document: the cosmological constant problem ( ccp ) is a biggest puzzle in particle physics  @xcite , and consists of several pieces . 
   the first piece is that the vacuum energy density @xmath4 can be the cosmological constant @xmath5 and various sources of @xmath4 exist , e.g. , a zero point energy of each particle and potential energies accompanied with phase transitions such as the breakdown of electroweak symmetry via higgs mechanism and the chiral symmetry breaking due to quark condensations . 
   the second one is that @xmath4 can receive large radiative corrections including a cutoff scale . 
   the third one is that the experimental value of @xmath5 is estimated as @xmath6gev@xmath7 where @xmath8 is the newton constant , and @xmath9 . ] from the observation that the expansion of present universe is accelerating  @xcite . 
   the energy density defined by @xmath10 is referred as @xmath11dark energy density  , and its existence has been a big mystery .    the pieces of puzzle are not fitted in the framework of the einstein gravity and the standard model of particle physics . 
   because a fundamental theory including gravity has not yet been established , it would be meaningful to give a suggestion based on an effective description of various experimental results concerning the vacuum energy . 
   it might be necessary to introduce a principle , assumptions and/or a framework beyond common sense of an accepted physics , and then the ccp is replaced by the problem to disclose the essence of new staffs .    in this article , we study physics on the ccp in the framework of effective field theory and suggest that a dominant part of dark energy can originate from gravitational corrections of vacuum energy under the assumption that the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . 
   the content of our article is as follows . in the next section , 
   we explain the pieces of puzzle on the ccp and their implications . in sect . 
   3 , we give an effective description for physics concerning the ccp and predict that gravitational corrections of vacuum energy can be a candidate of dark energy . in the last section , 
   we give conclusions and discussions . 
   _ the vacuum energy density @xmath4 can be the cosmological constant @xmath5 . _ 
   the energy - momentum tensor of perfect fluids is given by @xmath12 where @xmath13 , @xmath14 and @xmath15 are an energy density , a pressure and a four - velocity of fluids , respectively . 
   the energy - momentum tensor of vacuum is of the form , @xmath16 where @xmath17 is a constant .    from ( [ t - pf ] ) and ( [ t - v ] ) 
   , we obtain the relations , @xmath18 where @xmath19 and @xmath20 are an energy density and a pressure of vacuum , respectively . 
   the vacuum has a negative pressure with @xmath21 . 
   the einstein equation is given by @xmath22 where @xmath23 is a bare cosmological constant . from ( [ t - v ] ) and ( [ e - eq ] ) , the cosmological constant is given by @xmath24 , effectively . 
   _ there can be various sources of @xmath4 . 
   _    the first one is a zero point energy of each particle . for a relativistic bosonic particle with a mass @xmath25 , 
   the energy density @xmath26 and the pressure @xmath27 due to zero point fluctuations are given by @xmath28 respectively . 
   here @xmath29 is a momentum of particle . 
   the second one is the energy density from the higgs potential after the breakdown of electroweak symmetry , and its absolute value is estimated as @xmath30 where @xmath31gev ) is the vacuum expectation value on the neutral component of higgs doublet . 
   the third one is the energy density accompanied with the chiral symmetry breaking due to quark condensations , and its absolute value is estimated as @xmath32 where @xmath33 is the qcd scale . 
   _ the vacuum energy density can receive large radiative corrections including a cutoff scale . 
   _ the zero point energy density is calculated by using an effective potential at the one - loop level , and it naively contains quartic , quadratic and logarithmic terms concerning an ultra - violet ( uv ) cutoff parameter @xmath34 .    by imposing the relativistic invariance of vacuum ( [ rel - v ] ) on @xmath26 and @xmath27 , @xmath26 in ( [ zero - point ] ) should be of the form  , @xmath35 up to some finite terms . note that the terms proportional to @xmath36 and @xmath37 do not satisfy @xmath38 , and they can be regarded as artifacts of the regularization procedure . after the subtraction of logarithmic divergence , @xmath26 is given by @xmath39 where @xmath40 is a renormalization point .    for the higgs boson , its zero point energy is estimated as @xmath41 where we use @xmath42gev for the higgs boson mass and take @xmath43gev corresponding to the temperature of present universe @xmath44k .      _ the experimental value of cosmological constant is estimated as @xmath45gev@xmath7 , from the observation that the expansion of our present universe is accelerating . _ 
   the vacuum energy density of universe is theoretically given by @xmath46 where @xmath47 is the zero point energy density due to a particle labeled by @xmath48 and the ellipsis stands for other contributions containing unknown ones from new physics .    from ( [ rho - higgs ] ) , ( [ rho - qcd ] ) and ( [ rho - zero - higgs ] ) 
   , we have the inequalities , @xmath49 where @xmath50 is a dark energy density defined by @xmath10 . 
   there is a possibility that the magnitude of @xmath4 becomes the 4-th power of the terascale through a cancellation among various contributions from a higher energy physics based on a powerful symmetry such as supersymmetry . because supersymmetry can not work to reduce @xmath4 close to @xmath50 , an unnatural fine - tuning is most commonly required to realize @xmath50 .    from ( [ rho - th ] ) and ( [ rho - ineq ] ) 
   , we have a puzzle that consists of unfitted pieces . 
   nature proposes us a big riddle @xmath51why is the observed vacuum energy density so tiny compared with the theoretical one? and a big mystery @xmath51what is an identity of dark energy density? 
   to uncover a clue of ccp and probe into an identity of @xmath50 , let us start with the question whether @xmath4 in ( [ rho - th ] ) exists in physical reality , it gravitates or the classical gravitational field feels @xmath4 .    in the absence of gravity , the vacuum energy from matters 
   @xmath52 itself is not observed directly because there is a freedom to shift the origin of energy . 
   here , matters mean various fields including radiations such as photon except for graviton . for instance , the zero point energy of free fields is removed by taking a normal ordering in the hamiltonian . only energy differences can be physically meaningful , as suggested by the casimir effect .    in the presence of gravity , 
   if @xmath4 gravitates , the motion of the planets in our solar system can be affected by @xmath4  @xcite . from the non - observation of such an effect for mercury 
   , we have a constraint , @xmath53 as seen from ( [ rho - higgs ] ) , ( [ rho - qcd ] ) and ( [ rho - zero - higgs ] ) , the existence of @xmath54 , @xmath47 ( for particles heavier than 10ev ) and/or @xmath55 threatens the stability of our solar system . 
   hence , it seems to be natural to suppose that the classical gravitational field does not feel a large portion of @xmath52 . 
   in contrast , the ratios of the gravitational mass to the inertial mass stay for heavy nuclei , and hence it is reasonable to conclude that the equivalence principle holds with accuracy at the atomic level and the gravitational field couples to every process containing radiative corrections , accompanied by an emission and/or an absorption of matters  . 
   more specifically , the external matter - dependent part of energies must gravitate in both macroscopic and microscopic world . 
   now , let us move to the next step , as phenomenological ingredients of ccp are already on the table . 
   first , based on a standpoint that the einstein gravity is a classical effective theory , physics on the ccp can be described by @xmath56,~~ \rho_{\rm de } \equiv \varlambda_{\rm c(exp)}/(8 \pi g ) = 2.4 \times 10^{-47 } { \rm gev}^4 , \label{scl}\end{aligned}\ ] ] where @xmath57 is the ricci scalar made of the classical gravitational field @xmath58 , @xmath59 , @xmath60 is the lagrangian density of matters as classical objects and @xmath60 does not contain a constant term . because an effective theory is , in general , an empirical one 
   , it would not be so strange even if it can not answer the questions why a large portion of @xmath52 does not gravitate and what the identity of dark energy is . 
   those questions remain as subjects in a fundamental theory . 
   second , we explain why it is difficult to derive ( [ scl ] ) in the framework of ordinary quantum field theory , starting from the action , @xmath61 , \label{s}\end{aligned}\ ] ] where @xmath62 is the ricci scalar made of the graviton @xmath63 , @xmath64 , and @xmath65 and @xmath66 are the lagrangian densities for the standard model particles and other particles beyond the standard model , respectively . 
   the amplitudes representing the coupling between gravitons and the vacuum energy are evaluated by calculating green s functions , @xmath67 for example , on the background minkowski spacetime , two - point function is written as @xmath68 where @xmath69 is the quantum part of graviton in the interaction picture and @xmath70 is the interaction hamiltonian density . 
   the transition amplitude is obtained by removing the propagators on the external lines . 
   the vacuum expectation value @xmath71 corresponds to the vacuum energy density , and then a large cosmological constant term is derived after the identification of classical gravitational fields for external gravitons . here , 
   external gravitons mean gravitons in real states represented by wave functions . 
   third , to reconcile ( [ scl ] ) and ( [ g2 ] ) , we need a radical idea and take a big assumption that _ the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . 
   _ we expect that it stems from unknown features of external gravitons . 
   for example , if a kind of exclusion principle works , as a bold hypothesis , that _ external gravitons can not take the same place in the zero total four - momentum state _ , external gravitons would not feel the vacuum energy . 
   however , if it holds in the strong form , we would arrive at undesirable conclusions such as the violation of the equivalence principle for external matter fields with the zero total four - momentum , the vanishing scattering amplitudes among only gravitons and the absence of dark energy . to improve them , 
   we need another assumption such that _ the exclusive attribute of external gravitons is violated by the coupling of external matter fields ( at the same point and/or different ones ) , gravitons with derivatives or internal gravitons . 
   _ here , internal gravitons mean gravitons in virtual states represented by propagators . 
   note that the exclusion principle is merely an example of reasoning to justify the first assumption . 
   the point of the second one is that _ external gravitons can couple to gravitational corrections of vacuum energy involving internal gravitons . _    under the above assumptions , we give a conjecture on a candidate of dark energy for the case that the minkowski spacetime is taken as a background one , i.e. , @xmath72 . in this case , the full propagator of graviton is proportional to @xmath73 . here 
   , @xmath74 is the gravitational scale ( the reduced planck scale ) defined by @xmath75gev ) . 
   using the propagator , we obtain the gravitational corrections of @xmath76 , @xmath77 at one - loop level . by replacing @xmath76 into @xmath52 
   , we obtain the zero point energy density of @xmath78 , @xmath79 after the subtraction of logarithmic divergence . 
   if @xmath80 dominates @xmath50 , the magnitudes of @xmath81 and @xmath52 are estimated as @xmath82 and @xmath83 respectively . here 
   we take @xmath43gev . 
   then , we have a conjecture that physics around the terascale is relevant to the dark energy of our universe . 
   if the zero point energy of some scalar field dominates @xmath52 , such scalar field has a mass of @xmath84tev and become a candidate of dark matter called @xmath11wimp ( weakly interacting massive particle )  . 
   if superpartners appear around the terascale , they can produce zero point energies of @xmath84tev@xmath7 . in this case 
   , we obtain an interesting scenario that a vacuum acquires an energy of @xmath84tev@xmath7 from zero point energies of dark matter and/or superpartners , it does not gravitate directly and the zero point energy of graviton becomes a source of dark energy . 
   let us study the evolution of @xmath50 in the case with @xmath85 and @xmath86 . 
   if we identify @xmath40 with a temperature of the universe , @xmath50 is not constant but varying logarithmically such that @xmath87 , \label{rho - de}\end{aligned}\ ] ] where @xmath44k and @xmath88 @xmath89 is a scale factor of the universe ( the present one ) . here 
   , we also use the fact that the temperature is inversely proportional to the radius of our universe . 
   the evolution of several energy densities are depicted in figure [ f1 ] .    , @xmath90 and @xmath50 , respectively . here , 
   @xmath91 and @xmath90 are energy densities of radiations and non - relativistic matters ( except for dark matter ) , respectively.,title="fig:",width=377 ] -    in the appendix , we derive ( [ rho - de ] ) and show that the logarithmically changing energy density is described by the equation of state @xmath92 . 
   if @xmath50 evolves as ( [ rho - de ] ) , there are several possibilities for physics beyond the terasclae . 
   first one is that there is no sensitive physics to contribute the vacuum energy beyond the terascale , i.e. , no processes associated with a huge vacuum energy such as the breakdown of grand unified symmetry and no superheavy particles generating huge zero point energies . 
   second one is that there is a higher energy physics , but a miraculous cancellation can occur among various contributions based on a powerful symmetry such as supersymmetry . 
   then , the vacuum energies can be diminished in supersymmetric grand unified theories and/or supergravity theories , and the vacuum energy due to inflaton can vanish at the end of inflation in early universe . 
   third one is that there survives a huge vacuum energy originated from a higher energy physics in the absence of a powerful mechanism , but even virtual gravitons do not couple to such a vacuum energy beyond the terascale . 
   then , another graviton could be required to realize a higher energy physics . 
   finally , we pursue a last possibility in the presence of the zero point energy of inflaton with a mass of @xmath93gev . 
   we consider the action , @xmath94 , \label{tildes}\end{aligned}\ ] ] where @xmath95 is a coupling constant , @xmath96 is the ricci scalar made of another graviton @xmath97 , @xmath98 , and @xmath99 is the lagrangian density of inflaton . 
   we assume that the ordinary graviton does not couple to inflaton directly . in the same way as the ordinary graviton 
   , we obtain the zero point energy of @xmath97 , @xmath100 where @xmath101 . 
   if @xmath102 dominates @xmath50 with @xmath103gev@xmath7 , the magnitude of @xmath104 is estimated as @xmath105gev . 
   we have studied physics on the ccp in the framework of effective field theory and suggested that a dominant part of dark energy can originate from gravitational corrections of vacuum energy of @xmath84tev@xmath7 , under the following assumptions .    * the graviton @xmath78 couples to the potential of matter fields with the strength of @xmath106 , and couples to the vacuum energy of matters with the same strength in the virtual state . 
   * the classical gravitational fields do not couple to a large portion of the vacuum energy effectively . * 
   external gravitons can couple to gravitational corrections of vacuum energy involving internal gravitons . 
   the second one is beyond our comprehension , because it is difficult to understand it in the framework of ordinary local quantum field theory . 
   however , if the ccp is a highly non - trivial problem that can not be solved without a correct theory of gravity in a proper manner , it would not be so strange that it is not explained from the present form of quantum gravity theory . 
   it ease a major bottleneck of the ccp if realized with unknown features of gravity at a more fundamental level , and hence we have tried to step boldly from common sense . as an example of reasoning , we have presented a kind of exclusion principle that external gravitons can not take the same place in the zero total four - momentum state , unless they couple to external matter fields ( at the same point and/or different ones ) , gravitons with derivatives or internal gravitons . 
   it may provide a useful hint to disclose physics behind the ccp . 
   the universe dominated by our dark energy might cause instability , because it does not satisfy the energy condition such as @xmath107 for perfect fluids . 
   a phenomenologically viable model must be fulfilled the requirements that matters are stable and a time scale of instability should be longer than the age of universe . 
   if our speculation were correct , the ccp is replaced by the challenge to construct a microscopic theory of gravity compatible with the above assumptions . 
   we might need a new ingredient or a novel formalism . 
   it would be important to pursue much more features of graviton and the relationship between the classical gravitational field and the quantum one . 
   in such a case , the theory of fat graviton may provide a helpful perspective and clue . 
   we would like to thank prof . 
   t.  inami , prof . k.  izumi and prof . 
   m .  ho for useful comments on the bimetric theory and the area - metric theory . 
   according to them , we have returned the title and contents of our article to the original one . 
   we discuss the evolution of dark energy ( [ rho - de ] ) . 
   the effective potential including contributions of graviton at one - loop level is given by @xmath108 where @xmath76 is the potential of matters . from the feature that @xmath109 is independent of the renormalization point @xmath40 , 
   i.e. , @xmath110 , we obtain the relation , @xmath111 from ( [ dv - eff ] ) , we find that the magnitude of @xmath112 is much less than @xmath52 if @xmath52 is much less than @xmath113 . 
   then , the zero point energy density @xmath114 given by ( [ gr - loop ] ) varies on @xmath40 almost logarithmically . 
   hence , the relation ( [ rho - de ] ) is derived after @xmath114 is regarded as the dark energy density @xmath50 and @xmath40 is identified with the temperature , that is proportional to the radius of our universe . 
   the potential @xmath76 changes after the incorporation of zero point energies of particles and the breakdown of symmetry , and contains @xmath4 given in ( [ rho - th ] ) at present . from the above observation , the magnitude of @xmath76 is almost constant except for the period of the early universe . 
   next , we show that the logarithmically changing energy density is described by the van der waals type equation of state , @xmath115 where @xmath116 is a positive constant . 
   the energy conservation @xmath117 is rewritten as @xmath118 where @xmath88 is the radius of our universe . from ( [ eq - state ] ) and ( [ drho ] ) , we derive the differential equation @xmath119 and obtain the solution , @xmath120 where @xmath121 is a constant . by comparing ( [ rho - de ] ) with ( [ rho - b ] ) , @xmath116 and @xmath121 
   are determined as @xmath122 , \label{b - b0}\end{aligned}\ ] ] respectively . 
   it is pointed out that , for models of dark energy characterized by @xmath123 , the system can be unstable if @xmath124 , which is realized by a negative kinetic term  . 
   although our effective theory of dark energy is different from the case with @xmath124 , @xmath14 and @xmath13 yielding ( [ eq - state ] ) do not satisfy the condition @xmath107 derived from various energy conditions and might cause instability . 
   it is not clear whether they are phenomenologically viable or not , because it depends on the details of model at a microscopic level .
summary: we study physics concerning the cosmological constant problem in the framework of effective field theory and suggest that a dominant part of dark energy can originate from gravitational corrections of vacuum energy , under the assumption that the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . 
   our speculation is excellent with terascale supersymmetry . 
   dark energy from gravitational corrections    .45em 1.5 cm    yugo abe@xmath0 , masaatsu horikoshi@xmath1 and yoshiharu kawamura@xmath1    1.5em @xmath2_graduate school of science and engineering , shimane university , _   + matsue 690 - 8504 , japan + @xmath3_department of physics , shinshu university , _ 
   + matsumoto 390 - 8621 , japan    4.5em    = 1.0em = 1.0em = 0.5em = 0.5em