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dynamical breaking of a symmetry can be described by a quantity known as the order parameter . in the case of a broken chiral symmetry a commonly used order parameter is the chiral condensate which vanishes if the chiral symmetry is unbroken . the investigation of the chiral condensate plays a crucial role in attempts to describe phase transitions related to the dynamical chiral symmetry breaking .    in qcd chiral condensate is obtained by numerical studies , as a rule via lattice simulations . this approach is usually employed in studies of the chiral condensate in the presence of external factors such as temperature , chemical potentials , magnetic field , finite size effects , etc ( see [ 1 - 3 ] and references therein ) .    apart from numerical studies there is a lot of models employed for the investigation of the chiral condensate in various physical systems @xcite , @xcite . in the first paper on this subject @xcite nambu and jona - lasinio ( njl ) analyzed a specific field model in four dimensions . later , the chiral condensate was studied by gross and neveu ( gn ) in two - dimensional spacetime in the limit of a large number of fermion flavors @xmath0 @xcite . these two models are similar but in contrast to the njl model , the gn model is a renormalizable and asymptotically free theory . due to these properties the gn model is used for qualitative modeling of qcd . the relative simplicity of both models is a consequence of the quartic fermion interaction .    in our previous paper @xcite we investigated a system of a self - interacting massive scalar and a massless fermion field with the yukawa interaction in a @xmath1-dimensional spacetime . in the limit of a large mass of the scalar field the model becomes equivalent to the gn model . the chiral condensate was obtained by the functional integration using the method of the saddle point . for technical reasons the contribution of the scalar field fluctuations was not taken into account . this can be done if the coupling constant of the self - interacting scalar field is much less than the yukawa coupling constant . continuing our investigation of this model , we present in this paper a study of how the chiral condensate is affected by the quantum fluctuations of the scalar fields . the consideration of the quantum fluctuations allows one to find a phase transition in this model in a @xmath1-dimensional spacetime . apart from that , we also investigate the chiral condensate in a @xmath2-dimensional spacetime . in the limit of the massless scalar field in @xmath2 dimensions , the chiral condensate is obtained in analytic form . we also find a phase transition in this setting . all results are obtained in the one - loop approximation framework . we consider a model with the lagrangian density given by @xmath3 where @xmath4 is a real scalar field , @xmath5 is a massless fermion field and the index @xmath6 runs from 1 to @xmath0 . the potential of the scalar field @xmath7 includes the mass term and the self - interaction @xmath8 of the scalar field . the potential @xmath9 is the yukawa interaction . @xmath10 where @xmath11 is a coupling constant . the lagrangian ( [ eq1 ] ) is invariant under the discrete transformation @xmath12 the symmetry ( [ eq3 ] ) is broken by the chiral condensate . in this paper we extend the calculation of the chiral condensate in this model performed in ref . @xcite to include the quantum fluctuations of the scalar field . we formally define the chiral condensate using the functional integral @xmath13 where @xmath14 is a normalization constant . ( [ eq4 ] ) can be rewritten as @xmath15 @xmath16 to integrate over the scalar field one decomposes @xmath17 , where @xmath18 satisfies the classical equation of motion while @xmath19 describes small fluctuations around the classical background . in order to study the properties of the chiral condensate we choose a specific potential of the scalar field as @xmath20 where @xmath21 is a coupling constant . then we take into account the quadratic terms of the scalar fluctuation @xmath19 and integrate over them , getting @xmath22 where @xmath23 . integrating over the fermionic field , we obtain @xmath24 where @xmath25 is a fermionic ultraviolet cutoff , @xmath26 is a ground state value of the bosonic field which is determined by the equations @xmath27 where @xmath28 , whereas @xmath11 and @xmath21 are dimensional coupling constants . it is convenient to introduce dimensionless constants @xmath29 where @xmath30 is a bosonic ultraviolet cutoff . then eq . ( [ eq7 ] ) may be written as @xmath31 and the saddle point @xmath26 is given by the solution of the equation @xmath32 where all the quantities are dimensionless . the third term in this equation describes the contribution of the scalar field fluctuations . generally speaking , eq . ( [ eq9 ] ) is resolved numerically , however , some special cases lead to simplifications that allow one to solve it analytically . an analytic solution can be easily compared with the results of our previous work @xcite , where the quantum fluctuations of the scalar field were not accounted for . we consider two such possibilities . \i ) when @xmath33 , then @xmath34 and substituting @xmath26 in ( [ eq8 ] ) , we get @xmath35 to see that this solution coincides with that of ref . @xcite , we note that @xmath36 equals @xmath26 of that work ( see eq . ( 15 ) there ) . \ii ) when @xmath37 , then , rescaling @xmath38 in eq . ( [ eq9 ] ) , we get @xmath39 where @xmath40 using ( [ eq10 ] ) and ( [ eq8 ] ) , we obtain @xmath41 if @xmath42 , then the value of the chiral condensate given by eq . ( [ eq11 ] ) equals that of eq . ( 13 ) in ref . @xcite . the value of @xmath43 is renormalization invariant . indeed , the @xmath44-functions in the one - loop approximation are @xmath45 using ( [ eq12 ] ) one can verify that @xmath43 does not depend on the choice of the renormalization point , i.e. @xmath46 it is worth to note that in a @xmath1-dimensional spacetime there is only one @xmath14-factor that appears in the renormalization of the scalar field mass , therefore @xmath47 where @xmath48 and @xmath49 are the bare coupling constants , and @xmath50 , @xmath51 are the renormalized coupling constants . the relation in eq . ( 13 ) can be obtained directly from the @xmath44-functions of eq . ( [ eq12 ] ) .    using eq . ( [ eq13 ] ) @xmath43 is rewritten as @xmath52 one can see that @xmath43 and the @xmath44-functions tend to zero in the limit @xmath53 . on the other hand , if the @xmath44-function equals to zero then in the model there is a phase transition as shown in ref . therefore one can conclude that if @xmath54 is @xmath55 then there is a phase transition in the model . although @xmath43 is not a solution of the eq . ( [ eq9 ] ) or eq . ( [ eq10 ] ) , @xmath43 also tends to zero when the @xmath44-functions tend to zero therefore @xmath43 may be considered as a qualitative approximation of @xmath26 . the value of @xmath43 becomes a good enough approximation of @xmath26 provided @xmath56 . the results of our numerical studies of the chiral condensate are presented in fig . 1 . for the convenience the value @xmath57 is shown in this figure . we plot the value @xmath58 as a function of the yukawa coupling constant @xmath59 . the other parameters entering eq . ( [ eq9 ] ) take the values @xmath60 . since the chiral condensate falls sharply when @xmath54 tends to 1 , then @xmath58 is presented in different scales in fig . 1 ( a , b ) . the solid line corresponds to @xmath26 obtained from the solution of eq . ( [ eq9 ] ) . the dotted line corresponds to @xmath26 obtained without the contribution of the scalar field fluctuations . the dash - dotted line is obtained for @xmath61 . we see that the chiral condensate equals to zero and the chiral symmetry is unbroken provided @xmath54 is @xmath55 . the chiral condensate appears when this condition is violated and increases with @xmath48 . hence there is a phase transition in this model . with increasing @xmath48 the quantity @xmath43 becomes a good approximation of @xmath26 according to the corresponding condition @xmath56 .     as a function of the yukawa coupling constant @xmath59 the values of parameters are @xmath60 . the solid line shows the chiral condensate where @xmath26 is an exact solution of eq . ( [ eq9 ] ) . the dotted line is @xmath58 obtained without the contribution of the scalar field fluctuations . the dash - dotted line is @xmath58 obtained for @xmath61 . ] the method used above to calculate the chiral condensate is applicable in any dimension of space . therefore calculating @xmath62 we get @xmath63 @xmath64 the quadratically divergent terms which appear in @xmath65 are eliminated by the mass renormalization . further we consider the case of a massless scalar field @xmath66 . rescaling the ultraviolet cutoff @xmath67 we rewrite the effective potential ( [ eq15 ] ) as @xmath68 or @xmath69 where @xmath70 note that @xmath43 is not a renormalization - invariant quantity . it is possible to check that by direct calculation , but it can be done in a simpler way . namely , we consider the following limiting cases    \i ) @xmath71 , @xmath72 ,    \ii ) @xmath73 , @xmath74 is not a renormalization - invariant quantity , we recall that @xmath75 .    in four dimensions the chiral condensate is defined as @xmath76 where @xmath26 is determined by the equations @xmath77 the effective potential @xmath78 has a minimum at @xmath79 and @xmath80 . in this case the chiral condensate is given by @xmath81 in the limit @xmath82 , @xmath43 tends to zero and consequently @xmath83 . if @xmath84 then @xmath62 has a maximum , the chiral condensate does not exist , and the chiral symmetry is unbroken . in this paper we studied the dynamical symmetry breaking in models consisting of a fermion field and a self - interacting scalar field with the yukawa interaction between the two fields in the one - loop approximation . we believe that the dynamical symmetry breaking as exhibited by these models can help to analyze similar phenomena in more realistic models . the models are considered in two and four spacetime dimensions . we showed that each models has two phases , one with the preserved symmetry and the other with the broken symmetry . we conclude that both models have a phase transitions as a function of the coupling constants . in four spacetime dimensions , we obtain an analytic expression for the chiral condensate in the limit of a massless scalar field . the two - dimensional case was investigated numerically by solving of the algebraic equation for the scalar field . buividovich , m.n . chernodub , e.v . luschevskaya and m.i . polikarpov , phys . lett . * b682 * , 484 ( 2010 ) ; v.v . braguta , p.v . buividovich , t. kalaydzhyan , s.v . kuznetsov , m.i . polikarpov , arxiv : 1011.3795v3 [ hep - lat ] . d. ebert , n.v . gubina , k.g . klimenko , s.g . kurbanov and v. ch . zhukovsky , phys . rev . * d84 * , 025004 ( 2011 ) ; d. ebert . khunjua , k.g . klimenko and v. ch . zhukovsky , arxiv : 1306.4485v2 [ hep - th ] . g. thooft , nucl . phys . * b75 * , 461 ( 1974 ) ; a. zhitnitsky , phys . rev . * d53 * , 5821 ( 1996 ) ; l.ya . glozman , v.k . sazonov , m. shifman , r.f . wagenbrunn , arxiv : 1201.5814v1 [ hep - th ] ; s. bellucci , e.r . bezerra de mello , a.a . saharian , arxiv : 1312.1686v1[hep - th ] . y. nambu and g. jona - lasinio , phys . rev . * 122 * , 345 ( 1961 ) . gross and a. neveu , phys . rev . * d10 * , 3235 ( 1974 ) . v.g . ksenzov and a.i . romanov , phys . nucl . * 76 * , 534 ( 2013 ) ; arxiv : 1204.3051v2 [ hep - th ] . a.m. polyakov , pisma zh . * 12 * , 381 ( 1970 ) .
we investigate a class of models with a massless fermion and a self - interacting scalar field with the yukawa interaction between these two fields . 
 the models considered are formulated in two and four spacetime dimensions and possess a discrete symmetry . 
 we calculate the chiral condensates are calculated in the one - loop approximation . 
 we show that the models have a phase transitions as a function of the coupling constants .