String Resonances at Hadron Colliders

[Abridged] We consider extensions of the standard model based on open strings ending on D-branes. Assuming that the fundamental string mass scale M_s is in the TeV range and that the theory is weakly coupled, we discuss possible signals of string physics at the upcoming HL-LHC run (3000 fb^{-1}) with \sqrt{s} = 14 TeV, and at potential future pp colliders, HE-LHC and VLHC, operating at \sqrt{s} = 33 and 100 TeV, respectively. In such D-brane constructions, the dominant contributions to full-fledged string amplitudes for all the common QCD parton subprocesses leading to dijets and \gamma + jet are completely independent of the details of compactification, and can be evaluated in a parameter-free manner. We make use of these amplitudes evaluated near the first (n=1) and second (n=2) resonant poles to determine the discovery potential for Regge excitations of the quark, the gluon, and the color singlet living on the QCD stack. We show that for string scales as large as 7.1 TeV (6.1 TeV), lowest massive Regge excitations are open to discovery at 5\sigma in dijet (\gamma + jet) HL-LHC data. We also show that for n=1, the dijet discovery potential at HE-LHC and VLHC exceedingly improves: up to 15 TeV and 41 TeV, respectively. To compute the signal-to-noise ratio for n=2 resonances, we first carry out a complete calculation of all relevant decay widths of the second massive level string states. We demonstrate that for string scales M_s<~ 10.5 TeV (M_s<~ 28 TeV), detection of n=2 Regge recurrences at HE-LHC (VLHC) would become the smoking gun for D-brane string compactifications. Our calculations have been performed using a semi-analytic parton model approach which is cross checked against an original software package. The string event generator interfaces with HERWIG and Pythia through BlackMax. The source code is publically available in the hepforge repository.


Introduction
One of the most challenging problems in high energy physics today is to find out what is the underlying theory that completes the standard model (SM). Despite its remarkable success, the SM is incomplete with many unsolved puzzles -the most striking one being the huge disparity between the strength of gravity and of the other three known fundamental interactions corresponding to the electromagnetic, weak, and strong nuclear forces. Indeed, gravitational interactions are suppressed by a very high energy scale, the Planck mass M Pl = G −1/2 N ∼ 10 19 GeV, associated to a length l Pl ∼ 10 −35 m, where they are expected to become important. This hierarchy problem suggests that new physics could be at play above about the electroweak scale M EW ∼ G −1/2 F ∼ 300 GeV, and is arguably the driving force behind high energy physics for several decades.
In a quantum theory, the hierarchy implies a severe fine-tuning of the fundamental parameters in more than 30 decimal places in order to keep the masses of elementary particles at their observed values. The reason is that quantum radiative corrections to all masses generated by the Higgs vacuum expectation value (VEV) are proportional to the ultraviolet cutoff which in the presence of gravity is fixed by the Planck mass. As a result, all masses are "attracted" to about 10 16 times heavier than their observed values. A fine-tuned cancellation of the radiative corrections seems unnatural, even though it is in principle self-consistent. Naturalness implies that either the fundamental scale of gravity must be much smaller than the Planck mass, or else there should exist a mechanism which ensures this cancellation, perhaps arising from a new symmetry principle beyond the SM. Low energy supersymmetry (SUSY) with all superparticle masses in the TeV region is a textbook example. Indeed, in the limit of exact SUSY, quadratically divergent corrections to the Higgs self-energy are exactly cancelled, while in the softly broken case, they are cutoff by the SUSY breaking mass splittings. On the other hand, for low-mass-scale strings, quadratic divergences are cutoff by the string scale M s and low energy SUSY is not needed [1]. These two diametrically opposite viewpoints are experimentally testable at high-energy particle colliders, and in particular at the CERN Large Hadron Collider (LHC).
The recent discovery of a particle with a mass around 126 GeV [2,3], which seems to be the SM Higgs, has possibly plugged the final remaining experimental hole in the SM, cementing the theory further. The LHC data are so far compatible with the SM within 2σ and its precision tests. It is also compatible with low energy SUSY, although with some degree of fine-tuning in its minimal version. Indeed, in the minimal supersymmetric standard model (MSSM), the lightest Higgs scalar mass m h satisfies the following inequality: where the first term in the r.h.s. corresponds to the tree-level prediction and the second term includes the one loop corrections due to the top and stop loops. Here, m Z , m t , mt are the Z-boson, the top and stop quark masses, respectively, v = v 2 i + v 2 2 with v i the VEVs of the two higgses, tan β = v 2 /v 1 , and A t the trilinear stop scalar coupling. Thus, a Higgs mass around 126 GeV requires a heavy stop m t 3 TeV for vanishing A t , or A t 3mt 1.5 TeV in the "best" case scenario. These values are obviously consistent with the present LHC bounds on SUSY searches, but they are expected to be probed in the next run at double energy. Theoretically, they imply a fine-tuning of the electroweak scale at the percent to per mille level. This fine-tuning can be alleviated in supersymmetric models beyond the MSSM.
Low-mass-scale superstring theory provides a brane-world description of the SM, which is localized on membranes extending in p + 3 spatial dimensions, the so-called D-branes. Gauge   string theory should be weakly coupled, constrain the size of all parallel dimensions to be of order of the string length, while transverse dimensions remain unrestricted. Assuming an isotropic transverse space of n = 9 − p compact dimensions of common radius R ⊥ , one finds: where g s is the string coupling. It follows that the type I string scale can be chosen hierarchically smaller than the Planck mass [20,21] at the expense of introducing extra large transverse dimensions felt only by gravity, while keeping the string coupling small [20]. The weakness of 4d gravity compared to gauge interactions (ratio M W /M P ) is then attributed to the largeness of the transverse space R ⊥ compared to the string length l s = M −1 s . An important property of these models is that gravity becomes effectively (4 + n)-dimensional with a strength comparable to those of gauge interactions at the string scale. The first relation of Eq. (12) can be understood as a consequence of the (4 + n)-dimensional Gauss law for gravity, with M (4+n) * = M 2+n s /g 4 the effective scale of gravity in 4 + n dimensions. Taking M s ≃ 1 TeV, one finds a size for the extra dimensions R ⊥ varying from 10 8 km, .1 mm, down to a Fermi for n = 1, 2, or 6 large dimensions, respectively. This shows that while n = 1 is excluded, n ≥ 2 is allowed by present EPJ Web of Conferences actions emerge as excitations of open strings with endpoints attached on the D-branes, whereas gravitational interactions are described by closed strings that can propagate in all nine spatial dimensions of string theory (these comprise parallel dimensions extended along the (p + 3)-branes and transverse dimensions). For an illustration, consider type II string theory compactified on a six-dimensional torus T 6 , which includes a Dp-brane wrapped around p − 3 dimensions of T 6 with the remaining dimensions along our familiar (uncompactified) three spatial dimensions. We denote the radii of the internal longitudinal directions (of the Dp-brane) by R i , i = 1, . . . p − 3 and the radii of the transverse directions by R ⊥ j , j = 1, . . . 9 − p, see Fig. 1. The Planck mass, which is related to the string mass scale by determines the strength of the gravitational interactions. Here, is the volume of T 6 and g s is the string coupling. It follows that the string scale can be chosen hierarchically smaller than the Planck mass at the expense of introducing 9 − p large transverse dimensions felt only by gravity, while keeping the string coupling small. E.g. for a string mass scale M s ≈ O(1 TeV) the volume of the internal space needs to be as large as V 6 M 6 s ≈ O (10 32 ). On the other hand, the strength of coupling of the gauge theory living on the D-brane world volume is not enhanced as long as R i ∼ M −1 s remain small, The weakness of the effective 4 dimensional gravity compared to gauge interactions (ratio of v/M Pl ) is then attributed to the largeness of the transverse space radii R ⊥ i ∼ 10 32 l s compared to the string length l s = M −1 s . Should nature be so cooperative, a whole tower of infinite string excitations will open up at this low mass threshold, and new particles of spin J follow the well known Regge trajectories of vibrating strings: J = J 0 + α M 2 , where α is the Regge slope parameter that determines the fundamental string mass scale (1.5) Only one assumption will be necessary in order to set up a solid framework: the string coupling must be small for the validity of the above D-brane framework and of perturbation theory in the computation of scattering amplitudes. In this case, black hole production and other strong gravity effects occur at energies above the string scale, therefore at least the few lowest Regge recurrences are available for examination, free from interference with some complex quantum gravitational phenomena.
In a series of publications we have computed open string scattering amplitudes in D-brane models and have discussed the associated phenomenological aspects of low mass string Regge recurrences related to experimental searches for physics beyond the SM [4][5][6][7][8][9][10][11][12][13][14][15][16]. 1 We have shown that certain amplitudes to leading order in string coupling (but including all string α corrections) are universal. These amplitudes, which include 2 → 2 scattering processes involving 4 gluons or 2 gluons and 2 quarks, are 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. 2 This model independence makes it possible to compute the string corrections to γ + jet and dijet signals at the LHC, which, if traced to low-mass-scale string theory, could with 100 fb −1 of integrated luminosity (at √ s = 14 TeV) probe deviations from SM physics at a 5σ significance for M s as large as 6.8 TeV [5,8]. Indeed, the signal for string excitations is spectacularly dazzling: after operating for only a few months, with merely 2.9 inverse picobarns of integrated luminosity, the LHC7 CMS experiment ruled out M s < 2.5 TeV by searching for narrow resonances in the dijet mass spectrum [30]. In fact, LHC has the capacity of discovering strongly interacting narrow resonances in practically all range up to √ s LHC /2, and therefore, since no significance excess above background has been observed thus far, the ATLAS [31] and CMS [32,33] experiments have already excluded M s 4.5 TeV.
In this work we extend our previous studies in various directions. In all our previous analyses, the discovery reach was laid out processing the string amplitudes using a semi-analytic parton model approach. To confront technical detector challenges, however, the standard approach to 1 String Regge resonances in models with low-mass string scale are also discussed in [17][18][19][20][21][22][23][24], while Kaluza-Klein (KK) graviton exchange into the bulk, which appears at the next order in perturbation theory, is discussed in [25,26]. 2 The only remnant of the compactification is the relation between the Yang-Mills coupling and the string coupling. We take this relation to reduce to field theoretical results in the case where they exist, e.g., gg → gg. Then, because of the require correspondence with field theory, the phenomenological results are independent of the compactification of the transverse space. However, a different phenomenology would result as a consequence of warping one or more parallel dimensions [27][28][29]. data analysis is typically reliant on the existence of Monte Carlo event simulation tools that allow complete simulation of the signal. In this paper we are filling this gap by bringing the excitations of open strings into the ATLAS/CMS analysis software environment. A complete simulation with full Pythia treatment is quite a difficult task, because this event generator is set up in the same way perturbation theory works and consequently handles color flow lines of ordinary Feynman diagrams. Note that in string theory, there are processes (like gg → gγ) that in ordinary field theory work only at loop level and their color lines do not follow the normal lines of tree level Feynman diagrams. The proposed strategy here is to incorporate the string amplitudes into BlackMax [34,35], a comprehensive black hole event generator for LHC analysis that interfaces (via the Les Houches accord [36]) to HERWIG and Pythia. The parton evolution and hadronization will then be performed with the correct format for direct implementation in the official Monte Carlo packages for simulating an actual experiment at the LHC. The two-step approach advanced herein can circumvent the color line technicalities and, at the same time, facilitate the comparison with high-multiplicity events from gravitational collapse.
Recently the idea of building a 33 TeV and/or 100 TeV circular proton-proton collider has gained momentum, starting with an endorsement in the Snowmass Energy Frontier report [37], and importantly followed by the creation of two parallel initiatives: one at CERN [38] and one in China [39]. In this paper we study the discovery reach and exclusion limits of lowest massive Regge excitations for the following collider specifications: that are extensively discussed in the Snowmass Energy Frontier report [37]. For HE-LHC and VLHC, the second excited string states may also be within reach. The decay widths of n = 2 resonances into massless particles have been previously obtained in [22,23]. For a full treatment, however, one still need to compute the decay widths into one massive n = 1 particle and a massless particle. Herein we obatin all these widths by factorizing 4-point amplitudes with one massive (n = 1) and three massless particles.
The layout of the paper is as follows. We beging in Sec. 2 with an outline of the basic setting of intersecting D-brane models and we discuss general aspects of the effective low energy theory inhereted from properties of the overarching string theory. After that, we particularize the discussion to 3-and 4-stacks intersecting D-brane configurations that realize the SM by open strings. For completness, in Sec. 3 we provide a summary of previous results. In particular, we give an overview of all formulae relevant for the s-channel string amplitudes of lowest massive Regge excitations leading to γ + jet and dijets. Readers already familar with these topics may skip this section. In Secs. 4 and 5 we present a complete calculation of all relevant decay widths of the second massive level string states. The computation is performed in a model independent and universal way, and so our results hold for all compactifications. Armed with the full-fledged string amplitudes of all partonic subprocesses, in Sec. 6 we quantify signal and background rates of n = 1 and n = 2 Regge recurrences in the early LHC phase I, HL-LHC, HE-LHC, and VLHC. In Sec. 7 we describe the input and output of the string event generator interface (SEGI) with HERWIG 8 and Pythia through BlackMax and present some illustrative results. Finally in Sec. 8 we make a few observations on the consequences of the overall picture discussed herein.
A point worth noting at this juncture: The tensor-to-scalar ratio (r = 0.20 +0.07 −0.05 ) inferred from the excess B-mode power observed by the Background Imaging of Cosmic Extragalactic Polarization (BICEP2) experiment suggests in simple slow-roll models an era of inflation with energy densities of order (10 16 GeV) 4 , not far below the Planck density [40]. This presumably suggests that low-mass-scale string compactifications in connection with large extra dimension are quite hard to realize. However, one should keep in mind that there is an on going controversy concerning the effect of background on the BICEP2 result [41,42].

Intersecting D-brane string compactifications
D-brane low-mass-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 [43][44][45][46][47][48][49][50][51][52][53][54][55][56]. 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 U (1) field, so that a stack of N identical D-branes eventually generates a U (N ) theory with the associated U (N ) gauge group. In the presence of many D-brane types, the gauge group becomes a product form U (N i ), where N i 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 posses 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 non-oriented. 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 branes whose images under the orientifold are different from themselves, and also branes who are their own images under the orientifold procedure. Stacks of the first type combine with their mirrors and give rise to U (N ) gauge groups, while stacks of the second type give rise to only SO(N ) or Sp(N ) gauge groups.

Mass mixing effect
In three-stack intersecting brane models, one could have 1 or 2 massive U (1)'s, depending on using Sp (1) or U (2) to realize SU (2); while in four-stack models, one could have 2 or 3 massive U (1)'s. In general, one can have many U (1)'s in the intersecting brane model constructions including hidden sectors, and in these cases there will be many massive U (1)'s, which have been studied in [57][58][59]. Assuming no kinetic mixing, effectively the Lagrangian for all the U (1)'s from an n-stack model can be written as where ψ a denotes the matter fields charged under U (1) a (a, b, · · · label the stack of D-branes), g a are the gauge couplings, and Q a the charges. Note that the relation for U (N ) unification, g a = g a / √ 2N , holds only at M s because the U (1) couplings (g 1 , g 2 , g 3 , · · · ) run differently from the non-abelian SU (3) (g 3 ) and SU (2) (g 2 ) [60]. The U (1) mass-squared matrix is of the following form [58,61] where the integer-entry matrix K contains all the information of local model constructionswrapping numbers which give rise to correct family multiplicity and the (MS)SM spectrum -and G ij is the metric of the complex structure moduli space. 3 In general, the entries of the U (1) masssquared matrix are all of order of M 2 s . This U (1) mass-squared matrix is positive-semi-definite which has one zero eigenvalue that corresponds to the hypercharge. One could diagonalize M 2 ab using an orthogonal matrix O such that where the eigenvalues are sorted from small to large, i.e., λ i < λ j for i < j. λ 1 = 0 corresponds to the mass of the hypercharge gauge boson Y µ ≡ A (m) 1,µ . We can define the gauge boson corresponding to the lightest massive U (1) to be Z . Here we only discuss the case that there is only one massless U (1), thus D 2 contains only one zero eigenvalue (hypercharge) and all other U (1)'s are massive. 4 This transformation also takes the gauge fields from their original basis into the physical mass eigenbasis as (with an upper index (m) ): The column vectors of the orthogonal matrix O are the eigenvectors of M 2 . Since the eigenvalues are already sorted, the first column vector gives rise to the hypercharge combination 5) and the second column vector gives rise to and so on. Conversely, one could also write the gauge bosons in the original basis in terms of the mass eigenstates After the mass mixing, the Lagrangian in the U (1) gauge boson mass eigenbasis reads Since the elements in the orthogonal matrix O are in general irrational numbers (except for the first column, whose entrees are all fractional numbers which give rise to to the hypercharge), the gauge charges in the U (1) mass eigenbasis are not quantized. A matter field carrying Q a under U (1) a , with the gauge coupling g a , after the mass mixing couples to the gauge field A (m) i in the mass eigenbasis, with strengthḡ Thus, all the matter fields raised from D-brane can couple to all the anomalous U (1)'s. Since the elements of U (1) mass-squared matrix are around the same order, the entries of the orthogonal matrix O are in general of order O(1). Thus the anomalous U (1)'s could couple to all the SM particles with sizable strength [58].

Higgs mechanism and Z − Z mixing
The Higgs field(s) are also realized as open string(s) stretching between two stacks of D-branes and hence are charged under the two U (1)'s. After the mass mixing, the Higgs field(s) would be also charged under all the U (1)'s in the mass eigenbasis, and couple to all these massive U (1) gauge bosons. Thus after the electroweak symmetry breaking, all the gauge boson masses would be corrected. The covariant derivative reads where T a = σ a /2 is the SU (2) generator, and Y µ the hypercharge gauge boson. Effectively, the mass terms of all the U (1)'s take the form where v is the VEV of the Higgs. A 1 µ and A 2 µ give rise to W ± and the mass mixing only occurs within A 3 µ , A (m) i . One needs to perform another diagonalization to determine the mass eigenstates of all the massive U (1) gauge bosons. The special form of Eq. (2.10) ensures there is only one massless eigenstates which will be identified to be the photon. And the electric charge remains unchanged, i.e., e = g 2 g Y √ However, the Z-boson would be a mixture of i . The mass of the Z-boson is corrected by Hence, the mass of the Z gauge boson cannot be very light otherwise it would violate the constraints on Z − Z mixing from the electroweak precision test [64]. In addition, as mentioned earlier, all the anomalous U (1)'s could couple to all the SM particles with sizable strength. LEP II and LHC both set stringent bounds on them. In particular, the bound from LEP II on Z reads M Z /g Z l + l − > 6 TeV [65,66]. Due to the QCD background, LHC could set bounds on the Z by either examining the leptonic Drell-Yan processes pp → Z → l + l − [67,68], or examining the dijet resonances from a heavy Z [33]. These bounds are quite strong. Though it is difficult for LHC to distinguish low energy hadronic final states due to the QCD background, the LHC bound on a leptophobic Z (for example Z for U (1) B ) is not that strong [69]. However, it is very likely that the Z from D-brane models would couple to all the SM particles with sizable strength. Thus in general, unless some fine-tuning, this type of Z has to be quite massive ( 2 TeV) to pass all the current experimental constraints from colliders. We also would like to point out here that although in general Z (the lightest anomalous U (1)) can be much lighter than the string scale, this is a model-dependent question. For many cases, especially for intersecting brane models with less extra U (1)'s (e.g., the minimal D-brane model U (3) × Sp(1) × U (1) with only one additional (massive) U (1)), the mass of Z can also be closed to the string scale.

SM from D-brane constructs
While the existence of Regge excitations is a completely universal feature of string theory, there are many ways of realizing the SM in such a framework. Individual models utilize various D-brane configurations and compactification spaces. Consequently, these may lead to very different SM extensions, but as far as the collider signatures of Regge excitations are concerned, their differences boil down to a few parameters. The most relevant characteristics is how the U (1) Y hypercharge is embedded in the U (1) associated to D-branes. One U (1) (baryon number) comes from the "QCD" stack of three branes, as a subgroup of the U (3) group that contains SU (3) color but obviously, one needs at least one extra U (1). As noted in Sec 2.1, in D-brane compactifications the hypercharge always appears as a linear, non-anomalous combination of the baryon number with one, two or more U (1)s. The precise form of this combination bears down on the photon couplings, however the differences between individual models amount to numerical values of a few parameters.
The minimal embedding of the SM particle spectrum requires at least three brane stacks [70] leading to three distinct models of the type U (3) × U (2) × U (1) that were classified in [70,71]. In such minimal models the color stack a of three D-branes are intersected by the (weak doublet) stack b and by one (weak singlet) D-brane c [70]. For the two-brane stack b, there is a freedom of choosing physical state projections leading either to U (2) or to the symplectic Sp(1) representation of Weinberg-Salam SU (2) L .
In the bosonic sector, the open strings terminating on QCD stack a contain the standard SU (3) octet of gluons g a µ and an additional U (1) a gauge boson C µ , most simply the manifestation of a gauged baryon number symmetry: U (3) a ∼ SU (3) × U (1) a . On the U (2) b stack the open strings correspond to the electroweak gauge bosons A a µ , and again an additional U (1) b gauge field X µ . So the associated gauge groups for these stacks are SU (3) × U (1) a , SU (2) L × U (1) b , and U (1) c , respectively. We can further simplify the model by eliminating X µ ; to this end instead we can choose the projections leading to Sp(1) instead of U (2) [72]. The U (1) Y boson Y µ , which gauges the usual electroweak hypercharge symmetry, is a linear combination of C µ , the U (1) c boson B µ , and perhaps a third additional U (1) gauge field, X µ . 5 The fermionic matter consists of open strings located at the intersection points of the three stacks. Concretely, the left-handed quarks are sitting at the intersection of the a and the b stacks, whereas the right-handed u quarks comes from the intersection of the a and c stacks and the right-handed d quarks are situated at the intersection of the a stack with the c (orientifold mirror) stack. All the scattering amplitudes between these SM particles essentially only depend on the local intersection properties of these D-brane stacks.

Name
Representation The chiral fermion spectrum of the U (3) × Sp(1) × U (1) D-brane model is given in Table 1. In such a minimal D-brane construction, the coupling strength of C µ is down by root six when compared to the SU (3) C coupling g 3 and the hypercharge is free of anomalies. However, the Q a (gauged baryon number) is anomalous. This anomaly is canceled by the f-D version of the Green-Schwarz (GS) mechanism [73][74][75][76][77][78]. The vector boson Y µ , 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 [61].
In the U (3) × Sp(1) × U (1) D-brane model, the U (1) a assignments are fixed (they give the baryon number) and the hypercharge assignments are fixed by the SM. Therefore, the mixing angle θ P between the hypercharge and the U (1) a is obtained in a similar manner to the way the Weinberg angle is fixed by the SU (2) L and the U (1) Y couplings (g 2 and g Y , respectively) in the SM. The Lagrangian containing the U (1) a and U (1) c gauge fields is given by whereB µ = cos θ P Y µ + sin θ P Y µ andĈ µ = − sin θ P Y µ + cos θ P Y µ are canonically normalized. Substitution of these expressions into (2.13) leads to with We have seen that the hypercharge is anomaly free if From (2.15) we obtain the following relations (2.16) We use the evolution of gauge couplings from the weak scale M Z as determined by the one-loop beta-functions of the SM with three families of quarks and leptons and one Higgs doublet, We also use the measured values of the couplings at the Z pole α 3 (M Z ) = 0.118 ± 0.003, α 2 (M Z ) = 0.0338, α Y (M Z ) = 0.01014 (with the errors in α 2,Y less than 1%) [79]. Running couplings up to 5 TeV, which is where the phenomenology will be, we get κ ≡ sin θ P ∼ 0.14. When the theory undergoes electroweak symmetry breaking, because Y couples to the Higgs, one gets additional mixing. Hence Y is not exactly a mass eigenstate. The explicit form of the low energy eigenstates A µ , Z µ , and Z µ is given in [80].
We pause to summarize the degree of model dependency stemming from the multiple U (1) content of the minimal model containing 3 stacks of D-branes. First, there is an initial choice to be made for the gauge group living on the b stack. This can be either Sp (1) or U (2). In the case of Sp(1), the requirement that the hypercharge remains anomaly-free was sufficient to fix its U (1) a and U (1) c content, as explicitly presented in Eqs. (2.15) and (2.16). Consequently, the fermion couplings, as well as the mixing angle θ P between hypercharge and the baryon number gauge field are wholly determined by the usual SM couplings. The alternative selection -that of U (2) as the gauge group tied to the b stack -branches into some further choices. This is because the Q a , Q b , Q c content of the hypercharge operator is not uniquely determined by the anomaly cancelation requirement. In fact, as seen in [70], there are three possible embeddings with one more possibility for the hypercharge combination besides (2.12). This final choice does not depend on further symmetry considerations.
The SM embedding in four D-brane stacks leads to many more models that have been classified in [81,82]. In order to make a phenomenologically interesting choice, we focus on models where U (2) can be reduced to Sp (1). Besides the fact that this reduces the number of extra U (1)'s, one avoids the presence of a problematic Peccei-Quinn symmetry, associated in general with the U (1) of U (2) under which Higgs doublets are charged [70]. We then impose Baryon and Lepton number symmetries that determine completely the model U (3) C × Sp(1) L × U (1) L × U (1) R , as described in [47,82]. A schematic representation of the D-brane structure is shown in Fig. 2. The corresponding fermion quantum numbers are given in Table 2. The two extra U (1)'s are the Baryon and Lepton number, B and L, respectively; they are given by the following combinations: with c 1 = 1/2, c 3 = 1/6, and c 4 = −1/2; or equivalently by the inverse relations: As usual, the U (1) gauge interactions arise through the covariant derivative where g 1 , g 3 , and g 4 are the gauge coupling constants. We can define Y µ and two other fields Y µ , Y µ that are related to C µ , B µ ,B µ by the orthogonal transformation [83] with Euler angles θ, ψ, and φ. Equation (2.21) can be rewritten in terms of Y µ , Y µ , and Y µ as follows Now, by demanding that Y µ has the hypercharge Q Y given in Eq. (2.19) we fix the first column of the rotation matrix O and we determine the value of the two associated Euler angles (2.26) The couplings g 1 and g 4 are related through the orthogonality condition, with g 3 fixed by the relation g 3 (M s ) = √ 6 g 3 (M s ) [60]. The field Y µ then appears in the covariant derivative with the desired Q Y . The ratio of the coefficients in Eq. (2.24) is determined by the form of Eq. (2.19) and Eq. (2.21). The value of g Y is determined so that the coefficients in Eq. (2.24) are components of a normalized vector so that they can be a row vector of O. The rest of the transformation (the ellipsis part) involving Y , Y is not necessary for our calculation. The point is that we now know the first row of the matrix O and hence we can get the first column of O T , which gives the expression of Y µ in terms of C µ , B µ ,B µ , This is all we need when we calculate the interaction involving Y µ ; the rest of O, which tells us the expression of Y , Y in terms of C, X, B is not necessary. For later convenience, we define κ, η, ξ as The expression for the C − Y mixing parameter κ is the same as that of the Note that with the 'canonical' charges of the right-handed neutrino Q 1L = Q 1R = −1, the combination B − L is anomaly free, while for Q 1L = Q 1R = +1, both B and B − L are anomalous. 6 As mentioned already, anomalous U (1)'s become massive necessarily due to the GS anomaly cancellation, but non anomalous U (1)'s can also acquire masses due to effective six-dimensional anomalies associated for instance to sectors preserving N = 2 SUSY [85,86]. 7 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 Dp-branes with (p − 3)longitudinal compact dimensions the masses of the anomalous and, respectively, the non-anomalous U (1) gauge bosons have the following generic scale behavior: Here g a is the gauge coupling constant associated to the group U (1) a , given by g a ∝ g s / V where g s is the string coupling and V is the internal D-brane world-volume along the (p − 3) compact extra dimensions, up to an order one proportionality constant. Moreover, V 2 is the internal two-dimensional volume associated to the effective six-dimensional anomalies giving mass to the non-anomalous U (1) a . 8 E.g., for the case of D5-branes, whose common intersection locus is just 4-dimensional Minkowski-space, V = V 2 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 non-anomalous U (1)-gauge bosons are generically larger than the masses of the anomalous gauge bosons.
In principle, in addition to the orthogonal field mixing induced by identifying anomalous and non-anomalous U (1) sectors, there may be kinetic mixing between these sectors. In all the D-brane models discussed in this section, however, since there is only one U (1) per stack of D-branes, the relevant kinetic mixing is between U (1)'s on different stacks, and hence involves loops with fermions at brane intersection. Such loop terms are typically down by g 2 i /16π 2 ∼ 0.01 [87]. Generally, the major effect of the kinetic mixing is in communicating SUSY breaking from a hidden U (1) sector to the visible sector, generally in modification of soft scalar masses. Stability of the weak scale in various models of SUSY breaking requires the mixing to be orders of magnitude below these values [87]. For a comprehensive review of experimental limits on the mixing, see [88]. Moreover, none of the D-brane constructions discussed above have a hidden sector -all the U (1)'s (including the anomalous ones) couple to the visible sector. In summary, kinetic mixing between the nonanomalous and the anomalous U (1)'s in every basic model discussed in this paper will be small because the fermions in the loop are all in the visible sector. In the absence of electroweak symmetry breaking, the mixing vanishes.

Lowest massive Regge excitations of open strings
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 non-vanishing contributions can be then generated from a single, maximally helicity violating (MHV), amplitude -the so-called partial MHV amplitude [89]. Assume that two vector bosons, with the momenta k 1 and k 2 , in the U (N ) gauge group states corresponding to the generators T a 1 and T a 2 (here in the fundamental representation), carry negative helicities while the other two, with the momenta k 3 and k 4 and gauge group states T a 3 and T a 4 , respectively, carry positive helicities. (All momenta are incoming.) Then the partial amplitude for such an MHV configuration is given by [90,91] where g is the U (N ) coupling constant, ij are the standard spinor products written in the notation of Ref. [92,93], and the Veneziano form factor, is the function of Mandelstam variables, s = 2k 1 k 2 , t = 2k 1 k 3 , u = 2k 1 k 4 ; s + t + u = 0. (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 s-channel resonances [94] which exhibits s-channel poles associated to the propagation of virtual Regge excitations with masses √ nM s . Thus near the nth level pole (s → nM 2 s ): In specific amplitudes, the residues combine with the remaining kinematic factors, reflecting the spin content of particles exchanged in the s-channel, ranging from J = 0 to J = n + 1. 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 γ + jet couples at tree level [5]. This implies that there is an order g 2 3 contribution in string theory, whereas this process is not occuring until order g 4 3 (loop level) in 18 field theory. One can write down the total amplitude for this process projecting the gamma ray onto the hypercharge, where κ is the (model dependent) C-Y mixing coefficient.
Consider the amplitude involving three SU (N ) gluons g 1 , g 2 , g 3 and one U (1) gauge boson γ 4 associated to the same U (N ) stack: where I is the N ×N identity matrix and Q is the U (1) charge of the fundamental representation. The color factor where the totally symmetric symbol d abc is the symmetrized trace while f abc is the totally antisymmetric structure constant (see Appenix A).
The full MHV amplitude can be obtained [90,91] by summing the partial amplitudes (3.1) with the indices permuted in the following way: where the sum runs over all 6 permutations σ of {1, 2, 3} and i σ ≡ σ(i), N = 3. 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, V = 1 at the leading order of Eq.(3.5) and the amplitude vanishes due to the following identity: Similarly, the antisymmetric part of the color factor (3.8) cancels out in the full amplitude (3.9). As a result, one obtains: µ(s, t, u) 12 23 34 41 + µ(s, u, t) 12 24 13 34 , All non-vanishing amplitudes can be obtained in a similar way. In particular, 24 12 34 , (3.13) 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 gg → gγ, 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 (3.9) is: (3.14) 19 Taking into account all 4(N 2 − 1) 2 possible initial polarization/color configurations and the formula [95] a,b,c we obtain the average squared amplitude [5] |M(gg → gγ) Before proceeding, we need to make precise the value of Q. If we were considering the process gg → Cg, then Q = 1/6 due to the U (N ) normalization condition [70]. However, for gg → γg there are two additional projections given in (3.6): from C µ to the hypercharge boson Y µ , yielding a mixing factor κ; and from Y µ onto a photon, providing an additional factor cos θ W . This gives The two most interesting energy regimes of gg → gγ scattering are far below the string mass scale M s and near the threshold for the production of massive string excitations. At low energies, Eq. (3.16) becomes The absence of massless poles, at s = 0 etc., 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 s ≈ M 2 The general form of (3.9) for any given four external gauge bosons reads The modulus square of the four-gluon amplitude, summed over final polarizations and colors, and averaged over all 4(N 2 − 1) 2 possible initial polarization/color configurations follows from (3.21) and is given by [9] |M(gg → gg)| 2 = g 4 The average square amplitudes for two gluons and two quarks are given by and 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 KK and winding states with a model-dependent spectrum determined by the geometry of extra dimensions. Fortunately, they are suppressed, for two reasons: (i) the QCD SU (3) color group factors favor gluons over quarks in the initial state; (ii) the parton luminosities in protonproton 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 [14]. The collisions of valence quarks occur at higher luminosity; however, there are no Regge recurrences appearing in the s-channel of quark-quark scattering [9].
In the following we isolate the contribution from the first resonant state in Eqs. ) In order to factorize amplitudes on the poles due to the lowest massive string states, it is sufficient to consider s = M 2 s . In this limit, V s is regular while Thus the s-channel pole term of the average square amplitude (3.23) can be rewritten as (3.28) 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. 9 Before proceeding, we pause to present our notation. The first Regge excitations of the gluon g, the color singlet C, and quarks q will be denoted by G (1) , C (1) , Q (1) , respectively. Recall that C µ 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 C (1) is composed (approximately) of the anomalous mass of the C µ and M s added in quadrature, we would expect only a minor error in our results by taking the C (1) to be degenerate with the other resonances. The singularity at s = M 2 s needs softening to a Breit-Wigner form, reflecting the finite decay widths of resonances propagating in the s channel. Due to averaging over initial polarizations, Eq.(3.28) contains additive contributions from both spin J = 0 and spin J = 2 U (3) bosonic Regge excitations (G (1) and C (1) ), created by the incident gluons in the helicity configurations (±±) and (±∓), respectively. The M 8 s term in Eq. (3.28) originates from J = 0, and the t 4 + u 4 piece reflects J = 2 activity. Since the resonance widths depend on the spin and on the identity of the intermediate state (G (1) , C (1) ) the pole term (3.28) should be smeared as [8] |M(gg → gg) GeV are the total decay widths for intermediate states G (1) and C (1) , with angular momentum J [7]. The associated weights of these intermediate states are given in terms of the probabilities for the various entrance and exit channels and W gg→gg A similar calculation transforms Eq. (3.24) near the pole into and W gg→qq Near the s pole Eq. (3.25) becomes whereas Eq. (3.26) can be rewritten as The total decay widths for the Q (1) excitation are: Γ The s-channel poles near the second Regge resonance can be approximated by expanding the The associated scattering amplitudes and decay widths of the n = 2 string resonaces are discussed in Secs. 4 and 5. Roughly speaking, the width of the Regge excitations will grow at least linearly with energy, whereas the spacing between levels will decrease with energy. This implies an upper limit on the domain of validity for our phenomenological approach [15]. In particular, for a resonance R of mass M , the total width is given by where C > 1 because of the growing multiplicity of decay modes [7,22]. On the other hand, since For excitation of the resonance R via a + b → R, the assumption Γ tot (R) ∼ Γ(R → ab) (which underestimates the real width) yields a perturbative regime for n 40. This is to be compared with the n ∼ 10 4 levels of the string needed for black hole production. 11 Before discussing the decay widths of the second massive level string states, we note that the Breit-Wigner form for gluon fusion into γ + jet follows from (3.20) and is given by and the dominant s-channel pole term of the average square amplitude contributing to pp → γ + jet reads 4 Decay widths of the second massive level string states

Amplitudes and Factorization
The main goal of this section is to obtain the decay widths of the second massive level string states which will appear as resonances in scattering processes gg → gg, gq → gq and gg → qq in hadron colliders. In intersecting brane models, gluons g are the zeroth level massless strings attaching to the U (3) a stack of D-branes; left-handed quarks q L which participate in the weak interactions are massless strings stretching between U (3) a stack and the SU (2) stack (U (2) or Sp (1)); righthanded quarks q R could arise as either massless strings stretching between U (3) a stack and another U (1) stack, or massless strings attaching only to U (3) a stack and appearing as the anti-symmetric representation of U (3).
Let us firstly clarify our notation on various string states in different massive levels. We follow the notations in [9][10][11][12][13], and we will focus on the string states which contribute to gg → gg and gq → gq processes. The bosonic sector of the first massive level consists of two universal string states: a spin two field α and a complex scalar Φ. In addition, there is a spin one field d whose vertex operator involves the internal current J . This vector d can decay into qq which is a universal property of all N = 1 compactifications [11]. As the U (3) generators decompose to the SU (3) color generators plus the U (1) generator (color singlet), we have two copies of the string excitations. We will denote the color octets by G (n) , and the color singlets by C (n) , where n indicates the nth massive level. For the fermionic sector, the excited quark triplets Q (1) consists of one spin- 3 2 field χ and one spin-1 2 field a (and also their opposite chirality fieldsχ,ā). For the bosonic sector of the second massive level (G (2) , C (2) ), four universal states has been determined [12]: a spin three field σ, a spin two field π and two complex vector fields Ξ 1,2 .
The total decay width of a second massive level bosonic string state G (2) consists of four contributions: G (2) decays into two massless string states (G (2) → gg and G (2) → qq); G (2) decays into one first massive level string state plus one massless string state (G (2) → G (1) g and G (2) → Q (1) q); G (2) decays into a color singlet (anomalous U (1)'s) plus a massless gluon or an excited gluon (G (2) → gA a and G (2) → G (1) A a ); G (2) decays into the excitation of the color singlet C (1) plus one massless gluon. For a second massive level color singlet string state C (2) , its decay width also involves four contributions: C (2) decays into two massless string states (C (2) → gg and C (2) → qq); C (2) decays into one first massive level string state plus one massless string state (C (2) → C (1) g and C (2) → Q (1) q); C (2) decays into two anomalous U (1)'s; C (2) decays into the excitation of the color singlet C (1) plus one anomalous U (1). For a second massive level excited quark Q (2) , its total decay width could consist of five contributions: Q (2) decays into one massless gluon plus one massless quark (Q (2) → gq); Q (2) decays into one first massive level string states and one massless string state (Q (2) → G (1) q and Q (2) → Q (1) g); Q (2) decays into anomalous U (1)'s plus a massless quark or an excited quark (Q (2) → qA a and Q (2) → Q (1) A a ); Q (2) decays into the excitation of the color singlet C (1) plus one quark; finally for Q (2) which participates in weak interactions, it could also decay into SU (2) gauge bosons plus one quark. All above decay channels of the second massive level string states are summarized in Table 3. Most of these decay channels are universal to all compactifications, while there are also several model-dependent channels. We will comment on them in Section 4.7, 4.8 and 4.9.
The partial decay width of G (2) and Q (2) decaying into two massless string states were already obtained in [22,23] by using factorization. However, we realize that there are some mistakes in their results. The widths of G (2) decaying into gg in [22] should be reduced by one half. Moreover, 2 massless 1 first level string state involve 1 or 2 involve 1 first level string states plus 1 massless string state color singlet(s) color singlet excitation G (2) gg, qq G (1) g, Q (1)q ,Q (1) q gA a , G (1) A a C (1) g C (2) gg, qq Table 3: Possible decay channels for the second massive level string states G (2) , C (2) , Q (2) . Excited massive quarks which participate in weak interactions can also decay into SU (2) gauge bosons plus another quark.
there are in fact two distinct Q (2) (J = 3/2) states. They can decay into gq of helicities (+1, +1/2) and (−1, +1/2) respectively and do not mix with each other. So we need to consider their widths separately (instead of adding them up as in [23]). In this section, we will obtain the partial decay widths of G (2) , C (2) and Q (2) decaying into one first massive level string state (G (1) , C (1) or Q (1) ) plus one massless string state (g or q) using four-point amplitudes with one leg being the first massive level string state obtained in [11]. We will comment on other decay channels at the end of this section.
We have seen in Sec. 3 that four-point amplitudes A(g, g, g, g) and A(g, g, q,q) carry the form factor V (s, t, u) which can be expanded in terms of s-channel resonances. Recasting the expansion we can re-express the amplitudes as sums of Wigner d-matrices, one could then obtain two threepoint amplitudes of massive string states decaying into different final states with specific spin combinations [7]. Using this method, one could identify the contributions of various string states with different spins appearing as resonances in the s-channel pole at a certain massive level. Previous works only deal with the four-point amplitude with four massless string states, whereas in this work we consider the factorization of four-point amplitudes one of whose external legs is massive. More specifically, we consider four-point amplitudes A(G (1) , g, g, g), A(G (1) , g, q,q) and A(Q (1) , g, g,q) which were computed in [11]. By factorizing these amplitudes and using the known results (amplitudes that G (2) , Q (2) decaying into two massless string states), we could obtain the partial decay widths of one second massive level string state decaying into a first massive level string state plus a massless one.
For the four bosonic string states scattering, there is one subtlety which is the decomposition of the group factors. The structure constant of the gauge group f a 1 a 2 a 3 or the total symmetric trace d a 1 a 2 a 3 would arise when we combine the three-point amplitudes of two different orderings (1, 2, 3) and (1, 3, 2) on the worldsheet. This depends on the overall worldsheet parity (−1) N +1 where N is the sum of the overall massive level number of the three scattering string states. More specifically, the combined amplitudes have the following group factors When factorizing a four-point amplitude with one first massive level leg, on one side one gets a second massive level string state decaying into a first massive string state plus a zeroth level mode, and on the other side one gets the same second massive level string state decaying into two zeroth level massless string states. Thus one would get a group factor of d a 1 a 2 a on the left and f a 3 a 4 a on the right, see Fig. 3. Factorizing amplitudes involving two fermions is simpler since there are only two Chan-Paton factors involved. Our notation on these group factors is summarized in Appendix A.
T a4 d a1a2a f a3a4a G (2) or C (2) Figure 3: Factorization of the amplitude A(G (1) , g, g, g) gives different group factors on two sides. The doubled wavy line present the first massive level bosonic string state, whereas the single lines present massless bosonic string states. G (2) or C (2) are the second massive level intermediate string states obtained from factorization.
In this section, all the four-point amplitudes with one first massive level string states are taken from [11]. In [11], the massive string state was placed at the position 4, and the three massless ones took the position 1,2,3. For our convenience, in this work we prefer to place the massive string state at position 1, while the three massless string states were placed at 2,3,4. The corresponding amplitudes can be easily obtained by performing permutations of the original amplitudes.
The helicity wave function of a massive higher spin particle is specified by a pair of light-like vector p µ , q µ , which is a decomposition of the momentum of the particle k µ = p µ + q µ . 12 The spin quantization axis is along the direction of q in the rest frame, here it is most convenient to set q µ = k µ 2 , so that the spin axis of the first massive level string state (at position 1) is along the same direction as the spin axis of the massless string state at position 2, and we denote this direction to be + z. Due to angular momentum conservation, the spin axis of the intermediate second massive level string state (see Fig. 3) should also align to + z, and the corresponding helicity amplitudes of these three states with only specific j z combinations can survive. The reference momentums of particle 1 are chosen to be: The spinor products become, where s, t, u are Mandelstam variables. With this choice, we could extract the helicity amplitudes of the second massive level strings decaying into a first massive level string plus a massless one with their spin axes all along + z (the direction of the momentum of the massless string state), from the four-point amplitudes in [11]. In the next section, we will focus on the spin three and spin two universal string states from the second massive level, computing their scattering amplitudes and their partial decay widths, where we will also align the spins of the three interacting states in the direction of the momentum of the massless particle. Thus, we are expecting the helicity amplitudes we obtained from factorization in this section to match exactly with the string amplitudes from CFT computations in the next section.
We will discuss the factorization of the four-point amplitudes in the following order: We start from the amplitudes which involve the first massive level spin two field α, and obtain the decay widths of second massive level string states decaying into α plus another massless string state. Then we discuss the decays which involve the final states d, Φ, χ, a in order, which are obtained from the four-point amplitudes with d, Φ, χ, a plus three other massless string states. The full results of decay widths for n = 2 resonances are summarized in Table 4 at the end of this section.

α(J = 2)
The highest spin field from the first massive level is the spin two boson α with its vertex operator given in Eq. (5.4). We will need to use the following amplitudes (all particles are incoming) [11],

6)
A α(−1), The other nonvanishing amplitudes can be obtained by taking complex conjugate and permutation. where θ is the angle between − z and the spatial momentum of particle 3. It is related to the Mandelstam variables u, t by From (4.7) we can read off the matrix elements as, where we use F a,J λ 1 λ 2 a 1 a 2 to denote the amplitude of a spin-J particle with angular momentum j z = λ 1 + λ 2 (and gauge index a) decaying into particles 1, 2 with momenta along the z-axis. λ 1 , λ 2 are helicities of the two particles while a 1 , a 2 are gauge indices. Thus, the result of Eq. (4.9) presents the decay of a second massive level spin three string state with j z = −3 decaying into α 1 (j z = −2) and − 2 , which is exactly what we get in Eq. (5.48) in the next section. In Eq. (5.48), all particles are incoming and the corresponding outgoing particles are one α(−2) and one − . We would like to remind the reader that the definition of F a,J λ 1 λ 2 a 1 a 2 is in some sense different from what is used in the literature [7,22,23]. Previously the helicity λ 1 (of a massless particle) was usually defined with its spin axis along k 1 . In our convention the spin axis of every particle is along + z. Particle 1 is moving along − z and its spin axis is opposite to k 1 .
Similarly, we can do the factorization for amplitudes with other spin configurations: a 1 a 2 a d a 3 a 4 a , (4.10) 14) a 1 a 2 a d a 3 a 4 a , (4.16) We need to take into account both the channels into α + g + and into α + g − and the results are The spin one resonances arise from factorization of the amplitude A[α, −, +, +]: and we obtain which corresponds to the complex vectors found in [12]. Unlike G (2) (J = 3, 2), G (2) (J = 1) is not parity invariant, the matrix elements in (4.21) are for two different particles and should not be added together. Thus the corresponding partial decay width reads

Q (2) (J = 5/2, 3/2) → α + q
We could obtain the second massive level spin-5 2 and spin-3 2 resonances from factorizing amplitude A[α, + 1 2 , − 1 2 , +]: A α(+2), 13 Since the decay product includes a massive particle, the decay width is suppressed by M 2 s /s compared to the width of decaying into two massless particles. The suppression is due to the difference in | k 1 |/ √ s, which appears in phase space integration of the final states. In the case of two outgoing massless particles this ratio is 1 2 while in the current case it is Left-handed and right-handed fermions are stretching between different branes. As a result, lefthanded excited quarks cannot decay into right-handed quarks plus gluons. For example, we have F α,J=5/2 +2+ 1 2 a 1 α 2 = F α,J=5/2 −2− 1 2 a 1 α 2 but they are decay amplitudes for left and right-handed excited quarks and should not be combined. The corresponding decay widths are: (4.31)

d(J = 1)
The spin one field d is different from the universal bosonic fields α, Φ in that it is tied to spacetime SUSY. Although its vertex operator contains the worldsheet current J , the vector d does give rise to universal amplitudes into quark-antiquark pair [11]. The existence of this vector resonance is a universal property of all N = 1 SUSY compactifications. We will need the amplitude A[d 1 , u 2 ,ū 3 , 4 ], which reads These amplitudes will give rise to two channels of the second massive level string resonances. 14 4.3.1 Q (2) (J = 5/2, 3/2) → d + q We could obtain the second massive level spin-5 2 and spin-3 2 resonances from factorizing amplitude The corresponding partial decay widths read The second massive level spin-3 2 and spin-1 2 resonances arise from amplitude A[d, − 1 2 , +, + 1 2 ]: 14 Indeed, by factorizing A[d, +, − 1 2 , + 1 2 ] amplitudes, one can get the second massive level J = 2, 1 resonances where the states can decay into d + g. These states are not the same as the G (2) (J = 2, 1) we have discussed above. For N = 1 compactification, the vertex operator of this vector d involves internal current J [11]. It only couples to quark-antiquark pairs. While the G (2) (J = 2, 1) states, for whose vertex operators, c.f., [12], cannot decay into d + g. Thus the vertex operators of J = 2, 1 resonances which arise from this channel must also contain internal components. These J = 2, 1 states do not couple to a pair of gluons and thus play no role in processes gg → gg or gg → qq. Even though these states do couple to quark-antiquark pairs and may contribute to four-fermion amplitudes, we will not consider such processes as they are suppressed [8]. Thus we will not discuss these states in this work. and the corresponding partial decay widths read Similar to previous case, we identify the spin-3 2 fermion in this channel asQ (2) (J = 3/2).

Φ ± (J = 0)
Φ is a complex scalar field, which couples to only (anti)self-dual gauge field configurations, i.e., to gluons in (++) or (−−) helicity configurations. The vertex operator of Φ is given in Eq. (5.5). We will use the following amplitudes: [42] , (4.49) The second massive level spin three and spin two excitations arise from factorization of A[Φ + , +, +, −]: The G (2) (J = 1) that goes into Φ + + g + is not parity invariant. Instead, its partner decays into Φ − + g − . On the other hand, both channels of Φ + + g + and Φ + + g − are possible and we need to add them up.
The second massive level spin-5 2 and spin-3 2 resonances arise from The corresponding partial decay widths read The second massive level spin-3 2 and spin-1 2 resonances arise from The corresponding partial decay widths read Similar to previous cases, we identify the spin-3 2 fermion to beQ (2) (J = 3/2).

G (2) (J = 1) → χ +q
The second massive level spin one excitations arise from factorization of

a(J = 1/2)
The vertex operator of the spin-1 2 fermion a is given in Eq. (5.9). We will use the following amplitudes: The second massive level spin three and spin two resonances arise from

Excited quarks decay to SU (2) gauge bosons
For exited quarks which arise from the intersection of U (3) stack and U (2) (or Sp (1)) stack, it is easy to see that the massive quarks could decay into a SU (2) gauge boson plus a massless quark.
One could obtain the total decay width of the massive quark decaying into SU (2) gauge bosons A a by performing a factorization of the amplitude A(q, A a ,q, g) which was obtained in [9]. While in the broken electroweak symmetry, W and Z bosons are produced. Hence we need to translate the decay widths of the massive quarks to A a into the decay width of W and Z bosons.
For illustration, let us focus on the higher level excited quark u (n) . Effectively, its couplings can be written as where c W ≡ cos θ W , s W ≡ sin θ W , e = g 2 g Y / g 2 2 + g 2 Y and Since u (n) is very massive (∼ √ nM s ), we can simply treat all the gauge bosons after the electroweak symmetry breaking as massless. A simple calculation shows (4.126) and Since g 3 is not much greater than g 2 at 10 TeV-100 TeV, we should also include these contributions to the total decay widths of the massive quark excitations.
For the second massive level excited quarks, the decay channels Q (2) → A a + Q (1) also exist. Similar analysis gives the same front factor (1) ) . (4.129) For the massive string states decay into photon plus other string states, see the discussion of the next subsection on massive string states decaying to anomalous U (1)'s.

Massive string states decay to anomalous U (1)'s
We have seen that for intersecting D-brane brane models, the SM gauge group must be extended with new U (1) symmetries. These U (1)'s are in general anomalous. They couple to RR axions and would obtain a string scale mass [85]. These U (1)'s would mix with each other through the U (1) mass-squared matrix. The mass mixing effects have been discussed in Sec. 2.1. Massive string excitations carry the SM gauge charges and thus they could decay into anomalous U (1)'s if kinetically allowed. In this subsection, we will briefly study the possible decay channels of massive string excitations.
Let's first focus on the amplitude A(g, g, g, A a ), where A a denotes the U (1) from the U (3) a stack. Factorization gives rise to the resonances of excited massive gluons, and we have (4.130) Similarly, the factorization of amplitude A(g, g, A a , A a ) gives rise to a massive color singlet that and we also need to write this decay in terms of mass eigen fields. We can also consider amplitudes A(G (1) , g, g, A a ) and A(C (1) , g, g, A a ), for which factorization could give the following decay channels (4.133) Additionally, the factorization of the amplitude A(g, q,q, A a ) gives rise to higher level excited massive quarks decay into anomalous U (1)'s: if kinetically allowed. Also, factorization of the amplitudes A(g, q,q, C (1) ) and A(Q (1) , g,q, A a ) gives Since A a is not in the physical eigenbasis, we need to write it in terms of physical fields (fields in the mass eigenbasis). Using Eq. (2.7), we rewrite Eq. (4.130) as and similar for other decay channels. As long as kinetically allowed, the massive string excitations can decay also into heavier massive anomalous U (1)'s. This is a model-dependent issue, since the transformation matrix O depends on the details of the model building. Unless we know an explicit model construction, we cannot perform further studies for these decay channels.
In this work, we follow the treatment of [7] that we consider A a (the anomalous U (1) from U (3) a stack) as massless and do not consider the mass mixing effect of this U (1) with others (this field was referred as C 0 in [7]). The cases involving the excitation of the color singlet fields C (1) (as decay product) is simpler. It has a mass M s and we expect they do not couple to RR axions.

Comments on how to realize right-handed quarks in intersecting brane models
In intersecting brane models, right-handed quarks can be realized as either open string stretching between U (3) a stack and another U (1) stack (let us label this stack as c stack), or open string stretching between U (3) a stack and its orientifold image. In the former case, right-handed quarks are bi-fundamental representations under U (3) a and U (1) c ; whereas in the latter case, right-handed quarks are the anti-symmetric representation of U (3).
For the former case, U (1) B is a symmetry remaining unbroken at the perturbative level in the low energy effective theory [97], but it can be broken by non-perturbative effects, which are in principle sufficient to suppress proton decay. For the latter case that (one of the two) right-handed quarks are realized as anti-symmetric representation of U (3), U (1) B is not a symmetry. This is problematic since the leftover global U (1) of U (3) allows for baryon number violating couplings already at the lowest order. However, this might be cured by the implementation of discrete gauge symmetries [98,99] to forbid the unwanted couplings.
The difference between these two realizations is that we can have the scattering process A(g, q R ,q R , A c ) for the former case, but this process is absent for the latter case. Thus compare to the latter case, from factorization we know that, the second massive level right-handed quark excitations have several more decay channels Q (2)  4π M s . For the widths of G (2) , we have N = 3, N f = 6. On the other hand, Q (2) can decay into bosons on different stacks. For example the decay product G (1) of a left-handed Q (2) in (4.139) can be either an SU (3) or an SU (2) boson but for each channel the width is of the same form (with different coupling constant and N ). So the widths Γ Q (2) in the table should be understood as only for a particular channel and we need to sum over all possible channels to get the total widths.
However as we discussed in the previous subsection, A c is not in the physical eigenbasis and we need to rewrite it in terms of physical mass eigen fields. 15 These are all model-dependent issues. Unless we focus on a specific D-brane model, we cannot make any general statements on them.
Similarly for the left-handed quarks, if one uses Sp(1) type construction, there's no additional U (1) coming from this stack. Thus compare to the U (2) type constructions, decay channels b +q do not exist, since the amplitude A(g, q R ,q R , A b ) is absent for Sp(1) cases.

Summary of the results
Using factorization, for the second massive level bosonic string states, we have identified a spin three field, a spin two field, complex vector fields, which contribute to scattering processes gg → gg and gg → qq. For the second massive level fermionic states, we have identified a spin-5 2 field, two spin-3 2 fields and a spin-1 2 field which contribute to scattering process gq → gq. For a second massive level color octets, its total decay width includes For the second massive level color singlets, we have (4.138) For the second massive level excited quarks, we have In general left-handed and right-handed excited quarks have different decay channels and therefore different widths. We note that among the amplitudes contributing to dijet signal, Q L only appears as the intermediate state in the channel of gq L → gq L and similarly Q (2) R only appears in gq R → gq R . In the phenomenology analysis, we will take the average of |M(gq L → gq L )| 2 and |M(gq R → gq R )| 2 since the incoming quark is equally likely to be left-or right-handed.
The total decay widths of the second massive level string states are summarized in Table 4.

String computation of partial decay widths
In this section, we will focus on two second massive level universal string states: the spin three field σ µνρ and the spin two field π µν , computing their decays in various channels.
N -point tree level string amplitudes are obtained by calculating the N -point correlation functions 16 of associate vertex operators inserted on the boundary of the disk worldsheet, which read where the sum runs over all the cyclic ordering of the N (N ≥ 3) vertices on the boundary of the disk. The corresponding string vertex operators are constructed from the fields of the underlying superconformal field theory and contain explicit Chan-Paton factors. In order to cancel the total background ghost charge −2 on the disk, we should choose the vertex operators in the correlator in appropriate ghost "pictures" which makes the total ghost number to be −2. In addition, the factor V CKG is defined to be the volume of the conformal Killing group of the disk after choosing the conformal gauge, which would be canceled by fixing three vertices and introducing respective c-ghost fields into the vertex operators. Then we integrate over other N − 3 points and get the amplitude.
To obtain the decay widths of the second massive level string states we only need to compute the three-point amplitudes, in which all the positions of the vertex operators on the disk boundary are fixed. 16 The relevant worldsheet fields correlation functions can be found in [9,10].

Vertex operators of the second massive level universal string states
Before we compute the amplitudes, we summarize all the relevant vertex operators of the zeroth to the second massive level string states. For the zeroth level string, the vertex operator for massless gluon g (with the polarization vector µ ) in the −1 and 0 ghost picture read respectively: where µ · k µ = k 2 = 0. The Chan-Paton factor T a indicates the vertex operator is inserted on the segment of disk boundary on stack a, and α 1 , α 2 represent the two string ends. Massless quarks originated from brane intersections are given by where the u a ,ūȧ satisfy the Dirac equation u a / k aȧ =ūȧ/ k˙a a = 0, and Ξ a∩b is the boundary changing operator [9]. These vertex operators connect two segments of disk boundary, associate to two stacks of D-branes, with the indices α 1 and β 1 representing the string ends on the respective stacks.
The first massive level string states and their properties were comprehensively studied in [11,13]. For the bosonic sector, we only need the spin two field α µν and the complex scalar Φ ± : where α µν is symmetric, transverse and traceless.
Fermionic sector contains spin-3 2 and spin-1 2 fields which read which involve the excited spin field S µ and the derivative of the standard spin field, c.f., [13] for their OPEs. The spin-3 2 field satisfies χ a µ k µ = χ a µ σ µ aȧ = 0. Here, all the normalization factors for the vertex operators listed above were fixed by factorization as worked out in [11] and have also been checked from supersymmetry transformations in [13].
For the second massive level, we will focus on two bosonic universal states σ, π, whose vertex operators were obtained in [12] where in V (−1) π a we symmetrize only µ, ν indices. σ µνρ , π µν are spin three and spin two bosonic fields respectively, which are both symmetric, transverse and traceless. The normalization C σ , C π will be fixed later. Before we carry out the scattering amplitudes and obtain the partial decay widths of various channels, we pause and present the construction of helicity wave functions for higher spin massive bosonic fields.

Helicity wave functions for higher spin massive fields
In this subsection, we first review the helicity wave functions for spin one and spin two bosonic fields. Then we construct the helicity wave functions for higher spin massive bosonic fields. Helicity formalism for massless fields as well as massive fermions are briefly reviewed in Appendix B and C.

Review of helicity wave functions for spin one and spin two bosonic fields
Massive spin one boson A spin J particle contains 2J + 1 spin degrees of freedom associated to the eigenstates of J z . The choice of the quantization axis z can be handled in an elegant way by decomposing the momentum k µ into two arbitrary light-like reference momenta p and q: Then the spin quantization axis is chosen as the direction of q in the rest frame. The 2J + 1 spin wave functions depend of p and q, while this dependence would drop out in the squared amplitudes summing over all spin directions.

Massive spin two boson
The wave function (polarization tensor) of massive spin two boson α µν satisfies the following relations (symmetric, transverse, traceless), which read where λ denotes the helicity of α µν .
An arbitrary four by four tensor has 16 degrees of freedom. The first condition above reduces the degree of freedom to 10, and the second and third condition would further reduce degree of freedom 4 and 1 respectively. Thus we are left with 5 physical degrees of freedom as expected. Different helicity states of the spin two massive boson satisfy the relation The spin two boson helicity wave functions are constructed in [101], up to a phase factor,

Building helicity wave functions for higher spin massive bosons
This spin-n massive boson Φ µ 1 µ 2 ···µn n satisfies the following physical state conditions: In four dimensions, the first symmetric condition brings down the degrees of freedom from 4 n to 4 + n − 1 n , and the transversality and traceless eliminate further 4 + n − 2 n − 1 and n 2 conditions. Thus the Φ µ 1 µ 2 ···µn n has 4 + n − 1 n − 4 + n − 2 n − 1 − n 2 = 2n + 1 degrees of freedom.
Thus the helicity wave function of the highest helicity j z = +n of a spin-n massive boson Φ µ 1 µ 2 ···µn n can be written as, up a phase factor, µ 2ȧ2 a 2 q a 2 ) · · · (p * anσ µnȧnan q an ) , and as always, p µ + q µ = k µ . Now to obtain all the helicity wave functions of a spin-n boson Φ µ 1 µ 2 ···µn n , we can make use of angular momentum ladder operators J − . By acting J − on the the highest J z state successively, one can obtain all the helicity wave functions of Φ µ 1 µ 2 ···µn More specifically, we have the following relations (5.27) One could write these relations in a simpler form as (5.28) These formulas allow us to get all the wave functions of an arbitrary spin massive boson. By applying J − operator on Φ µ 1 µ 2 ···µn n (n, n) successively, one can obtain wave functions of all the helicities.
Indeed, this J − operator is extremely useful in the computation of the helicity amplitudes involving massive states. Since the wave function of the highest helicity state Φ µ 1 µ 2 ···µn n (n, n) has the simplest form, one could relatively easily obtain the helicity amplitude A[Φ n (n, n), · · · ] that Φ µ 1 µ 2 ···µn n (n, n) interacts with other states, and it is usually in a simple form. One could then apply J − successively to the amplitude A[Φ n (n, n), · · · ] to obtain all the helicity amplitudes A[Φ n (n, m), · · · ], which is much simpler than plugging in explicit forms of the Φ n helicity wave functions of lower j z . 17 There is another way of constructing the helicity wave functions of a spin-n massive boson, that we can treat the spin-n boson as a spin-(n − 1) and a spin one boson coupling. Thus, given the helicity wave function of a spin-(n − 1) boson, one can write down an arbitrary J z = m state o f the spin-n boson as where the CG coefficients read , (5.30) Thus Eq. (5.29) can be written as As a simple example, we consider the amplitudes Eqs. (4.5) obtained in [11]. We have p4 3 4q 23 34 42 , which just reproduce the desired result. Using this method, one could then check all the results in [11], where all the helicity amplitudes were computed using the explicit forms of the helicity wave functions in different j z , for example Eqs. (5.20).

Decay of the second massive level string states
We need to first fix the normalization of vertex operators for σ µνρ and π µν . To this end, we compute the amplitude that σ µνρ , π µν decay into two massless gluons, and the result reads Applying the helicity formalism, we obtain Extract the second level pole information from the Veneziano amplitude A(g, g, g, g), we obtain (up to a phase factor) Thus we obtain C σ = g 3 /2 √ α , where we have used C D 2 = 1/(g 2 3 α 2 ) and Eq. (A.3). For π µν decay to two massless gluons, we have Similarly, by applying the helicity formalism, we match the helicity amplitude with the amplitude we extract from Veneziano amplitude and we obtain C π = g 3 /4 √ 3.
The partial decay widths of second massive level string states to two massless string states were already obtained in [22,23]. We are now the most interested in computing the partial decay widths of a second massive level string states decay into one first massive level string state plus a massless one.

Partial decay widths of the spin three state σ µνρ
We now focus on the spin three bosonic string states σ µνρ . It has four possible decay channels whose final states consist of one first massive level string states and one massless string states. which read σ → α +g, σ → Φ ± +g, σ →χ+u, σ →ā+u (the decay widths of σ → χ+ū, σ → a+ū are the same as the last two channels). Straightforward computation gives

39)
We place the second massive level string state, the first massive level string state and the massless string at position 1,2,3 with corresponding momentum k 1 , k 2 , k 3 , and thus we have To obtain the partial decay widths of the above channels, again we apply the helicity formalism. In principle, by plugging in directly the helicity wave functions of the fields participating in the processes, e.g. Eqs. (5.20) and (5.34), we could obtain the helicity amplitudes. Then by summing over their squares we can achieve the final results. However, specially treatment is needed here. For example, for the amplitude A(σ 1 , α 2 , 3 ), σ has 7 degrees of freedom, α has 5 and has 2. Thus we need to compute total 7 × 5 × 2 = 70 helicity amplitudes, and the computation would be very tedious. First of all, we observe that This would reduce the total number of the amplitudes we need to compute by half. In addition, as we mentioned, the helicity wave functions of massive bosonic fields are built by decomposing their momentum into two light-like momentums k µ → p µ + q µ , and the spin axis of the field aligns to the q direction. Hence if we align the spin axes of all the scattering fields to one same direction, we only need to compute very few helicity amplitudes, the others should vanish automatically because of the angular momentum conservation.
The most clever choice of reference momentums read: 18 where r is the reference momentum for the massless gluon 3 (k 3 ) with r 2 = 0. It can be easily verified that (p 1 + q 1 ) 2 = 2(p 2 + q 2 ) 2 . (5.47) Then by using the mass shell condition Eq. (5.41), we fix the reference momentum r as r · k 3 = −1/(2α ). This particular choice of reference momentums not only simplify the computation dramatically, but also align the spins of all the interacting particles in one same direction (the direction of k 3 ) and thus we are expecting the results we obtained from this section to match exactly with the results we obtained in the previous section using factorization.
Using massive helicity wave functions and the above choice of reference momentums, we compute the helicity amplitudes of A(σ 1 , α 2 , + 3 ). Only five survive, which read All other helicity amplitudes are checked to vanish. These results match exactly with the results obtained from factorization Eqs. (4.9)-(4.17), as expected.
With the same choice of the reference momentums, for A(σ 1 , Φ 2± , 3 ), we obtain: For the decay channels that final states being fermions. The scattering amplitudes read, For scattering amplitudes involving two fermionic fields, a factor ofC D 2 would appear and we have usedC D 2 = e −φ 10 /(2α 2 ) [11].
For the fermionic decay channels, again we align the spin axes of the three interacting states into the direction of k 3 . We will use exactly the same reference momentums Eq. (5.45) as we did for the bosonic decay channels. Here we also need to introduce an addition reference momentum r with r · k 3 = −1/(2α ). Using the massive fermion helicity wave functions summarized in Appendix C, we obtain the following helicity amplitudes

57)
A σ 1 (+1), , (6.14) The total decay widths for n = 2 string resonances can be computed using the formulas in Table 4. We note that the widths of Q (2)  The dijet signal-to-noise ratio for n = 2 is shown in Fig. 6. For M s 10.5 TeV the second massive Regge excitations could also be observed with a statistical significance ≥ 5σ at HE-LHC, and for M s 28 TeV at the VLHC. Measurement of both resonant peaks would constitute definitive evidence for string physics.

Angular distributions
In what follows we briefly comment on the angular distributions. QCD parton-parton cross sections are dominated by t-channel exchanges that produce dijet angular distributions which peak at small center of mass scattering angles. In contrast, non-standard contact interactions or excitations of resonances result in a more isotropic distribution. In terms of rapidity variable for standard transverse momentum cuts, dijets resulting from QCD processes will preferentially populate the large rapidity region, while the new processes generate events more uniformly distributed in the entire rapidity region. To analyze the details of the rapidity space the DØ Collaboration introduced a new parameter [105], R = dσ/dM | (|y 1 |,|y 2 |<0.5) dσ/dM | (0.5<|y 1 |,|y 2 |<1.0) , (6.18) the ratio of the number of events, in a given dijet mass bin, for both rapidities |y 1 |, |y 2 | < 0.5 and both rapidities 0.5 < |y 1 |, |y 2 | < 1.0. The ratio R is a genuine measure of the most sensitive part of the angular distribution, providing a single number that can be measure as a function of the dijet invariant mass. An illustration of the use of this parameter in a heuristic model where standard model amplitudes are modified by a Veneziano form factor has been presented in [106].
It is important to note that although there are no s-channel resonances in qq → qq and qq → qq scattering, KK modes in the t and u channels generate calculable effective 4-fermion contact terms. These in turn are manifest in a small departure from the QCD value of R outside the resonant region [14]. In an optimistic scenario, measurements of this modification could shed light on the Dbrane structure of the compact space. It could also serve to differentiate between a stringy origin for the resonance as opposed to an isolated structure such as a Z , which would not modify R outside the resonant region. While the signal of quark scattering is suggestive, the analysis in [14] did not take into account all of the potential detector effects, which is necessary to be confident that the effect is real. In the next the section we describe the first steps towards a more realistic description of the string physics processes.

SEGI
SEGI is a modification of the original BlackMax event generator [34,35], which is extensively used by ATLAS and CMS collaborations in search for exotic physics. At its inception, BlackMax could simulate only black hole production in particle collisions (including all the greybody factors known to date) [107][108][109][110][111][112][113]. Then it gradually grew into a very comprehensive generator that can accommodate different signatures of quantum gravity, e.g., stringball evaporation in two body final state [114]. With the current modification, BlackMax will be able to simulate production and decay of lowest massive Regge excitations yielding γ + jet, Z + jet, and dijet events.
A necessary input for the event generator are the amplitudes for perturbative string mediated processes. The parton-parton suprocesses of lowest massive Regge excitations decaying to dijets are given in Eqs. (3.29), (3.33), (3.36), and (3.37), whereas those decaying into γ + jet are giving in Eqs. (3.41) and (3.42). 19 The cross section can be written as a convolution of (6.2) with PDFs e.g., for dijets, whereŝ max andŝ min are the maximum and minimum square center of mass energy of the colliding partons. The code iterates 10 6 times to calculate the Monte Carlo integral. As an illustration, in Fig. 7 we show a comparison of the invariant mass distribution, setting M s = 5 TeV, as obtained by SEGI and with the semi-analytic (parton model) approach adopted in the preceding section. The input parameters for the generator are read from the file parameter.txt. In the following bulleted list we provide an explanation for the relevant input parameters.
• Number_of_simulations This parameter is the number of events to be generated.
• Type_of_incoming_particles This parameter determines the type of incoming particles: • Choose_a_pdf_file (200_to_240_CETQ6_or_>10000_for_LHAPDF) This parameter determines which PDF is used in the simulation. The code includes CETQ6 PDFs by default. In that case, this parameter should be set from 200 to 240. For different PDFs, one must install LHAPDF. The impact of the different PDFs and induced systematics in the production and decay of Regge recurrences is shown in Fig. 8.
• Minimum_mass The minimum mass that one wants to include in the simulation in units of GeV • Maximum_mass The maximum mass that one wants to include in the simulation in units of GeV • String_scale This parameter is the string scale M s in units of GeV.
• string_coupling This parameter is the string coupling; default is set to g s = 0.1.
• kappa This is the C − Y mixing parameter, default is set to κ = 0.14.
All the other BlackMax parameters are irrelavant for simulation of Regge recurrences.
The generator gives the output.txt file. This file contains the cross sections and the energy momentum distributions of the incoming and outgoing particles (pseudorapidity distributions are displayed in Fig. 9 for illustrative purposes only). The incoming particles are marked as Parent.
The outgoing particles are marked as Elast. The meaning of each column is the same as in the original BlackMax event generator [34,35]. The most up-to-date source code and TarBall can be downloaded from:

Conclusions
We have explored the discovery potential of existing and proposed hadron colliders to unmask excitations of the string. We have studied the direct production of Regge recurrences, focusing on the first and second excited levels of open strings localized on the world-volume of D-branes. In this framework, U (1) B and SU (3) C appear as subgroups of U (3) associated with open strings ending on a stack of 3 D-branes. In addition, the minimal models contain two other stacks to accommodate the electro-weak SU (2) L ⊂ U (2) and the hypercharge U (1) Y . For such D-brane models, the resonant parts of the relevant string theory amplitudes are universal to leading order in the gauge coupling. As a consequence, it is feasible to extract genuine string effects which are independent of the compactification scheme. In this paper we have made use of the amplitudes evaluated near the first and second resonant poles to report on the discovery potential for Regge excitations of the quark, the gluon, and the color singlet living on the QCD stack of D-branes.
To calculate the string signal for n = 1 resonances, we used the partial decay widths obtained elsewhere [7]. To compute the signal for n = 2 resonances, we have presented here a complete calculation of all relevant decay widths of the second massive level string states, including decays into massless particles and a massive n = 1 and a massless particle.
Our phenomenological study among the various processes indicates that: • For M s 7.1 TeV, the HL-LHC will be able to discover (with statistical significance > 5σ) lowest massive Regge excitations in dijet events. For string scales as high as 6.1 TeV, observations of resonant structures in pp → γ + jet can provide interesting corroboration (with statistical significance > 5σ) of low-mass-scale string physics.
• The dijet discovery potential exceedingly improves at the HE-LHC and VLHC. For n = 1, the HE-LHC will be able to discover string excitations up to M s ≈ 15 TeV, whereas the VLHC will attain 5σ discovery up to M s ≈ 41 TeV. Moreover, for n = 2, the HE-LHC will reach 5σ discovery for M s 10.5 TeV, while the VLHC will be able to discover Regge excitations for M s 28 TeV.
• Keeping only transverse Z's and assuming that cross sections × branching into lepton pairs are large enough for complete reconstruction of pp → Z + jet, the D-brane contribution to the signal is suppressed relative to pp → γ + jet by a factor of tan 2 θ W = 0.29. This differs radically from stringball evaporation in two-body final state. In such a case, emission of γ + jet and Z + jet are comparable. The suppression of Z +jet production, whose origin lies in the particular structure of the D-brane model, will hold true for all the low-lying levels of the string.
Our calculations have been performed using a semi-analytic parton model approach which is cross checked against an original software package. The string event generator interfaces with HERWIG and Pythia through BlackMax. The source code is publically available in the hepforge repository.
In summary, in this paper we have provided a concrete starting point for understanding the string physics potential of proposed machines that would collide protons at energies approaching the boundary of what (wo)mankind can daydream to achieve. The results presented herein will help to lay out opportunities, connections, and challenges for future LHC upgrades.
where T ab or T cd presents either [T a , T b ] or {T a , T b }. For massless spin-1 2 spinors, we use the notation following [11]:

B.2 Helicity wave functions for massless spin one gauge boson
The gauge transformation for a spin one gauge boson reads µ → µ + Λk µ . The massless spin one gauge boson only has two degrees of freedom, which are helicity up (+) and down (−). The helicity wave functions (polarization vectors) of a massless spin one gauge boson can be written as