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the anomalous magnetic moment of the muon has served over many years as an important test of the standard model , see the reviews  @xcite . it is also sensitive to potential contributions from new physics . the current status of the muon @xmath0 is summarized in table  [ tab : status_g-2 ] where we list the different contributions in theory ( qed , weak , hadronic ) from various recent sources and compare with the experimental value . more references to earlier work can be found in the quoted papers and in refs . the experimental world average is dominated by the final result of the brookhaven muon @xmath0 experiment  @xcite , corrected for a small shift in the ratio of the magnetic moments of the muon and the proton  @xcite . we observe a difference between experiment and theory of more than three standard deviations @xmath2 unfortunately , the theoretical uncertainties  @xcite from hadronic vacuum polarization ( hvp ) and hadronic light - by - light scattering ( hlbl ) make it difficult to interpret this discrepancy as a clear sign of new physics . most recent evaluations  @xcite , which differ slightly in the treatment of the hadronic contributions , obtain deviations of @xmath3 . in ref . @xcite the hadronic cross - section data below @xmath4 was fitted to a ( broken ) hidden local symmetry ( hls ) model and a discrepancy in the muon @xmath0 between @xmath5 was observed , depending on the selected data . .standard model contributions to @xmath6 and comparison of theory and experiment . [ cols="<,>,>,^",options="header " , ]     a selection of estimates for hlbl is presented in table  [ tab : hlbl_summary ] . note that the refs . @xcite are the only full calculations of hlbl to date , using , as much as possible , one model for all the contributions ( hls model in ref . @xcite , extended nambu - jona - lasinio ( enjl ) model in ref . both calculations showed that the exchanges of the lightest pseudoscalar states , @xmath7 , dominate numerically , which can be understood from the large-@xmath8 counting . the contributions from the ( dressed ) pion - loop and the ( dressed ) quark - loop are subdominant , but not negligible , and they happen to largely cancel each other numerically . the final results for the total hlbl contribution were rather close in both models . in ref . @xcite an ansatz for the pion - photon transition form factor with a minimal number of narrow vector resonances in large-@xmath8 qcd ( lowest meson dominance ( lmd , lmd+v ) ) was matched to short - distance constraints from the ope . the reevaluation of the pion - pole contribution to hlbl in ref . @xcite with the ansatz from ref . @xcite then revealed a sign error in the earlier calculations  @xcite . furthermore , for the neutral pion contribution , a 2-dimensional integral representation was derived in ref . @xcite for a certain class of vector meson dominance ( vmd)-like form factors . in the end , one integrates over the lengths of the two independent loop - momenta @xmath9 in euclidean space . schematically , @xmath10 with universal weight functions @xmath11 and where the ( model ) dependence on the form factors resides in the functions @xmath12 , see ref . @xcite for details . note that the expressions in ref . @xcite with on - shell form factors are in general not valid as they stand . as pointed out in ref . @xcite , one needs to set the form factor at the external vertex to a constant to obtain the pion-_pole  _ contribution . but the expressions are valid for the constant wess - zumino - witten ( wzw ) form factor and the off - shell vmd form factor . the weight functions @xmath11 are plotted in figure  [ fig:2d_weightfunctions ] . one can see that the relevant momentum regions are about @xmath13 , see also the numerical analysis in ref . as long as the form factors in different models lead to a damping , we expect comparable results for @xmath14 at the level of 20% , which is indeed the case , see the papers quoted in table  [ tab : hlbl_summary ] and ref . @xcite . for the general case with off - shell form factors , a 3-dimensional integral representation was derived in ref . @xcite , where the integrations run over @xmath15 and @xmath16 , where @xmath17 .     in eq . ( [ 2d_integral ] ) from ref . @xcite.,scaledwidth=80.0% ]    later , ref . @xcite derived new short - distance constraints from the four - point function on the pion - pole and axial - vector - pole contributions , which do not allow for any form factors at the external vertex . reference  @xcite also included the mixing of two axial - vector nonets and studied the pion - loop within the hls model in more detail . all this lead to a substantial enhancement of these contributions to hlbl . more recently , a new short - distance constraint on the off - shell form factor at the external vertex in pion - exchange was derived in ref . @xcite , which yielded , again in the framework of the lowest meson dominance approximation to large-@xmath8 qcd , a value for this contribution about half - way between the results in refs . @xcite and those in ref . note that the compilations  @xcite and ref . @xcite are largely based on the full calculations  @xcite , with revised or newly calculated values for some of the contributions . more recent estimates , mostly for the pseudoscalar contribution , can be found in ref . while most of these evaluations agree at the level of 15% , if one takes the extreme values , there is a spread of @xmath18 .    until 2010 , a consensus had been reached about the central value @xmath19 , but there was a discussion on how to estimate the error , more progressively , @xmath20 , in ref . @xcite and more conservatively , @xmath21 , in refs . @xcite . in view of the precision goal of future @xmath0 experiments at fermilab and j - parc  @xcite with @xmath22 and the continued progress in improving the error in hvp , the hlbl contribution might soon be the main uncertainty in the theory prediction , if it can not be brought under better control  @xcite .    in the last few years , several works have appeared which yield much larger ( absolute ) values for some of the contributions , see table  [ tab : hlbl_summary ] . in ref . @xcite the quark - loop was studied using a dyson - schwinger equation approach . in contrast to refs . @xcite , no damping compared to the bare constituent quark - loop result was seen , when a dressing was included . note that this calculation of the quark - loop is not yet complete and that earlier , very large results for the quark - loop seem to have been affected by some errors in the numerics in certain parts . the large size of the quark - loop contribution in ref . @xcite was questioned in the papers  @xcite , using different quark - models and approaches , see also the ballpark prediction for hlbl in ref . the pion - loop contribution was analyzed in ref . the authors stressed the importance of the pion - polarizability effect and the role of the axial - vector resonance @xmath23 , which are not included in the models used in refs . @xcite . depending on the value of the pion - polarizability and the model for the @xmath23 resonance used , a large variation was seen . the issue was taken up in ref . @xcite where different models for the pion - loop were studied . the inclusion of the @xmath23 resonance was attempted , but no finite result for @xmath0 could be achieved . with a cutoff of @xmath4 , a result close to the earlier estimate in ref . @xcite was obtained . reference  @xcite also pointed out that the very small ( absolute ) value for the pion - loop in ref . @xcite could be due to the fact that the hls model used in ref . @xcite has a wrong high - energy behavior and that there is some cancellation between positive and negative contributions in the pion - loop in hlbl . concerning the future , maybe lattice qcd will provide a reliable calculation of hlbl at some point , see ref . @xcite for some promising recent results . in the meantime , only a close collaboration between theory and experiment can lead to a better controlled estimate for hlbl . on the theory side , the hadronic models can be improved by short - distance constraints from perturbative qcd to have a better matching at high momenta . one can also use dispersion relations to connect the theory with experimental data , e.g. in @xmath24  @xcite . also the issue about whether the dressing of the bare constituent quark - loop leads to a suppression or an enhancement needs to be studied further . this problem is also related to the question whether there is any double counting involved .    on the experimental side , the information on various processes ( decays , form factors , cross - sections ) of hadrons interacting with photons at low and intermediate momenta , @xmath25 , can help to constrain the models . important experiments which should be pursued include more precise measurements of the ( transition ) form factors of light pseudoscalars with possibly two off - shell photons in the process @xmath26 and the two - photon decay width and the ( double ) dalitz decays of these mesons . this could further reduce the error of the dominant pseudoscalar exchange contribution  @xcite . concerning the pion - loop contribution , in addition to studying @xmath27 , measurements of the pion - polarizability in various processes , e.g.  in radiative pion decay @xmath28 , in radiative pion photoproduction @xmath29 or with the hadronic primakoff effect @xmath30 or @xmath31 ( with some nucleus @xmath32 ) , can help to improve the models  @xcite . for the development of models with the axial - vector resonance @xmath23 and estimates of the sizable axial - vector contribution , information about the decays @xmath33 would be useful as well . finally , to extract the needed quantities from experiment will also require the development of dedicated monte carlo programs for the relevant processes  @xcite . if the recent results for the quark - loop and pion - loop are taken at face value , one obtains the range @xmath34 . while the new approaches raise some important issues and point to potential shortcomings in the previously used models , these calculations are also still preliminary and further studies are needed . therefore , the estimate @xmath35 from refs . @xcite still seems to give a fair description of the current situation . i would like to thank the organizers of the _ international workshop on @xmath36 collisions from @xmath37 to @xmath38 2013 ( phipsi13 ) _ , of the _ working group on radiative corrections and monte carlo generators for low energies _ and of the _ lc13 workshop : exploring qcd from the infrared regime to heavy flavour scales at b - factories , the lhc and a linear collider _ for the opportunity to present reviews on the hlbl in the muon @xmath0 . generous support from the organizers , the infn , the ect@xmath39 , trento , italy , and the heinrich - greinacher - stiftung , physics institute , university of bern , switzerland , is gratefully acknowledged . the aec is supported by the `` innovations- und kooperationsprojekt c-13 '' of the `` schweizerische universittskonferenz suk / crus . 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we give an update on the status of the hadronic light - by - light scattering contribution to the muon @xmath0 . 
 we review recent work by various groups , list some of the open problems and give an outlook on how to better control the uncertainty of this contribution . 
 this is necessary in order to fully profit from planned future muon @xmath0 experiments to test the standard model . 
 despite some recent developments , we think that the estimate @xmath1 still gives a fair description of the current situation . 
 december 17 , 2013     + andreas nyffeler + _ regional centre for accelerator - based particle physics , harish - chandra research institute , chhatnag road , jhusi , allahabad - 211 019 , india _ + _ albert einstein center for fundamental physics , institute for theoretical physics , university of bern , sidlerstrasse 5 , ch-3012 bern , switzerland _ 
 pacs numbers : 14.60.ef , 13.40.em , 12.38.lg