You are an expert at summarizing long articles. Proceed to summarize the following text: 
in field theories we often encounter infrared ( ir ) divergences of different kind . some of them have physical meaning while others are just artifacts of the perturbation theory . a common feature , however , is that the ir divergences appear at each order of perturbation theory , usually being more and more severe at higher loop orders . in order to consistently define the theory , these ir divergences must be summed up . 2pi resummations provide a consistent resummation framework known for a long time @xcite . the basic idea is to replace the free propagator in the perturbation theory with the exact one which is approximated self - consistently with fixed - loop skeleton diagrams . the so - defined perturbation theory is renormalizable @xcite-@xcite , and can be applied to study different physical questions from non - equilibrium @xcite , @xcite , thermodynamics @xcite , @xcite,@xcite,@xcite and different systems like o(n ) model @xcite , @xcite or gauge theories @xcite . although the 2pi approximation is constructed by physical arguments , and we expect better results ( ie . closer to the exact one ) after 2pi resummation , a priory it is not sure that one really achieves this goal . probably the finite lifetime effects are well represented by 2pi resummation both in equilibrium @xcite as well in non - equilibrium , where the 2pi is close to the boltzmann - equation approximation @xcite . but if the deep ir regime is important where multi - particle excitations also play crucial role , the picture is far to be so clean . to make the case even worse , in most gauge theory models there is hard to make exact statements about the ir behavior of the model .    in this paper we aim to study the bloch - nordsieck model , which is an exactly solvable 3 + 1d gauge theory @xcite . it is the eikonal approximation model of the qed , and one can argue @xcite that in the deep ir regime it describes correctly qed . therefore it is used to estimate ir properties of qed from this model , for example the soft photon creation processes @xcite or finite temperature electron damping @xcite . this model is therefore a motivated case study where the accuracy of the 2pi resummation can be assessed . we therefore perform a series of approximations for the bloch - nordsieck model : a one - loop perturbation theory , a 2pi resummation and finally the solution of the schwinger - dyson equations with an ansatz for the vertex function . in this model all levels of the approximations can be treated analytically . we show that the last method is exact in the model under consideration  although that is not expected in general . this observation , however , leads us to a proposal how the 2pi resummation can be improved in a generic model in order to catch the ir physics correctly . the structure of the paper is as follows . we first overview the standard solution of the bloch - nordsieck propagator at zero temperature in section [ sec:1 ] . then we compute the one loop level fermion propagator in section [ sec : oneloop ] . next , in section [ sec:2pi ] we write up the 2pi equations for this model , perform renormalization on that , and give the analytical solution as far it can be done . then we study the problem also numerically , determine the fermion propagator and compare the result with the exact one . finally , in section [ sec : sd ] we study the truncated schwinger - dyson equations , renormalize it , and show that for the bloch - nordsieck model the so - defined approximation scheme is exact . for conclusion ( section [ sec : conc ] ) we summarize our approach again and speculate about the possible generalizations . the bloch - nordsieck model is a simplification of the qed lagrangian where the dirac matrices @xmath0 are replaced by a four - vector @xmath1 @xmath2 the singled - out four vector @xmath3 represents the velocity of the rest frame of the fermion . the fermion wave function here has only one component and @xmath4 . we are interested in the fermion propagator which reads in the path integral representation @xmath5}\psi(x)\bar\psi(0).\ ] ] at the tree level it reads : @xmath6 since it has a single pole , there is no antiparticles in the model , and also the feynman propagator is the same as the retarded propagator . the lack of antiparticles also means that all closed fermion loops are zero . in a closed loop @xmath7 , therefore the product of theta functions is zero almost everywhere . ] . as a consequence the photon self - energy is zero , the free photon propagator is the exact one . in feynman gauge therefore the _ exact _ photon propagator is @xmath8    now we shortly review the exact solution for the fermion propagator , cf . @xcite , @xcite . we first define the gauge field dependent propagator : @xmath9}\psi(x)\bar\psi(0).\ ] ] this satisfies the schwinger - dyson equation @xmath10 we will need the solution in an exponential form for @xmath11 , and this is achieved by introducing the auxiliary quantity @xmath12 which satisfies @xmath13 by integration of the above equation by @xmath14 and assuming @xmath15 ( for which we need a convergence factor @xmath16 ) we see that @xmath17 we perform fourier transformation and separate the free time dependence @xmath18 , then we obtain @xmath19 the linear @xmath20 behavior is the consequence of the linearity in the kernel . if the kernel is non - linear or it is not scalar ( has a matrix structure ) then this form is not true any more . from here an inverse fourier transformation yields @xmath21 once we have an exponential representation for the background - dependent propagator , we can perform the gaussian @xmath11-integration . as a result we obtain in the exponent the factor @xmath22 this integral is uv divergent ; in dimensional regularization one finds the result @xmath23 where @xmath24 and @xmath25 . then the fermion propagator reads @xmath26 where @xmath27 . this is uv divergent which means that we need a wave function renormalization . the renormalized propagator reads @xmath28 where @xmath29 is a finite quantity . we can determine the discontinuity of this formula , for simplicity choosing @xmath30 : @xmath31 with this spectral function the sum rule @xmath32 , which is the consequence of the equal time anticommutation relations , can not be fulfilled , since the integral is divergent . this divergence should be compensated with the choice @xmath33 , but then we are faced with a @xmath34 expression . therefore one should always use a regularized version of the spectral function ( or propagator ) , maintaining the sum rule , and only at the end of the calculation is one allowed to release the regularization . the lagrangian is lorentz - invariant in the sense that we must also transform @xmath35 . so we can choose a lorentz - transformation where @xmath36 . if @xmath1 is a 4-velocity then @xmath37 ; if it is of the form @xmath38 , then it is @xmath39 . after rescaling the field @xmath40 and the mass as @xmath41 , the lagrangian reads @xmath42 this lagrangian will be used mostly in this work later . if necessary , the complete @xmath35 dependence is easily recoverable . the goal of our investigations is to see , how the different levels of resummations improve the result . thus first we start with the one loop perturbation theory . here we need the renormalized lagrangian ; in feynman gauge it reads ( using the fact that the photon self - energy is zero ) : @xmath43 for the fermion self - energy the one loop diagram is the bubble with the contribution : @xmath44 moreover we have wave function and mass renormalization counterterms @xmath45 the complete one loop self - energy is @xmath46 . in the calculation we have to take care of the non - standard form of the free fermion propagator . the details of the computation can be found in the appendix , as a result we obtain @xmath47,\ ] ] where @xmath24 and @xmath48 for renormalization we have to subtract the divergences with help of the counterterms , the finite parts are fixed by the renormalization scheme . in the @xmath49  scheme we choose the counterterms like @xmath50 this results in @xmath51    the discontinuity of the renormalized self - energy reads @xmath52 for the one - loop propagator we obtain @xmath53 this is consistent with the exact result in the leading order of @xmath54 . the spectral function @xmath55 reads @xmath56 this spectral function is normalizable , since @xmath57    on the other hand the one - loop result is not reliable when @xmath58 , ie . in the vicinity of the mass shell as well as in the large @xmath59 regime . in order to have a better description of these kinematical regimes , we need resummation of certain class of diagrams . as it is discussed in the introduction , the next level of our approximations is the 2pi resummation . the idea is to use the exact propagators in the perturbation theory , this propagator is determined self - consistently using skeleton diagrams as resummation patterns . the one - loop bubble diagram in the present case generates the resummation of all the `` rainbow '' diagrams . to obtain an expression for the 2pi resummation we use the technique of @xcite : we use the 1loop formula , interpret the appearing propagators as full propagators , and finally perform renormalization with the same _ form _ of divergent parts of the counterterms as in the 1-loop case ( the actual values will be different ) . the tree level photon propagator is exact , therefore we can write @xmath60 using a spectral representation for the fermion propagator ( using that now the feynman propagator is the retarded one and that the fermion spectral function is @xmath61 ) we find @xmath62 from this form it is clear that we obtain the weighted one - loop result , ie . @xmath63 in particular , if @xmath64 , then we get back the one - loop result .    at this point it is worth to examine the uv divergence structure of the 2pi approximation . uv divergences may occur in for large values of @xmath65 : using we find that the large @xmath65 behavior of the one - loop self - energy reads : @xmath66 since @xmath67 is integrable for large @xmath65 values , therefore the @xmath68 is already finite . therefore the divergence structure of the self - energy is @xmath69 , just like for the free case , and so the same type of counterterms are needed ( although the values are different ) . this is a manifestation of the general case of counterterm renormalizability of 2pi resummations @xcite . first we try to analyze with analytic methods . we differentiate it with respect to @xmath59 to find @xmath70 since @xmath71 , we find for @xmath72 : @xmath73 to solve the equation we first should realize that the @xmath74 and @xmath75 cases are very different . if @xmath74 then @xmath76 and the propagator behaves as @xmath77 with some wave function renormalization constant @xmath78 and mass @xmath79 . this agrees with the free case . we also see that the integration constants correspond to the renormalization scheme ( here the wave function and mass renormalization ) . if @xmath80 then we can redefine the variables with an arbitrary @xmath81 scale as @xmath82 then we find @xmath83 this equation does not depend on the coupling any more . the coupling constant dependence shows up in the integration constants which are the manifestation of the renormalization scheme . we shall also note that the equation does not give information about the sign of @xmath84 and @xmath85 , because for @xmath86 or @xmath87 the equation remains the same . the chosen signs in turn out later to be the physical choice . we introduce @xmath88 this means that we can write for @xmath89 : @xmath90 with an integration constant @xmath91 . therefore @xmath92 there could appear an integration constant also here on the right hand side : @xmath93 . but recalling that @xmath94 , we see that @xmath95 corresponds to a mass shift : if the mass remains the tree level @xmath96 then @xmath97 . this is the ( implicit ) solution of the 2pi equations . we see that for real @xmath85 the left hand side is real and positive , moreover for @xmath98 . the @xmath99 part corresponds to imaginary values of @xmath85 . since the equation itself is real , if @xmath85 is a solution , it is @xmath100 , too . this means that the imaginary part is in fact the ( half ) discontinuity of the solution . we see that irrespective of the value of @xmath91 , at @xmath101 , ie . on the mass shell @xmath102 and so @xmath103 finite . this yields difficulties when we try to apply renormalization conditions on the self - energy . namely , if we keep the mass shell unchanged ( this would correspond to the choice of @xmath95 above ) , then the renormalization of the self - energy would mean @xmath104 and @xmath105 finite . then , however , near the mass shell the propagator should always behave as @xmath106 , ie . _ infinite _ at the mass shell . this means that the physical renormalization process requires @xmath107 . in this case the propagator behaves near the mass shell as : @xmath108 because if @xmath85 is close to @xmath109 then the log term can be neglected in , and we find @xmath110 .    for large values of @xmath85 , on the other hand , @xmath91 can be neglected . then the integral can be evaluated as @xmath111 for large @xmath85 values it behaves as @xmath112      now let us turn to the numerical study of the system , based on @xcite and @xcite : we determine the discontinuity of the self - energy self - consistently . the discontinuity of now reads @xmath113 knowing the discontinuity of the self - energy , we can use the kramers - kronig relation to restore the complete self - energy : @xmath114 while is a completely finite expression , in the kramers - kronig relation we will find divergences . this corresponds to the divergences of the self - energies which must be made finite by applying the appropriate counterterms . technically one can regularize the integral in and then make it finite with counterterms , or use the ( twice ) subtracted form of the kramers kronig relation . to see how it works , we determine the one - loop result from the tree level spectral function and the dimensional regularization of the kramers - kronig equations ( interpreting @xmath115 ) : @xmath116.\ ] ] the divergence structure is the same , and also the @xmath49  scheme result is the same as in ( the different finite parts are due to the different regularization method ) . now we can set up an algorithm to solve . we choose an arbitrary spectral function as a starting one ( practically the free spectral function ) , then follow the following steps :    step 1 : : :    compute the discontinuity of the self - energy using step 2 : : :    compute the complete self - energy using the kramers - kronig relation step 3 : : :    renormalize the self - energy with local counterterms . to fix the    counterterms we used on - mass - shell ( om ) renormalization scheme , ie . the real part of the self - energy at the mass shell is zero and its    derivative is also zero @xmath117 we    note here that releasing the first condition yields a mass shift ,    releasing the second condition yields a finite wave function    renormalization . but in all renormalization schemes it will remain    true that near the ( renormalized ) mass shell the propagator behaves as    @xmath118 . step 4 : : :    construct the new spectral function from the discontinuity of the    propagator knowing the real and imaginary part of the self - energy as    @xmath119 step 5 : : :    continue with step 1 until the process converges . integrations in the above algorithm are performed numerically . this strategy was applied successfully for the @xmath120 model in @xcite . the direct application of this strategy , however , this times fails . numerically what we can observe is that the spectral function becomes more and more shallow , and pointwise it goes to zero @xmath121 . in order to see a convergence , we had to use a supplementary step in the iteration after step 4 :    step 4 : : :    use a rescaling of the generated spectral function :    @xmath122 with appropriate    @xmath11 and @xmath123 which can ensure convergence . the appropriate values can be found by inspection , but the actual values are not too important ( we used @xmath124 and @xmath125 in our numerics ) . in this way finally we succeeded to see convergence in the spectral function . the numerical reason of this behavior is that the exact spectral function has a discontinuity at the mass shell , and apart from this single point it has always negative derivative . numerically , however , we can not have a jump , since in all regularizations equation yields @xmath126 where @xmath127 . since the exact curve starts to bend downwards , the recursion tries to lower the spectral function in order to have smaller derivative near the mass shell . since the spectral function has to be positive , these requirements can be satisfied only with @xmath128 . with the continuous rescaling we can achieve that the numerically badly conditioned part , the vicinity of the mass shell , becomes smaller and smaller . the numerical results can be seen on figure [ fig:1 ] . the expected asymptotics can be nicely identified on the calculation ( cf . [ fig:2 ] ) . this also proves implicitly that the strategy to resolve the aforementioned numerical problem with the 2pi equation was correct . if we compare the 1-loop , the 2pi and the exact results we see that there is not too much improvement . the ir problem near the mass shell which made the 1-loop calculation unreliable , _ seems _ to be cured , but in fact the result is not closer to the exact one as the one - loop result . the physics of the deep infrared photons can not be described by the 2pi approximation . the next level of the approximations is based on the schwinger - dyson equations . for the bloch - nordsieck model in feynman gauge it can be written as @xmath129 where @xmath130 is the vertex function . for the vertex function there is another exact equation , coming from the current conservation . this results in the ward - identity analogous to the qed case @xcite : @xmath131    in this model , however , the vertex function is proportional to @xmath1 . in principle the lorentz - index in this model can come from @xmath1 or from any of the momenta . but , since the fermion propagator depends on the 4-momentum in the form @xmath132 , the fermion - photon vertex does not depend on the momentum components which are orthogonal to @xmath3 . therefore the lorentz - index which comes from @xmath133 in fact comes from the longitudinal part of @xmath133 , ie . proportional to @xmath1 . so we can write @xmath134 . this gives us the possibility that from the ward identities we _ exactly _ determine the vertex function . the ward - identity for the current conservation yields then in case when @xmath135 : @xmath136 therefore we find @xmath137 this is an exact equation in the bloch - nordsieck model . now we will solve this equation in the renormalized theory , and demonstrate that the solution is indeed identical with the bolch - nordsieck solution presented in section [ sec:1 ] . in the second term @xmath138 drops out , resulting in an integral @xmath139 because of @xmath140 symmetry . what remains is @xmath141    this form is true in the original model , we shall now find the renormalized form . first we adapt the wave function renormalization for the fermionic fields which changes the bare propagator to @xmath142 where @xmath143 . we will assume that the mass shell remains the same , then @xmath144 , and the free propagator will be @xmath145 . we will use also the notation @xmath146 . the full propagator then reads @xmath147 using we find the equation @xmath148 where @xmath149 @xmath150 can be interpreted as a running wave function renormalization constant .    with a spectral representation @xmath151 in the appendix we evaluate @xmath152 , and we find @xmath153.\ ] ] we rewrite it into , then , assuming normalizable spectral function , after some algebraic manipulation we find @xmath154 we may assume that the explicit integral is not uv divergent ( it can be checked a posteriori , or , as in the present case , knowing the exact solution ) . then the above equation can be made finite by requiring @xmath155 where @xmath156 and @xmath157 are finite . this form can be interpreted physically as the appearance of the renormalized coupling @xmath156 and the finite wave function renormalization @xmath157 . we note that the coupling constant renormalization equation agrees with the nonperturbative coupling constant renormalization in the o(n ) models @xcite . now we find @xmath158 this function depends on the arbitrary scale @xmath159 , but the physics , of course , must be @xmath159 independent . this can be achieved by appropriately changing the @xmath157 and @xmath156 constant when we change @xmath159 . the @xmath159-independence of @xmath150 requires ( using the sum rule for @xmath67 ) : @xmath160 this can be satisfied if @xmath161 the second equation means @xmath162 where @xmath163 is a constant ; the first equation then reads @xmath164    using the normalizability of @xmath67 we finally find @xmath165 the @xmath163 and the scale @xmath166 are renormalization group independent quantities ( ie . independent of the scale @xmath159 ) , these characterize the renormalization scheme . the appearance of a scale @xmath166 is the manifestation of dimensional transmutation . now , instead of that scale @xmath166 it is worth to use @xmath167 for which @xmath168 . clearly @xmath169 if @xmath170 . then with differentiating @xmath29 with respect to @xmath59 we find @xmath171 this gives finally @xmath172 by differentiation with respect to @xmath59 we find @xmath173 where @xmath174 is an arbitrary constant . this is indeed the solution of bloch and nordsieck , now in terms of the renormalized quantities . but we also see that the condition @xmath175 can be satisfied only with @xmath176 . this is in close relation with the fact that at the mass shell @xmath177 , the propagator ( and its discontinuity ) is not integrable . the lesson of this analysis is that the deep ir physics is well describable by the schwinger - dyson equation , truncated in a way which respects the ward - identities . as we have seen , this strategy is renormalizable and exact in case of the bloch - nordsieck model . a big advantage of this approach is that , besides being exact in the ir , it can be easily generalized to other theories . so we expect that in qed the schwinger - dyson equations truncated in the way we have done it in the bloch - nordsieck theory will represent the exact result well in the problematic deep ir regime . in this paper we examined the exactly solvable bloch - nordsieck model from the point of view of different perturbative methods . we first reviewed the known method to obtain the exact solution @xcite . then the different levels of approximations , like the one - loop level perturbation theory , the 2pi resummation and the truncated schwinger - dyson equations were studied . the 1-loop result exhibits an ir sensitivity when we approach the mass shell which renders the theory ill - defined . the self - energy ( 2pi ) resummation reorganizes the perturbative series in a way that this ir problem disappears . this does not mean , however , that the result itself would be closer to the exact one , only the explicit ir sensitivity can not be seen . on the other hand , the schwinger - dyson equations , truncated in a way that the ward - identities are satisfied yield the exact result in the bloch - nordsieck model . this is a new way of obtaining the exact solution in the bloch - nordsieck model . and , while the original solution method is very hard to generalize to other theories , the generalization of the specially truncated schwinger - dyson equations is straightforward . the authors thank useful discussions with t.s . br , f. csikor and a. patks . this work is supported by the hungarian research fund ( otka ) under contract no . the one - loop contribution to the self - energy reads , with a generic @xmath35 vector in feynman gauge : @xmath178 this is lorentz - invariant , if we a lorentz transformation both on @xmath35 and @xmath179 . so we may choose a special frame where @xmath180 . if @xmath35 is a proper 4-velocity , then @xmath37 ; if it is @xmath38 , then @xmath39 , but still constant , since @xmath181 is a parameter of the theory . we find then @xmath182 thus it is enough to consider @xmath183 only . there we transform to positive frequency integrals @xmath184 where in the last step we performed wick rotation ( the choice of the imaginary part of @xmath185 is crucial for the direction of the rotation on the complex plane ) . now we can write up the integral in @xmath186 and @xmath187 space , in the latter using @xmath188 dimensions : @xmath189 we use the relation @xmath190 to proceed as @xmath191.\end{aligned}\ ] ] we write it as @xmath192,\ ] ] where @xmath193 therefore @xmath194.\ ] ]        j. m. luttinger and j. c. ward , phys . 118 ( 1960 ) 1417 . g. baym , phys . 127 ( 1962 ) 1391 . j. m. cornwall , r. jackiw and e. tomboulis , phys . d 10 ( 1974 ) 2428 . h. van hees an j. knoll , phys . d65 ( 2002 ) 025010 [ hep - ph/0107200 ] ; phys . d65 ( 2002 ) 105005 [ hep - ph/0111193 ] ; phys . d66 ( 2002 ) 025028 [ hep - ph/0203008 ] j .- p . blaizot , e. iancu and u. reinosa , phys . b568 ( 2003 ) 160 [ hep - ph/0301201 ] ; nucl . a736 ( 2004 ) 149 [ hep - ph/0312085 ] j. berges , sz . borsanyi , u. reinosa and j. serreau , annals phys . 320 ( 2005 ) 344 [ hep - ph/0503240 ] u. reinosa , j. serreau , jhep 0607 ( 2006 ) 028 [ hep - th/0605023 ] ; a.  jakovac , phys . rev .   * d76 * , 125004 ( 2007 ) . [ hep - ph/0612268 ] . a.  patkos , z.  .szep , nucl . phys .   * a811 * , 329 - 352 ( 2008 ) . [ arxiv:0806.2554 [ hep - ph ] ] . j. berges and j. cox , phys . lett . b 517 ( 2001 ) 369 . j. berges , s. borsanyi and c. wetterich , phys . 93 ( 2004 ) 142002 . j. berges , s. roth , nucl.phys.b847:197-219,2011 , [ arxiv:1012.1212 ] m.e . carrington , e. kovalchuk , phys.rev.d76:045019,2007 , [ arxiv:0705.0162 ] n.n . bogoliubov and d.v . shirkov , _ introduction to the theories to the quantized fields _ ( john wiley & sons , inc . , 1980 ) h.m . fried , _ green s functions and ordered exponentials _ ( cambridge university press , 2002 ) j.  -p . blaizot and e.  iancu , phys . d  * 56 * , 7877 ( 1997 ) [ hep - ph/9706397 ] , j.  -p . blaizot and e.  iancu , phys . d  * 55 * , 973 ( 1997 ) [ hep - ph/9607303 ] . weldon , phys . d44 ( 1991 ) 3955 ; phys d49 ( 1994 ) 1579 . m. le bellac , _ thermal field theory _ , ( cambridge univ . press , 1996 . ) p. mati diplomawork j.o . andersen , d. boer , h.j . warringa , phys . d70 ( 2004 ) , 116007 [ hep - ph/0408033 ] , a.  jakovac , phys .  rev .  d  * 78 * , 085013 ( 2008 ) [ arxiv:0808.1800 [ hep - th ] ] , g.  fejos , a.  patkos and z.  .szep , phys . d * 80 * , 025015 ( 2009 ) [ arxiv:0902.0473 [ hep - ph ] ] . peskin , d.v . schroeder , _ an introduction to quantum field theory _ , ( perseus books publishing , 1995 . )
we studied different levels of resummations of the exactly solvable bloch - nordsieck model in order to be able to compare the approximations with an exact result . 
 we studied one - loop perturbation theory , 2pi resummation and schwinger - dyson equations truncated in a way to maintain ward - identities . 
 at all levels we carefully performed renormalization . 
 we found that although the 2pi resummation does not exhibit infrared ( ir ) sensitivity at the mass shell , as the one - loop perturbation theory does , but it is still far from the exact solution . 
 the method of truncated schwinger - dyson equations , however , is exact in this model , so it provides a new way of solving the bloch - nordsieck model . 
 this method can also be generalized to other , more complicated theories .