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Advances in High Energy Physics
Volume 2012 (2012), Article ID 129879, 43 pages
Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA
Received 30 August 2011; Revised 17 October 2011; Accepted 20 October 2011
Academic Editor: Ira Rothstein
Copyright © 2012 Luis Alfredo Anchordoqui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We outline the basic setting of the gauge theory and review the associated phenomenological aspects related to experimental searches for new physics at hadron colliders. In this construction, there are two massive Z′-gauge bosons, which can be naturally associated with baryon number being lepton number). We discuss the potential signals which may be accessible at the Tevatron and at the Large Hadron Collider (LHC). In particular, we provide the relevant cross sections for the production of Z′-gauge bosons in the TeV region, leading to predictions that are within reach of the present or the next LHC run. After that we direct attention to embedding the gauge theory into the framework of string theory. We consider extensions of the standard model based on open strings ending on D-branes, with gauge bosons due to strings attached to stacks of D-branes and chiral matter due to strings stretching between intersecting D-branes. Assuming that the fundamental string mass scale is in the TeV range and the theory is weakly coupled, we explore the LHC discovery potential for Regge excitations.
1. General Idea
The recent development in high energy physics has put a great emphasis on gauge theories; indeed the general theory of fundamental interactions, rather unimaginatively named the Standard Model (SM), is completely formulated in this framework. The SM agrees remarkably well with current data but has rather troubling weaknesses and appears to be a somewhat ad hoc theory. It is thought that the SM may be a subset of a more fundamental gauge theory. Several models have been explored, using the fundamental principle of gauge invariance as guidepost. The purpose of this paper is to outline the main phenomenological aspects of one such models: . The first aim is to survey the basic features of the gauge theory’s prediction regarding the new mass sector and couplings. These features lead to new phenomena that can be probed using data from the Tevatron and the Large Hadron Collider (LHC). In particular the theory predicts that additional gauge bosons that we will show are accessible at LHC energies. Having so identified the general properties of the theory, we focus on the potential to embed this model into a string theory. We show this can be accomplished within the context of D-brane TeV-scale compactifications. Finally, we explore predictions inherited from properties of the overarching string theory.
The SM is a spontaneously broken Yang-Mills theory with gauge group . Matter in the form of quarks and leptons (i.e., triplets and singlets, resp.) is arranged in three families () of left-handed fermion doublets (of ) and right-handed fermion singlets. Each family contains chiral gauge representations of left-handed quarks and leptons as well as right-handed up and down quarks, and , respectively, and the right-handed lepton . The hypercharge is shown as a subscript of the gauge representation . The neutrino is part of the left-handed lepton representation and does not have a right-handed counterpart.
The SM Lagrangian exhibits an accidental global symmetry , where is the baryon number symmetry and () are three lepton flavor symmetries, with total lepton number given by . It is an accidental symmetry because we do not impose it. It is a consequence of the gauge symmetries and the low energy particle content. It is possible (but not necessary), however, that effective interaction operators induced by the high energy content of the underlying theory may violate sectors of the global symmetry.
The electroweak subgroup is spontaneously broken to the electromagnetic by the Higgs doublet which receives a vacuum expectation value in a suitable potential. Three of the four components of the complex Higgs are “eaten” by the and bosons, which are superpositions of the gauge bosons of and of , with masses , , and at . The fourth vector field, persists massless, and the remaining Higgs component is left as a neutral real scalar. The measured values GeV and GeV fix the weak mixing angle at and the Higgs vacuum expectation value at GeV .
Fermion masses arise from Yukawa interactions, which couple the right-handed fermion singlets to the left-handed fermion doublets and the Higgs field, where is the antisymmetric tensor. In the process of spontaneous symmetry breaking, these interactions lead to charged fermion masses, , but leave the neutrinos massless . (One might think that neutrino masses could arise from loop corrections. This, however, cannot be the case, because the only possible neutrino mass term that can be constructed with the SM fields is the bilinear which violates the total lepton symmetry by two units (). As mentioned above, total lepton number is a global symmetry of the model and therefore -violating terms cannot be induced by loop corrections. Furthermore, the subgroup is nonanomalous, and therefore violating terms cannot be induced even by nonperturbative corrections. It follows that the SM predicts that neutrinos are strictly massless.) Experimental evidence for neutrino flavor oscillations by the mixing of different mass eigenstates implies that the SM has to be extended . The most economic way to get massive neutrinos would be to introduce the right-handed neutrino states (having no gauge interactions, these sterile states would be essentially undetectable) and obtain a Dirac mass term through a Yukawa coupling.
The SM gauge interactions have been tested with unprecedented accuracy, including some observables beyond even one part in a million . Nevertheless, the saga of the SM is still exhilarating because it leaves all questions of consequence unanswered. The most evident of unanswered questions is why there is a huge disparity between the strength of gravity and of the SM forces. This hierarchy problem suggests that new physics could be at play at the TeV-scale and is arguably the driving force behind high energy physics for several decades. Much of the motivation for anticipating the existence of such new physics is based on considerations of naturalness. The nonzero vacuum expectation value of the scalar Higgs doublet condensate sets the scale of electroweak interactions. However, due to the quadratic sensitivity of the Higgs mass to quantum corrections from an arbitrarily high mass scale , with no new physics between the energy scale of electroweak unification and the vicinity of the Planck mass, the bare Higgs mass and quantum corrections have to cancel at a level of one part in ~1030. This fine-tuned cancellation seems unnatural, even though it is in principle self-consistent. Thus either the scale of new physics is much smaller than the Planck scale or there exists a mechanism which ensures this cancellation, perhaps arising from a new symmetry principle beyond the SM; minimal supersymmetry (SUSY) is a textbook example . In either case, an extension of the SM appears necessary.
In this paper we examine the phenomenology of a newfangled extension of the gauge sector, , which has the attractive property of elevating the two major global symmetries of the SM ( and ) to local gauge symmetries . (The fundamental principles of the model are summarized in [6–8]. Herein though we replace at full length the doublets by doublets. Besides the fact that this reduces the number of extra ’s, one avoids the presence of a problematic Peccei-Quinn symmetry [9–11], associated in general with the of under which Higgs doublets are charged . A point worth noting at this juncture: the compact symplectic group is equivalent to ; our choice of notation will become clear in Section 5.) The boson , which gauges the usual electroweak hypercharge symmetry, is a linear combination of the of gauge boson , the boson , and a third additional field . The , , content of the hypercharge operator is given by with , and . The corresponding fermion and Higgs doublet quantum numbers are given in Table 1. The criteria we adopt here to define the Higgs charges are to make the Yukawa couplings (, , , ) invariant under all three ’s. From Table 1, has the charges ( and has (); therefore, the Higgs has , , , whereas has opposite charges , , . The two extra ’s are the baryon and lepton number; they are given by the following combinations: or equivalently by the inverse relations Even though is anomalous, with the addition of three fermion singlets , the combination is anomaly free. One can verify by inspection of Table 1 that these have the quantum numbers of right-handed neutrinos, that is, singlets under hypercharge. Therefore, this is a first interesting prediction of the gauge theory: right-handed neutrinos must exist.
Before discussing the favorable phenomenological implications of the model, we detail some desirable properties which apply to generic models with multiple symmetries.
2. Running of the Abelian Gauge Couplings
We begin with the covariant derivative for the fields in the “flavor” basis in which it is assumed that the kinetic energy terms containing are canonically normalized: The relations between the couplings and any nonabelian counterparts are left open for now. We carry out an orthogonal transformation of the fields . The covariant derivative becomes where for each Next, suppose we are provided with normalization for the hypercharge (taken as ) hereafter we omit the bars for simplicity. Rewriting (2.3) for the hypercharge and substituting (2.4) into (2.5), we obtain
One can think about the charges as vectors with the components labeled by particles . Let us first take the charges to be orthogonal, that is, for . Multiplying (2.6) by , we obtain or equivalently Orthogonality of the rotation matrix, , implies Then, the condition encodes the orthogonality of the mixing matrix connecting the fields coupled to the flavor charges and the fields rotated, so that one of them, , couples to the hypercharge . Therefore, for orthogonal charges, as the couplings run with energy, the condition needs to stay intact .
A very important point is that the couplings that are running are those of the fields; hence the functions receive contributions from fermions and scalars, but not from gauge bosons. As a consequence, if we start with a set of couplings at a high mass scale satisfying , this condition will be maintained at one loop as the couplings run down to lower energies (). The one-loop correction to the various couplings is where with and indicating contribution from fermion and scalar loops, respectively.
Should the charges not be orthogonal, it is instructive to write (2.6) as , where Certainly still holds as a possible solution. But as the charges do not form a mutually orthogonal basis, one can ask whether other solutions exist. This will be the case if, for nonzero , for each , where is the charge of the particle . In the gauge group of , the right-handed electron is charged only with respect to one of the abelian groups. From (2.19), this sets one of the ’s (say ) equal to zero. For , there remain at least 4 additional equations satisfied by the remaining components and . The resulting overcompleteness leads to .
Although in most models the condition holds in spite of the nonorthogonality of the ’s, the RG equations controlling the running of the couplings lose their simplicity. In particular, since the RG equations become coupled. In addition, kinetic mixing is generated at one loop level even if it is absent initially [14, 15]. Removal of the mixing term in order to restore canonical gauge kinetic energy requires an additional rotation, greatly complicating the analysis.
Here, we are considering models where the underlying symmetry at high energies is rather than . Following , we normalize all generators according to and measure the corresponding charges with respect to the coupling , with as the coupling constant. Hence, the fundamental representation of has charge unity. Another important element of the RG analysis is that the couplings () run different from the nonabelian () and (). This implies that the previous relation for normalization of abelian and nonabelian coupling constants, , holds only at the scale of unification . The SM chiral fermion charges in Table 1 are not orthogonal as given (,). Orthogonality can be completed by including a right-handed neutrino.
An obvious question is whether each of the fields on the rotated basis couples to a single charge . Let be the Lagrangian in the basis, with and vectors and a diagonal matrix in -dimensional “flavor” space. Now rotate to new orthogonal basis () for : equation (2.22) becomes As it stands, each does not couple to a unique charge ; hence we rotate , to obtain We wish to see if, for given and , we can find an so that This allows each to couple to a unique charge with strength . To see the problem with this, we rewrite (2.27) in terms of components for , (2.28) leads to In general, in (2.29) there are equations, but only independent generators in ; therefore the system is overdetermined . Of course, if , the equation becomes and so .
We illustrate with the case ; let then, From the off-diagonal terms, we obtain which implies that or equivalently that is a multiple of the unit matrix. Next, we consider the diagonal elements using to obtain Note that the matrix has one independent variable, and there are two independent homogeneous equations.
Any vector boson , orthogonal to the hypercharge, must grow a mass in order to avoid long range forces between baryons other than gravity and Coulomb forces. The anomalous mass growth allows the survival of global baryon number conservation, preventing fast proton decay . It is this that we now turn to study.
3. Premises of the Anomalous Sector
Outside of the Higgs couplings, the relevant parts of the Lagrangian are the gauge couplings generated by the covariant derivatives acting on the matter fields and the matrix of the anomalous sector where are the three gauge fields in the D-brane basis (), is a diagonal coupling matrix , and are the 3 charge matrices.
Again, perform a rotation and require that one of the ’s (say ) couples to hypercharge. We then obtain the constraint on the first column of given in (2.9). However, there is now an additional constraint: the field is an eigenstate of with zero eigenvalue. Under the rotation, the mass term becomes with . We know that at least is an eigenstate with eigenvalue zero. We also know that Poincare invariance requires the complete diagonalization of the mass matrix in order to deal with observables. However, further similarity transformations will undo the coupling of the zero eigenstate to hypercharge. There seems no way of eventually fulfilling all these conditions except to require that the same which rotates to couple to hypercharge simultaneously diagonalizes so that This implies that the original in the flavor basis is given by which results in the following baroque matrix: where
We turn now to discuss the phenomenological aspects of anomalous gauge bosons related to experimental searches for new physics at the Tevatron and at the CERN’s Large Hadron Collider (LHC).
4. Search for New Gauge Bosons at Hadron Colliders
Taken at face value, the disparity between CDF [18–21] and DØ  results insinuates a commodious uncertainty as to whether there is an excess of events in the dijet system invariant mass distribution of the associated production of a boson with 2 jets (hereafter production). The excess showed up in 4.3 fb−1 of integrated luminosity collected with the CDF detector as a broad bump between about 120 and 160 GeV [18, 19]. The CDF collaboration fitted the excess (hundreds of events in the channel) to a Gaussian and estimated its production cross-section times, the dijet branching ratio, to be 4 pb. This is roughly 300 times the SM Higgs rate . For a search window of 120–200 GeV, the excess significance above SM background (including systematics uncertainties) has been reported to be 3.2σ [18, 19]. Recently, CDF has included an additional 3 fb−1 to their data sample, for a total of 7.3 fb−1, and the statistical significance has grown to ~4.8σ (~4.1σ including systematics) [20, 21]. More recently, the DØ collaboration released an analysis (which closely follows the CDF analysis) of their data finding “no evidence for anomalous resonant dijet production’’ . Using an integrated luminosity of 4.3 fb−1, they set a 95% CL upper limit of 1.9 pb on a resonant production cross section.
Although various explanations have been proposed for the CDF anomaly [23–34], perhaps the simplest is the introduction of a new leptophobic gauge boson [35–44]. The suppressed coupling to leptons (or more specifically, to electrons and muons) is required to evade the strong constraints of the Tevatron searches in the dilepton mode [45–47] and LEP-II measurements of above the -pole [48–50]. In complying with the precision demanded of our phenomenological approach, it would be sufficient to consider a 1% branching fraction to leptons as consistent with the experimental bound. This approximation is within a factor of a few of model independent published experimental bounds. In addition, the mixing of the with the SM boson should be extremely small to be compatible with precision measurements at the -pole by the LEP experiments .
All existing dijet-mass searches via direct production at the Tevatron are limited to GeV [52–55] and therefore cannot constrain the existence of a with GeV. The strongest constraint on a light leptophobic comes from the dijet search by the UA2 collaboration, which has placed a 90% CL upper bound on in this energy range [56, 57]. A comprehensive model independent analysis incorporating Tevatron and UA2 data to constrain the parameters for predictive purposes at the LHC was recently presented . (Other phenomenological restrictions on -gauge bosons were recently discussed in .) As of today, the ATLAS and CMS experiments are not sensitive to the signal [60, 61]. However, LHC will eventually weigh in on this issue: if new physics is responsible for the CDF anomaly, an excess in should become statistically significant in ATLAS and CMS by the end of the year .
As usual, the gauge interactions arise through the covariant derivative where , , and are the gauge coupling constants. The fields are related to , and by the rotation matrix, with Euler angles , , and . Equation (4.1) can be rewritten in terms of , and as follows: Now, by demanding that has the hypercharge given in (1.4) we fix the first column of the rotation matrix : and we determine the value of the two associated Euler angles The couplings and are related through the orthogonality condition (2.11), with fixed by the relation for unification: . In what follows, we take 5 TeV as a reference point for running down to 150 GeV the coupling using (2.13), that is, ignoring mass threshold effects. This yields . We have checked that the running of the coupling does not change significantly by varying the scale of unification between 3 TeV and 10 TeV.
The phenomenological analysis thus far has been formulated in terms of the mass-diagonal basis set of gauge fields . As a result of the electroweak phase transition, the coupling of this set with the Higgses will induce mixing, resulting in a new mass-diagonal basis set . It will suffice to analyze only the system to see that the effects of this mixing are totally negligible. We consider simplified zeroth and first order (mass)2 matrices where is the mass of the gauge field, is the usual tree level formula for the mass of the particle in the electroweak theory (before mixing), is the electroweak coupling constant, is the vacuum expectation value of the Higgs field, , and , are of .
Standard Rayleigh-Schrodinger perturbation theory then provides the (mass)2 (to second order in ) and wave functions (to first order) of the mass-diagonal eigenfields corresponding to , From (4.8) the shift in the mass of the is given by , so that . The admixture of in the mass-diagonal field is Since all effects go as , all further discussion will be, with negligible error, in terms of . By the same token, the admixture of in the eigenfield is negligible, so that the discussion henceforth will reflect and .
The Lagrangian is of the form where each is a fermion field with the corresponding matrices of the Dirac algebra, and , with and , the vector and axial couplings, respectively. The (precut) production rate at the Tevatron pb, for arbitrary couplings and GeV, is found to be  where is the hadronic branching fraction. The presence of a in the process shown in Figure 1 restricts the contribution of the quarks to be purely left-handed. Since and the required branching to quarks is above about 99% (after selection cuts are accounted for), the coupling strength is fixed by the production rate. Below, we avoid reference to specific experimental selection cuts and present results for a generous range of possibilities consistent with existing data.
The dijet production rate at the UA2 GeV can be parametrized as follows : (our numerical calculation  using CTEQ6  agrees within 5% with the result of ). The maximum allowed value of the and couplings consistent with the UA2 upper limit is shown in Figure 2. The dilepton production rate at UA2 energies is given by where is the leptonic branching fraction. From (4.3) and (4.10), we obtain the explicit form of the chiral couplings in terms of and :
The second strong constraint on the model derives from the mixing of the and the through their coupling to the Higgs doublet. The last two terms in the covariant derivative are conveniently written as where and . The Higgs field kinetic term together with the anomalous mass terms () yields the following mass square matrix (we note in passing that two “supersymmetric” Higgses and , with charges , , and , , , would also be sufficient to give masses to all the chiral fermions. Here, , , , and . It is easily seen that the corresponding mass square matrix is independent of ):
Next, taking GeV we use the two degrees of freedom of the model to demand the shift of the mass to lie within 1 standard deviation of the experimental value and leptophobia. This occurs for , and TeV, corresponding to a suppression . This also corresponds to , and . The and couplings to the chiral fields are fixed and given in Table 2. The accompanying values of and are shown in Figure 2. Now, substituting the above figures into (4.4), we obtain the projections over : Using (1.6) it is straightforward to see that and become essentially and , respectively.
The couplings to quarks lead to a large (precut) production (≃6 pb) at the Tevatron and at GeV, a direct (precut) production (≃700 pb) in the region excluded by UA2 data. However, it is worthwhile to point out that the UA2 collaboration performed their analysis in the early days of QCD jet studies. Their upper bound depends crucially on the quality of the Monte Carlo and detector simulation which are primitive by today’s standard. They also use events with two exclusive jets, where jets were constructed using an infrared unsafe jet algorithm . In view of the considerable uncertainties associated with the UA2 analysis, we remain skeptical of drawing negative conclusions. Instead we argue that our model  could provide an explanation of the CDF anomaly if acceptance and pseudorapidity cuts reduce the production rate by about 35–66% and the UA2 90% CL bound is taken as an order-of-magnitude limit (Similar arguments have been previously advocated by Nelson et al. in [23–34] and by Liu et al. in [35–44]).
Since the CDF signal is in dispute, it is of interest to study the predictions of the model for a leptophobic at energies not obtainable at the Tevatron, but within the range of the LHC. The ATLAS collaboration has searched for narrow resonances in the invariant mass spectrum of dimuon and dielectron final states in event samples corresponding to an integrated luminosity of 1.21 fb−1 and 1.08 fb−1, respectively, . The spectra are consistent with SM expectations and thus a lower mass limit of 1.83 TeV on the sequential SM has been set. (In the sequential SM the has the same couplings to fermions as the boson.) Therefore, for TeV, we scan the parameter space demanding the shift of the mass to lie within 1 standard deviation of the experimental value and small (≲1%) branching to leptons. We find that for , and , the ratio . The chiral couplings to the and gauge bosons for these fiducial values are given in Table 3. Again, we see that and are essentially and .
The decay width of is given by  where is the Fermi coupling constant, , is the strong coupling constant at the scale , , and are the vector and axial couplings, and or 1 if is a quark or a lepton, respectively. Using the fiducial values of and fitted in Table 2, for TeV, we obtain TeV. Hence, to compare our predictions (at the parton level) with LHC experimental searches in dilepton and dijets, it is sufficient to consider the production cross section in the narrow width approximation, where the -factor represents the enhancement from higher order QCD processes estimated to be . After folding with the CTEQ6 parton distribution functions , we determine (at the parton level) the resonant production cross section. In Figure 3 we compare the predicted (a) and (b) production rates with 95% CL upper limits recently reported by the ATLAS  and CMS  collaborations. Selection cuts will probably reduce event rates by factors of 2 to 3. Keeping this in mind, we conclude that the 2012 LHC7 run will probe , whereas future runs from LHC14 will provide a generous discovery potential of up to about .
We turn now to discuss the string origin and the compelling properties of the gauge group.
5. Perturbative D-Brane Models in a Nutshell
At the time of its formulation and for years thereafter, Superstring Theory was regarded as a unifying framework for Planck-scale quantum gravity and TeV-scale SM physics. Important advances were fueled by the realization of the vital role played by D-branes [69, 70] in connecting string theory to phenomenology. This has permitted the formulation  of string theories with compositeness setting in TeV scales and large extra dimensions, see Appendix A. There are two paramount phenomenological consequences for TeV scale D-brane string physics: the emergence of Regge recurrences at parton collision energies and the presence of one or more additional gauge symmetries, beyond the of the SM.
D-brane TeV-scale string compactifications provide a collection of building block rules that can be used to build up the SM or something very close to it [72–82]. The details of the D-brane construct depend a lot on whether we use oriented string or unoriented string models. The basic unit of gauge invariance for oriented string models is a field, so that a stack of identical D-branes eventually generates a theory with the associated gauge group. In the presence of many D-brane types, the gauge group becomes a product form , where reflects the number of D-branes in each stack. Gauge bosons (and associated gauginos in a SUSY model) arise from strings terminating on one stack of D-branes, whereas chiral matter fields are obtained from strings stretching between two stacks. Each of the two strings end points carries a fundamental charge with respect to the stack of branes on which it terminates. Matter fields thus possess quantum numbers associated with a bifundamental representation. In orientifold brane configurations, which are necessary for tadpole cancellation, and thus consistency of the theory, open strings become in general nonoriented. For unoriented strings, the above rules still apply, but we are allowed many more choices because the branes come in two different types. There are the branes whose images under the orientifold are different from themselves and their image branes, and also branes which are their own images under the orientifold procedure. Stacks of the first type combine with their mirrors and give rise to gauge groups, while stacks of the second type give rise to only or gauge groups.
The minimal embedding of the SM particle spectrum requires at least three brane stacks  leading to three distinct models of the type that were classified in [12, 83]. Only one of them (model C of ) has baryon number as symmetry that guarantees proton stability (in perturbation theory) and can be used in the framework of TeV strings. Moreover, since (associated to the of ) does not participate in the hypercharge combination, can be replaced by leading to a model with one extra , the baryon number, besides hypercharge [84, 85]. Since baryon number is anomalous, the extra abelian gauge field becomes massive by the Green-Schwarz mechanism [86–90], behaving at low energies as a with a mass in general lower than the string scale by an order of magnitude corresponding to a loop factor [91, 92]. Lepton number is not a symmetry creating a problem with large neutrino masses through the Weinberg dimension-five operator suppressed only by the TeV string scale.
The SM embedding in four D-brane stacks leads to many more models that have been classified in [13, 93]. In order to make a phenomenologically interesting choice, we focus on models where can be reduced to . The minimal SM extension build up out of four stacks of D-branes is . A schematic representation of the D-brane structure is shown in Figure 4. The corresponding fermion quantum numbers are given in Table 1. Recall that the combination is anomaly free. As mentioned already, anomalous ’s become massive necessarily due to the Green-Schwarz anomaly cancellation, but nonanomalous ’s can also acquire masses due to effective six-dimensional anomalies associated for instance to sectors preserving SUSY [91, 92]. (In fact, also the hypercharge gauge boson of can acquire a mass through this mechanism. In order to keep it massless, certain topological constraints on the compact space have to be met.) These two-dimensional “bulk” masses become therefore larger than the localized masses associated to four-dimensional anomalies, in the large volume limit of the two extra dimensions. Specifically for D-branes with -longitudinal compact dimensions the masses of the anomalous and, respectively, the nonanomalous gauge bosons have the following generic scale behavior: Here is the gauge coupling constant associated to the group , given by , where is the string coupling and is the internal D-brane world-volume along the compact extra dimensions, up to an order one proportionality constant. Moreover, is the internal two-dimensional volume associated to the effective six-dimensional anomalies giving mass to the nonanomalous . For example, for the case of D5-branes, whose common intersection locus is just 4-dimensional Minkowski-space, denotes the volume of the longitudinal, two-dimensional space along the two internal D5-brane directions. Since internal volumes are bigger than one in string units to have effective field theory description, the masses of nonanomalous -gauge bosons are generically larger than the masses of the anomalous gauge bosons. Since we want to identify the light gauge boson with baryon number, which is always anomalous, a hierarchy compared to the second -gauge boson can arise, if we identify with the anomaly free combination , and take the internal world-volume a bit larger than the string scale. (In  a different (possibly T-dual) scenario with D7-branes was investigated. In this case the masses of the anomalous and nonanomalous ’s appear to exhibit a dependence on the entire six-dimensional volume, such that the nonanomalous masses become lighter than the anomalous ones.) In principle, this hierarchy can be advocated to explain the mass ratio required to explain the CDF anomaly. (It is important to stress that in SUSY models derived from D-brane compactifications there can be a light even if the string scale is [95, 96].)
Particles created by vibrations of relativistic strings populate Regge trajectories relating their spins and masses , where is the Regge slope parameter. Thus, if is of order few TeVs, a whole tower of infinite string excitations will open up at this low mass threshold. Should nature be so cooperative, one would expect to see a few string states produced at the LHC. The leading contributions of Regge recurrences to certain processes at hadron colliders are universal. This is because the full-fledged string amplitudes which describe parton scattering subprocesses involving four gauge bosons as well as those with two gauge bosons and two chiral matter fields are (to leading order in string coupling, but all orders in ) independent of the compactification scheme. Only one assumption will be necessary in order to set up a solid framework: the string coupling must be small for the validity of perturbation theory in the computations of scattering amplitudes. In this case, black hole production and other strong gravity effects occur at energies above the string scale (see Appendix A); therefore, at least the few lowest Regge recurrences are available for examination, free from interference with some complex quantum gravitational phenomena. We discuss this next.
6. Regge Recurrences
The most direct way to compute the amplitude for the scattering of four gauge bosons is to consider the case of polarized particles because all nonvanishing contributions can be then generated from a single, maximally helicity violating (MHV), amplitude—the so-called partial MHV amplitude . Assume that two vector bosons, with the momenta and , in the gauge group states corresponding to the generators and (here in the fundamental representation), carry negative helicities while the other two, with the momenta and and gauge group states and , respectively, carry positive helicities. (All momenta are incoming.) Then the partial amplitude for such an MHV configuration is given by [98, 99] where is the coupling constant, are the standard spinor products written in the notation of [100, 101], and the Veneziano form factor is the function of Mandelstam variables, , , ; . (For simplicity we drop carets for the parton subprocess.) The physical content of the form factor becomes clear after using the well-known expansion in terms of -channel resonances  which exhibits -channel poles associated to the propagation of virtual Regge excitations with masses . Thus, near the th level pole (), In specific amplitudes, the residues combine with the remaining kinematic factors, reflecting the spin content of particles exchanged in the -channel, ranging from to . (There are resonances in all the channels, that is, there are single particle poles in the and channels which would show up as bumps if or are positive. However, for physical scattering and are negative, so we do not see the bumps.) The low-energy expansion reads
Interestingly, because of the proximity of the 8 gluons and the photon on the color stack of D-branes, the gluon fusion into + jet couples at tree level . This implies that there is an order contribution in string theory, whereas this process is not occurring until order (loop level) in field theory. One can write down the total amplitude for this process projecting the gamma ray onto the hypercharge, The mixing coefficient evaluated at the scale for unification follows from (4.4) and is given by It is quite small, around for couplings evaluated at the mass, which is modestly enhanced to as a result of RG running of the couplings up to ~5 TeV.
Consider the amplitude involving three gluons , , and one gauge boson associated to the same stack: where is the identity matrix and is the charge of the fundamental representation. The color factor is where the totally symmetric symbol is the symmetrized trace while is the totally antisymmetric structure constant [100, 101].
The full MHV amplitude can be obtained [98, 99] by summing the partial amplitudes (6.1) with the indices permuted in the following way: where the sum runs over all 6 permutations of and . Note that in the effective field theory of gauge bosons there are no Yang-Mills interactions that could generate this scattering process at the tree level. Indeed, at the leading order of (6.5) and the amplitude vanishes due to the following identity: Similarly, the antisymmetric part of the color factor (6.9) cancels out in the full amplitude (6.10). As a result, one obtains where All nonvanishing amplitudes can be obtained in a similar way. In particular, and the remaining ones can be obtained either by appropriate permutations or by complex conjugation.
In order to obtain the cross section for the (unpolarized) partonic subprocess , we take the squared moduli of individual amplitudes, sum over final polarizations and colors, and average over initial polarizations and colors. As an example, the modulus square of the amplitude (6.10) is Taking into account all possible initial polarization/color configurations and the formula  we obtain the average squared amplitude : where Before proceeding, we need to make sure of the value of . If we were considering the process , then due to the normalization condition (2.21). However, for there are two additional projections given in (6.6): from to the hypercharge boson , yielding a mixing factor and from onto a photon, providing an additional factor . This gives
The two most interesting energy regimes of scattering are far below the string mass scale and near the threshold for the production of massive string excitations. At low energies, (6.17) becomes The absence of massless poles, at , and so forth, translated into the terms of effective field theory, confirms that there are no exchanges of massless particles contributing to this process. On the other hand, near the string threshold , see Appendix B for details.
The general form of (6.10) for any given four external gauge bosons reads where The modulus square of the four-gluon amplitude, summed over final polarizations and colors and averaged over all possible initial polarization/color configurations, follows from (6.22) and is given by 
The amplitudes involving two gluons and two quarks are also independent of the details of the compactification, such as the configuration of branes, the geometry of the extra dimensions, and whether SUSY is broken or not. This model independence makes it possible to compute all the string corrections to dijet signals at the LHC. The corresponding scattering amplitudes, computed at the leading order in string perturbation theory, are collected in . The average square amplitudes are given by the following:
The amplitudes for the four-fermion processes like quark-antiquark scattering are more complicated because the respective form factors describe not only the exchanges of Regge states but also of heavy Kaluza-Klein and winding states with a model-dependent spectrum determined by the geometry of extra dimensions. Fortunately, they are suppressed, for two reasons. First, the QCD color group factors favor gluons over quarks in the initial state. Second, the parton luminosities in proton-proton collisions at the LHC, at the parton center of mass energies above 1 TeV, are significantly lower for quark-antiquark subprocesses than for gluon-gluon and gluon-quark, see Figure 5. The collisions of valence quarks occur at higher luminosity; however, there are no Regge recurrences appearing in the -channel of quark-quark scattering .
In the following, we isolate the contribution from the first resonant state in (6.24)–(6.27). For partonic center of mass energies , contributions from the Veneziano functions are strongly suppressed, as , over standard model processes, see (6.20). (Corrections to SM processes at are of order , see (6.5)). In order to factorize amplitudes on the poles due to the lowest massive string states, it is sufficient to consider . In this limit, is regular while Thus the -channel pole term of the average square amplitude (6.24) can be rewritten as Note that the contributions of single poles to the cross section are antisymmetric about the position of the resonance and vanish in any integration over the resonance. (As an illustration, consider the amplitude in the vicinity of the pole, where and are real, , , and . Then, since Re, the cross section becomes . Integrating over the width of the resonance, one obtains , because , , and .) Before proceeding, we pause to present our notation. The first Regge excitations of the gluon , the color singlet , and quarks will be denoted by , respectively. Recall that has an anomalous mass in general lower than the string scale by an order of magnitude. If that is the case, and if the mass of the is composed (approximately) of the anomalous mass of the and added in quadrature, we would expect only a minor error in our results by taking the to be degenerate with the other resonances. The singularity at needs softening to a Breit-Wigner form, reflecting the finite decay widths of resonances propagating in the channel. Due to averaging over initial polarizations, (6.29) contains additive contributions from both spin and spin bosonic Regge excitations ( and ), created by the incident gluons in the helicity configurations () and (), respectively. The term in (6.29) originates from , and the piece reflects activity. Since the resonance widths depend on the spin and on the identity of the intermediate state (), the pole term (6.29) should be smeared as