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the origin and nature of quark and lepton masses and mixings remains one of the most intriguing questions left unanswered by the standard model ( sm ) of particle physics . within the sm , quark and lepton masses and mixings arise from yukawa couplings which are essentially free and undetermined . in extensions such as grand unified theories ( guts ) , the yukawa couplings within a particular family may be related , but the mass hierarchy between different families is not explained and supersymmetry ( susy ) does not shed any light on this question either . indeed , in the sm or guts , with or without susy , a specific structure of the yukawa matrices has no intrinsic meaning due to basis transformations in flavour space . for example , one can always work in a basis in which , say , the up quark mass matrix is taken to be diagonal with the quark sector mixing arising entirely from the down quark mass matrix , or _ vice versa _ , and analogously in the lepton sector ( see e.g.  @xcite ) . this is symptomatic of the fact that neither the sm or guts are candidates for a theory of flavour . the situation changes somewhat once these theories are extended to include a family symmetry spontaneously broken by extra higgs fields called flavons . this approach has recently received a massive impetus due to the discovery of neutrino mass and approximately tri - bimaximal lepton mixing @xcite whose simple pattern strongly suggests some kind of a non - abelian discrete family symmetry might be at work , at least in the lepton sector , and , assuming a gut - type of structure relating quarks and leptons at a certain high energy scale , within the quark sector too . the observed neutrino flavour symmetry may arise either directly or indirectly from a range of discrete symmetry groups @xcite . examples of the direct approach , in which one or more generators of the discrete family symmetry appears in the neutrino flavour group , are typically based on @xmath3 @xcite or a related group such as @xmath4 @xcite or @xmath5 @xcite . models of the indirect kind , in which the neutrino flavour symmetry arises accidentally , include also @xmath6 @xcite as well as @xmath7 @xcite and the continuous flavour symmetries like , e.g. , @xmath8 @xcite or @xmath9 @xcite which accommodate the discrete groups above as subgroups @xcite . theories of flavour based on a spontaneously broken family symmetry are constructed in a particular basis in which the vacuum alignment of the flavons is particularly simple . this then defines a preferred basis for that particular model , which we shall refer to as the `` flavour basis . '' in such frameworks , the resulting low energy effective yukawa matrices are expected to have a correspondingly simple form in the flavour basis associated with the high energy simple flavon vacuum alignment . this suggests that it may be useful to look for simple yukawa matrix structures in a particular basis , since such patterns may provide a bottom - up route towards a theory of flavour based on a spontaneously broken family symmetry . unfortunately , experiment does not tell us directly the structure of the yukawa matrices , and the complexity of the problem , in particular , the basis ambiguity from the bottom - up perspective , generally hinders the prospects of deducing even the basic features of the underlying flavour theory from the experimental data . we are left with little alternative but to follow an _ ad hoc _ approach pioneered some time ago by fritzsch @xcite and currently represented by the myriads of proposed effective yukawa textures ( see e.g.  @xcite ) whose starting assumption is that ( in some basis ) the yukawa matrices exhibit certain nice features such as symmetries or zeros in specific elements which have become known as `` texture zeros . '' for example , in his classic paper , fritzsch pioneered the idea of having six texture zeros in the 1 - 1 , 2 - 2 , 1 - 3 entries of the hermitian up and down quark yukawa ( or mass ) matrices @xcite . unfortunately , these six - zero textures are no longer consistent with experiment , since they imply the bad prediction @xmath10 , so texture zerologists have been forced to retreat to the ( at most ) four - zero schemes discussed , for example , in @xcite which give up on the 2 - 2 texture zeros allowing the good prediction @xmath11 . however , four - zero textures featuring zeros in the 1 - 1 and 1 - 3 entries of both up and down hermitian mass matrices may also lead to the bad prediction @xmath12 unless @xmath13 results from the cancellation of quite sizeable up- and down - type quark 2 - 3 mixing angles , leading to non - negligible induced 1 - 3 up- and down - type quark mixing @xcite . another possibility is to give up on the 1 - 3 texture zeros , as well as the 2 - 2 texture zeros , retaining only two texture zeros in the 1 - 1 entries of the up and down quark matrices @xcite . here we reject both of these options , and instead choose to maintain up to four texture zeros , without invoking cancellations , for example by making the 1 - 1 element of the up ( but not down ) quark mass matrix nonzero , while retaining 1 - 3 texture zeros in both the up and down quark hermitian matrices , as suggested in @xcite . in this paper we discuss phenomenologically viable textures for hierarchical quark mass matrices which have both 1 - 3 texture zeros and negligible 1 - 3 mixing in both the up and down quark mass matrices . such textures clearly differ from the textures discussed in @xcite and @xcite , but include some cases discussed in @xcite , as remarked above . our main contribution in this paper is to derive quark mixing sum rules applicable to textures of this type , in which @xmath0 is generated from @xmath1 as a result of 1 - 2 up - type mixing , in direct analogy to the lepton sum rules derived in @xcite . another important result of our study is to use the sum rules to show how the right - angled unitarity triangle , i.e. , @xmath2 , can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element of either the up or down quark mass matrix being purely imaginary . fritzsch and xing have previously emphasized how their four - zero scheme with 1 - 1 and 1 - 3 texture zeros in the hermitian up and down mass matrices can be used to accommodate right unitarity triangles @xcite , but since their scheme involves large 2 - 3 and non - negligible 1 - 3 up and down quark mixing , our sum rules are not applicable to their case . therefore , the textures in refs . @xcite and @xcite do not allow us to explain @xmath2 by simple structures with a combination of purely real and purely imaginary matrix elements . recently , it has become increasingly clear that current data is indeed consistent with the hypothesis of a right unitarity triangle , with the best fits giving @xmath14 @xcite , and this provides additional impetus for our scheme . the phenomenological observation that @xmath15 has also motivated other approaches ( see e.g. @xcite ) which are complementary to the approach developed in this paper . the layout of the rest of the paper is as follows . in section 2 , we derive the quark mixing sum rules , assuming zero up and down quark 1 - 3 mixing angles . in section 3 , using the sum rules , we discuss the phenomenological viability of quark mass matrix textures with 1 - 3 texture zeros , show how modifications in the up sector can achieve viability , and show how @xmath2 allows each matrix element to be either real or purely imaginary . in section 4 , in the framework of guts , we discuss the implications of zero 1 - 3 mixing for the charged lepton and neutrino sectors , and show that the quark mixing sum rules may be used to yield an accurate prediction for the reactor angle . finally , section 5 concludes the paper . appendix a shows that textures with nonzero 1 - 3 elements in the up sector are disfavoured . the mixing matrix in the quark sector , the cabibbo - kobayashi - maskawa ( ckm ) matrix @xmath16 , is defined as the unitary matrix occurring in the charged current part of the sm interaction lagrangian expressed in terms of the quark mass eigenstates . these mass eigenstates can be determined from the mass matrices in the yukawa sector , namely @xmath17 where @xmath18 and @xmath19 are the mass matrices of the up - type and down - type quarks , respectively . the change from the flavour into the mass basis is achieved via bi - unitary transformations @xmath20 where @xmath21 , @xmath22 , @xmath23 and @xmath24 are unitary @xmath25 matrices . the ckm matrix @xmath26 ( in the `` raw '' form , i.e.  before the `` unphysical '' phases were absorbed into redefinitions of the quark mass eigenstate field operators ) is then given by @xmath27    in this paper we shall use the standard ( or so - called particle data group ( pdg ) @xcite ) parameterisation for the ckm matrix ( after eliminating the `` unphysical '' phases ) with the structure [ updg ] u_ckm = r_23u_13r_12 , where @xmath28 denote real ( i.e.  orthogonal ) matrices , and the unitary matrix @xmath29 contains the observable phase @xmath30 . for more details , see e.g.  @xcite . other alternative parametrisations , motivated by the observation of @xmath2 ( see e.g.  @xcite ) have been suggested , but we prefer to stick to the standard one here . let us now suppose that @xmath55 . from the sm point of view , this corresponds to just a convenient choice of basis but , as discussed in the introduction , it becomes a nontrivial assumption at the level of a specific underlying model of flavour . for models where zero 1 - 3 mixing is realised with flavour symmetries , see e.g.  @xcite . for @xmath55 , eq .  ( [ eq : param3 ] ) simplifies to [ eq : param5 ] u_ckm = u^u_l_12^^u^d_l_23 u^d_l_12 . then , by equating the right - hand sides of eqs . ( [ eq : param4 ] ) and ( [ eq : param5 ] ) and expanding to leading order in the small mixing angles , we obtain the following relations ( up to cubic terms in the physical quark mixing angles ) : [ f1 ] _ 23e^-i_23&= & _ 23^de^-i_23^d -_23^ue^-i_23^u , + [ f2 ] _ 13e^-i_13&= & -_12^ue^-i_12^u ( _ 23^de^-i_23^d - _ 23^ue^-i_23^u ) , + [ f3 ] _ 12e^-i_12&= & _ 12^de^-i_12^d -_12^ue^-i_12^u . let us first consider eq . ( [ f2 ] ) , which can be transformed into @xmath56 where @xmath57 and @xmath58 stand for the measurable 1 - 3 and 2 - 3 mixing angles in the quark sector , respectively . taking the modulus of eq . ( [ eq : theta13withphases ] ) , the 1 - 2 angle entering the up - sector rotation ( @xmath31 ) in the flavour basis obeys @xmath59 where the 1@xmath60 errors are displayed @xcite . similarly , combining eq . ( [ f3 ] ) with eq . ( [ eq : theta13withphases ] ) one receives @xmath61 this , together with the identification eq . ( [ eq : deltafromparam1 ] ) gives rise to the quark sector sum rule , the sum rule may be further simplified to @xmath62 . for similar considerations in the lepton sector , see e.g.  @xcite . ] @xmath63 which is valid up to higher order corrections . the present best - fit value and the 1@xmath60 errors are also displayed .    needless to say , the relations ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) apply at the scale at which the flavour structure emerges , often close to the scale of grand unification . thus , in principle , the renormalisation group ( rg ) effects should be taken into account . however , due to the smallness of the mixing angles in the quark sector and the hierarchy of the quark masses , the rg corrections to the above relations are very small and can be neglected to a very good approximation . it is interesting that , with the 1 - 2 mixing angles in the up and down sector derived from the physical parameters , the 1 - 2 phase difference in the up and down sectors can also be determined . indeed , combining all three equations ( [ f1])-([f3 ] ) , one obtains @xmath64 using eqs .  ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) we can solve eq . ( [ eq : deltadu ] ) for @xmath65 and obtain ( at 1@xmath60 level ) @xmath66 which is remarkably close to @xmath67 . we emphasise that this is a consequence of zero 1 - 3 mixing in the up and down sectors , @xmath68 . we now show that , assuming quark textures with negligible 1 - 3 up and down quark mixing , corresponding to 1 - 3 texture zeros for hierarchical quark mass matrices , @xmath65 is approximately equal to @xmath69 . this comes from the definition of the unitarity triangle angle @xmath69 : @xmath70 for the second term in the argument , we can use eqs . ( [ f1 ] ) , ( [ f2 ] ) and ( [ f3 ] ) @xmath71 thus , one can see that the angle @xmath69 is nothing but the phase difference @xmath72 , corresponding to a very simple phase sum rule [ phase ] _ 12^d - _ 12^u . according to the phase sum rule in eq . ( [ phase ] ) , the experimental observation that @xmath2 , or the equivalent determination in eq . ( [ phasediff ] ) , suggests looking at quark mass matrices with 1 - 3 texture zeros and with @xmath74 or @xmath75 at the special values @xmath76 . this would correspond to a set of rather specific textures of the quark mass matrices with , for example , purely imaginary 1 - 2 elements in either @xmath18 or @xmath19 while the 2 - 2 elements remain real . for example , in @xcite the relation between the phases of the mixing angles and the phases of the matrix elements is discussed . for instance , the following patterns naturally emerge : @xmath77 and/or @xmath78 where @xmath79 and @xmath80 are real parameters , and the elements marked by `` * '' are irrelevant as long as the hierarchy of the mass matrix is large enough , or , equivalently , as long as the mixing angles in @xmath81 and @xmath82 are small . these textures are all phenomenologically viable , and consistent with @xmath83 , and their simple phase structure provides a post justification of our assumption of 1 - 3 texture zeros and negligible 1 - 3 up- and down - type quark mixing . however , the above textures are clearly not the most predictive ones and , for example , do not relate the up and down quark 1 - 2 mixing angles to masses . this requires additional assumptions , such as additional texture zeros and hermitian or symmetric matrices , as we now discuss .      under the additional assumptions of symmetric or hermitian mass matrices in the 1 - 2 block and zero textures in the 1 - 1 positions of the quark mass matrices , i.e. , @xmath84 we obtain as additional predictions the gatto - sartori - tonin ( gst ) relations @xcite with 1@xmath60 errors displayed , @xmath85 here we already see a conflict in the up sector . the prediction for @xmath86 from the sum rule in eq . ( [ eq : quarkrelation ] ) is quite different ( several @xmath60 away ) from the gst relation above . that suggests that the texture in the up sector should be modified to be in good agreement with experiment . by contrast the prediction from the sum rule in eq . ( [ eq : quarksumrule ] ) for @xmath87 is in very good agreement ( within the errors ) with the gst result in eq . ( [ eq : m_d_over_m_s ] ) , and therefore it is quite plausible to keep the simple texture ansatz for the down sector .    combining eqs . ( [ eq : m_u_over_m_c ] ) and ( [ eq : m_d_over_m_s ] ) with the sum rules in eqs . ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) , the two relations [ eq : qsplusgst1 ] |_12 - e^- i _ | = and [ eq : qsplusgst2 ] = emerge . we emphasize that these results do not hold for the textures in @xcite where the 2 - 3 up and down quark mixings are large and the 1 - 3 up and down quark mixings are non - negligible .    ) . the blue lines indicate the predicted values of @xmath88 for given @xmath89 under the assumptions of section [ sec : sumruleplusgst ] , and the dashed horizontal and vertical black lines ( and solid black lines ) show the @xmath90 errors ( and best - fit values ) for @xmath88 and @xmath89 , respectively . ] the compatibility of eq . ( [ eq : qsplusgst1 ] ) with the experimental results for the down - type quark masses and mixing parameters @xcite is illustrated in fig .  1 . we note that rg running for the quark masses , as well as their potential susy threshold corrections , are very similar for the first two generations and thus cancel out in their ratio . for our estimates , we have considered the running quark masses at the top mass scale @xmath91 @xcite . @xmath88 is extracted for given @xmath89 . the solid blue line shows the relation for best - fit values of the parameters while the dashed blue lines indicate the range with @xmath90 errors included . the dashed horizontal and vertical black lines ( and solid black lines ) show the @xmath90 errors ( and best - fit values ) for @xmath88 and @xmath89 , respectively . the relation of eq . ( [ eq : qsplusgst1 ] ) is well compatible with the present data . future more precise experimental measurements ( for instance at lhcb or @xmath92 factories ) and , in particular , an improved knowledge on @xmath93 ( e.g.  from lattice qcd ) are required to test it more accurately .    in the following , we consider some examples of possible modifications to the textures in the up sector which are phenomenologically acceptable , while leaving the successful down sector texture unchanged , and retaining the successful real and imaginary scheme which leads to the right unitarity triangle . as discussed in appendix a , the idea of relaxing the up quark 1 - 3 texture zero is disfavoured , so we restrict ourselves to either relaxing the up quark 1 - 1 texture zero , or relaxing symmetry in the 1 - 2 up quark sector , as discussed below . one possible modification is to introduce a nonzero element in the 1 - 1 position of the up quark mass matrix , i.e.@xmath94 as a result , we obtain the up sector relation @xmath95 which allows to adjust @xmath96 , which is of the order of the up quark mass , while @xmath97 has to be equal to the value obtained in eq . ( [ eq : quarkrelation ] ) using the sum rule . for the down sector , there is still the successful prediction from eq . ( [ eq : m_d_over_m_s ] ) leading to the successful sum rule relation of eq . ( [ eq : qsplusgst1 ] ) . furthermore , as discussed in section [ sec : deltackm ] , the dirac phase of the ckm matrix is correct . we note that there exist several variants of the texture . for example , we can choose the 1 - 2 element of @xmath19 real , the 1 - 2 element of @xmath18 purely imaginary and all the other elements also real . these variants are valid as long as @xmath98 is real and @xmath99 is purely imaginary , or _ vice versa_.    we emphasize that the elements marked by `` * '' are irrelevant as long as the hierarchy of the mass matrix is large enough , so they may be replaced by zeros or , if the matrices are hermitian , the 3 - 1 elements may be zero while the 3 - 2 elements are determined by hermiticity . since the sum rule in eq . ( [ f1 ] ) shows that @xmath1 is determined only by the difference in 2 - 3 mixing angles in the up and down sectors , it is also possible to set either @xmath100 or @xmath101 equal to zero without changing the physical predictions . in this way it is possible to arrive at some of the four - zero textures discussed , for example , in @xcite . however , we emphasize that here we are additionally assuming the real and imaginary structures consistent with the right unitarity triangle and this was not discussed in @xcite .      a second option for a texture consistent with experimental data consists in relaxing the symmetry of the 1 - 2 block in the up sector , while keeping the texture zero in the 1 - 1 position : @xmath102 the two up - sector relations @xmath103 can be simultaneously fulfilled by choosing @xmath104 and @xmath105 appropriately . the prediction from eq . ( [ eq : m_d_over_m_s ] ) and the prediction for @xmath30 do not change and remain compatible with data . we note that there exist several variants of the texture . as before , it is sufficient to have @xmath98 real and @xmath99 purely imaginary , or _ vice versa_. extending the notion of zero 1 - 3 mixing to the lepton sector ( i.e.  under the assumption of @xmath106 ) , the presently unknown mixing angle @xmath107 of the leptonic ( mns ) mixing matrix satisfies the relation ( analogous to eq . ( [ eq : quarkrelation ] ) ) ^mns_13 = ^mns_23 ^e_12 , where @xmath108 is the 1 - 2 mixing in the charged lepton mass matrix @xmath109 . this relation has emerged before , for example , in the context of lepton sum rules in @xcite .    in many classes of gut models of flavour , the 1 - 2 mixing angles corresponding to @xmath109 and @xmath19 are related by a group theoretical clebsch factor , for example @xmath110 @xcite . in general , it is usually assumed that @xmath111 is of the order of the cabibbo angle , leading to a prediction @xmath112 @xcite . however , in the context of fritzsch - type textures , which are based on hermitian matrices with 1 - 1 , 2 - 2 and 1 - 3 texture zeros , this prediction can be made more precise by using the sum rule which relates @xmath111 to down - type quark masses . thus , applying eq . ( [ eq : quarksumrule ] ) at low energies and taking the present experimental data for the quark mixing angles , and for @xmath113 ( taken from @xcite ) , one can make the rather precise prediction [ eq : mns13mixing ] ^mns_13 = ( 2.84^+0.22_-0.18 ) ^which gives @xmath114 and holds under the assumption of texture zeros in the 1 - 3 elements of the mass matrices ( or more precisely @xmath115 ) and @xmath110 . of course , eq .  ( [ eq : mns13mixing ] ) is only a single example out of a larger variety of predictions which may arise in unified flavour models ( see e.g.  @xcite ) . we emphasise that the main use of eq . ( [ eq : quarksumrule ] ) in this context is that it allows to `` determine '' the down quark mixing @xmath111 , which is generically involved in relations between quark and lepton mixing angles , from measurable quantities . we note that in the lepton sector , the rg corrections ( see e.g.  @xcite ) can be significant , depending on the absolute neutrino mass scale ( and on @xmath116 in a susy framework ) and other effects such as canonical normalisation on the mixing angles can also be sizeable @xcite . furthermore , relaxing the 1 - 1 texture zero in the up quark sector may switch on a nonzero 1 - 3 mixing angle in the neutrino sector via partially constrained sequential dominance @xcite . in this paper we have discussed phenomenologically viable textures for hierarchical quark mass matrices which have both 1 - 3 texture zeros and negligible 1 - 3 mixing in both the up and down quark mass matrices . such textures differ from the textures discussed in @xcite and @xcite . our main contribution in this paper has been to derive quark mixing sum rules applicable to textures of this type , in which @xmath0 is generated from @xmath1 as a result of 1 - 2 up - type mixing , in direct analogy to the lepton sum rules derived in @xcite . an important result of our study is to show how the right - angled unitarity triangle , i.e. , @xmath2 , can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element of either the up or down quark mass matrix being purely imaginary . the experimental result that @xmath2 therefore provides an impetus for having hierarchical textures compatible with negligible 1 - 3 mixing in both the up and down quark mass matrices . this is probably the most important take - home message of this paper . the quark mixing sum rules in eqs . ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) relate the up and down quark 1 - 2 mixing angles to observable parameters in the ckm matrix . using these sum rules the four - zero texture with 1 - 1 and 1 - 3 texture zeros and a 2 - 1 symmetric or hermitian structure , is shown to be viable for the down quark sector but not for the up quark sector . however , it is possible to have four - zero textures compatible with our sum rules by , for example , filling in the up quark 1 - 1 texture zero , then having hermitian matrices with either of the 2 - 3 elements in the up or down sector set equal to zero as in @xcite . however , we emphasize that here we are additionally assuming the real and imaginary structures consistent with the right unitarity triangle and this was not discussed in @xcite .    in the framework of guts , it is natural to have 1 - 3 texture zeros for both the quark and charged lepton sectors , and in such a case we have shown that the quark mixing sum rules may be used to yield an accurate prediction for the reactor angle ; see eq . ( [ eq : mns13mixing ] ) . however , we caution that this prediction is subject to considerable theoretical uncertainty due to the model dependence of the quark - lepton mixing angle relations , rg and canonical normalisation effects , as well as the assumption that the underlying neutrino 1 - 3 mixing angle is zero . indeed , relaxing the 1 - 1 texture zero in the up quark sector will switch on a nonzero 1 - 3 mixing angle in the neutrino sector via partially constrained sequential dominance @xcite . finally , we emphasise that the strategy of exploring particular textures for yukawa matrices , though necessarily rather _ ad hoc _ , is meaningful from the perspective of theories of flavour based on spontaneously broken family symmetry , where , in the flavour basis defined by the high energy theory , simple yukawa matrix structures are expected . indeed , the study of simple yukawa textures may provide the only bottom - up way of deducing a high energy theory of flavour from experimental data . we have shown that @xmath2 may provide a clue towards such a high energy theory of flavour via rather simple yukawa matrices involving 1 - 3 texture zeros whose nonzero elements are either real or purely imaginary . such patterns could be achieved , in principle , by appropriate alignment of the vacuum expectation values of flavour symmetry breaking flavon fields , and in particular their phases . it would be interesting to build a theory of flavour along these lines . we would like to thank jonathan flynn for discussions and christoph luhn for carefully reading the manuscript and providing helpful comments . we are indebted to nordita for the hospitality and support during the programme `` astroparticle physics  a pathfinder to new physics '' held in stockholm in march 30 - april 30 , 2009 during which part of this study was performed . sfk is very grateful for the support and hospitality provided by the max - planck - institute of physics during his stay in munich while this paper was being finalized . sa and ms acknowledge partial support by the dfg cluster of excellence `` origin and structure of the universe . '' the work of mm is supported by the royal institute of technology ( kth ) , contract no . sii-56510 . sfk acknowledges partial support from the stfc rolling grant no . st / g000557/1 and a royal society leverhulme trust senior research fellowship . with nonzero 1 - 3 elements , @xmath30 depends not only on @xmath117 but also on other parameters ( in particular @xmath118 and @xmath119 ) and the simple quark mixing sum rules in eqs . ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) are no longer valid . examples of this type of texture include ( with real parameter @xmath120 ) @xmath121 and @xmath122 but also variations with different elements chosen either purely imaginary or real . we will demonstrate our approach for this case by means of the texture in eq . ( [ eq : futexture1 ] ) . the starting point is here ( similar to eqs . ( [ f1])-([f3 ] ) ) [ g1 ] _ 23e^-i_23&= & _ 23^de^-i_23^d -_23^ue^-i_23^u , + [ g2 ] _ 13e^-i_13&= & -_13^ue^-i_13^u -_12^u _ 23e^-i(_12^u + _ 23 ) , + [ g3 ] _ 12e^-i_12&= & _ 12^de^-i_12^d -_12^ue^-i_12^u , where we have also neglected terms of order @xmath123 . from our texture ansatz , we know the phases @xmath124 , @xmath125 and @xmath126 . for the values of @xmath127 , we take the values from the gst relations , i.e.  eq .  ( [ eq : m_u_over_m_c ] ) , which hold here because of the zeros in the 1 - 1 position and the symmetric structure for the first two generations . then we can calculate @xmath128 and @xmath30 in terms of the known quantities and obtain @xmath129 . this result is several standard deviations away from the measurements . beyond the particular example discussed above , we found that the inconsistency of the prediction for @xmath30 also appears in all other cases with @xmath130 , @xmath124 and @xmath131 @xmath132 . furthermore , the same happens for textures with @xmath133 and @xmath134 , where @xmath135 denotes the 1 - 3 element of @xmath19 . we conclude that under these conditions textures with nonzero 1 - 3 elements are disfavoured .        s.  f.  king and c.  luhn , arxiv:0908.1897 [ hep - ph ] . c.  s.  lam , arxiv:0907.2206 [ hep - ph ] . e.  ma and g.  rajasekaran , phys . d * 64 * ( 2001 ) 113012 [ arxiv : hep - ph/0106291 ] . g.  altarelli , arxiv : hep - ph/0611117 ; g.  altarelli , f.  feruglio and y.  lin , nucl . b * 775 * ( 2007 ) 31 [ arxiv : hep - ph/0610165 ] ; 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in analogy with the recently proposed lepton mixing sum rules , we derive quark mixing sum rules for the case of hierarchical quark mass matrices with 1 - 3 texture zeros , in which the separate up and down type 1 - 3 mixing angles are approximately zero , and @xmath0 is generated from @xmath1 as a result of 1 - 2 up type quark mixing . using the sum rules 
 , we discuss the phenomenological viability of such textures , including up to four texture zeros , and show how the right - angled unitarity triangle , i.e. , @xmath2 , can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element being purely imaginary . in the framework of grand unified theories , 
 we show how the quark and lepton mixing sum rules may combine to yield an accurate prediction for the reactor angle .