\bm{Z^{\prime}}-gauge Bosons as Harbingers of Low Mass Strings

-gauge Bosons as Harbingers of Low Mass Strings

Luis A. Anchordoqui Department of Physics,
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
   Ignatios Antoniadis Department of Physics,
CERN Theory Division, CH-1211 Geneva 23, Switzerland
   Haim Goldberg Department of Physics,
Northeastern University, Boston, MA 02115, USA
   Xing Huang School of Physics and Astronomy,
Seoul National University, Seoul 141-747, Korea
   Dieter Lüst Max–Planck–Institut für Physik,
Werner–Heisenberg–Institut, 80805 München, Germany
Arnold Sommerfeld Center for Theoretical Physics Ludwig-Maximilians-Universität München, 80333 München, Germany
   Tomasz R. Taylor Department of Physics,
Northeastern University, Boston, MA 02115, USA

Massive -gauge bosons act as excellent harbingers for string compactifications with a low string scale. In D-brane models they are associated to gauge symmetries that are either anomalous in four dimensions or exhibit a hidden higher dimensional anomaly. We discuss the possible signals of massive -gauge bosons at hadron collider machines (Tevatron, LHC) in a minimal D-brane model consisting out of four stacks of D-branes. In this construction, there are two massive gauge bosons, which can be naturally associated with baryon number and ( being lepton number). Here baryon number is always anomalous in four dimensions, whereas the presence of a four-dimensional anomaly depends on the -charges of the right handed neutrinos. In case is anomaly free, a mass hierarchy between the two associated -gauge bosons can be explained. In our phenomenological discussion about the possible discovery of massive -gauge bosons, we take as a benchmark scenario the dijet plus signal, recently observed by the CDF Collaboration at Tevatron. It reveals an excess in the dijet mass range , beyond SM expectations. We show that in the context of low-mass string theory this excess can be associated with the production and decay of a leptophobic , a singlet partner of gluons coupled primarily to baryon number. Even if the CDF signal disappears, as indicated by the more recent D0 results, our analysis can still serve as the basis for future experimental search for massive -gauge bosons in low string scale models. We provide the relevant cross sections for the production of -gauge bosons in the TeV region, leading to predictions that are within reach of the present or the next LHC run.

preprint: MPP–2011–86 LMU-ASC 32/11 CERN-PH-TH/2011-180 thanks: On leave of absence from CPHT Ecole Polytechnique, F-91128, Palaiseau Cedex.

I Introduction

Very recently, the CERN Large Hadron Collider (LHC) has fired mankind into a new era in particle physics. The Standard Model (SM) of electroweak and strong interactions was once again severely tested with a dataset corresponding to an integrated luminosity of of collisions collected at  TeV. The SM agrees remarkable well with LHC7 data, but has rather troubling weaknesses and appears to be a somewhat ad hoc theory.

It has long been thought that the SM may be a subset of a more fundamental gauge theory. Several models have been proposed, using the fundamental principle of gauge invariance as guidepost. A common thread in most of these proposals is the realization of the SM within the context of D-brane TeV-scale string compactifications Antoniadis:1998ig (). Such D-brane constructions extend the SM with several additional symmetries Blumenhagen:2001te ().111See also Lebed:2011zg (). The basic unit of gauge invariance for these models is a field, so that a stack of identical D-branes eventually generates a theory with the associated gauge group. (For the gauge group can be rather than .) Gauge interactions emerge as excitations of open strings with endpoints attached to 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 D-branes and transverse large extra dimensions).

In this paper we study the main phenomenological aspects of one particular D-brane model that contains two additional symmetries, which can be chosen to be mostly baryon number and , where is lepton number. This choice is very natural from the point of view of the SM. Moreover, with this choice of the two additional gauge symmetries, one can obtain a natural mass gap between the light anomalous gauge boson and the heavier non-anomalous gauge boson . Our first goal is to survey the basic features of the gauge theory’s prediction regarding the new mass sector and couplings. These features lead to new phenomena that can be probed using data from the Tevatron and the LHC. In particular the theory predicts additional gauge bosons that we will show are accessible at the hadron colliders.

The layout of the paper is as follows. In Sec. II we detail some desirable properties which apply to generic models with multiple symmetries. We perform a renormalization group analysis for the running of the gauge couplings, pointing out that the gauge couplings of the two group factors run differently towards low energies below the string scale. This observation has some interesting phenomenological consequences. Having so identified the general properties of the theory, in Sec. III we outline the basic setting of TeV-scale string compactifications and discuss general aspects of the intersecting D-brane configuration that realize the SM by open strings. In Secs. IV and V we discuss the associated phenomenological aspects of anomalous gauge bosons related to experimental searches for new physics at the Tevatron and at the LHC. Finally, in Sec. VI we explore predictions inhereted from properties of the overarching string theory. Concretely, we study the LHC discovery potential for Regge excitations within the D-brane model discussed in this work. Our conclusions are collected in Sec. VII.

Ii Abelian gauge couplings at low energies

We begin with the covariant derivative for the fields in the basis in which it is assumed that the kinetic energy terms containing are canonically normalized


The relations between the couplings and any non-abelian counterparts are left open for now. We carry out an orthogonal transformation of the fields . The covariant derivative becomes


where for each


Next, suppose we are provided with normalization for the hypercharge (taken as )


hereafter we omit the bars for simplicity. Rewriting (3) for the hypercharge


and substituting (4) into (5) we obtain


One can think about the charges as vectors with the components labeled by particles . Assuming that these vectors are linearly independent, Eq.(6) reduces to


or equivalently


Orthogonality of the rotation matrix, , implies


Then, the condition


encodes the orthogonality of the mixing matrix connecting the fields coupled to the stack charges , , and the fields rotated, so that one of them, , couples to the hypercharge .

A very important point is that the couplings that are running are those of the fields; hence the functions receive contributions from fermions and scalars, but not from gauge bosons. The one loop correction to the various couplings are




with and indicating contribution from fermion and scalar loops, respectively.

Let us assume that the charges are orthogonal, for . Then Eq.(4) implies


and the same thing for fermions, hence


On the other, the RG-induced change of defined in Eq.(10) reads


Thus, stays valid to one loop if the charges are orthogonal. An example of orthogonality is seen in the D-brane model of Antoniadis:2004dt (); Berenstein:2006pk (), for which the various assignments are given in Table 1. In the 3-stack D-brane models of Antoniadis:2000ena (), the charges are linearly independent, but not necessarily orthogonal. If the charges are not orthogonal, the RG equations controlling the running of couplings associated to different charges become coupled. One-loop corrections generate mixed kinetic terms for gauge fields delAguila:1988jz (), greatly complicating the analysis.

Name   Representation      
Table 1: Quantum numbers of chiral fermions and Higgs doublet for .

Another important element of the RG analysis is that the relation for unification, holds only at because the couplings () run differently from the non-abelian () and ().

Figure 1: Pictorial representation of the D-brane model.

In this paper we are interested in a minimal 4-stack model , which has the attractive property of elevating the two major global symmetries of the SM (baryon number and lepton number ) to local gauge symmetries. A schematic representation of the D-brane structure (to be discussed in detail in Sec. III) is shown in Fig. 1. The chiral fermion charges in Table 2 are not orthogonal as given (). Orthogonality can be completed by including a right-handed neutrino with charges , , . We turn now to discuss the string origin and the compelling properties of this model.

Iii Generalities of

The generic perturbative spectrum in intersecting D-brane models consists of products of unitary groups associated to stacks of coincident D-branes and matter in bi-fundamental representations. In the presence of orientifolds which are necessary for tadpole cancellation, and thus consistency of the theory, open strings become in general non oriented allowing for orthogonal and symplectic gauge group factors, as well as for symmetric and antisymmetric matter representations.

Name   Representation         
Table 2: Chiral fermion spectrum of the D-brane model.

The minimal embedding of the SM particle spectrum requires at least three brane stacks Antoniadis:2000ena () leading to three distinct models of the type that were classified in Antoniadis:2000ena (); Antoniadis:2004dt (). Only one of them (model C of Antoniadis:2004dt ()) has Baryon number as symmetry that guarantees proton stability (in perturbation theory), and can be used in the framework of TeV strings. Moreover, since (associated to the of ) does not participate in the hypercharge combination, can be replaced by leading to a model with one extra , the Baryon number, besides hypercharge Berenstein:2006pk (). The quantum numbers of the chiral SM spectrum are given in Table 1. Since baryon number is anomalous, the extra abelian gauge field becomes massive by the Green-Schwarz (GS) mechanism, behaving at low energies as a with a mass in general lower than the string scale by an order of magnitude corresponding to a loop factor Antoniadis:2002cs (). Given the three SM couplings and the hypercharge combination, this model has no free parameter in the coupling of to the SM fields. Moreover, lepton number is not a symmetry creating a problem with large neutrino masses through the Weinberg dimension-five operator suppressed only by the TeV string scale. We therefore proceed to models with four D-brane stacks.

The SM embedding in four D-brane stacks leads to many more models that have been classified in Antoniadis:2002qm (); Anastasopoulos:2006da (). In order to make a phenomenologically interesting choice, we first focus on models where can be reduce to . Besides the fact that this reduces the number of extra ’s, one avoids the presence of a problematic Peccei-Quinn symmetry, associated in general with the of under which Higgs doublets are charged Antoniadis:2000ena (). We then impose Baryon and Lepton number symmetries that determine completely the model , as described in Anastasopoulos:2006da () (see subsection 4.2.4). The corresponding fermion quantum numbers are given in Table 2, while the two extra ’s are the Baryon and Lepton number, and , respectively; they are given by the following combinations:


or equivalently by the inverse relations:


Note that with the ‘canonical’ charges of the right-handed neutrino , the combination is anomaly free, while for , both and are anomalous. Actually, both choices guarantee orthogonality of the charges discussed in the previous section. As mentioned already, anomalous ’s become massive necessarily due to the Green-Schwarz anomaly cancellation, but non anomalous ’s can also acquire masses due to effective six-dimensional anomalies associated for instance to sectors preserving supersymmetry Antoniadis:2002cs ().222In fact, also the hypercharge gauge boson of can acquire a mass through this mechanism. In order to keep it massless, certain topological constraints on the compact space have to be met. These two-dimensional ‘bulk’ masses become therefore larger than the localized masses associated to four-dimensional anomalies, in the large volume limit of the two extra dimensions. Specifically for D-branes with -longitudinal compact dimensions the masses of the anomalous and, respectively, the non-anomalous gauge bosons have the following generic scale behavior:


Here is the gauge coupling constant associated to the group , given by where is the string coupling and is the internal D-brane world-volume along the compact extra dimensions, up to an order one proportionality constant. Moreover, is the internal two-dimensional volume associated to the effective six-dimensional anomalies giving mass to the non-anomalous . 333It should be noted that in spite of the proportionality of the masses to the string scale, these are not string excitations but zero modes. The proportionality to the string scale appears because the mass is generated from anomalies, via an analog of the GS anomaly cancellations: either 4 dimensional anomalies, in which case the GS term is equivalent to a Stuckelberg mechanism, or from effective 6 dimensional anomalies, in which case the mass term is extended in two more (internal) dimensions. The non-anomalous can also grow a mass through a Higgs mechanism. The advantage of the anomaly mechanism versus an explicit vev of a scalar field is that the global symmetry survives in perturbation theory, which is a desired property for the Baryon and Lepton number, protecting proton stability and small neutrino masses. E.g. for the case of D5-branes, whose common intersection locus is just 4-dimensional Minkowski-space, denotes the volume of the longitudinal, two-dimensional space along the two internal D5-brane directions. Since internal volumes are bigger than one in string units to have effective field theory description, the masses of non-anomalous -gauge bosons are generically larger than the masses of the anomalous gauge bosons. Since we want to identify the light gauge boson with baryon number, which is always anomalous, a hierarchy compared to the second -gauge boson can arise, if we identify with the anomaly free combination , and take the internal world-volume a bit larger than the string scale.444In Conlon:2008wa () a different (possibly T-dual) scenario with -branes was investigated. In this case the masses of the anomalous and non-anomalous ’s appear to exhibit a dependence on the entire six-dimensional volume, such that the non-anomalous masses become lighter than the anomalous ones. In summary, this model has two free parameters: one coupling and one angle in the two-dimensional space orthogonal to the hypercharge defining the direction of the corresponding . Tuning the later, it can become leptophobic, while the former controls the strength of its interactions to matter. As discussed already, one can distinguish two cases: (i) when and have 4d anomalies, the mass ratio of the two extra gauge bosons ( and ) is fixed by the ratio of their gauge couplings, up to order one coefficients; (ii) when is anomaly free and gets a mass from effective six-dimensional anomalies, the mass ratio of the leftover anomalous compared to the non-anomalous is suppressed by the two-dimensional volume.

To summarize, we will analyze the phenomenology of two D-brane constructions with three mutually orthognal charges, in which the combination is either anomalous or anomaly free. In the next section, we analyze these situations and study the regions of the parameter space where is leptophobic and can accommodate the recent Tevatron data.

Iv Leptophobic at the Tevatron

Taken at face value, the disparity between CDF Aaltonen:2011mk (); Punzi () and D0 Abazov:2011af () results insinuates a commodious uncertainty as to whether there is an excess of events in the dijet system invariant mass distribution of the associated production of a boson with 2 jets (hereafter production). The excess showed up in of integrated luminosity collected with the CDF detector as a broad bump between about 120 and 160 GeV Aaltonen:2011mk (). The CDF Collaboration fitted the excess (hundreds of events in the channel) to a Gaussian and estimated its production cross section times the dijet branching ratio to be . This is roughly 300 times the SM Higgs rate . For a search window of , the excess significance above SM background (including systematics uncertainties) has been reported to be  Aaltonen:2011mk (). Recently, CDF has included an additional to their data sample, for a total of , and the statistical significance has grown to ( including systematics) Punzi (). More recently, the D0 Collaboration released an analysis (which closely follows the CDF analysis) of their data finding “no evidence for anomalous resonant dijet production” Abazov:2011af (). Using an integrated luminosity of they set a 95% CL upper limit of on a resonant production cross section.

In a previous work Anchordoqui:2011ag () we presented an explanation of the CDF data by identifying the resonance with a inherent to D-brane TeV-scale string compactifications Antoniadis:1998ig (). In this section we repeat our analysis but with two highly significant changes. First, we allow for the experimental uncertainty by focusing on a wide range () of the (pre-cut) resonant production cross section. This interpolates between the CDF and D0 results. Second, we turn our attention to a different D-brane model which has the attractive property of elevating the two major global symmetries of the SM (baryon number and lepton number ) to local gauge symmetries.

Related explanations for the CDF anomaly based on an additional leptophobic gauge boson have been offered Buckley:2011vc (). Alternative new physics explanations include technicolor, new Higgs sectors, supersymmetry with and without parity violation, color octect production, quirk exchange, and more Eichten:2011sh (). There are also attempts to explain this puzzle within the context of the SM He:2011ss ().

The suppressed coupling to leptons (or more specifically, to electrons and muons) is required to evade the strong constraints of the Tevatron searches in the dilepton mode Acosta:2005ij () and LEP-II measurements of above the -pole Barate:1999qx (). In complying with the precision demanded of our phenomenological approach it would be sufficient to consider a 1% branching fraction to leptons as consistent with the experimental bound. This approximation is within a factor of a few of model independent published experimental bounds. In addition, the mixing of the with the SM boson should be extremely small Langacker:1991pg (); Umeda:1998nq () to be compatible with precision measurements at the -pole by the LEP experiments :2005ema ().

All existing dijet-mass searches via direct production at the Tevatron are limited to  Abe:1993kb () and therefore cannot constrain the existence of a with . The strongest constraint on a light leptophobic comes from the dijet search by the UA2 Collaboration, which has placed a 90% CL upper bound on in this energy range Alitti:1990kw (). A comprehensive model independent analysis incorporating Tevatron and UA2 data to constrain the parameters for predictive purposes at the LHC has been recently presented Hewett:2011nb ().555Other phenomenological restrictions on -gauge bosons were recently presented in Williams:2011qb ().

In the D-brane model the , , content of the hypercharge operator is given by,


with , and .

The covariant derivative (1) can be re-written as


The fields are related to and by the rotation matrix,


with Euler angles , and . Equation (21) can be rewritten in terms of , , and as follows

Now, by demanding that has the hypercharge given in Eq. (20) we fix the first column of the rotation matrix


and we determine the value of the two associated Euler angles




The couplings and are related through the orthogonality condition (10),


with fixed by the relation . In what follows, we take as a reference point for running down to 150 GeV the coupling using (12), that is ignoring mass threshold effects of stringy states. This yields . We have checked that the running of the coupling does not change significantly within the LHC range, i.e.,

The phenomenological analysis thus far has been formulated in terms of the mass-diagonal basis set of gauge fields . As a result of the electroweak phase transition, the coupling of this set with the Higgses will induce mixing, resulting in a new mass-diagonal basis set . It will suffice to analyze only the system to see that the effects of this mixing are totally negligible. We consider simplified zeroth and first order (mass) matrices


where is the mass of the gauge field, is the usual tree level formula for the mass of the particle in the electroweak theory (before mixing), is the electroweak coupling constant, is the vacuum expectation value of the Higgs field, and are of .

Standard Rayleigh-Schrodinger perturbation theory then provides the (mass) (to second order in ) and wave functions (to first order) of the mass-diagonal eigenfields corresponding to .




From Eqs. (29) and (30) the shift in the mass of the is given by so that The admixture of in the mass-diagonal field is


where we have taken  Nakamura:2010zzi (). Interference effects which are proportional to are present in processes with fermions (e.g. Drell-Yan). However, these vanish at the peak of the resonance. Because of the smallness of , modifications of SM partial decay rates of the are negligible. (See e.g. Langacker:1991pg (), for an analysis of such effects.) Remaining effects are order , and therefore all further discussion will be, with negligible error, in terms of . By the same token, the admixture of in the eigenfield is negligible, so that the discussion henceforth will reflect and .

The third Euler angle and the coupling are determined by requiring sufficient suppression () to leptons, a (pre-cut) production rate at , and compatibility with the 90%CL upper limit reported by the UA2 Collaboration on at  Alitti:1990kw ().

The Lagrangian is of the form


where each is a fermion field with the corresponding matrices of the Dirac algebra, and , with and the vector and axial couplings respectively. The (pre-cut) production rate at the Tevatron , for arbitrary couplings and , is found to be Hewett:2011nb ()


where is the hadronic branching fraction. The presence of a in the process shown in Fig. 2 restricts the contribution of the quarks to be purely left-handed.

Figure 2: Feynman diagram for .

The dijet production rate at the UA2 can be parametrized as follows Hewett:2011nb ()


(Our numerical calculation using CTEQ6 Pumplin:2002vw () agrees within 5% with the result of Hewett:2011nb ().) The dilepton production rate at UA2 energies is given by


where is the leptonic branching fraction. From (IV) and (32) we obtain the explicit form of the chiral couplings in terms of and


Using (33), (34), (35), and (36) the ratio of branching ratios of electrons to quarks is minimized within the parameter space, subject to sufficient production and saturation of the 90%CL upper limit. For a (pre-cut) production varying between , one possible allowed region of the parameter space is found to be and .

iv.1 Anomalous

Let us first consider a reference point of the parameter space consistent with the recent D0 limit Abazov:2011af (). For and , corresponding to a suppression , we obtain at . From Eqs. (25) and (26), this also corresponds to , .666The UA2 data has a dijet mass resolution  Alitti:1990kw (). Therefore, at 150 GeV the dijet mass resolution is about 15 GeV. This is much larger than the resonance width, which is calculated to be  Barger:1996kr (). All the couplings of the (or equivalently ) gauge boson are now detemined and contained in Eq. (IV). Numerical values are given in Table 3 under the heading of .

In Fig. 3 we show a comparison of at and the UA2 90% CL upper limit on the production of a gauge boson decaying into two jets. One can see that for our fiducial values, and , the single production cross section saturates the UA2 90% CL upper limit.

The Tevatron rate for the associated production channels Hewett:2011nb ()




is always substantially smaller. (In (37) the SM leptonic branching fractions have been included to ease comparison with the experiment.) It is straightforward to see that these processes should not yet have been observed at the Tevatron.

Figure 3: Comparison of the total cross section for the production of at and the UA2 90% CL upper limit on the production of a gauge boson decaying into two jets Alitti:1990kw (). We have taken and .
Table 3: Chiral couplings of and gauge bosons for and .

The second strong constraint on the model derives from the mixing of the and the through their coupling to the two Higgs doublets and . The criteria we adopt here to define the Higgs charges is to make the Yukawa couplings (, , ) invariant under all three ’s. From Table 2, has the charges ( and has (). So the Higgs has , , , whereas has opposite charges , , .

Let us consider first the case in which the right-handed neutrino has the following charges . As explained before, is then anomalous, and there is no hierarchy among the masses of and . For simplicity we can assume that and , with . For the second Higgs field the charges are , , , .777Note that cannot correspond to an elementary open string excitation, since it has . One possibility is to regard as a composite scalar field, built from two elementary open string scalars, a SM singlet and another Higgs doublet , , with the following -charges: and . In case is a composite scalar field so that the corresponding Yukawa coupling arises from a dimension-5 effective operator, one expects that its vacuum expectation value is somewhat suppressed compared to the vev of , i.e. . Here,