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on a lattice left - handed weyl fermions @xmath0 are always accompanied by their right - handed counterparts @xmath1 , unless certain mild conditions for the action are broken @xcite . such a doubling is present both in the wilson @xcite and in the domain wall formulations @xcite ( for a review , see @xcite ) , being the main obstacle to the non - perturbative definition of chiral gauge theories . so far all attempts to make the chiral counterpart @xmath1 sterile , in a way that does not use a hard breaking of gauge symmetry and leaves an interacting chiral theory , have failed ( see , for example , reviews @xcite and references therein ) .    another conceivable way to define a theory with only fermion fields @xmath0 is to decouple @xmath1 by giving them majorana masses of the order of the inverse lattice spacing . it can be done directly if the fermions belongs to real representation of the gauge group . if they belong to complex representation , a generalized higgs mechanism is to be employed in order the majorana mass not to break the gauge invariance . in this case model must have strong coupling paramagnetic ( pms ) phase , where fermions acquire masses of the order of the inverse lattice spacing , while the gauge symmetry is not spontaneously broken , i.e. no chiral condensate or no vacuum expectation value of the higgs fields arise ( for a review and further references , see @xcite ) .    in the wilson formulation , the @xmath0 and @xmath1 are coupled through the wilson term . therefore the introduction of a majorana mass for @xmath1 generates a majorana coupling for @xmath0 . so in the gauge theory fine tuning becomes necessary not only for dirac mass but also for majorana mass of @xmath0 . furthermore , in the case of complex representations , there arises a serious problem with the properties of the model within the pms phase , where its spectrum either becomes vectorlike , or consists only of neutral , i.e. singlet under the gauge group , chiral fermions , whose gauge interactions very likely vanish in the continuum limit @xcite . in the domain wall formulation the chiral fermions @xmath0 and @xmath1 appear as collective states of coupled five - dimension fermions . these states are localized at two surfaces formed by mass defects in the five - dimensional system @xcite or by free boundaries of the five - dimensional space @xcite . these surfaces are separated in the fifth dimension and the overlap between these states is suppressed exponentially with this distance . this gives rise to a hope that the above problems can be avoided in such a formulation . this , in fact , is underlying idea of the recent proposal @xcite for lattice formulation of the standard model . in order to generate the majorana mass for the collective state @xmath1 , it has been suggested to introduce on the surface at which @xmath1 is localized certain gauge invariant four - fermion interactions motivated by the approach @xcite . a similar idea is followed by the proposal @xcite for lattice formulation of @xmath2 supersymmetric theories . in this case the majorana mass is introduced on the same surface directly , for the fermions belong to real representation . thus , yet more questions , which should be answered first , arise in such an approach : ( @xmath3 ) whether the generation of the majorana mass on one of the surfaces leads to the chiral spectrum of the model ? this question is common to both proposals , @xcite and @xcite , and requires special investigation , since the chiral states in the domain wall formulation do not present in the action explicitly . ( @xmath4 ) whether the pms phase exists in the systems employing the higgs mechanism ? although some indirect arguments in favour of the presence of the pms phase has already been given in @xcite , it seems interesting to look at the problem from a more general point of view . ( @xmath5 ) does the model has chiral spectrum in the pms phase , and if so , what fermions , charged or neutral , form it ? this is a crucial question to all formulations of chiral gauge theories that employ the pms phase . the aim of this paper is to answer these questions . we consider the variant of the domain wall formulation with free lattice boundaries @xcite and the majorana mass or higgs - majorana coupling introduced on one of the boundaries . these models are introduced in section 2 . in section 3 we analyze the fermion propagators in the model with the majorana mass and gauge fields switched off , and show that the low - lying spectrum of such a model is chiral . in section 4 using a mean - field technique we demonstrate the existence of the pms phase in the systems with the higgs - majorana coupling . in section 5 we consider such systems within the pms phase and argue that they may have chiral spectrum only for the fermions that are singlets under the gauge group . section 6 contains a summary and a discussion of possible applications of such models . our conventions are the following . we consider euclidean hypercubic @xmath6-dimensional lattice with spacing @xmath7 , which is set to one unless otherwise indicated , and volume @xmath8 with @xmath9 even . the lattice sites numbered by @xmath6-dimensional vectors @xmath10 , where @xmath11 , and @xmath12 ; @xmath13 are unit vectors along positive directions in four - dimensional space . fermion ( boson ) variables obey antiperiodic ( periodic ) boundary conditions for the first four directions and free boundary conditions in the fifth direction . we consider the variant of the lattice domain wall fermions proposed in @xcite . the action of such a model can be written in the form : @xmath14 = \sum_{m , n , s , t}[\overline{\psi}_{m , s } \delta_{s t }   d_{m n } \psi_{n , t }   + \overline{\chi}_{m , s } \delta_{s t } \overline{d}_{m n } \chi_{n , t } \cr & & \quad + \overline{\psi}_{m , s } ( \delta_{s-1 \ , t}\delta_{m n } -   \delta_{s t } w_{m n } ) \chi_{n , t } + \overline{\chi}_{m , s } ( \delta_{s+1 \ , t}\delta_{m n } -   \delta_{s t } w_{m n } ) \psi_{n , t } ] ,                \label{eq : dmf}\end{aligned}\ ] ] where @xmath15 and @xmath16 are two - component weyl fermions in the five - dimensional space , transforming under the four - dimensional rotations as left- and right - handed spinors , respectively , @xmath17 @xmath18 are the pauli matrices , @xmath19 are four - dimensional gauge variables , and @xmath20 , but without loss of generality we can restrict it as indicated above . ] is intrinsic mass parameter of the formulation ( ` domain wall ' mass ) . in ( [ eq : dmf ] ) it is implied that @xmath21 . both @xmath0 and @xmath1 belong to the same representation @xmath22 of the gauge group , so the action ( [ eq : dmf ] ) is gauge invariant . without loss of generality we consider unitary groups . the chiral fermions in such a formulation arise as surface modes : the left- and right - handed fermions are low momentum states localized at the @xmath23 and @xmath24 boundaries , respectively . so these states are separated in the fifth dimension and overlap between them is suppressed exponentially with @xmath25 .    the idea of refs . @xcite is to introduce on the surface @xmath24 certain gauge invariant terms which might generate majorana masses @xmath26 for right - handed fermions @xmath1 . such terms can always be represented in the form bilinear in the fermions , so we consider the following action @xmath27 = a_0[u ] + a_m[h ] , \cr & & a_m[h ] = \sum_{n } ( \chi_{n , n_s}^{t }    h_{n } \chi_{n , n_s } + \overline{\chi}_{n , n_s } h^{\dagger}_{n }   \overline{\chi}_{n , n_s}^{t } ) . \label{eq : dmf1}\end{aligned}\ ] ]    if representation @xmath22 is ( pseudo)real , one can simply put @xmath28 where @xmath29 is a certain constant symmetric matrix whose form depends on the group and ensures the gauge invariance of the mass term requires more than one fermion generations . ] .    if @xmath22 belongs to complex representation , @xmath30 has the form @xmath31 where @xmath32 is the higgs - majorana coupling , and @xmath33 is the higgs field transforming under the gauge group as @xmath34 . so the action ( [ eq : dmf1 ] ) is gauge invariant . in this case the majorana mass @xmath26 is to be generated without spontaneous breaking of the gauge symmetry . it is achieved in the pms phase , provided the system has such a phase . let us examine first the case of the explicit mass ( [ eq : mr ] ) . complete information about euclidean system is contained in the correlators , or propagators , of the fields involving in the action . however , before we shall analyse them , it seems to be instructive to get a qualitative idea about its spectrum that can be easily obtained in a certain limiting case from the equations of motion .    for our system with the gauge interactions switched off ( @xmath35 ) these equations can be written in the momentum space as @xmath36 , where @xmath37 is @xmath38 matrix , and @xmath39 is the column constructed from the fields @xmath0 , @xmath1 , @xmath40 , and @xmath41 . the solutions to this equations are determined by zeros of det@xmath37 . since the formulation is reflection positive , one can safely continue the equations to the minkowski space , so that @xmath42 , where @xmath43 is three - vector . then these zeros will determine the energy spectrum of the system after its quantization .    to make the analysis as simple as possible , consider four dimensional space continuous and take the limit @xmath44 . then @xmath45 , and the above equation can be reduced to two independent sets of equations : @xmath46 , and @xmath47 , where @xmath48 are diagonal @xmath49 matrices @xmath50 from these expressions we immediately can read off that in the @xmath0 sector there are one massless left - handed fermion ( @xmath51 ) localized on the surface @xmath23 , and @xmath25 massive solutions with the unit masses ( @xmath52 ) , that generally are not localized . in the @xmath1 sector there are no massless fermions . instead , in addition to non - localized solutions with the unit masses , which form with their @xmath0 counterparts @xmath25 massive dirac fermions , we have right - handed fermion with majorana mass @xmath53 localized on the surface @xmath24 . in the limit @xmath54 it turns to the mirror massless mode of the domain wall formulation . thus , we can conclude that such simplified system has desirable chiral spectrum . of course , one can easily find also the fermion propagators in this limiting case . returning to the euclidean space we get @xmath55 , \cr & & \langle \chi_s \overline{\chi}_t\rangle = - d\left[\delta_{s t } \frac{1}{p^2 + 1 } + \delta_{s n_s } \delta_{t n_s}\left(\frac{1}{p^2 + 4 m^2 }   - \frac{1}{p^2 + 1}\right)\right ] , \cr & & \langle \psi_s \overline{\chi}_t\rangle = \delta_{s \ , t+1}\frac{1 } { p^2 + 1 } , \quad s \neq 0 , \ , t \neq n_s , \cr & & \langle \psi_0 \overline{\chi}_t\rangle = \langle \psi_s \overline{\chi}_{n_s}\rangle = 0.\end{aligned}\ ] ]    let us now consider the fermion propagators of our lattice model . the most simple way to do that is to express them in terms of the propagators determined by the original action ( [ eq : dmf ] ) . we shall use for them notation @xmath56 with @xmath57 and @xmath58 . they have been calculated in @xcite and in the four - dimensional momentum space have the following form : @xmath59 where @xmath60 , @xmath61 , and @xmath62 and @xmath63 are fourier transforms of the operators @xmath64 and @xmath65 in ( [ eq : def1 ] ) at @xmath35 : @xmath66 , @xmath67 . in the approximation specified bellow one has @xmath68 with @xmath69 . here @xmath70 , @xmath71 and @xmath72 are functions of @xmath73 , but only @xmath70 has a pole at @xmath74 : @xmath75 where @xmath76 . ( [ eq : g ] ) and ( [ eq : ab ] ) are given in the approximation where all terms @xmath77 are neglected . since the @xmath78 is positive definite , the larger @xmath25 , the better such an approximation is justified . at @xmath79 one has @xmath80 , so this quantity can be considered as the accuracy with which this formulation defines chiral fermions at finite @xmath25 . now the propagators for the system ( [ eq : dmf1 ] ) with the majorana mass can be represented as @xmath81\right ) \overline\psi_b \rangle_0,\ ] ] and similarly for @xmath82 and @xmath83 . thus after some algebra we arrive at the following expressions @xmath84    consider now the low - momentum structure of the propagator @xmath85 . in the approximation specified above we have : @xmath86 , @xmath87 , and @xmath88 . then , from ( [ eq : prop ] ) it follows that the introduction of the majorana mass modifies only function @xmath70 [ cf . ( [ eq : prop0 ] ) , ( [ eq : g ] ) ] : @xmath89 , \cr & & a^m_l = a_l - \frac{4m^2 \overline p^2 ( a_l+b)^3}{1 + 4m^2 \overline p^2 ( a_l+b)^2},\end{aligned}\ ] ] and this modification exactly cancels the pole in @xmath90 : @xmath91 note that this result matches well with our analysis in the beginning of this section , for in the limit considered there @xmath92 . on the other hand , in the same approximation the propagator @xmath93 takes the form @xmath94,\end{aligned}\ ] ] with @xmath95 since @xmath96 , we get @xmath97 therefore , the pole structure of the propagator of the field @xmath0 is not affected by the majorana mass acquired by the field @xmath1 .    finally , in order to see the effect of the fermion number violation in the physical left - handed sector @xmath0 , caused by the majorana mass term for @xmath1 , we consider the propagator @xmath98 . in the above approximation we get @xmath99 thus , for physical left - handed fermions the effect is suppressed exponentially with @xmath25 . we should note that the action of the original domain wall model @xcite can be taken as the action @xmath100 in ( [ eq : dmf ] ) . the analysis of the spectrum and the propagators in this case , though is more complicated , leads essentially to the same conclusions . let us now consider the question of the existence of the pms phase in the systems with the higgs - majorana coupling . whether a system has the pms phase can be examined within mean - field approximation . we consider the case of radially frozen higgs field , @xmath101 , using the technique developed in @xcite . for that , it is sufficient to consider the gauge interactions turned off . we write down the field @xmath102 as @xmath103 , with @xmath104 , and @xmath105 matrices determined by the group and its representation . then the action for the @xmath33 takes the form @xmath106 where @xmath107 is a hopping parameter . not to break the reflection positivity of the model , we shall consider it at @xmath108 . the partition function of the system reads as @xmath109 ) \cr & & \quad = \int \prod_n d \phi_n \exp ( -a_{\phi } + \ln z_f[y \phi ] ) , \label{eq : sys1}\end{aligned}\ ] ] where @xmath110 is the haar measure on the group , and @xmath111 $ ] is the fermion partition function in the external field @xmath33 . according to @xcite , the critical lines separating symmetric ( paramagnetic ) phases , where the vacuum expectation value of the higgs field @xmath112 , from the broken ( ferromagnetic ) phases , where @xmath113 , are determined by the expression @xmath114   \rangle_h \right| _ { h = 0 } ,                                                \label{eq : kkr}\ ] ] where @xmath115^{1/2}$ ] is the mean field , @xmath116 \rangle_h = \frac{\int \prod_n d \phi_n   \ln z_f[y \phi ] \exp \sum_{n , a } h^a \phi^{a}_{n}}{\int \prod_n d \phi_n   \exp \sum_{n , a } h^a \phi^{a}_{n } } ,                          \label{eq : exv}\ ] ] and @xmath117 and @xmath118 are group dependent positive numbers . for instance , for u(1 ) one has @xmath119 .    integrating in ( [ eq : sys1 ] ) over @xmath0 , and then over @xmath1 successively slice by slice in the fifth direction , we find @xmath120 = \mbox{det}^{n_s+1}[d ] \ : \prod_{s = 0}^{n_s-1 }   \mbox{det}[b_s ] \ :   \mbox{det}^{1/2 } [ 4 y^2 + \phi^{\dagger } \sigma_2 b_{n_s}^{t }   \phi \sigma_2 b_{n_s } ] ,                             \label{eq : z1}\ ] ] where det means the determinant in the space of four - dimensional lattice and spinors indices , and the operators @xmath121 are determined by the recursive relations @xmath122 with @xmath123 , and @xmath124 .    since one is interested in the paramagnetic phase at strong coupling @xmath32 , it is sufficient to know expectation value ( [ eq : exv ] ) in the leading order in @xmath125 and @xmath126 . from ( [ eq : exv ] ) , ( [ eq : z1 ] ) , and ( [ eq : rea ] ) , using formulae of ref . @xcite , we get @xmath116 \rangle_h = - \frac{c_3}{32 y^2 } h^2   \mbox{tr } [ \sigma_2 b_{n_s}^{t } \sigma_2 b_{n_s } ] + o(h^0 ) + o(\frac{1}{y^4 } , h^3 ) ,                    \label{eq : exp2}\ ] ] where tr means the trace over the spatial and spinor indices , and @xmath127 is group dependent positive number ; in the case of u(1 ) it is equal to one , as well . finally , from ( [ eq : kkr ] ) , ( [ eq : rea ] ) , and ( [ eq : exp2 ] ) , we get : @xmath128 where @xmath129 and @xmath130 . we find @xmath131 numerically practically independent of @xmath25 . for instance at @xmath132 we have : @xmath133 , @xmath134 , @xmath135 . thus , the system is in the pms phase at @xmath136 , and @xmath137 . the corresponding phase diagram is shown in fig . 1 . this confirms the indirect arguments of ref . @xcite . let us assume now that the system is within such a phase and consider what may happen to it there . since the higgs field fluctuates strongly in this phase , and @xmath138 , standard methods of examination of such systems , working in the broken phases , is inapplicable here . we however still can get some idea of what may happen in this phase making use the method proposed in @xcite . let us represent the higgs field in ( [ eq : dmf1 ] ) as @xmath139 where the group valued field @xmath140 , @xmath141 , transforms under the gauge group as @xmath142 . following the arguments of @xcite , we assume that adequate nonzero parameter in the pms phase is the link expectation value @xmath143    further results will depend on the dynamical variables in terms of which the consideration is performed . therefore we consider all the three possibilities . introduce the gauge group singlet fields @xmath144 then , with taking into account ( [ eq : vev ] ) , the system ( [ eq : dmf1 ] ) in the pms phase at tree level can be described by the action @xmath145 where we omit summation over four - dimensional indices . we see that in this case the massive neutral fermion @xmath146 is decoupled , for the wilson term at @xmath24 vanishes . this made @xmath146 and @xmath147 naive , and although at the first glance this should not affect the chiral properties of the system , since both these fermions still have masses of the order of the cutoff , it turns out that it gives rise to species doubling of a massless mode . it is clearly seen from the structure of the fermion determinant , which in this case takes the form [ cf . ( [ eq : z1 ] ) ] @xmath148 \ : \prod_{s = 0}^{n_s-1 }   \mbox{det}[b_s ] \ :   \mbox{det}^{1/2 } [ 4 \frac{y^2}{z^4 } - d \overline{d } ] ,          \label{eq : zn1}\ ] ] where the functions @xmath121 are determined in ( [ eq : rea ] ) . since @xmath149 , the determinant has zero at the corners of the brillouine zone . so this scenario leads to the failure of the model . * 2 . * together with the neutral fields ( [ eq : chin ] ) one can also introduce the neutral fields @xmath150 in terms of these variables the system in the pms phase takes the form @xmath151 where in the four - dimensional momentum space @xmath152 in this case the slice @xmath24 decouples from the rest of the lattice , so that one has massive neutral fermions at @xmath24 and the original massless model on the rest of the lattice . the neutral fermions @xmath153 , @xmath146 can still be made massless , by setting the domain wall mass to @xmath154 , but this does not rescue the situation with the rest of the lattice . so in this case the model fails , too . * 3 . * finally , in terms of the singlet fields like ( [ eq : chin ] ) and ( [ eq : psin ] ) introduced at all @xmath155 , the action reads as @xmath156 where @xmath157    we see that these system is equivalent to that considered in the previous section , only if @xmath158 . however one can hardly expect that it is so on the ground of the estimations of @xmath159 in pure scalar models @xcite . for instance , in @xmath160 and @xmath161 models it has been found @xmath162 . therefore our previous analysis is not directly applicable to this system . however we can made some qualitative conclusions about the effects of such renormalization . an analysis similar to that of the beginning of the previous section , shows that in the continuum case , when the wilson term is equal to zero , @xmath163 simply renormalizes the masses of the massive excitations : now they become @xmath164 , rather than 1 . the situation on the lattice is more complicated . indeed , making the small @xmath73 expansion in ( [ eq : w3 ] ) one can see that the role of the factor @xmath165 , which provides the localization of the massless states at the boundaries , now is played by @xmath163 dependent combination @xmath166 , that may considerably shift the range of admissible values of @xmath167 . this requires a priori knowledge of the @xmath163 , that complicates considerably the numerical study of such models . but the main problem in this variant is that all the fermions in the system now are neutral . the question of which of these scenarios is realized actually in the system requires special investigation , and the answer may depend on the strength of the wilson parameter @xmath168 , set to 1 in this paper , as well as on the couplings @xmath32 and @xmath107 . however none of them leads to the chiral spectrum of charged , i.e. gauge non - singlet , fermions which one tends to describe . so we can conclude that in this respect the domain wall fermions have no appreciable advantages compared with the four dimensional wilson fermions . the lattice boundary formulation of chiral fermions with majorana mass introduced for the field @xmath1 on the boundary @xmath24 indeed yields the low - lying chiral spectrum at the tree level : there exists only one massless left - handed fermion @xmath0 localized at the boundary @xmath23 . our results also show that effects caused by the introduction of such majorana mass is strongly suppressed in physical sector . for instance , the fermion number violation for the @xmath0 is suppressed exponentially as the size of the fifth dimension @xmath25 increases . since the majorana mass does not break the gauge invariance for real representations of the gauge group , the most immediate implementation of such systems is the lattice formulation of the @xmath2 supersymetric models @xcite . our results speaks in favour of this idea , since the exponential suppression of the undesirable effects in the physical sector is an indication that the problem of fine tuning can be avoided in such a formulation , at least in the perturbation theory . this however should be demonstrated explicitly , and this work now is in progress . the prospect for chiral gauge theories is less certain , for they have to employ the generalized higgs mechanism in the pms phase . although we have demonstrated that the pms phase can exist in these models , their possible dynamics within such a phase does not give a ground for optimism . indeed , we argued that the spectrum of charge fermions within the pms phase is vectorlike , and that only spectrum of the neutral states , by a certain tuning of the domain wall mass , can be made chiral . thus the crucial question is what is the gauge interactions of such neutral chiral states . previous studies of similar states appearing in the models with the wilson - yukawa couplings give a strong evidence that such states in the continuum limit become non - interacting @xcite . it is this point that leads to very plausible failure of those models , as well as the models with multifermion couplings @xcite . such screening of the chiral charges appears to be an universal phenomenon pursuing any models within the pms phase . therefore we consider this point as the main problem on the way of implementation of the domain wall fermions to exactly gauge invariant nonperturbative formulation of chiral gauge theories . +    * acknowledgements * + s.v.z . is grateful to the staff of the center for computational physics , where this work has been done , and particularly to y. iwasaki and a. ukawa , for their kind hospitality . s.a . was supported in part by the grants - in - aid of the ministry of education ( no.08640350 ) ; s.v.z . was supported by `` coe foreign researcher program '' of the ministry of education of japan and by the russian basic research fund under the grant 95 - 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we examine the lattice boundary formulation of chiral fermions with either an explicit majorana mass or a higgs - majorana coupling introduced on one of the boundaries . 
 we demonstrate that the low - lying spectrum of the models with an explicit majorana mass of the order of an inverse lattice spacing is chiral at tree level . within a mean - field approximation 
 we show that the systems with a strong higgs - majorana coupling have a symmetric phase , in which a majorana mass of the order of an inverse lattice spacing is generated without spontaneous breaking of the gauge symmetry . 
 we argue , however , that the models within such a phase have a chiral spectrum only in terms of the fermions that are singlets under the gauge group . 
 the application of such systems to nonperturbative formulations of supersymmetric and chiral gauge theories is briefly discussed .    -0.6 
 cm    uthep-360 + ut - ccp - p21 + april 1997 +    1.0truecm    s. aoki ,    institute of physics , university of tsukuba    tsukuba , ibaraki 305 , japan    k. nagai , and s. v. zenkin    center for computational physics , university of tsukuba    tsukuba , ibaraki 305 , japan    1.0truecm    1