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there are reasons to believe that at the planck scale , the scale at which gravity becomes a quantum phenomenon , the very structure of space - time may change . that this may happen is suggested even by general relativity . a quantum mechanical particle of momentum @xmath2 in the presence of a classical gravitational field ( the latter described by einstein s equations ) causes the metric @xmath3 to fluctuate . this induces an additional uncertainty in position , given by @xmath4 , where @xmath5 is the planck length . thus the usual quantum mechanical uncertainty relation gets modified to @xmath6 at high energies , the second term can become significant and lead to important deviations from the usual quantum mechanics . for example , in the usual quantum mechanics , @xmath7 is large at low values of momenta but can become small at high momenta which can provide higher resolution . with the modified uncertainty relation even at high momenta @xmath7 is limited in resolution because of strong curvature effects . in other words , independent of momentum , @xmath7 is always larger than a minimal length scale @xmath5 . the appearance of the minimal length in the classical theory of gravity should tell us that it is no surprise to expect that such a conclusion becomes even more inevitable in a quantum theory of gravity . indeed , a whole range of quantum gravity models predict the existence of a minimal length @xcite . however , because the planck length is so minuscule it is of consequence only to the physics of the early universe , if at all @xcite . there is a twist in the plot , however , and that is due to the development of brane - world physics . this new paradigm in physics has far - reaching implications but of our immediate interest is the fact that there are models of brane - world which allow a realisation of quantum gravity effects at very low energy scales  as low as a tev . in this paper , we will deal with the simplest of such realizations which is a model due to arkani - hamed , dimopoulos and dvali .    in the model of arkani - hamed , dimopoulos and dvali , the so - called add model @xcite , extra space dimensions are compactified to have large volume moduli and thereby it becomes possible to lower the scale of quantum gravity from the planck scale to the tev scale . in the add model , the gauge interactions are confined to a 3-brane while gravity propagates in all the available dimensions . the higher dimensional planck scale @xmath8 in the add model is related to the usual planck scale by @xmath9 , where @xmath10 is the radius of compactification and @xmath11 is the number of extra space dimensions . for appropriate choices of @xmath10 and @xmath11 , the fundamental ( higher dimensional ) planck or string scale can be brought down to @xmath12 1 tev . this scale can be probed at present and future colliders . various signals of graviton production and virtual graviton exchange have been studied @xcite and reviewed @xcite in recent years . in the context of add model with @xmath13 tev , the minimal length hypothesis is phenomenologically interesting if we take it to be around an inverse tev , viz . @xmath14 . for a review , see ref . @xcite . different applications of the minimal length scenario ( mls ) have been discussed in @xcite . in an earlier analysis @xcite , we had studied how the mls hypothesis influences the dilepton production process in hadron colliders . in the add model , for such processes involving virtual gravitons , one has to sum over an infinite tower of graviton kaluza - klein ( kk ) states at the amplitude level . the result is divergent and to cure the divergence an _ ad - hoc _ cutoff of the order of @xmath15 is used . however , in the mls scenario , the minimal length acts as an ultraviolet regulator and allows one to sum over the entire kk graviton tower by smoothly cutting off the contribution of higher energy kk states rendering the amplitude finite . another important modification comes from the rescaling of momentum measure leading to an alteration in the phase space integration . these , as we observed in @xcite , lead to a significant deviation of the bound on @xmath8 from the one obtained in the conventional add picture without the mls hypothesis .    in the present analysis , we consider processes involving emission of real gravitons in hadron colliders , namely , the upgraded tevatron and the future large hadron collider ( lhc ) . the partonic sub - processes are @xmath16 , where @xmath17 are the appropriate quarks / gluons . here @xmath18 means a real graviton with kk index @xmath19 , where @xmath11 is the number of extra dimensions . for a given process , one must sum over such external modes at the cross section level . this kk summation gets modified in the mls scenario . the phase space integration also gets modified in the mls scenario . again , as we shall see towards the end , the bounds that one obtains on @xmath8 in the conventional add model from the jet plus missing energy mode in the hadron colliders are considerably weakened if we implement the mls hypothesis . specifically , by mls scenario we mean the add model with a minimal length @xmath5 of the order of tev@xmath20 incorporated in it . since the uncertainty in position measurement now can not be smaller than @xmath5 , now the standard commutation relation between position and momentum has to be modified @xcite . in the mls scheme , the compton wavelength ( @xmath21 ) of a particle can not be arbitrarily small . we suppose that even though the wave vector @xmath22 is bounded from above , the momentum @xmath2 can be as large as possible . in the same way , the frequency @xmath23 is restricted from above while the energy @xmath24 can go up arbitrarily . this means that the standard relations @xmath25 and @xmath26 need to be modified . this can be realised by introducing the unruh ansatz @xcite @xmath27 ,   \nonumber \\ l_{p } \omega(e)&=&\tanh^{1/\gamma } \left[\left(\frac{e}{m_{s}}\right)^{\gamma}\right],\end{aligned}\ ] ] where @xmath28 is a positive constant . ( [ dispersion ] ) captures the essence of a minimal length scale : as @xmath2 ( or @xmath24 ) becomes very large , @xmath22 ( or @xmath23 ) approaches the upper bound @xmath29 and hence one can not probe arbitrarily small length and time scales . the generalized position - momentum and energy - time uncertainty principle can now be written in a lorentz covariant form as @xmath30=i   \,\,\frac{\partial p_{\mu}}{\partial k_{\nu}}.\ ] ] as explained in @xcite , the effect of eq . ( [ dispersion ] ) can conveniently be accounted for by a simple redefinition of the momentum measure , which in one dimension is given by @xmath31 the lorentz invariant 4-momentum measure is modified as @xmath32 where the jacobian matrix @xmath33 can be kept diagonal . in this paper we study how processes involving real graviton emissions in hadron colliders are going to be influenced by the mls hypothesis . the emitted gravitons will disappear carrying missing energy , so the events that we look for are jets plus missing energy . the partonic sub - processes that contribute to this channel are @xmath34 , @xmath35 , @xmath36 , and @xmath37 . the differential cross section for a particular sub - process ( labelled @xmath38 ) for a given kk graviton mode has the form @xmath39 where @xmath40 is the mass of @xmath18 , and @xmath41 are functions of the mandelstam variables @xmath42 whose explicit expressions are given in ref . these cross sections have to be convoluted with the density of kk states given by @xcite @xmath43 noting that one can express @xmath44 , the differential cross section summed over the kinematically allowed kk graviton tower is given by @xmath45 where the lower limit of integration is zero and the upper limit corresponds to the maximum allowed graviton energy as @xmath46 . an important thing that has to be noticed here is that while the usual 4-dimensional graviton emission cross section goes like @xmath47 , the effect of kk summation jacks up the contribution which goes effectively as @xmath48 . this is the add effect . now , how does the mls hypothesis modify the kk summation ? now for analytic simplicity we take @xmath49 and set @xmath50 . the departure from these assumptions will be discussed later . the unruh relations ( [ dispersion ] ) will modulate the integral in ( [ dscon ] ) by a factor @xmath51 . the mls - influenced differential cross section is given by @xmath52 for large argument , the sech - square function falls exponentially , thus the contributions from high mass kk states will be smoothly cut off . another important modification due to the mls is the change in the cross section due to the rescaling of the momentum measure given in eq . ( [ scale ] ) . it is straightforward to check , as derived in @xcite , that the phase space integration in the total cross section picks up the following modification factor : @xmath53 where @xmath54 runs over the four initial and final states in a @xmath55 process , and @xmath56 and @xmath57 are the total initial and final four momenta in a @xmath58 process . we work out this modification factor for the process we are studying viz . note , while @xmath17 are massless , @xmath18 s are massive . using eqs . ( [ dispersion ] ) and ( [ scale ] ) , setting @xmath60 and @xmath61 , we can easily show that @xmath62 above , @xmath63 and @xmath64 are the momentum fractions of the hadrons carried by the initial state partons , @xmath65 and @xmath66 are the rapidities of the final state parton and the graviton , @xmath67 is the transverse momentum of the final state parton , and @xmath68 is the transverse mass of the graviton @xmath18 . it is instructive to check that in the decoupling limit @xmath69 , the phase space correction factor goes to unity . the sech - square modulation in eq . ( [ dsmls ] ) together with the phase space modifications in eqs . ( [ mod ] ) and ( [ ebyomega ] ) constitute the complete correction . if we keep @xmath28 and @xmath70 as free parameters the analytic expressions of the modifications in eqs . ( [ dsmls ] ) and ( [ ebyomega ] ) will look more complicated , and we do not display them . instead , we have numerically demonstrated the sensitivity of cross section due to variations of @xmath28 and @xmath71 in the next section . we have numerically computed the cross section for the production of the graviton kk tower in association with a jet at the run ii of tevatron ( @xmath72 1.96 tev ) and at the lhc ( @xmath72 14 tev ) in both the add model and in the mls scenario . there is an irreducible sm background to the signal coming from @xmath73jet production followed by the invisible decay of the @xmath74 into neutrinos . we have computed the sm background and the signal in the add and the mls cases using the mrst 2001 lo parton distributions @xcite ( the mls modification of the parton densities is numerically insignificant ) . since the signal peaks at larger missing-@xmath75 as compared to the background the purity of the sample can be improved by employing a strong cut on the missing-@xmath75 . we require the missing-@xmath75 to be greater than 80 gev for the tevatron run ii analysis . the cut on the jet pseudorapidity is @xmath76 2.4 . our results are shown for the case @xmath77 . we display the results of this computation in figure 1 by plotting the cross section in the add and the mls scenarios as a function of @xmath8 . experimentally , the @xmath73jet cross section is not known very precisely . to have a rough estimate , we multiplied the sm cross section by the appropriate luminosity to obtain the number of events , and twice the square - root of those events ( back to pb unit ) correspond to the 2-@xmath78 spread around the sm value . thus , no detector effect has been taken into consideration . in other words , assuming purely statistical errors on the sm cross section , we have plotted the 2-@xmath78 allowed bands in the figure , which may be considered as a rough estimate of the allowed zone . as expected , the mls cross section has a steeper fall - off with @xmath8 . consequently , the lower bound on @xmath8 in the mls scenario is diluted by about 60 gev compared to what one obtains in the conventional add model at the @xmath79 confidence level .    for the lhc analysis , we have used the much stronger cut of 200 gev on the missing @xmath75 and a jet pseudorapidity cut of @xmath80 3.5 . the cross section for the signal in the two cases and the sm - allowed band are shown in figure 2 . the effects of the suppression of cross section due to the minimal length is stronger at the lhc and hence the dilution of the bound ( by a few hundred gev ) is greater than at the tevatron .    figures 1 and 2 have been drawn setting @xmath60 and @xmath61 . in figure 3 , we have displayed the variation of the total cross section , for tevatron run ii , with @xmath28 for different choices of @xmath81 . the value of @xmath82 has been fixed at 1.5 tev for this figure . in figure 4 , we have presented the analogous variation for the lhc energy for @xmath83 tev . we recall that the dependence of the cross section on @xmath28 and @xmath71 arise from the use of eq . ( [ dispersion ] ) while calculating the mls modulation factors . the existence of a minimal length is a generic prediction of quantum gravity theories . the model of large extra dimensions , the add model , is an example where quantum gravity effects become manifest at energy scales as low as a tev making it accessible to collider experiments . we incorporate the idea of a minimal length in the add model , which we now call the mls scenario , and study what impact the existence of a minimal length has on the collider effects of the conventional add model . in particular , we have considered graviton production in association with jets at hadron colliders to address this particular issue . by analysing the jet@xmath84missing energy channel at the upgraded tevatron and the large hadron collider , we demonstrate that the add model bounds get diluted when the minimum length hypothesis is invoked . finally , we make a few comments on the possible implications of minimal length on the astrophysical constraints on the add model . first we note that gravitons produced in supernovae core by nucleon nucleon bremsstrahlung process can carry away missing energy . but neutrino fluxes from sn1987a detected by the imb and kamiokande experiments have been observed to carry away most of the energy ( @xmath85 ergs ) released during core collapse . this has been used to obtain tight bounds on @xmath8 of the order of 50 tev for @xmath86 and about 4 tev for @xmath87 @xcite in the conventional add model . since the production process of the gravitons is the same as the one studied in the present paper , it is important to ask as to what extent the mls scenario dilutes the supernova bounds . note , the high nucleon density at the supernova core and the large multiplicity of kk graviton modes for low values of @xmath11 is responsible for the enhancement of the effects , which in turn places strong constraints on @xmath8 . the mls scenario would indeed suppress the graviton production cross section and tend to dilute the bounds . but the numerical impact of this dilution would not be large compared to the other uncertainties , e.g. strong core temperature dependence , inherited from pure supernova dynamics . a string theory calculation of this cross section has been done in @xcite , where the amplitude is multiplied by a form factor . it is worth comparing the string theory result with the unruh ansatz that we have used . in the string theory case the form factor goes like @xmath88 in the asymptotic limit . in our case , the modulation depends on the choice of @xmath28 . for @xmath60 , the high mass kk states are cut off by @xmath89 since the argument of sech - square in eq . ( [ dsmls ] ) has a linear dependence on graviton mass . for @xmath90 , the dependence on the graviton mass is quadratic , and we obtain a string - like modulation . we note at this point that the string calculation as done in @xcite is based on a toy scenario with an embedding of qed in string theory , and the calculation of jet plus missing energy relies on a rough extrapolation from this toy scenario . because of this model dependence , a more detailed comparison of the results in @xcite and our analysis is not possible . gb and ks acknowledge support by the indo - french centre for the promotion of advanced research , new delhi ( ifcpar project no : 2904 - 2 ) , and thank cern ph / th division for hospitality during the work . gb also thanks lpt - orsay and spht - saclay for hospitality during the final stage of the work . gb s research has also been supported , in part , by the dst , india , project no : sp / s2/k-10/2001 .          n. arkani - hamed , s. dimopoulos and g. dvali , phys . b * 429 * ( 1998 ) 263 [ arxiv : hep - ph/9803315 ] ; i. antoniadis , n. arkani - hamed , s. dimopoulos and g. dvali , phys . b * 436 * ( 1998 ) 257 [ arxiv : hep - ph/9804398 ] ; n. arkani - hamed , s. dimopoulos and g. dvali , phys . d * 59 * ( 1999 ) 086004 [ arxiv : hep - ph/9807344 ] ; n. arkani - hamed , s. dimopoulos and j. march - russell , phys . d * 63 * ( 2001 ) 064020 [ arxiv : hep - th/9809124 ] . u.  harbach , s.  hossenfelder , m.  bleicher and h.  stocker , phys . b * 584 * ( 2004 ) 109 [ arxiv : hep - ph/0308138 ] , m.  cavaglia and s.  das , arxiv : hep - th/0404050 , m.  cavaglia , s.  das and r.  maartens , class . * 20 * ( 2003 ) l205 [ arxiv : hep - ph/0305223 ] ; s.  hossenfelder , phys . d * 70 * ( 2004 ) 105003 [ arxiv : hep - ph/0405127 ] ; phys . b * 598 * ( 2004 ) 92 [ arxiv : hep - th/0404232 ] ; arxiv : hep - ph/0409350 ; mod . a * 19 * ( 2004 ) 2727 [ arxiv : hep - ph/0410122 ] ; arxiv : hep - ph/0412265 ; arxiv : hep - th/0510245 ; c.  barcelo , s.  liberati and m.  visser , arxiv : gr - qc/0505065 .
theories of quantum gravity suggest the existence of a minimal length scale . 
 we study the consequences of a particular implementation of the idea of a minimal length scale in the model of large extra dimensions , the add model . to do this 
 we have looked at real graviton production in association with a jet at hadron colliders . in the minimal length scenario , 
 the bounds on the effective string scale are significantly less stringent than those derived in the conventional add model , both at the upgraded tevatron and at the large hadron collider .    ` 
 pacs nos :   11.25.wx , 13.85.qk ` + ` key words :  extra dimension , minimal length , hadron collider `    ` hep - ph/0512050 ` + sinp / tnp/05 - 28 + tifr / th/05 - 46 +    * studying the effects of minimal length in large extra dimensional models in the jet @xmath0 missing energy channels at hadron colliders * +    , kumar rao @xmath1 , k. sridhar @xmath1   +    \1 . 
 saha institute of nuclear physics , 1/af bidhan nagar , kolkata 700064 , india + 2 . 
 department of theoretical physics , tata institute of fundamental research , + homi bhabha road , mumbai 400005 , india