISE expressed for necessarily is in reflect managed this the paper by policies are ORAU and the under views authors’ con- of and DOE, do ORAU, or ORISE. Acknowledgments We thank Stephen ParkeDepartment for of useful Energy conversations. under GrantRP PBD did Contract acknowledges was desc0012704. the supported The by United the work States U.S. presented Department here of Energy, that Office of Science, Office of Work- mixing matrix fall intopresent, sharp the Jarlskog relief, is used, we whichinstead hope depends on of that all just four when parameters quantifying in any the parameterization, amount of CPV terization, the statements in thisThat article is, apply if in a a givenzation, straightforward model that fashion predicts range to a would each certain beor model. range not given for the by sum figure rulescertain connected parameterizations; with see various appendix models exhibited any useful simplifications in rotation) when discussing the matterrecommend effect, that see in e.g. most [ associating applications, the the phase conventional with PDG the parameterization, including given symmetry structure suchwith tribimaximal, their trimaximal, modifications [ golden ratio, and others along as the PDG parameterizationis (without the the location Majorana of phases). thedoes oscillation The not phase only lead on remaining to one any choice important of redefinition the than of three the the rotations: order phase other ofputting than rotations. the possibly phase Nonetheless, a there sign, on are its location some is theoretical less benefits to The only parameterization that satisfies all of these conditions is JHEP05(2021)139 0 ], is δ α 91 ]. In , ], the (B.1) (A.1) (A.2) and 89 matrix, 84 – 90 – , α 4 87 82 PDG ], so we can δ . 92 ◦ 53 . = 68 parameterization where all PDG η . We found that the presence , δ 1 ◦ , 323 , or the matrix is upper triangular, or there is . 1 matrix wherein the non-unitary mixing α ττ 0 ∼ U, α = 2 α ) values of the parameters in the α ] 23 τµ µµ 0 0 4 parameterization. In this parameterization, − α α any (1 , θ ee – 17 – τe 123 µe ◦ V α α α ≡ ) wherein the ], and LSND and MiniBooNE anomalies [ N 2068 ≡ . and is often parameterized in the same way as the usual 86 unitary and thus is the product of a unitary matrix and A.2 α , 9 = 0 V 85 )–( 13 A.1 , θ ◦ ] matrix 09 3 . , 2 = 13 for more on the CKM matrix, in general. 12 θ B , is presumably mostly is a unitary matrix parameterized as one chooses with four parameters, and term such that the diagonal elements are N matrix, ) U 3 − Given a UV scheme, one can then evaluate how the parameters (in the unitary part or A common parameterization for UV is via the × Note that this parameterization is not perturbative; it completely covers the parameter space of uni- (1 9 3 the four parameters are well measured to be [ tary violation. B Quark mixing For completeness, we briefly commenterned on by the the same CKM [ issuePDG for parameterization quarks. for Quark mixing leptons, is the gov- large or small, realin or the complex, various uniform different parameterizations orthat is structured, unaffected in the by mapping the the between presencematrix presence of can of UV. be This completely UV distinguished means measurements the from this the complex may rest phase or of may the associated not matrix, with although be with the the realistic case. unitary part of the in the UV part) changeso, as we one performed changes the theof same parameterization UV numerical of is exercise the completely unitary asthe decoupled matrix. in matrix. from figure To In do the particular, issue we of found parameterizations that of for the unitary part of there are variations on eqs. ( no elements are populated andrelated the to matrix each is other Hermitian.fix without The ourselves any UV to effect parameters this on are parameterization the all without unitary trivially loss part of of generality. the matrix [ with real elements ondiagonal the elements diagonal introduce and three complex additional elements complex off phases. the Note diagonal. that Thus in the the off- literature another matrix describing the violation from unitarity, where a reactor anti-neutrino anomaly [ addition, there are somesee anomalies appendix related to UV in the quarkmatrix, sector as well [ hints of unitary violation in the neutrino sector known as the Galium anomaly [ JHEP05(2021)139 4 = − = 0 J 10 (C.2) (C.3) (C.1) is the J × 1 and thus . leading to CKM ◦ V ◦ = 3 9 . which leads to , max where 024 depending on one’s . /J 2 = 178 1 0 2 ◦ √ 1 0 0 √ BM − δ − 34 U CKM = J 2 , subject to experimental 2 1 1 2 1 ]. If one parameterizes the √ J . J † CKM 2 1 2 1 2 73 5 V 1 , √ , − − ' and φ 72 c and is relatively close to 90 C = 78 2 1 c . . as an indication of CP violation, one we find that is related to the complex phase in the that vary by C √ φ PMNS s δ 0 s BM which presumably implies that φ PDG U δ = 0 C δ 212 ) determines − 1 c V φ . C − c 2 C 2 ,U s C.3 c 4 2 < + 1 √ 0 0 – 18 – φ 4 c 2 C 4 ] wherein iφ s − − C e = = 51 c , C in light of the fact that the other three parameters in 2 J | s is a parameterization dependent quantity, we point out 50 2 e iφ δ is the bimaximal matrix [ U e | and then eq. ( PDG , we find in this model in various different parameterizations. Using C 0 0 1 δ C c 0 s φ BM δ 225 − . U . Then we find that ' = 0 60 . in all parameterizations. We instead consider the more interesting C could be as large as 0.35, more than one thousand times larger, subject s . This leads to predictions for = 0 CKM , π is close to one. Yet it is well known that the total amount of CP violation | V 69 which is nearly zero. While these two choices of parameterizations describe 2 max . ) determines e 10 and 0 93 = 0 U . /J − | 0 C.2 55 δ is approximately the Cabibbo angle and . 0197 = . = 0 C , we can determine PMNS 0 θ J δ = 0 = 0 0 While this example is specific to one particular model, the general conclusions remain One then finds that To further emphasize that PDG PDG | In order to actually have CP violation in the quark matrix the other two quark mixing angles must be δ δ δ 2 sin e 10 0 r U value (or range ofthe values) mixing of matrix arevalue relatively and well the range measured would now. change as In shown a in differentnon-zero, figures but parameterization their the impact on the leptonic parameters is small. and thus case where sin choice of parameterization, as shown in table the same. Flavor models often provide predictions that can be interpreted as a specific Thus eq. ( uncertainties and higher orderJ effects such as| renormalization running. Then using quark matrix. where We also investigate the impactexample, of one’s we choice consider of the parameterizationquark model on mixing flavor [ models. matrix and Asquark an and bimaximal matrices as that if thesin quark matrix isthe parameterized same as underlying physics,will if come one to only very considers different conclusions. C Flavor models sin in the quark sectorwhile is quite smallto as upcoming the measurements. Jarlskog coefficient is It is interesting that, in this parameterization, JHEP05(2021)139 , ] ]. (2018) ]. 98 SPIRE IN SPIRE IN (1963) 531 [ arXiv:1105.5936 ]. [ 10 Phys. Rev. D (1957) 429 [ , SPIRE 6 IN ][ (1985) 1039 (2011) 036 New CP-violation in neutrino 55 ]. Non-Standard Interactions at a 08 ) ◦ ]. ( Phys. Rev. Lett. 0 224 223 229 201 195 201 228 228 200 200 222 222 , δ SPIRE JHEP IN ]. SPIRE , [ IN Sov. Phys. JETP [ , hep-ph/0105159 [ SPIRE ]. Phys. Rev. Lett. 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