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in this report we assume that a signature for new physics is observed at a linear collider in the photon + missing energy channel , e^+e^- + .
[ p311signal ] we analyze two possible interpretations of this signal .
one possibility is in the context of the proposal of arkani - hamed , dimopoulos and dvali ( add )  @xcite that we live in a world with extra spatial dimensions . in this proposal ,
the gauge hierarchy problem is resolved by assuming that the fundamental , higher - dimensional gravitational scale is around a tev .
the apparent high value of the planck scale in the effective four - dimensional theory can be explained if the volume of the extra dimensions is large .
the standard model ( sm ) degrees of freedom are confined to a four - dimensional manifold ( a `` brane '' ) in the full space - time , while gravitons can propagate in the extra dimensions .
if an extra - dimensional graviton is emitted in a particle collision , it would not be detected , leading to the signature of  .
another possibility is in the context of an effective supersymmetric theory obtained from a spontaneously broken supergravity theory .
if the scale of supersymmetry breaking is sufficiently low such theories will contain an extremely light gravitino , @xmath0  ev .
gravitino pair production in conjunction with a photon will again produce the signature of  .    in this work
we ask how well we can interpret a photon + missing energy signal .
in particular , we would like to know whether one can distinguish between the two possibilities discussed above .
we will also study if it is possible to distinguish between the add - type models with various numbers of extra dimensions .
in theories with large extra dimensions , the missing energy in @xmath1 could be carried by the kaluza - klein(kk ) excitations of the gravitons .
the differential cross - section is given by  @xcite@xcite : ( e^+e^- g ) =  s_n -1   ( ) ^n + 2 f(x _ , ) [ p311sezgg ] f(x , y)= . [ p311funz ] here @xmath2 is the fundamental mass scale , @xmath3 is the number of extra dimensions , @xmath4 is the surface area of an @xmath3-dimensional sphere of unit radius , @xmath5 , @xmath6 is the photon energy , and @xmath7 is the angle between the photon and beam directions .
in supersymmetric theories with a superlight gravitino , photon + missing energy events could arise due to the pair - production of gravitinos , @xmath8 .
the differential cross - section is given by  @xcite : = ( ) f_(x _ , ) [ p311eq : ggg1 ] f_(x , ) = 2(1-x)^2 [ p311eq : ggg ] where @xmath9 is the fine structure constant , @xmath10 is the gravitational constant , @xmath11 is the gravitino mass , @xmath12  is the photon scaled energy ( @xmath13 ) and @xmath7 is the polar angle .
single photon plus missing energy events also occur within the sm due to escaping neutrinos . the sm background is @xmath14 , and can be obtained from numerous monte carlo packages , including pandora  @xcite .
previous analyses  @xcite have carefully outlined the parameter spaces that produce a discernible excess of signal events over sm background in extra dimensional gravity models and in superlight gravitino models .
we do not reproduce those results here , but are more interested in how similar or dissimilar the graviton - induced signal is to the gravitinos - induced signal .    to help us
see the difference between the possible signal interpretations of single photon plus missing energy events at a linear collider , we have plotted in fig .
[ p311diffcs1 ] the differential cross - section as a function of the photon missing energy , @xmath5 , obtained by integrating the cross - sections   and   with respect to @xmath15 .
the three solid lines are for the graviton kk signal with 2 , 4 and 6 extra dimensions . at @xmath16
the ordering of the solid lines is @xmath17 highest , and @xmath18 lowest .
the dashed line is the gravitino signal .
the scales ( @xmath2 for the add - type models and @xmath11 for the supersymmetric model ) have been chosen so that the total cross sections within the applied kinematic cuts , @xmath19 and @xmath20 , are equal in all four cases .
similarly , fig .
[ p311diffcs2 ] shows the normalized differential cross - sections of the four models as a function of @xmath15 , obtained by integrating with respect to @xmath21 .
the ordering of the solid lines at @xmath22 is @xmath17 highest , and @xmath18 lowest .     and
@xmath11 are tuned so that the total cross - section is the same for each model .
the three solid lines are for the graviton kk signal with 2 , 4 and 6 extra dimensions . at @xmath16
the ordering of these solid lines is @xmath17 highest , and @xmath18 lowest .
the dashed line is the gravitino signal . ]    .
the scales @xmath2 and @xmath11 are tuned so that the total cross - section is the same for each model .
the three solid lines are for the graviton kk signal with 2 , 4 and 6 extra dimensions . at @xmath22
the ordering of these solid lines is @xmath17 highest , and @xmath18 lowest .
the dashed line is the gravitino signal . ]
it is clear from the figures that both the recoil energy and angular distributions of the events are quite similar in all four cases .
it is still possible , however , to use the photon energy distribution to discriminate between the add models with different values of @xmath3 . to do this
, we perform a harder cut on the photon energy , @xmath23 .
this is advantageous because in this region , the differences between cross sections with different @xmath3 do not change sign .
we also perform an additional cut on @xmath21 to get rid of the peak in the standard model background associated with on - shell @xmath24 production . assuming that electron and positron beam polarizations of 80 and 60% respectively are achieved at a 500 gev linear collider , and that the cross section can be measured with 1% precision
, we find that , for example , the case @xmath17 can be distinguished from @xmath25 for values of @xmath26 up to 4.6 tev .
( if the uncertainty is purely statistical , 1% error on the cross section measurement corresponds to integrated luminosity of about 270 fb@xmath27 . )
note , however , that the photon energy distributions for the cases of gravitino emission and @xmath18 add model are extremely close .
it is not likely that the ambiguity between these two cases can be resolved by this measurement .
we have also studied whether performing a more restrictive angular cut could help in distinguishing between the cases of gravitino and @xmath18 graviton emission . from fig .
[ p311diffcs2 ] it appears that the two signals have discernible differences at small values of @xmath15 .
we have found , however , that with the assumed 1% precision on the cross section measurement , this analysis will not be able to conclusively separate these two cases .
another way to distinguish between models is by varying the collider center of mass energy @xmath28 .
the tesla study  @xcite has shown that measuring cross sections at two design center of mass energies , @xmath29 and @xmath30 , allows to determine the number of extra dimensions in the add models for a wide range of parameters .
for example , if there are two extra dimensions , the @xmath25 case can be excluded at 99% cl for values of @xmath26 up to 6.1 tev .
the beam polarizations assumed in  @xcite coincide with the values chosen for our analyses here .
while upgrading the linear collider energy from 500 gev to 800 gev , for example , will take much time , effort and money , the beam energy can be changed by as much as 5% without significantly disrupting the operation of the collider . therefore , it is interesting to study if different models can be distinguished by performing measurements at two center of mass energies separated by 5% .
we have performed such a study using pandora .
we have chosen the scales of each model so that their cross sections ( with the same cuts and polarizations as before ) are identical at @xmath31 gev , and evaluated their differences at @xmath32 gev . taking the standard model background into account and assuming that the cross sections at each energy can be measured with 1% precision , we find that the @xmath17 and @xmath25 cases can be distinguished at the 3@xmath33 level for @xmath34 tev .
the sensitivity of this method is thus somewhat lower than for studying photon energy distributions .
surprisingly , the scaling of the gravitino pair production cross section with @xmath28 is again identical to the @xmath18 add model .
changing collider energy does not help in separating these two cases .
we conclude that while the add models with @xmath35 can be distinguished from the supersymmetric models with light gravitino by carefully studying the photon + missing energy signal alone , this can not be done for the case of 6 extra dimensions .
( let us note in passing that the missing energy signal in the @xmath18 add model is also identical to the one due to the emission of scalar states associated with the brane coordinates  @xcite . ) in this case , distinctions between the models will only occur with additional phenomena .
the next - best observables in the extra - dimensional graviton model include the dimension - eight contact interactions  @xcite induced by e^+e^-g^(n ) , f|f , etc . , or from string regge states  @xcite .
the superlight gravitino models are supersymmetric models , and the superpartner spectrum must satisfy the same naturnalness criteria as other supersymmetry ideas .
the expectation is that some superpartner states , most notably charginos and neutralinos , should be accessible and well - studied at a 500 gev linear collider  @xcite .
the kinematic reach for superpartners then approaches @xmath36 in @xmath37 collisions .
an increase of the reach of selectrons is possible in @xmath38 collisions , since selectron masses approaching @xmath39 are kinematically accessible  @xcite .
the full complement of observables will be needed to distinguish the precise underlying theory being discovered at the colliders .
we would like to thank the organizers of the snowmass 2001 workshop where this work was initiated .
we are grateful to many of the workshop s participants and especially to the members of the p3 and e3 working groups for interesting discussions .
we are indebted to michael peskin for his invaluable help with the pandora event generator .
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_ [ american linear collider working group collaboration ] , slac - r-570 _ resource book for snowmass 2001 , 30 jun - 21 jul 2001 , snowmass , colorado_.
the photon plus missing energy signature is a primary handle on two important classes of theories 
 . theories with large extra dimensions predict the production of photons in association with kaluza - klein excitations of the graviton . in supersymmetric theories with superlight gravitinos 
 , photons can be produced in association with gravitino pairs . 
 the signatures for these two theories are compared , and it is found that they can be distinguished by studying the photon energy distributions and scaling of the cross section with center - of - mass energy . 
 both these methods fail , however , if there are six extra dimensions . in that case 
 , additional phenomena predicted by the theories would be required to narrow down the underlying causes of the photon plus missing energy signal . 
 we also study the ability of these measurements to determine the number of extra dimensions .    
 hep - ph//0110339 + lbnl-49069 +