We consider the high-energy behaviour of processes involving Kaluza-Klein (KK) gravitons of weak-scale string theories. We discuss how form-factors derived within string theory modify the couplings of KK gravitons and thereby lead to an exponential fall-off of cross sections in the high-energy limit. Further, we point out that the assumption of Regge behaviour for a scattering amplitude in the high energy limit, $T\propto s^{\alpha(t)}$, combined with a linear growth of the total cross-section, $\sigma_{tot}(s)\propto s$, violates elastic unitarity. Regge behaviour leads to a stringent bound on the growth of the total cross-section, $\stot (s) \leq 32\pi \alpha' \ln(s/s_0)$

Kachelriess, M

Plümacher, Michael

arXiv

We consider the high-energy behaviour of processes involving Kaluza-Klein (KK) gravitons of weak-scale string theories. We discuss how form-factors derived within string theory modify the couplings of KK gravitons and thereby lead to an exponential fall-off of cross sections in the high-energy limit. Further, we point out that the assumption of Regge behaviour for a scattering amplitude in the high energy limit, $T\propto s^{\alpha(t)}$, combined with a linear growth of the total cross-section, $\sigma_{tot}(s)\propto s$, violates elastic unitarity. Regge behaviour leads to a stringent bound on the growth of the total cross-section, $\stot (s) \leq 32\pi \alpha' \ln(s/s_0)$.We consider the high-energy behaviour of processes involving Kaluza-Klein (KK) gravitons of weak-scale string theories. We discuss how form-factors derived within string theory modify the couplings of KK gravitons and thereby lead to an exponential fall-off of cross sections in the high-energy limit. Further, we point out that the assumption of Regge behaviour for a scattering amplitude in the high energy limit, $T\propto s^{\alpha(t)}$, combined with a linear growth of the total cross-section, $\sigma_{tot}(s)\propto s$, violates elastic unitarity. Regge behaviour leads to a stringent bound on the growth of the total cross-section, $\stot (s) \leq 32\pi \alpha' \ln(s/s_0)$

Kachelriess, M.

Plumacher, M.

CERN Document Server

CERN-TH 2001-251OUTP-01-48-PRemarks on the high-energy behaviour of cross-sections in weak-scale string theoriesM. Kachelrie1 and M. Plu¨macher21TH Division, CERN, CH-1211 Geneva 23, Switzerland2Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, UK(September 21, 2001)We consider the high-energy behaviour of processes involving Kaluza-Klein (KK) gravitons ofweak-scale string theories. We discuss how form-factors derived within string theory modify the cou-plings of KK gravitons and thereby lead to an exponential fall-o of cross sections in the high-energylimit. Further, we point out that the assumption of Regge behaviour for a scattering amplitude inthe high energy limit, T ∝ sα(t), combined with a linear growth of the total cross-section, σtot(s) ∝ s,violates elastic unitarity. Regge behaviour leads to a stringent bound on the growth of the totalcross-section, σtot(s) ≤ 32piα′ ln(s/s0).PACS numbers: 98.70.Sa, 14.60.Lm, 11.25.MjI. INTRODUCTIONNeutrinos are the only known stable particles thatcan traverse extragalactic space without attenuation evenat energies E  1020 eV, thus avoiding the Greisen{Zatsepin{Kuzmin cuto [1]. Since neutrinos are notdeflected by (extra-) galactic magnetic elds, this pri-mary candidate could also explain possible correlationsbetween the arrival directions of observed ultrahigh en-ergy cosmic rays and astrophysical objects at cosmologi-cal distances [2]. Therefore, it has been speculated thatthe ultrahigh energy primaries initiating the observed airshowers are not protons, nuclei or photons but neutri-nos [3{5]. However, in the Standard Model (SM) neu-trinos are deeply penetrating particles producing onlyhorizontal not vertical extensive air showers. Conse-quently, either the neutrino has to be converted locallyinto strongly interacting particles [6] or one has to postu-late new interactions that enhance the ultrahigh energyneutrino-nucleon cross-section.A particular realization of the latter possibility arestring theories with δ large extra dimensions [7]. Ifthe SM particles are conned to the usual 3 + 1-dimensional space and only gravity propagates in thehigher-dimensional space, the compactication radius Rof the extra dimensions can be large, corresponding to asmall scale 1/R of new physics. From a four-dimensionalpoint of view the higher dimensional graviton in thesetheories appears as an innite tower of Kaluza-Klein(KK) excitations with masses squared m2~n = ~n2/R2.Since the weakness of the gravitational interaction is par-tially compensated by the large number of KK states andcross-sections of reactions mediated by spin 2 particlesare increasing rapidly with energy, it has been arguedin Refs. [4,5] that neutrinos could initiate the observedvertical showers at the highest energies.In Refs. [8,9], the neutrino-nucleon cross-section viathe exchange of KK gravitons was calculated withinthe eective eld-theoretic model valid below the stringscale Mst. Since amplitudes involving virtual exchangeof KK gravitons diverge for more than one large ex-tra dimension, a form-factor suppressing the KK modesabove Mst was employed in [9]. Moreover, the boundσtot(s) / ln2(s/s0) was derived with the eikonalizationmethod assuming the validity of the Regge picture as aneective theory above Mst [9]. Recently, both the use ofa form-factor in the eld-theoretic framework as well asthe Froissart-like bound in the Regge picture was criti-cised in Ref. [10]. It is the purpose of this short articleto discuss these criticisms. In Sec. II, we review howform-factors suppressing the couplings of KK gravitonswith large squared four-momentum appear in string the-ory, and in Sec. III we discuss bounds on the neutrino-nucleon cross-section if the validity of the Regge pictureabove Mst is assumed1.II. FORM-FACTORS FOR KK GRAVITONCOUPLINGS FROM STRING THEORYThe amplitude for the exchange of KK gravitons in thet channel between two particles with four momentum pand k is1We do not consider the production of black holes in neutrino-nucleon scattering in this paper [11].1T (s, t) = 1M2Pl1∑~n=0Tµν(k)f2~nt−m2~nT µν(p) , (1)where Tµν = Tµν − 12ηµνT αα , Tµν is their energy-momentum tensor, ηµν is the metric tensor, M2Pl = G−1Nis the reduced Planck mass and s, t are the usual Man-delstam variables.In the calculation of Ref. [9], an exponential suppres-sion of the eective coupling of the level ~n KK mode tofour-dimensional elds was used as form-factor f~n,f~n = exp(− cm2~n2M2st), (2)where c is a constant of order 1. Then the summationover n which for f~n = 1 only converges in the case ofone extra dimension becomes well-dened for all δ. How-ever, a legitimate question to ask is whether this modi-cation is well-motivated. To address this point, we rstnote that the suppression of the couplings sets in onlyfor −t >M2st. Thus, the form-factor (2) does not modifythe eective theory in its range of validity, −t < M2st.On the other hand, string theories generically predictan exponential suppression of the coupling to higher KKmodes in the regime −t  M2st [12]. This suppressionis also consistent with the idea that recoil eects due tothe nite tension of D-branes become important in theemission of KK gravitons with large momentum transfer,−tM2st [13].Let us now discuss in detail how form-factors to thecoupling of KK gravitons appear as string corrections tothe well-known eld theoretical result [14]. The eectivelow-energy coupling of standard model particles to KKgravitons can be obtained by computing the amplitudefor the scattering of four open strings. At tree-level thestring amplitude is given byAtree = g2A(1, 2, 3, 4) tr[t1t2t3t4 + t4t3t2t1] S(s, t)+ g2A(1, 3, 2, 4) tr[t1t3t2t4 + t4t2t3t1] S(s, u)+ g2A(1, 2, 4, 3) tr[t1t2t4t3 + t3t4t2t1] S(t, u) , (3)where the ti are Chan-Paton matrices, g2A(i, j, k, l) arethe amplitudes evaluated in the low-energy eld theoryframework, and S(s, t) is the Veneziano amplitude,S(s, t) =Γ(1− sM2st)Γ(1− tM2st)Γ(1− sM2st− tM2st) . (4)The Veneziano amplitude has the usual Regge-pole struc-ture. In the hard-scattering limit, s ! 1 with xedscattering angle t/s = − sin2 φ2 , the amplitude falls oexponentially,S(s, t) / exp[sM2st(sin2φ2ln sin2φ2+ cos2φ2ln cos2φ2)].(5)It was pointed out in Ref. [15] that this exponentialfall-o can be interpreted as an eective thickness of D-branes of the order of the string scale, which would giverise to a form-factor of the kind (2).However, this tree-level result is due to string Reggeexcitations. Contributions from KK gravitons rst ap-pear at the one-loop level [16{18]. The Type I string di-agram which contains s-channel gravitational exchangein the low-energy limit is the non-planar cylinder dia-gram, which gives rise to contributions to the amplitudeof the formA1−loop = −12s tA(1, 2, 3, 4) tr(t1t2)tr(t3t4) f (1)T (s, t) + . . . ,(6)where the dots indicate permutations of the indices1, . . . , 4 and the Mandelstam variables s, t and u.The non-planar cylinder amplitude f (1)T (s, t) has re-cently been analysed in detail [19,20]. It is given byf(1)T (s, t) =g2sM10st1∫0dl1∫0dν2ν2∫0dν11∫0dν3 (7)(~ψT13 ~ψT24~ψ12 ~ψ34)s/M2st ( ~ψT13 ~ψT24~ψT14 ~ψT23)t/M2stF6(l, R) ,whereF6(l, R) =M6st(RMst)6ϑ63(0,12il(RMst)2)(8)are the winding mode contributions from the toroidalcompactication. For simplicity we have assumed thatall six extra dimensions have the same compacticationradius R. Further,~ψij =1lϑ1(νj − νi, il)η3(il), (9)~ψTij =1lϑ4(νj − νi, il)η3(il), (10)where ϑi are the usual Jacobi ϑ functions and η is theDedekind function.In Eq. (7), l is the modulus of the cylinder and in thelow-energy limit the contribution from exchange of KKgravitons in the s-channel can be extracted by consider-ing the l ! 1 limit of the integrand in Eq. (7). In thislimit one nds [19]:f(1)T (s, t) 2−s/M2stppiMPlΓ(12 − s2M2st)Γ(1− s2M2st)2(11) 1M2st1∫0dl∑~nexp[−pil(− s2M2st+~n22(RMst)2)].2The integral in the second line is just the proper-timeparametrization of the sum over the winding mode prop-agators. The l ! 0 limit of the integration is ill-dened,since we had replaced the integrand in Eq. (7) by itsasymptotic expansion for l ! 1. However, this diver-gence corresponds only to a harmless IR divergence of abox diagram [19]. By the modular transformation l = 1τthe amplitude (7) can be represented as a sum of boxdiagrams giving a well-dened result in the limit l! 0.Here we are mainly interested in the KK graviton con-tributions. The prefactor in Eq. (11) can be interpretedas the squared form factor for the emission of a KK gravi-ton with momentum-squared s > 0,f~n =2−s/M2stppiMPlΓ(12 − s2M2st)Γ(1− s2M2st) . (12)In the hard scattering limit this gives an exponential sup-pression of the KK graviton couplingf~n / exp(− sM2stln 2)for sM2st . (13)This form-factor is valid for KK gravitons emitted eitheron-shell or for virtual graviton exchange in the s-channel.In these cases we recover Eq. (2) with s = m2~n.Next, we discuss the t-channel exchange of KK gravi-tons which is important for neutrino-nucleon scattering.Contributions from the t-channel exchange are containedin the planar cylinder diagram, which is a one-loop cor-rection to the tree-level amplitude (3) of the formA1−loop = −12s tA(1, 2, 3, 4) tr(t1t2t3t4) f (1)(s, t) + . . .(14)The planar cylinder amplitude f (1)(s, t) is given by[18,21]f (1)(s, t) =g2sM10st1∫0dl1∫0dν2ν2∫0dν11∫0dν3 (15)(~ψ13 ~ψ24~ψ12 ~ψ34)s/M2st ( ~ψ13 ~ψ24~ψ14 ~ψ23)t/M2stF6(l, R) .Again considering the l ! 1 limit of the integrand oneobtains a contribution to the amplitude which can bewritten as a derivative of the tree-level Veneziano ampli-tude [18],f (1)(s, t)  M5st4pi2stM2Pl(∂S(s, t)∂Mst)(16) 1M2st1∫0dl∑~nexp[−pil~n22(RMst)2].It is easy to see that the prefactor in Eq. (16) has thesame Regge-pole structure as the Veneziano amplitude.In the hard-scattering limit, s ! 1, t/s xed, the am-plitude again falls o exponentially, as in Eq. (5). Thus,form-factors derived within string theory indeed modifythe couplings of virtual KK gravitons and, hence, alsothe cross-sections in the high-energy limit.III. UNITARITY LIMITS IN THE REGGEPICTUREThe authors of Ref. [8] pointed out that the Reggepicture is a reasonable approximation to string theoryvalid above Mst. As motivation for this assumption wenote that the Regge picture takes into account not onlythe KK modes of the graviton but also those from lowerlying trajectories and misses only genuine string modeslike winding modes. In the following, we will thereforealso use this assumption and derive bounds on the totalcross-sections valid within this framework.A general Regge amplitude TR can be represented byTR(s, t) = β(t)(ss0)α(t), (17)where the exponent α(t) is given by the Chew-Frautschiplot of the spin against the mass of the particles lyingon the leading Regge trajectory contributing to the reac-tion. In our case, the intercept α(0) of this trajectory isequal to the spin j of the massless graviton, α(0) = 2.We rst note that a Regge amplitude with interceptα(0) = 2 gives via the optical theorem a total cross-section growing linearly with s,σtot(s) =1sIm fTR(s, 0)g / sα(0)−1 . (18)Thus the assumed Regge-behaviour alone, without anyunitarization, reduces the growth of the total cross-section by one power of s compared to the naive expec-tation σ / sj = s2. On the other hand, the elasticcross-sectionσel(s) =116pis2∫ 0−sdt jTR(s, t)j2 / s2ln(s/s0)(19)increases even faster than the total cross-section. There-fore, elastic unitarity, σtot  σel, is violated above a cer-tain energy for any Regge amplitude with α(0) > 1 | asis well-known from the case of the pomeron. These nd-ings are in clear contradiction to the ones of Ref. [8,10]:there, it was claimed that a linear growth of σtot(s) forTR(s, 0) / s2 respects unitarity.Next, we derive the maximal total cross-section al-lowed for an arbitrary Regge amplitude by elastic uni-tarity. Following Leader [22], we rewrite TR(s, t) as3TR(s, t) = β(ss0)α(t)= TR(s, 0)(ss0)α(t)−α(0)(20)and expand the amplitude around t = 0,(ss0)α(t)−α(0)= expfα0t ln(s/s0) +O(α00t2)g . (21)Here, α0 denotes the derivative of α(t) evaluated at t = 0and we have neglected for clarity possible non-linearterms in t and the subdominant t dependence of β. Thenwe evaluate σel,σel =116pis2∫ 0−sdt jTR(s, t)j2 = jTR(s, 0)j216pis212α0 ln(s/s0).(22)Requiring now elastic unitarityσel  σtot = 1sIm fTR(s, 0)g < 1sjTR(s, 0)j , (23)it follows132piα0 ln(s/s0)jTR(s, 0)j2s2 σtot (24)orσ2tot32piα0 ln(s/s0) σtot (25)and nallyσtot(s)  32piα0 ln(s/s0) . (26)Thus the assumption of a Regge amplitude results in astronger bound for the total cross-section than the Frois-sart bound. A more general derivation of such a boundcan be found in Ref. [23].Some remarks are now in order: First, we have alwaysused formulae valid for d = 4 dimensions. This is appro-priate because the main contribution to the cross-sectionscomes from the small t region and therefore does notprobe the extra dimensions. Second, we note that thisbound applies on the parton not the hadron level. Third,the bound (26) contains two parameters, the slope of theRegge trajectory α0 and the unknown scale s0, and istherefore still not useful for a numerical evaluation.To proceed, we use thatσNνtot (s) = [N(s) + δN ]σtot(s)  N(s)σtot(s) , (27)where N(s) / s0.4 [24] takes into account the increasingnumber of target partons in the nucleon. The term δN <0 corrects that each parton carries only a fraction x < 1of the nucleon momentum, i.e. that ln(xs/s0) < ln(s/s0).A numerical value for the bound (27) can now be deter-mined by joining the eld-theoretic result and the Reggeresult on the hadron level at that scale s0  M2st, wherethe eld-theoretic result starts to violate s-wave unitarityon the parton level. We nd that the KK contributionto the total cross-section at UHE is at most of the sameorder of magnitude as the SM cross-section.Finally, we want to comment briefly on the suggestionof Ref. [4] that the exponential increase of (lepto-quarklike) KK resonances in the s channel could enhance theneutrino-nucleon cross-section. Since the Horn-Schmidduality [25] connects s and t channel Regge/String ampli-tudes, our discussion above can be applied immediatelyto this case. The n = 0 lepto-quarks can have either spinj = 0 or 1. Even in the later case, the intercept willbe smaller than 1 and the partonic cross-section will beasymptotically decreasing with s.IV. CONCLUSIONSCouplings of KK states derived within a eld-theoreticmodel valid below Mst are modied by form-factors cal-culable in string theory [15,19]. The use of these form-factors makes the sums over KK states well-dened with-out simply cutting-o KK modes withmn >Mst. For thecase of neutrino-nucleon scattering, the tree-level stringcross-section was calculated in Ref. [26]: even for a stringscale as low as 1 TeV, the cross-section found there isonly of the same order as the SM cross-section at ener-gies s >M2st. 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Remarks on the high-energy behaviour of cross-sections in weak-scale string theories

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