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after decades of neutrino oscillation experiments , it is generally taken for granted that neutrinos are massive particles that can vary among all the three flavors through the oscillation process described by neutrino mixing .
one of the most important issues concerning neutrino mixing is the determination of the neutrino mixing matrix , i.e. , the pontecorvo - maki - nakagawa - sakata  ( pmns ) matrix  @xcite , which is the lepton sector counterpart of the quark sector mixing matrix , the cabibbo - kobayashi - maskawa  ( ckm ) matrix  @xcite .
the pmns matrix is defined as the correlation matrix linking neutrino flavor eigenstates @xmath0 and mass eigenstates @xmath1 , @xmath2 this mixing matrix is conventionally represented in the standard chau - keung ( ck ) scheme  @xcite as    @xmath3    where three mixing angles are denoted by @xmath4 , @xmath5 , and @xmath6 , with their trigonometric functions @xmath7 , @xmath8 , etc . represented by @xmath9 , @xmath10 , etc . respectively .
the _ cp_-violating phase is denoted by @xmath11 , meanwhile @xmath12 and @xmath13 represent the other two phases in the case of majorana neutrinos . in the case of dirac neutrinos ,
the latter two phases @xmath12 and @xmath13 can be removed by redefinition , thus there remain only four independent parameters , i.e. , three mixing angles together with one _ cp_-violating phase .
if the neutrinos are of majorana type , the two phases @xmath12 and @xmath13 are needed for a full determination of the mixing matrix .
as the majorana phases do not manifest themselves in the oscillation , we ignore these two phases @xmath12 and @xmath13 and take only the first term on the right - hand side of eq .
( [ cke ] ) in this article . by now , the quark - sector mixing matrix has been measured with good precision . in the lepton sector , the values of the three mixing angles have been measured after years of neutrino oscillation experiments , though with relatively lower precision compared to the quark case .
the explicit form of the fermion mixing matrix is not unique and an alternative scheme is the kobayashi - maskawa  ( km ) scheme  @xcite @xmath14 in the km scheme , as will be mentioned later , the _ cp_-violating phase of the quark sector is quite near @xmath15 , leading to the hypothesis of `` maximal _ cp _ violation ''  @xcite .
besides the ck and km schemes , in ref .
@xcite all the other possible schemes of the mixing matrix are considered and presented .
there are actually 12 schemes of mixing matrix . among them
, 3 schemes can be transformed into others through straightforward redefinition of mixing angles , thus leaving 9 different schemes  @xcite , whose forms are provided in sec .
[ sec2 ] .
quark - lepton complementarity  ( qlc )  @xcite and self - complementarity  ( sc )  @xcite are phenomenological relations of quark and lepton mixing angles .
they provide a novel connection to link quarks and leptons .
they can be expressed in a more clear way as ( [ qlcsc1])-([qlcsc4 ] ) : @xmath16 ( from now on we use @xmath17 to represent lepton sector mixing angles to distinguish them from quark sector mixing angles @xmath18 . )    here we have marked the two qlc relations by qlc1 and qlc2 , and the two slightly different sc relations by sc1 and sc2 respectively .
originally these phenomenological relations are observed only in the ck scheme and fit the experimental results within small errors .
however , a question naturally arises , i.e. , whether these relations still hold in schemes other than the ck scheme since we can not find any justification for the priority of the ck scheme .
there are already some researches on qlc and sc in the nine schemes  @xcite .
however , all of these examinations of qlc and sc are carried out under some fixed phase choices . since the _ cp_-violating phase of the lepton sector
is not determined from current experiment , in this article we examine qlc and sc with the whole range variation of the lepton _ cp_-violating phase .
these will be treated in sec .
[ sec3 ] and sec .
[ sec4 ] .
we purpose to make a detailed re - analysis of qlc and sc in all the nine schemes , emphasizing on the influences due to the variation of the lepton _ cp_-violating phase . in sec .
[ sec2 ] we use the latest experiment results to do calculations on mixing angles and _ cp_-violating phases in all the nine schemes . in sec .
[ sec3 ] we focus on qlc , examine these complementarity relations and make some analyses . in sec .
[ sec4 ] we similarly examine and analyze sc . in sec .
[ sec5 ] we discuss some properties of _
cp_-violating phases among different schemes with a suggestion of convention redefinition , and suggest some empirical relations among quark _ cp_-violating phases in different schemes .
first we list all the nine schemes mentioned in sec .
[ sec1 ] in table  [ tabscheme ] .    [ cols="^,^",options="header " , ]
some empirical relations we can easily read out are listed here : @xmath19 in fact , ( [ 1 ] ) are satisfied by the similarities between their mixing angles . from table
[ tab1 ] , we have these relations approximately  ( here @xmath20 represents @xmath21 in p@xmath22 scheme ) : @xmath23 then using the scheme - independent jarlskog invariant , we get @xmath24 justifying the relation ( [ 1 ] ) . by the same way , ( [ 3 ] )
is justified through the similarities among mixing angles in p6 , p8 , and p9 .
relation ( [ 2 ] ) states possible maximal _ cp _ violation as we discussed before , and relation ( [ 4 ] ) merely reflects the relative largeness of the three mixing angles in p7 . with the existence of _ cp _
violation confirmed , @xmath25 can not be exactly @xmath26 though close to it  @xcite .
we are willing to find out some similar empirical relations on lepton _ cp_-violating phases .
unfortunately , similar relations can not be easily found , because the lepton mixing angles are quite different from each other in different schemes , thus invalidating our method used for the quark sector .
from the results of sec .  [ sec3 ] and sec .
[ sec4 ] , the validation of qlc and sc depends on the choices of schemes and lepton _ cp_-violating phases , and careful inspections should be carried out when we consider the generalization of qlc and sc from the standard ck scheme to the other eight schemes . on the issues of _ cp_-violating phases , restrictions on lepton _ cp_-violating phases in p6@xmath27p9 are recognized .
simultaneous maximal _
violation in both the quark and the lepton sector is possible in the p2 and p3 scheme .
a redefinition of _ cp_-violating phases for unification is suggested and some empirical relations on the quark _ cp_-violating phases are explored .
all of these results may enrich our knowledge of qlc , sc relations , and _ cp_-violating phases , helping us understand the mystery of lepton mixing .
this work is supported by the principal fund for undergraduate research at peking university .
it is also partially supported by the national natural science foundation of china ( grants no .  11021092 , no .
10975003 , no .
11035003 , and no .
11120101004 ) , by the national fund for fostering talents of basic science ( grants no .
j1030310 ) , and by the research fund for the doctoral program of higher education ( china ) .
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with the progress of increasingly precise measurements on the neutrino mixing angles , phenomenological relations such as quark - lepton complementarity  ( qlc ) among mixing angles of quarks and leptons and self - complementarity  ( sc ) among lepton mixing angles have been observed . using the latest global fit results of the quark and lepton mixing angles in the standard chau - keung scheme , we calculate the mixing angles and _ cp_-violating phases in the other eight different schemes . 
 we check the dependence of these mixing angles on the _ cp_-violating phases in different phase schemes . the dependence of qlc and sc relations on the _ cp _ phase in the other eight schemes is recognized and then analyzed , suggesting that measurements on _ cp_-violating phases of the lepton sector are crucial to the explicit forms of qlc and sc in different schemes .