Intersecting Branes, SUSY Breaking and the 2TeV Excess at the LHC

Intersecting D-brane models in string theory can naturally support the gauge and matter content of left-right symmetric extensions of the Standard Model with gauge symmetry SU(3)_c x SU(2)_L x SU(2)_R x U(1)_{B-L}. Considering such models as candidates for explaining the 2TeV excesses seen in Run-1 by both ATLAS and CMS, the minimal possible scale of supersymmetry breaking is determined by the requirement of precise one-loop gauge coupling unification. For the vector-like, bifundamental and (anti-)symmetric Higgs content of such brane configurations, this comes out fairly universally at around 19TeV. For the SU(2)_R gauge coupling one finds values 0.48<g_R(M_R)<0.6. Threshold corrections can potentially lower the scale of supersymmetry breaking.


Introduction
In the 8 TeV run of the LHC the Higgs particle was discovered, which completes the particle spectrum of the Standard Model. Concerning physics beyond the SM, the hope was that supersymmetry would be found at a scale not far above the weak scale, thus providing a solution to the hierarchy problem. Even though no direct indication of supersymmetry has appeared yet, some anomalies were reported indicating with 2-3σ a resonance of around 2 TeV in the di-boson decay channel by ATLAS [1] and in the e + e − j j, W h 0 and j j final states by CMS [2]. The most significant one is the ATLAS 3.4σ excess in the hadronic decay of a W Z pair of electro-weak gauge bosons.
Promising candidates to explain these excesses are left-right symmetric extensions of the Standard Model (LRSM) [3] with gauge symmetry SU (3) c ×SU (2) L × SU (2) R × U (1) B−L in which a bi-doublet Higgs field naturally generates mixings between the left and right SU (2) W-bosons. The breaking of SU (2) R × U (1) B−L to the Standard Model hypercharge at the scale M R ∼ 2 TeV can be performed e.g. by an SU (2) R triplet or simply by a doublet. The phenomenological potential of such models to explain all the excesses seen by ATLAS and CMS were e.g. analyzed in [4]. In these papers, the SU (2) R gauge coupling was restricted to be in the regime 0.4 < g R (M R ) < 0.6.
Taking the precision measurements for dilepton channels into account, some authors came to the conclusion that one should better use a leptophobic version of the LRSM, where the right handed leptons are not sitting in a doublet of SU (2) R . Formally, this makes the model less natural so that our attitude is that both experimentally and phenomenologically this issue is not yet completely settled so that here we proceed to consider the standard version of an LRSM. Then, right-handed neutrinos and the Z gauge boson should have masses larger than M W .
In fact, in this letter we are not so much concerned with such phenomenological fine-print, but would like to evaluate the potential of string theory inspired LRSM-like models to provide the overarching structure in the regime between M R and a unification scale M U . Let us mention that a string inspired explanation for the 2 TeV excesses was already proposed in terms of an anomalous Z gauge boson in [5] (see also [6] for application to LHCb b → s + − anomalies). Note that there the mass of the anomalous Z is of the order of the string scale M S , which then is bound to be small.
In this letter we follow a different route and consider stringy realizations of LRSMs. One option is to embed the LRSM into a grand unified gauge group SO(10), as it can be realized in heterotic or F-theory compactifications. In these constructions, one often gets a bunch of extra particles that can generate fast proton decay and one needs to solve the problem of mass hierarchies inside the Higgs multiplets (like the doublet-triplet splitting problem for SU(5) GUTs).
Since all the fields in the LRSM sit in bifundamental and (anti-)symmetric representations of the gauge group, it is also very natural to realize such models from intersecting D-branes [7] (see [8] for reviews), thereby avoiding some of the above mentioned problems present in GUT models. Baryon number is known to be a perturbative global symmetry in these constructions protecting the proton from a too fast decay. Therefore, in this paper we consider LRSM realizations by intersecting branes focusing on D7-branes in type IIB orientifolds. Since in controllable models of string theory, supersymmetry needs to be broken at a scale M SUSY smaller than the string scale M S ∼ M U , we consider the hierarchy of mass scales M Z < M R < M SUSY < M U and analyze the issue of gauge coupling unification. Such an analysis has been performed in a field theory context (see e.g. [9,10] for more recent studies), but to our knowledge it was always assumed either that one does not have supersymmetry below the GUT scale at all or that the intermediate hierarchy of scales is reversed, i.e. M Z < M SUSY < M R < M U . Thus, the observations made in this letter can be considered as complementary to earlier results reported in the literature.
This letter is organized as follows: In section 2, we review a few important aspects of intersecting D-brane models where in particular we discuss under what circumstance one can still get gauge coupling unification at the string scale. Section 3 provides two simple prototype examples of possible realizations of a LRSM.
In section 4 the one-loop running of the four gauge couplings is analyzed. Note that this analysis only depends on the matter content of the models and is therefore generically valid, i.e. without necessarily referring to an intersecting D-brane scenario. After fixing the Higgs sector, requiring precise one-loop gauge coupling unification and using M R = 2 TeV, one can uniquely determine the unification scale, the supersymmetry breaking scale and the SU (2) R gauge coupling g R . It is observed that the minimal possible value for the supersymmetry breaking scale, 19 TeV, shows a certain universal behavior. Moreover, the value of the SU (2) R gauge coupling come out as 0.48 < g R (M R ) < 0.6, depending on the vector-like Higgs sector.
Note added: The latest announcement of physics results by the ATLAS+CMS collaborations on 15.12.2015 did not show any 2 TeV excesses at Run-2. This of course diminishes the experimental motivation for the analysis done in this letter, but we think that the general results on the relation of the scales in LRSMs and supersymmetry that are obtained are nevertheless interesting.

Models of intersecting D7-branes
Let us review some of the main ingredients for the construction of intersecting D7-brane models in orientifolds of the Type IIB superstring. For more details we refer to the reviews [8] and in particular to [11].
One considers the Type IIB superstring compactified on a Calabi-Yau threefold X and performs the quotient by an orientifold projection Ωσ(−1) F L , where Ω is the world-sheet parity reversal and σ denotes a holomorphic involution of the Calabi-Yau satisfying σ(J) = J and σ(Ω 3 ) = −Ω 3 . Here J is the Kähler form and Ω 3 is the holomorphic (3, 0) form on the threefold. At the fixed locus of σ, one gets O7 and O3-planes, whose R-R charges (tadpoles) need to be cancelled by the introduction of D7 and D3-branes.
Stacks of N a D7-branes can wrap holomorphic four-cycles Σ a in the homology class H 4 (X, Z) of the threefold. Moreover, these D7-branes can carry a non-trivial gauge flux determining a line bundle L a on Σ a . Depending on whether this D7brane configuration is invariant under the orientifold projection or not, one gets SO(N )/SP (N )-gauge groups or U (N ) gauge groups, respectively. Note that one gets The most simple and phenomenologically useful configurations for SM-like model building arise for branes wrapping orientifold invariant, rigid four-cycles Σ a of SP -type with the different stacks being only distinguished by their gauge flux L a . If c 1 (L a ) ∈ H 2 − (Σ a , Z) then such a stack carries gauge group SP (N a ) and for a flux c 1 (L a ) ∈ H 2 + (Σ a , Z) one gets an orientifold image line bundle L a so that together they support a U (N a ) gauge group.
The massless modes between two such stacks of branes transform in bifundamental or (anti)-symmetric representations, where the chiral index is given by  For an SP -stack, chiral fields can only arise from the intersection with a U (N ) stack.
The non-chiral (vector-like) part of the spectrum can be determined by computing certain line bundle cohomology groups on Σ 1 . For branes wrapping the same four-cycle Σ, this was determined in [11] and for completeness we provide the result For branes wrapping two different 4-cycles intersecting over a curve C = Σ a ∩ Σ b , one has to compute the cohomology classes For more details, we refer to [11]. Moreover, the U (1) factors in the total gauge group are often anomalous. In string theory, these anomalies are cancelled by the Green-Schwarz mechanism, where a shift symmetry of an axion gets gauged so that it becomes the longitudinal mode of a massive abelian vector field. The abelian gauge symmetry gets broken but survives as a perturbative global symmetry that is only broken by D-brane instantons. We recall that an abelian gauge field can even gain a mass without having an anomaly.
Let us consider now the tree-level gauge couplings at the string scale where τ a = 1 2 4 s Σa J ∧ J denotes the volume of the four-cycle (in Einstein frame) in units of the string length s and g s = e ϕ is the string coupling constant. Here κ a = 1 for a U (N ) stack and κ a = 2 for an SP (2N )/SO(2N ) stack 2 . As usual in string theory, these couplings will receive one-loop threshold corrections. Note that for all branes wrapping the same 4-cycle, the differences among the gauge couplings g a come only from the line bundles and in fact only from the value of Σa c 2 1 (L a ) ∈ Z 3 . In contrast to single GUT groups, the gauge couplings at the string scale are not necessarily equal. This situation and its consequences for gauge coupling unification for intersecting D-brane models and F-theory models have been discussed in [12]. However, one can still design situations where precise unification can occur.
can contain homological two-cycles that are not closed in the ambient threefold X. One can always twist a line bundle L a by such a trivial bundle R a without changing the chiral spectrum. However, the vector-like part computed via (2.2) for the bundle L a ⊗ R a will change.
2 The factor κ = 1, 2 is due to the "doubled" embedding of U (N ) into SO(2N )/SP (2N ) 3 These values also depend on the twist line bundle R a .
If all branes are of U (N ) type, one can wrap all stacks around a single 4-cycle Σ of size τ and distinguish them solely by the line-bundles L a .
• In the regime τ g −1 s > 1 with the differences of the flux integrals not being too large, the gauge couplings approximately unify with 4π g 2 a = τ where the flux dependent corrections can be considered as small threshold corrections. The values of τ and g s are determined by moduli stabilization.
• Since the second term in (2.4) only depends on the topological quantity ch 2 (L a ), all gauge couplings can still be degenerate even if the L a themselves are different.
If some of the branes are of SP -type then due to the factor κ = 2 in (2.4), the SP branes should wrap a different four-cycle Σ SP than the U (N ) branes Σ U with τ SP = 2τ U . In this case, one also gets approximate gauge coupling unification in the sense just explained for U (N ) stacks. Thus, in the following we will work in a scheme where we realize the LRSM on such intersecting D7-branes and we will assume that we have gauge coupling unification at the string or unification scale up to small threshold corrections. Moreover, the initial brane realization should be supersymmetric at the string scale, where, as usual, supersymmetry breaking will be mediated to the observable sector by generating soft masses of the order M SUSY .

Brane realizations of LRSM
In this section two principal realizations of the LRSM in terms of intersecting D7-branes are presented. We do not provide fully fledged global string compactifications, but instead only local brane configurations that satisfy the consistency conditions following from string theory. The realization of the SM itself in terms of intersecting branes is also known as the Madrid quiver, first presented in [13]. That model is very similar to the LRSM quivers [14] to be discussed below.

LRSM quiver A
First we consider the simplest quiver A, in which the SU (2) L × SU (2) R sector is realized on orientifold invariant branes supporting SP (2) SU (2) gauge group. We introduce four stacks of D7-branes carrying appropriate line bundles such that one gets the initial gauge symmetry U (3) × SP (2) × SP (2) × U (1) and that the massless spectrum is the one shown in table 2.
One has two abelian gauge factors U (1) B = 1 3 U (1) a ⊂ U (3) and U (1) L = U (1) d , whose charges can be identified with baryon and lepton number, respectively. However, the only anomaly-free combination is U (1) B−L = 1 3 U (1) a −U (1) d . Therefore, the orthogonal combination receives a mass via the Green-Schwarz number field mechanism. This in particular means that baryon and lepton number survive as global symmetries and can protect the proton from decaying. In particular, the unification scale can be smaller than the usual GUT-scale M GUT = 2 · 10 16 GeV.
The gauge coupling of a linear combination U (1) assuming stringy gauge coupling unification, i.e. α s = α a = α d . Note that this is the same relation as for SO (10) GUTs. The bi-doublet Higgs field Φ originates from a vector-like intersection between the two SP (2)-branes. As indicated in the table, there could be more than just a single such Higgs field, but its minimal non-vanishing number is really N Φ = 1. It is clear that for intersecting branes, one cannot get an SU (2) R triplet with Q B−L = ±2. The open string of such a massless mode would need four instead of two ends. Therefore, the breaking of the SU (2) R × U (1) B−L gauge symmetry to U (1) Y has to proceed via a Higgs field in the doublet representation of SU (2) R with Q B−L = 1. Note that anomaly cancellation forces us here to introduce such Higgs fields in vector-like pairs H u,d R . We also added vector-like pairs S u,d of fields transforming in the symmetric representation of U (1) d and SU (2) R triplets with Q B−L = 0. Note that in contrast to other approaches, parity symmetry P : SU (2) L ↔ SU (2) R is broken explicitly in this Higgs sector.

LRSM quiver B
One can also realize the SU (2) L,R gauge symmetries on U (2) type of branes. In this case one has four U (1) factors, of which we assume that only the anomalyfree combination U (1) B−L = 1 3 U (1) a − U (1) d stays massless after the Green-Schwarz mechanism has been employed. A configuration consistent with the stringy constraints is presented in table 3. Note that the generation of all possible SM Yukawa couplings and anomaly cancellation forces one to introduce an even number of bi-doublet Higgses Φ. In the next section, we analyze the running of the four gauge couplings in the regime M R < M SUSY < M U . Even though the matter content of the models was motivated by D-brane constructions, the upcoming analysis only depends on the former and is therefore generically valid. At the scale M R the hypercharge coupling splits into the SU (2) R and the U (1) B−L coupling according to

Gauge Coupling Unification
The running beyond M R is evaluated under the following two assumptions 1. Following the extended survival hypothesis [15], in the regime M R < µ < M SUSY there is just the minimal particle content of the non-supersymmetric LRSM, i.e. in particular one scalar Higgs bi-doublet Φ and one scalar Higgs doublet H u R . Due to supersymmetry breaking, they are expected to gain soft masses of the order of M SUSY . Since for M SUSY M R one cannot refer to supersymmetry to solve the hierarchy problem, one needs some fine-tuning or string landscape argument to achieve this.
2. In the regime M SUSY < µ < M U all the supersymmetric states that the intersecting D-brane model provides contribute to the running. This includes N Φ chiral fields in the bi-doublet representation, as well as N H vector-like Higgs fields H R . We also allow for N ∆ vector-like fields ∆ 0 and N S vectorlike fields S.
The one-loop running of the four gauge couplings occurs according to 2 3 α B−L ) and the β-function coefficients in the non-supersymmetric and supersymmetric region are given by 4 (4.4) Here we were leaving the number n l H of light scalar Higgs fields in the doublet representation of SU (2) R after supersymmetry breaking as an open parameter. A fermion from the superfield H R that remains light contributes like n l H = 2. The extended survival hypothesis means that we have n l H = 1. Taking the relation (4.2) into account, we have three unspecified parameters namely α R (M R ), M SUSY and M U that can be uniquely determined by requiring that all four gauge couplings unify at a scale M U . For self-consistency one needs M R < M SUSY < M U and that all couplings remain in the perturbative regime. Let us discuss the two classes N Φ =even/odd separately.    For N H ≤ 3, 6 one gets M SUSY < M R which is in conflict with the assumption. The lowest possible values are therefore N H = 3, 6, for which the supersymmetry breaking scale comes out one order of magnitude larger than M R and is actually the same for the two choices N Φ = 1 and N Φ = 3. This value would be out of reach of the LHC Run-2, but threshold corrections at the high scale and two-loop corrections to the running might lower this value. This issue will be discussed below. The value of the SU (2) R gauge coupling does only vary slightly in the region 0.5 < g R < 0.55. Recall that baryon number is still a perturbative global symmetry so that for M U < M GUT fast proton decay is not an issue. In figure 1, the running coupling constants are shown for N Φ = 1 and N H = 3. We observed that the minimal value of the supersymmetry breaking scale M SUSY = 19.2 TeV was appearing for both choices of the number of bi-doublet Higgses N Φ = 1, 3. Clearly, this asks for an explanation. First, one notices that the values of M SUSY and M U do only depend on the combination (N H +N ∆ +N S ) with only g R depending on their individual values. For the (minimal) choice (N H + N ∆ + N S ) min = 3 2 (N Φ + 1), the supersymmetry breaking scale can be generically determined as . (3, 0, 0) 0.532 (2,1,0) 0.507 (2,0,1) 0.560 (1,2,0) 0.485 (1,1,1) 0.532 (1,0,2) 0.594 Table 5: Values of g R for N Φ = 1 and (N H + N ∆ + N S ) min = 3.

LRSM quivers A,B for even N Φ
The same computation can be performed for an even number of bi-doublet Higgs fields. Recall that for quiver B, anomaly cancellation enforced these fields to come in vector-like pairs. In  What one can do though is to add N T vector-like pairs of massless modes in the representation (3, 1, 1, ±2/3). These arise from open strings stretched between the U (3) stack and the U (1) d stack. The β-function coefficients in the susy regime M R < µ < M U then read Then it is clear that choosing N Φ even and N T = 1, the three values M SUSY , g R (M R ) and M U remain the same as for N Φ − 1 bi-doublet Higgses and N T = 0. Only the value of the unified gauge coupling changes. Hence, one is back to the discussion for odd N Φ . In figure 2, the one-loop running coupling constants are shown for N Φ = 2, N T = 1 and N H = 3.

Comment
The value of M SUSY = 19 TeV is a bit too large to be detected directly at the LHC. However, our analysis was very strict in the sense that we were assuming that all masses at the scales M R and M |rmSU SY are the same, respectively. Moreover, the computation is performed only at one-loop level, where two-loop corrections are usually expected to give a 4% correction to the couplings at the unification scale. A correction of the same order is expected if the gauge fluxes on the stacks of branes lead to string threshold corrections, as discussed in section 2. Just to get a first impression, for quiver A let us assume that e.g. the supersymmetry breaking scale is 3 TeV, with g R = 0.53 and just run the couplings up to the GUT scale. The result is shown in figure 3. In the left picture one does not even see that the couplings do not unify exactly. From the right picture one can estimate a 2% failure in doing so. Therefore, we conclude that for the minimal value N H = 3, the inclusion of threshold corrections and probably also two-loop corrections can make a supersymmetry breaking scale of just a bit above M R still consistent with stringy unification at around the GUT scale.

Conclusions
We presented possible realizations of LRSMs in terms of intersecting D7-brane quivers. Since only bifundamental and (anti-)symmetric representations can occur, the SM and LRSM Higgs representations were fairly constrained. Employing this point, we were considering the minimal matter content and were studying the running coupling constants in the regime between the LR-breaking scale of 2 TeV, as suggested by excesses in the LHC data, and a potential unification scale.
We found that for a not too large number of bi-doublet Higgses H Φ and in each case a sufficiently large number of vector-like fields {H R , ∆ 0 , S} one indeed achieves a precise one-loop unification scenario in which the minimal value of the supersymmetry breaking scale was determined as 19 TeV. This value was shown to be universal for an odd number of bi-doublet Higgs fields. For an even number of such Higgses, by adding a further vector-like colour triplet state, the same universality could be achieved 5 . Threshold corrections at the high scale might lower this value so that it can be detectable at the LHC Run-2.
Even though one can contemplate various variations and extensions of such stringy LRSM models, it is satisfying that with some stringy input and a number of reasonable assumptions it was possible to derive such a universal and in this scheme predictive result. Moreover, the value of the gauge coupling constant g R was not varying much, either, and came out as 0.48 < g R < 0.6. This is in the regime that was also suggested by a more phenomenological analysis of the LRSM to fit the various, of course still not significant, 2 TeV excesses observed in Run-1 at the LHC.
It would be interesting to generalize the computation in various directions like e.g. to D-brane realizations of leptophobic models. One should also include 2-loop effects and consider threshold effects both at the small and the large scale.