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an anomaly in field theory occurs if a symmetry of the action or the corresponding conservation law , valid in the classical theory , is violated in the quantized version . this surprising feature of quantum theory discovered by adler @xcite , bell and jackiw @xcite , and by bardeen @xcite in 1969 plays a fundamental role in physics ( for details see refs.@xcite @xcite ) . physically there is a difference between external and internal symmetries . the breakdown of an external symmetry is not dangerous for the consistency of the theory , on the contrary , it provides for instance the physical explanation for the @xmath1-decay @xcite , @xcite or the solution to the @xmath2 problem in qcd @xcite . on the other hand , the breakdown of an internal symmetry ( i.e. gauge symmetry ) leads to an inconsistency of the quantum theory , the anomalous ward identities destroy the renormalizability of the theory @xcite , and also the unitarity of the @xmath3-matrix may be lost @xcite . to avoid such anomalies imposes severe restrictions to the physical content of a theory . for instance , in the famous @xmath4 standard theory for electroweak interactions one had to demand the existence of the top quark long before it was discovered . gravitation regarded as a gauge theory also suffers from anomalies . the gauges are the general coordinate transformations ( diffeomorphisms ) or the rotations in the tangent frame ( lorentz transformations ) or the conformal transformations ( weyl transformations ) . then in the quantum case the classical conservation law of the energy - momentum tensor can be broken  an einstein anomaly occurs  or an antisymmetric part of the energy - momentum tensor can exist  a lorentz anomaly occurs  or the trace of the tensor is nonvanishing  a weyl anomaly arises ( for details see e.g. refs.@xcite,@xcite , @xcite ) . whereas the anomaly in the tensor trace has been found already in the seventies @xcite  @xcite the study of the gravitational anomalies started with the pioneering work of alvarez - gaum and witten @xcite in the eighties . the anomalies have been first found within perturbation theory , they are local polynomials in the connection and curvature . the authors @xcite  @xcite have calculated ( ultraviolet divergent ) feynman diagrams where the external gravitational field couples to a fermion loop via the energy - momentum tensor . of course , the anomaly  reflecting the deep laws of quantum physics  must show up within other approaches too . so they have been calculated by the heat kernel method @xcite , @xcite , by fujikawa s path integral approach @xcite , @xcite , and by modern mathematical techniques such as differential geometry and cohomology @xcite  @xcite and topology ( index theorems ) @xcite , @xcite , @xcite , ( for an overview see ref.@xcite ) .    deeply related to anomalies are the socalled schwinger terms @xcite  @xcite ( for an introduction see e.g. refs . @xcite , @xcite ) . in a yang - mills gauge theory schwinger terms ( st ) show up as additional terms ( extensions ) in the canonical algebra of the equal time commutators ( etc ) of the gauss law operators ( see e.g. @xcite cohomologically they are described by the faddeev - mickelsson cocycle @xcite , @xcite , and geometrically they can be related to a berry phase in the vacuum functional @xcite  @xcite . st are frequently calculated within perturbation theory where the bjorken - johnson - low limit @xcite , @xcite works very well . however , the definition of a point - splitting method turns out to be more subtle and might not lead to the correct result @xcite , @xcite .    in gravitation schwinger terms occur in the etc of the energy - momentum tensors , c - number terms that are proportional to derivatives of the @xmath0-function . they can be related to the gravitational anomalies @xcite , and they have been calculated explictly via the invariant spectral function and via cohomological techniques @xcite . furthermore there exists an interesting relation of the st to the curvature of the determinant line bundle @xcite , @xcite . our work deals with the calculation of the gravitational anomalies , specifically the einstein anomaly and the weyl anomaly . the lorentz anomaly is not independent of the einstein anomaly , both types of anomalies can be shifted into each other by a suitable counterterm @xcite . for convenience we choose the case where the lorentz anomaly is vanishing . we also calculate the gravitational schwinger terms . the purpose of our work is to show that all these anomalous features are easily obtained by the method of dispersion relations , a less familiar but very useful approach . some of our results we have briefly presented in refs . @xcite , @xcite . already since their first introduction into quantum field theory @xcite dispersion relations ( dr ) proved to be a very valuable tool . in connection with anomalies dr have been fomulated by dolgov and zakharov @xcite and also by kummer @xcite . in the following several authors @xcite  @xcite used successfully dr to determine the anomalies in the chiral current . recently hoej ' i and schnabl @xcite have applied the method to the well - known trace anomaly @xcite , @xcite which is related to the broken dilatation ( or scale ) invariance . we extend in our work the method of dr to the case of pure gravitation . so we consider chiral fermion loops coupled to gravitation  for their evaluation it is enough to use gravitation as an external or nonquantized field  and we show that the gravitational anomalies and the schwinger terms are , in fact , completely determined by dispersion relations . all calculations are performed in two dimensions . conceptuelly the dr approach is an independent and complementary view of the anomaly phenomenon as compared to the ultraviolet regularization procedures . within dr the anomaly manifests itself as a very peculiar infrared feature of the imaginary part of the amplitude . but as we shall show there is a link between the two approaches , the dr method and the n - dimensional regularization precedure .    our paper is organized as follows . in section 2 we present the general structure of the considered ( pseudo- ) tensor amplitude and we discuss the ward identities which we have to study . in section 3 we introduce the dispersion relations for the relevant formfactors of the amplitude and calculate their imaginary parts via the cutkosky rule . in order to reproduce the dr results in a definite ultraviolet regularization scheme we have worked out in detail the t  hooft - veltman regularization procedure in section 4 and we have compared the several amplitudes with the results of tomiya @xcite and alvarez - gaum and witten @xcite . the equivalence between the dispersive approach and the dimensional regularization procedure is given in section 5 . in section 6 we derive the anomalous ward identities and explain the source of the anomaly in the dr approach . from the ward identities we deduce the linearized gravitational anomalies the einstein- and the weyl anomaly  and we also determine their covariant versions , a comparison with the exact results is given . the gravitational schwinger terms occuring in the etc of the energy - momentum tensors we calculate in section 7 , where we adapt the dispersive approach to a method proposed by klln @xcite . finally we summarize our main results in section 8 . in two dimensions the lagrangian describing a weyl fermion in a gravitational background field can be written as    @xmath5    where @xmath6 is the zweibein and @xmath7 its inverse @xmath8 . the determinant of the zweibein is @xmath9 and @xmath10 is the covariant derivative with the spin connection @xmath11 . we use the following conventions in 2 dimensions :    * for the flat metric @xmath12 * for the epsilon tensor @xmath13 * for the dirac matrices @xmath14    the einstein and the weyl anomaly are determined by the one - loop diagram in fig.[loop ] and it is sufficient to use the linearized gravitational field    @xmath15    since in two dimensions the spin connection @xmath11 does not contribute ( see e.g. ref.@xcite ) we find the following linearized interaction lagrangian ( for convenience @xmath16 is absorbed into @xmath17 , @xmath18 acts only on @xmath19 )    @xmath20    and @xmath21 from this expression follow the feynman rules for the vertices in the loop diagram    @xmath22    and the explicit form of the ( symmetric ) energy - momentum tensor    @xmath23    where we have dropped the terms proportional to @xmath24 as they do not contribute to the amplitude . + then the whole amplitude is given by the two - point function    @xmath25    due to lorentz covariance and symmetry the general structure of the amplitude can be written in the following way    @xmath26    @xmath27    @xmath28 t_{8}(p^{2 } ) \ , , \label{formfactors3}\end{aligned}\ ] ]    where we have separated the amplitude into its pure tensor part @xmath29 ( coming from the vector piece of the chirality projection in eq.([energy - momentum - tensor ] ) ) and into its pseudo - tensor part @xmath30 ( coming from the axial piece in eq.([energy - momentum - tensor ] ) ) . the functions @xmath31 are the formfactors that are to be evaluated . + classically the energy - momentum tensor has the following properties :    1 . @xmath32 , symmetric 2 . @xmath33 , conserved 3 . @xmath34 , traceless    which lead to the canonical ( naive ) ward identities :    1 . @xmath35 2 . @xmath36 3 . @xmath37 . we are interested in the pure einstein anomaly therefore we demand the quantized energy - momentum tensor to be symmetric . this is always possible due to the bardeen  zumino theorem @xcite which states that the gravitational anomaly can be shifted from pure lorentz type to pure einstein type , and vice versa . ( we disregard here leutwyler s point of view @xcite who emphasizes his preference for the lorentz anomaly ) . thus the symmetry property 1 . ) of the amplitude is fulfilled and an other symmetry @xmath38 is trivially satisfied . however , the naive ward identities 2 . ) and 3 . ) need not be satisfied , they can be broken by the einstein and weyl anomalies respectively . the canonical ward identities we re - express by the formfactors    @xmath39 { } \nonumber\\{}&&+ \varepsilon_{\rho \tau } p^{\tau}\left[p_{\nu}p_{\sigma } ( p^{2 } t_{6 } + t_{7 } + t_{8 } ) + g_{\nu \sigma } p^{2 } t_{8}\right ] { } \nonumber\\{}&&+   \varepsilon_{\sigma \tau } p^{\tau}\left[p_{\nu}p_{\rho } ( p^{2 } t_{6 } + t_{7 } + t_{8 } ) + g_{\nu \rho } p^{2 } t_{8}\right ] \label{pta}\end{aligned}\ ] ]    and @xmath40 where @xmath41 is the dimension . we call from now on ward identity ( wi ) the property 2 . ) and trace identity ( ti ) the property 3 . ) . for the pure tensor part of the amplitude the wi may be written as    @xmath42    and the ti as    @xmath43    of course , relations ( [ ti2 ] ) and ( [ ti1 ] ) are not independent of each other , ( [ ti2 ] ) follows from ( [ ti1 ] ) and ( [ vwi1 ] )  ( [ vwi3 ] ) .    in the following we shall use a renormalization procedure which keeps the wi in the pure tensor part ( [ vwi1 ] )  ( [ vwi3 ] ) so that the anomaly occurs only in the axial pieces  representing the pseudotensor part of the amplitude . + calculating the amplitude with massive fermions ( the loop in fig . [ loop ] ) with help of the feynman rules gives    @xmath44 \frac{1\pm\gamma_{5}}{2}{}\nonumber\\&&{}\times \frac{/\hspace{-6.5pt}p + /\hspace{-6.5pt}k+m}{(p+k)^{2}-m^{2}+i \varepsilon } \left [ \gamma_{\rho } ( p+2k)_{\sigma } + \gamma_{\sigma } ( p+2k)_{\rho}\right ] \frac{1\pm\gamma_{5}}{2 } \frac{/\hspace{-6pt}k+m}{k^{2}-m^{2}+i \varepsilon } \biggr\}.\end{aligned}\ ] ]    for convenience we split the pure tensor piece of the loop in the following way    @xmath45    where @xmath46 represents the loop with the identity instead of the chirality projectors and @xmath47 denotes the part proportional to @xmath48 . we separate the no interchange amplitudes as    @xmath49    finally , the axial part of the amplitude is connected to the vector part due to relation    @xmath50    ( valid only in 2 dimensions for our conventions ( [ definition of the flat metric ] ) - ( [ definition of the gamma matrices ] ) ) . a symmetric decomposition turns out to be useful @xmath51 the formfactors of the amplitude ( [ formfactors1 ] )  ( [ formfactors3 ] ) are analytic functions in the complex @xmath52 plane except a cut on the real axis starting at @xmath53 . due to cauchy s theorem they can be expressed by dispersion relations which relate the real part of the amplitude to its imaginary part . the imaginary parts of the amplitude can be easily calculated via cutkosky s rule @xcite . it states to replace in the amplitude each propagator by its discontinuity on mass shell    @xmath54 { } \nonumber\\ { } & & \!\!\!\times \delta(k^{2}-m^{2 } ) \delta\bigl((p+k)^{2}-m^{2}\bigr ) \theta(-k^{0 } ) \theta(k^{0}+p^{0 } )   \label{abstract i m tni}\\ i m t^{dv , ni}_{\mu \nu \rho \sigma}(p ) & = & \frac{1}{32 } m^{2 } g_{\mu\rho } \int \!d^{2}k ( p+2k)_{\nu } ( p+2k)_{\sigma}{}\nonumber\\ { } & & \!\!\!\times \delta(k^{2}-m^{2 } ) \delta\bigl((p+k)^{2}-m^{2}\bigr ) \theta(-k^{0 } ) \theta(k^{0}+p^{0 } ) \ , . \label{abstract i m tdvni}\end{aligned}\ ] ]    the explicit integration gives    @xmath55 p_{\mu}p_{\nu}p_{\rho}p_{\sigma}{}\nonumber\\ { } & &   \qquad + \left [ -\frac{1}{6}p^{2}+\frac{4}{3}m^{2}-\frac{8}{3}\frac{m^{4}}{p^{2}}\right ] \left(p_{\mu}p_{\nu } g_{\rho \sigma } + p_{\mu}p_{\sigma } g_{\nu \rho } + p_{\nu}p_{\rho } g_{\mu \sigma } + p_{\rho}p_{\sigma } g_{\mu \nu}\right )   { } \nonumber\\ { } & & \qquad + \left [ \frac{1}{3}p^{2}-\frac{2}{3}m^{2}-\frac{8}{3}\frac{m^{4}}{p^{2}}\right ] p_{\mu}p_{\rho } g_{\nu \sigma } + \left[\frac{1}{3}p^{2}-\frac{5}{3}m^{2}+\frac{4}{3}\frac{m^{4}}{p^{2}}\right ] p_{\nu}p_{\sigma } g_{\mu \rho } { } \nonumber\\ { } & &   \qquad + \left[\frac{1}{6}(p^{2})^{2}-\frac{4}{3}p^{2}m^{2}+\frac{8}{3}m^{4}\right ] \left(g_{\mu \nu } g_{\rho \sigma } + g_{\mu \sigma } g_{\nu \rho}\right ) { } \nonumber\\{}&&\qquad + \left[-\frac{1}{3}(p^{2})^{2}+\frac{5}{3}p^{2}m^{2}-\frac{4}{3}m^{4}\right ] g_{\mu \rho } g_{\nu \sigma } \biggr\ } \\ & & i m t^{dv , ni}_{\mu \nu \rho \sigma}(p ) = \frac{1}{32 } j_{0 } m^{2 } g_{\mu\rho } \biggl\ { \left ( -p^{2}+4m^{2 } \right ) g_{\nu\sigma } + \left(1 - 4\frac{m^{2}}{p^{2 } } \right ) p_{\nu}p_{\sigma } \biggr\},\end{aligned}\ ] ]    with the threshold function @xmath56 from which we quickly find all imaginary parts of the formfactors in the total amplitude @xmath57 :    @xmath58    considering in addition the amplitude @xmath46 we have the following imaginary parts of the formfactors which we denote by :    @xmath59    clearly , the imaginary parts ( [ i m a1 ] )  ( [ i m a5 ] ) of the amplitude @xmath46 satiesfy the wi ( [ vwi1])([vwi3 ] ) with @xmath60 , and the subtraction procedure we choose in the following keeps this property for the entire formfactors @xmath61 . + now we start with an unsubtracted dispersion relation ( dr ) for the formfactors    @xmath62    and we observe that , for instance , the integral for @xmath63 is convergent whereas for @xmath64 it is logarithmically divergent and needs to be subtracted once , and for @xmath65 it is linearly divergent and needs to be subtracted twice . we can infer already from the @xmath52 behaviour of the imaginary parts which kind of dispersion relation we have to use , see table  [ subtraktionen ] . .the formfactors and the used type of dispersion relations [ cols= " < , > " , ]     so for the formfactors @xmath66 an unsubtracted dr is sufficient and we get    @xmath67 \label{t1 in dr}\end{aligned}\ ] ]    with    @xmath68    a once subtracted dr defined by    @xmath69    we use for the following formfactors ( see table 1 )    @xmath70    for the remaining formfactors a twice subtracted dr defined by    @xmath71    is necessary and we find    @xmath72\\ t^{r}_{5}(p^{2 } ) & = & \frac{p^{4}}{\pi } \int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2 } } \frac{1}{t^{3}}\left(1-\frac{4m^{2}}{t}\right)^{\hspace{-3pt}-\frac{1}{2 } } \hspace{-3pt } \left ( -\frac{1}{96}t^{2 } + \frac{1}{48}t m^{2}+\frac{1}{12 } m^{4 } \right ) { } \nonumber\\{}&= & p^{2}\left[-\frac{5}{288\pi } - \frac{1}{24\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{48\pi } \bigl(1 + 2 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right]\\ a^{r}_{5}(p^{2 } ) & = & \frac{p^{4}}{\pi } \int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2 } } \frac{1}{t^{3}}\left(1-\frac{4m^{2}}{t}\right)^{\hspace{-3pt}-\frac{1}{2 } } \hspace{-3pt } \left ( -\frac{1}{48}t^{2 } - \frac{1}{12}t m^{2}+\frac{2}{3}m^{4}\right ) { } \nonumber\\{}&= & p^{2}\left[-\frac{1}{72\pi } - \frac{1}{3\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{24\pi } \bigl(1 + 8 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right ] . \label{a5r in dr}\end{aligned}\ ] ]    with these explicit expressions for the formfactors we have determined the whole amplitude @xmath57 , eqs.([amplitude expressed by t - product])([formfactors3 ] ) , from which the correct ward identities will follow . although the method of dispersion relations appears quite different to the methods of regularizations we can reproduce its results in a definite regularization scheme , namely in the n - dimensional regularization procedure of t hooft  veltman @xcite . it is instructive to work it out in more detail . we start with amplitude ( [ amplitude by dimensional regularization with massive fermions ] ) , calculate the @xmath73-matrices and follow the standard procedure by inserting the feynman parameter integral @xmath74^{2 } } \ , \ , .\ ] ] then we obtain for the ` no interchange ' amplitudes of the pure tensor pieces    @xmath75^{2 } } { } \nonumber\\{}&&\quad \times \bigl [ ( l+px)_{\mu}(l - p(1-x))_{\rho } + ( l+px)_{\rho}(l - p(1-x))_{\mu } - g_{\mu\rho } ( l+px)^{\lambda}(l - p(1-x))_{\lambda } \bigr]\end{aligned}\ ] ]    and    @xmath76^{2 } } \ , \ , , \ ] ]    with    @xmath77    calculating the t hooft  veltman integrals @xmath78^{\alpha } } = \frac{(-1)^{\alpha}i } { ( 4\pi)^{\omega } } \frac{\gamma(\alpha - \omega)}{\gamma(\alpha ) } \delta^{\omega-\alpha } \\ p_{1}^{\mu\nu } & = & \int\!\ ! \frac{d^{2\omega}l}{(2\pi)^{2\omega } } \frac{l^{\mu}l^{\nu}}{[l^{2}-\delta]^{\alpha } } = \frac{\delta}{2(\omega - \alpha + 1 ) } g^{\mu\nu } p_{0 } \label{integral p1}\\ p_{2}^{\mu\nu\rho\sigma } & = & \int\!\ ! \frac{d^{2\omega}l}{(2\pi)^{2\omega } } \frac{l^{\mu } l^{\nu } l^{\rho } l^{\sigma } } { [ l^{2}-\delta]^{\alpha } } { } \nonumber\\{}&= & \frac{\delta^{2}}{4(\omega - \alpha + 1)(\omega - \alpha + 2 ) } ( g^{\mu\nu } g^{\rho\sigma } + g^{\mu\rho } g^{\nu\sigma } + g^{\mu\sigma } g^{\nu\rho } ) p_{0 } \label{integral p2}\end{aligned}\ ] ] with @xmath79 provides the following expressions @xmath80 p_{\nu}p_{\sigma } g_{\mu \rho } { } \nonumber\\ { } & & \qquad + 2 \frac{\delta^{2 } } { \omega(\omega-1 ) } ( g_{\mu \nu } g_{\rho \sigma } + g_{\mu \sigma } g_{\nu \rho } ) + \left [ -2\omega \frac{\delta^{2 } } { \omega(\omega-1 ) } + 2x(1-x)p^{2 } \frac{\delta}{\omega-1 } \right ] g_{\mu \rho } g_{\nu \sigma } \biggr\}\label{explicit tni in massive dim reg}\\   & & t^{dv , ni}_{\mu \nu \rho \sigma}(p ) = - i \frac{2^{\omega}}{32 } m^{2 } g_{\mu\rho } \int\limits_{0}^{1 } \!dx p_{0 } \biggl\ { 2 \frac{\delta}{\omega-1 } g_{\nu\sigma } + ( 1 - 2x)^{2 } p_{\nu}p_{\sigma } \biggr\}\label{explicit tdvni in massive dim reg}.\end{aligned}\ ] ] these expressions we compare now with the formfactor decompositions ( [ formfactors2 ] ) and ( [ formfactors3 ] ) ( recall eqs.([without interchange ] ) and ( [ axial part with gamma5 symmetrically distributed ] ) ) then we obtain all formfactors of the amplitude @xmath57 explicitly @xmath81 the formfactors of the amplitude @xmath46 follow via eq.([pure vector amplitude ] ) @xmath82 as we can see the formfactors @xmath83 and @xmath84 ( eqs.([t1 in massive dim reg])and ( [ a1 in massive dim reg ] ) ) are finite in the limit @xmath85 and explicitly we find @xmath86 \label{t1r in massive dim reg}\end{aligned}\ ] ] with @xmath87 given by eq.([definition of a ] ) . however , the remaining formfactors are divergent in the limit @xmath85 and we have to renormalize them in an appropriate way . we separate the formfactors into a divergent part and a finite part @xmath88 so that we can extract the finite result by a suitable prescription . the formula we need for this procedure is @xmath89 - \frac{1}{2\pi}\left.\frac{df}{d\omega}\right|_{\omega=1}+ o(1-\omega ) \ , .\end{aligned}\ ] ] it is the pole part @xmath90 of the formfactor which tells us which kind of renormalization we have to choose , for instance , for a constant a simple subtraction is sufficient , for a term proportional to @xmath91 a double subtraction .    in this way the following formfactors are determined by a simple subtraction @xmath92 whereas for the remaining formfactors a double subtraction is needed @xmath93 { } \nonumber\\ { } & = & p^{2}\left [ \frac{1}{18\pi } - \frac{1}{6\pi } \frac{m^{2}}{p^{2 } } - \frac{1}{24\pi } \bigl(1 - 4 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right]\label{t4r in massive dim reg}\\ t^{r}_{5}(p^{2 } ) & = &   t_{5}(p^{2 } ) - t_{5}(0 ) - p^{2 } \left.\frac{d}{dp^{2}}t_{5}(p^{2})\right|_{p^{2}=0 } { } \nonumber\\ { } & = & \frac{1}{16\pi } \int\limits^{1}_{0}\!dx \ p^{2}x(1-x ) \ln \frac{m^{2}-p^{2}x(1-x)}{m^{2 } } { } \nonumber\\{}&= & p^{2}\left[-\frac{5}{288\pi } - \frac{1}{24\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{48\pi } \bigl(1 + 2 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right]\\ a^{r}_{5}(p^{2 } ) & = &   a_{5}(p^{2 } ) - a_{5}(0 ) - p^{2 } \left.\frac{d}{dp^{2}}a_{5}(p^{2})\right|_{p^{2}=0 } { } \nonumber\\ { } & = & \frac{1}{8\pi } \int\limits^{1}_{0}\!dx \biggl[\bigl ( m^{2 } + p^{2}x(1-x ) \bigr ) \ln \frac{m^{2}-p^{2}x(1-x)}{m^{2 } } + p^{2}x(1-x)\biggr ] { } \nonumber\\{}&= & p^{2}\left[-\frac{1}{72\pi } - \frac{1}{3\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{24\pi } \bigl(1 + 8 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right ] . \label{a5r in massive dim reg}\end{aligned}\ ] ] of course , the formfactors @xmath94 of the tensor amplitude @xmath46 satisfy the wi ( [ vwi1 ] )  ( [ vwi3 ] ) and as the difference @xmath47 to the pure tensor amplitude is proportional to @xmath48 ( recall eqs.([pure vector amplitude]),([explicit tdvni in massive dim reg ] ) ) the amplitude @xmath29 will also satisfy the wi ( [ vwi1 ] )  ( [ vwi3 ] ) in the limit @xmath95 . this procedure works in complete analogy to the dispersion relation approach ( see table  1 ) and we clearly obtain the same results for the formfactors as one can see by comparing eqs.([t1r in massive dim reg])([a5r in massive dim reg ] ) with ( [ t1 in dr])([a5r in dr ] ) . even more , as we shall show in the next chapter , the two approaches are equivalent . but before we want to consider the case of massless fermions , the limit @xmath95 , in order to compare our results with the ones of other authors . taking the limit @xmath95 in the @xmath96 expansion ( eq.([pole terms ] ) ) of the formfactors we find the following results @xmath97   \label{t2 for wi } \\ t_{3}(p^{2 } ) & = & -\frac{1}{p^{2}}t_{5}(p^{2 } ) = \mp 4\ t_{8}(p^{2 } ) { } \nonumber\\{}&= & -\frac{1}{96\pi } \biggl [ \frac{1}{\omega-1 } + \gamma - \frac{2}{3 } + \ln\left|\frac{p^{2}}{2\pi\mu^{2}}\right| - i\pi \biggr ]   \label{t3 for wi}.\end{aligned}\ ] ] which agree with the expressions we obtain when starting with @xmath98 from the very beginning ( we also introduced here the mass @xmath99 to keep the correct dimensionality , see ref.@xcite ) . now , for comparison tomiya @xcite in his work on the gravitational anomaly defines the amplitude with the covariant @xmath100-product and calculates the following formfactors @xmath101 they also satiesfy the wi ( [ vwi1 ] )  ( [ vwi3 ] ) , however , they ( some of them ) differ from ours ( [ t1 for wi ] )  ( [ t3 for wi ] ) , which is not surprising as the covariant @xmath100-product differs from the ordinary @xmath102-product by suitable seagull terms . but as we shall show below his and our results lead to the same amplitudes and consequently to the same anomaly expression .    using on one hand the wi ( [ vwi1 ] )  ( [ vwi3 ] ) for the formfactors @xmath103 in the massless limit , @xmath95 , and on the other the fact that @xmath104 we find for the several amplitudes ( [ formfactors1 ] )  ( [ formfactors3 ] ) the following expressions @xmath105 as we can see all amplitudes depend only on the ( convergent ) formfactor @xmath63 , explicitly given by eq.([t1 for wi ] ) , and are independent of a second ( divergent ) formfactor , say @xmath64 , which one might have expected to contribute too ( since we have six restrictive relations between the eight formfactors @xmath103 ) . so the amplitudes do not determine the formfactors uniquely ! therefore @xmath64 can be chosen at will and tomiya @xcite in his work renormalized the formfactors conveniently such that @xmath64 came out to be zero . this is in accordance with the amplitude result of deser and schwimmer @xcite , @xcite who consider fields without chirality .    on the other hand , alvarez - gaum and witten @xcite consider the gravitational anomaly within light - cone coordinates and use a specific regularization prescription to calculate the following amplitude @xmath106 their result is consistent with our calculations as we have @xmath107 { } \nonumber\\{}&= & \frac{1\mp1}{24\pi p^{2}}p_{+}^{4 } \ , \ , .\end{aligned}\ ] ] the dimensional regularization procedure is a method which satisfies the wi ( [ vwi1 ] )  ( [ vwi3 ] ) . on the other hand , within the dispersion relation approach we also have effectively renormalized in a way which keeps this wi property ( [ vwi1 ] )  ( [ vwi3 ] ) . if we had subtracted more often or at some other point than @xmath108 we would have lost this property . the dispersive approach and the dimensional regularization procedure are equivalent in the following sense . for the formfactors that were convergent or logarithmically divergent the corresponding integrals in both approaches do not have just identical values but we even can transform them into each other by a suitable substitution . more precise , we transform the feynman parameter integrals over x in the dimensional regularization procedure into the corresponding dispersion integrals over t. for the formfactors that were linearly divergent the situation is as follows . accidentially for @xmath109 there exists such a substitution , however , not for @xmath110 and @xmath111 . but anyway , for these formfactors the integrals are identical , what is actually sufficient . the substitutions which link the two approaches are @xmath112 to show the equivalence we need the following relations which are valid for any function @xmath113 with suitable differentiation properties @xmath114 now we turn to the calculation of the ward identities and gravitational anomalies . we consider the massless limit , @xmath95 , where the formfactors @xmath115 ( i = 1, ... ,5 ) fulfill the wi ( [ vwi1 ] )  ( [ vwi3 ] ) . this means that the wi for the pure tensor part ( [ ptv ] ) is satiesfied @xmath116 calculating next the wi for the pseudo tensor part ( [ pta ] ) we use the formfactor identities ( [ axial  vector ] ) and we find @xmath117 in the flat space - time limit @xmath118 we have the relation @xmath119 and we finally obtain the anomalous result @xmath120 as we discovered already before when calculating the amplitudes , the anomalous wi depends only on the finite formfactor @xmath121 with its explicit result ( [ t1 for wi ] ) . so the anomaly is independent of a specific renormalization procedure ( as long as it preserves the wi ( [ vwi1 ] )  ( [ vwi3 ] ) ) and we clearly agree with the anomaly results of tomiya @xcite and alvarez - gaum and witten @xcite . we want to emphasize that our subtraction procedure is _ the _ ` natural ' choice dictated by the @xmath122behaviour of the imaginary parts @xmath123 of the formfactors . since on general grounds the imaginary parts of the amplitudes fulfill the wi ( [ vwi1 ] )  ( [ vwi3 ] ) ( in the limit @xmath95 ) the naturally chosen dispersion relations @xmath124=0 \\ & & \frac{1}{\pi}\int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2}}\frac{p^{4}}{t^2}\left[t\ , im\,t_{2}(t)+im\,t_{4}(t)\right]=0 \\ & & \frac{1}{\pi}\int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2}}\frac{p^{4}}{t^2}\left[t\ , im\,t_{3}(t)+im\,t_{5}(t)\right]=0\end{aligned}\ ] ] imply the pure tensor wi ( [ vwi1 ] )  ( [ vwi3 ] ) for the renormalized formfactors ( in the limit @xmath95 ) @xmath125 therefore our subtraction procedure automatically shifts the total anomaly into the pseudotensor part of the wi ( [ pta ] ) . + what is the origin of the anomaly in this dispersive approach ? the source of the anomaly is the existence of a superconvergence sum rule for the imaginary part of the formfactor @xmath63 @xmath126 the anomaly originates from a @xmath0-function singularity of @xmath127 when the threshold @xmath128 approaches zero ( the infrared region ) @xmath129 the limit must be performed in a distributional sense . + then the unsubtracted dispersion relation for @xmath63 , eq.([unsubtracted dr ] ) , provides the result ( [ t1 for wi ] ) . this threshold singularity of the imaginary part of the relevant formfactor is a typical feature of the dispersion relation approach for calculating the anomaly ( see e.g. refs.@xcite , @xcite @xcite , and the appendix b in ref . + next we turn to the energy - momentum tensor . from the anomalous wi @xmath130 we can deduce the linearized consistent einstein ( or diffeomorphism ) anomaly @xmath131 for comparison we demonstrate that result ( [ linearized consistent einstein anomaly ] ) is indeed the linearization of the exact result that follows from differential geometry and topology ( see for instance ref.@xcite ) . the exact einstein anomaly in two dimensions is given by @xmath132 where @xmath133 therefore we get @xmath134 considering the linearized gravitational field , eq.([linearization of the vielbein with upper indices ] ) , the christoffel symbols become ( with explicit @xmath16-dependence ) @xmath135 so that we find as linearization of the exact result ( [ exact consistent einstein anomaly ] ) @xmath136 this agrees with our result ( [ linearized consistent einstein anomaly ] ) since in two dimensions we have the identity @xmath137    now what about the covariant einstein anomaly ? it arises when considering the covariantly transforming energy - momentum tensor @xmath138 which is related to our tensor definition ( [ energy - momentum - tensor ] ) by the bardeen - zumino polynomial @xmath139 @xcite @xmath140 this polynomial is calculable and explicitly we find ( in two dimensions ) @xcite @xmath141 ( @xmath142 denotes the ricci scalar ) which leads to the covariant einstein anomaly @xmath143 linearizing the ricci scalar ( with explicit @xmath16-dependence ) @xmath144 and using ( [ linearization of the christoffel symbols ] ) , ( [ epsilon identity ] ) we get @xmath145 so that we find for the linearized covariant einstein anomaly @xmath146 it is twice the linearized consistent result ( [ linearized consistent einstein anomaly ] ) as it should be . + finally we also calculate the trace identity , eqs.([gtv ] ) and ( [ gta ] ) . using again relations ( [ axial  vector ] ) we find @xmath147,\ ] ] and taking into account the wi ( [ vwi1 ] ) provides us the anomalous result @xmath148.\ ] ] the anomalous ti depends only on the finite formfactor @xmath121 what we know already from the previous discussion so that it is independent of a specific renormalization procedure ( as long as it preserves the wi ( [ vwi1 ] )  ( [ vwi3 ] ) ) .    inserting the formfactor , eq.([t1 for wi ] ) , gives @xmath149,\ ] ] which implies the following linearization of the weyl ( or trace ) anomaly @xmath150.\ ] ]    again , we compare our calculation with the exact result for the weyl anomaly which is given by ( see for instance ref.@xcite ) @xmath151 and consequently we have @xmath152 considering the linearizations of the spin connection ( with explicit @xmath16-dependence ) @xmath153 and of the ricci scalar eq.([linearization of r ] ) , we see that our result ( [ linearized weyl anomaly ] ) is indeed the linearization of the exact expression ( [ exact consistent weyl anomaly ] ) . adding last but not least the bardeen - zumino polynomial @xmath139 @xmath154 with its linearization @xmath155 we find for the covariant trace anomaly @xmath156 and @xmath157 for its linearized version . clearly these results are in agreement with ref . in quantum field theory the energy - momentum tensors form an algebra which is generally not closed but has central extensions , socalled schwinger terms , for example @xmath158 the schwinger terms , the c - number terms @xmath159 , can be determined by considering the vacuum expectation value of the etc .    to evaluate the st we work with a technique that has been introduced by klln @xcite and is closely related to the dispersive approach used before . this technique has been applied already by skora @xcite to compute the st for currents in yang - mills theories . our aim is to generalize this procedure to the case of gravitation , where the current is replaced by the energy - momentum tensor .    from the lagrangian ( [ lagrangian ] ) describing a weyl fermion in a gravitational background field in two dimensions we get the following classical energy - momentum tensor @xmath160 where @xmath161 means normal ordering . using relation ( [ elimination of gamma5 ] ) and the equations of motions we can express the pseudo tensor part of the enery - momentum tensor by the pure tensor part ( recall that the tensor is symmetric ) @xmath162 in two dimensions we have the identity @xmath163 so that we find @xmath164\vert 0\rangle = \frac{1}{4 } \biggl\ { \langle 0\vert [ t^{v}_{\mu\nu}(x),t^{v}_{\rho\sigma}(0)]\vert 0\rangle + \langle 0\vert [ t^{v}_{\rho\nu}(x),t^{v}_{\mu\sigma}(0)]\vert 0\rangle { } \nonumber\\ { } & & \qquad- g_{\mu\rho } \langle 0\vert [ t^{v\lambda}_{\nu}(x),t^{v}_{\lambda\sigma}(0)]\vert 0\rangle   \mp \varepsilon_{\mu}^{\ \lambda}\langle 0\vert [ t^{v}_{\lambda\nu}(x),t^{v}_{\rho\sigma}(0)]\vert 0\rangle \mp \varepsilon_{\rho}^{\ \lambda } \langle 0\vert [ t^{v}_{\mu\nu}(x),t^{v}_{\lambda\sigma}(0)]\vert 0\rangle \biggr\}.\end{aligned}\ ] ] let us define @xmath165 by inserting the completeness relations @xmath166 and using the translation invariance @xmath167 we obtain @xmath168 where the sum runs over many - particle states @xmath169 with positive energy and momentum @xmath170 . + we may write @xmath171 where @xmath172 from lorentz covariance and symmetry we get the same decompositon into formfactors as for @xmath29 ( see eq . ( [ formfactors2 ] ) @xmath173 making use of @xmath174 provides the ward identity @xmath175 that can be expressed by the formfactors @xmath176 now let us explicitly evaluate @xmath177 . as we are considering the energy - momentum tensor as a free ( non interacting ) tensor  analogously to the case of free currents  we only need to sum up states that consist of one fermion - antifermion pair in eq . ( [ symbolic gmunurhosigma ] ) . we get @xmath178 where @xmath179 let us assume that the fermion fields are canonically quantized @xmath180\\ \bar\psi(x)&= & \frac{1}{(2\pi)^{1/2}}\int\!dp\sum_{s=1}^{2}\sqrt{\frac{m}{e_{p}}}\left [ b^{\dagger(s)}(p)\bar u^{(s)}(p)e^{ipx}+d^{(s)}(p)\bar v^{(s)}(p)e^{-ipx}\right],\end{aligned}\ ] ] then we find @xmath181 this provides @xmath182 without the interchanges we call this @xmath183 . + if we use the completeness relations @xmath184 and the integral equation @xmath185 we obtain @xmath186.\end{aligned}\ ] ] integrating next over the first @xmath0-function and evaluating the trace gives @xmath187.\end{aligned}\ ] ] if we compare this with ( [ abstract i m tni ] ) and ( [ abstract i m tdvni ] ) ( @xmath188 ) , we see that @xmath177 is given by the imaginary part of the amplitude of the pure vector loop @xmath189 this is the important relation which links the schwinger terms to the gravitational anomalies .    from eq . ( [ i m a1 ] )  ( [ i m a5 ] ) we read off the formfactors @xmath190 now we consider the commutator @xmath191 in terms of g } \langle 0\vert [ t^{v}_{\mu\nu}(x ) , t^{v}_{\rho\sigma}(0 ) ] \vert 0\rangle = f_{\mu\nu\rho\sigma}(x)-f_{\rho\sigma\mu\nu}(-x ) = \int\!d^{2}p\ e^{-ipx } \varepsilon(p^{0 } ) g_{\mu\nu\rho\sigma}(p).\ ] ] if we remove the mass , @xmath192 , that acted as an infrared cutoff , we get @xmath193 from eq . ( [ [ t , t ] in terms of g ] ) we explicitly find @xmath194 \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{1}^{4}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) = 0 \\ \langle 0\vert [ t^{v}_{11}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}^{4}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) = 0 \\ \langle 0\vert [ t^{v}_{00}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } & = & \langle 0\vert [ t^{v}_{01}(x ) , t^{v}_{01}(0 ) ] \vert 0\rangle_{et } { } \nonumber\\{}&= &   \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}^{2}p_{1}^{2}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) = 0 \\ \langle 0\vert [ t^{v}_{00}(x ) , t^{v}_{01}(0 ) ] \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}p_{1}^{3}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) \\ \langle 0\vert [ t^{v}_{01}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}^{3}p_{1}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) \ , .\label{[t01,t11 ] no interchange } \end{aligned}\ ] ] the first three expressions vanish because @xmath195 is antisymmetric . to evaluate the next two we use the pauli - jordan function @xmath196 with the properties @xmath197 and we find @xmath198 so we conclude @xmath199 \vert 0\rangle_{et } & = & \langle 0\vert [ t^{v}_{01}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } { } \nonumber\\{}&= & \lim_{x_{0}\rightarrow 0}-\frac{1}{12\pi}\partial_{1}^{3}\partial_{0}\triangle(x ) = \frac{i}{12\pi}\partial_{1}^{3}\delta(x^{1 } ) \ , .\end{aligned}\ ] ] with relation ( [ symbolic vev of commutator ] ) we finally obtain the schwinger terms in the etc of the energy - momentum tensors @xmath200 \vert 0\rangle_{et } & = & \langle 0\vert [ t_{11}(x ) , t_{11}(0 ) ] \vert 0\rangle_{et } = \mp\frac{i}{24\pi}(\partial_{1})^{3}\delta(x^{1})\\   \langle 0\vert [ t_{00}(x ) , t_{11}(0 ) ] \vert 0\rangle_{et } & = & \langle 0\vert [ t_{01}(x ) , t_{01}(0 ) ] \vert 0\rangle_{et } = \mp\frac{i}{24\pi}(\partial_{1})^{3}\delta(x^{1})\\ \langle 0\vert [ t_{00}(x ) , t_{01}(0 ) ] \vert 0\rangle_{et } & = & \langle 0\vert [ t_{01}(x ) , t_{11}(0 ) ] \vert 0\rangle_{et } = \frac{i}{24\pi}(\partial_{1})^{3}\delta(x^{1 } ) \ , .\end{aligned}\ ] ] our result agrees with the one of tomiya @xcite who works with an invariant spectral function method and in addition uses a cohomological approach . it also coincides with the result of ebner , heid and lopes - cardoso @xcite who derive the schwinger terms directly from the gravitational anomaly . in our approach ( [ g and tpv ] ) is the basic relation . it connects the schwinger terms , determined by @xmath201 , with the ( linearized ) gravitational anomalies given by @xmath202 in our dispersion relations procedure . we have investigated the gravitational anomalies , specifically the pure einstein anomaly and the weyl anomaly . so we demanded the quantized energy - momentum tensor to be symmetric  no lorentz anomaly occurs  which is a possible choice . the relevant amplitude , the two - point function of the energy - momentum tensors @xmath57 , we have separated into its pure tensor part @xmath29 and into its pseudo - tensor part @xmath30 , eqs.([amplitude expressed by t - product ] )  ( [ formfactors3 ] ) , and we have decomposed the amplitudes into a general structure of tensors containing 8 formfactors @xmath31 . these formfactors we have expressed by dispersion relations where we had to calculate only the imaginary parts via the cutkosky rules . our subtraction procedure for the formfactors  no subtraction for @xmath83 , one subtraction for @xmath203 and two subtractions for @xmath204  is _ the _ natural choice dictated by the @xmath122behaviour of the imaginary parts @xmath123 . it implies that the pure tensor wi ( [ vwi1 ] )  ( [ vwi3 ] ) for the renormalized formfactors is satisfied ( in the limit @xmath95 ) , so that the total anomaly is automatically shifted into the pseudotensor part of the wi ( [ pta ] ) . it turns out that the anomalous ward identity and the anomalous trace identity depend only on the finite formfactor @xmath121 , with its explicit result ( [ t1 for wi ] ) , demonstrating such the independence of a special renormalization procedure . from the anomalous ward identity and the anomalous trace identity we could deduce the linearized gravitational anomalies the linearized einstein- and weyl anomaly  and determine their covariant versions . the origin of the anomaly is the existence of a superconvergence sum rule for the imaginary part of the formfactor @xmath63 . in the zero mass limit the imaginary part of the formfactor approaches a @xmath0-function singularity at zero momentum squared , exhibiting in this way the infrared feature of the gravitational anomalies this is an independent and complementary view of the anomalies as compared to the ultraviolet regularization procedures . if we compare , however , the dr approach with the n - dimensional regularization procedure of t  hooft  veltman we find an equivalence . the two approaches are linked by the substitutions ( [ substitution dis rel - dim reg ] ) . we have also calculated the gravitational schwinger terms which occur in the etc of the energy - momentum tensors . we have adapted our dispersive approach to the method of klln . as a result all gravitational schwinger terms are determined by the formfactor @xmath205 which in the zero mass limit approaches a @xmath0-function singularity at zero momentum squared  as in the case of anomalies . so also the schwinger terms show this peculiar infrared feature of the anomalies . we have performed all calculations in two dimensions , where already all essential features of the dr approach show up , analogously to the chiral current case . from a practical point of view the method appears quite appealing . all one has to calculate is the imaginary part of an amplitude ( formfactor ) , which is an easy task . however , this computational simplicity is a special ( and convenient ) feature of the two space - time dimensions . in higher dimensions the amplitude will contain more formfactors and we have to calculate dispersion relations for higher loop diagrams , which is a much more delicate task , but nevertheless we expect the method to work here too . r. jackiw , _ field theoretic investigations in current algebra , topological investigations of quantized gauge theories _ , in : _ current algebra and anomalies _ , treiman , r. jackiw , b. zumino and e. witten ( eds . ) , p.81 , and p.211 , world scientific , singapore ( 1985 ) . p. van nieuwenhuizen , _ anomalies in quantum field theory : cancellation of anomalies in @xmath206 supergravity . leuven notes in mathematical and theoretical physics _ , vol .  3 , leuven university press ( 1988 ) .
we are dealing with two - dimensional gravitational anomalies , specifically with the einstein anomaly and the weyl anomaly , and we show that they are fully determined by dispersion relations independent of any renormalization procedure ( or ultraviolet regularization ) . 
 the origin of the anomalies is the existence of a superconvergence sum rule for the imaginary part of the relevant formfactor . in the zero mass limit the imaginary part of the formfactor approaches a @xmath0-function singularity at zero momentum squared , exhibiting in this way the infrared feature of the gravitational anomalies . 
 we find an equivalence between the dispersive approach and the dimensional regularization procedure . 
 the schwinger terms appearing in the equal time commutators of the energy momentum tensors can be calculated by the same dispersive method . 
 although all computations are performed in two dimensions the method is expected to work in higher dimensions too .