Spin-identification of the Randall-Sundrum resonance in lepton-pair production at the LHC
The determination of the spin of the quantum states exchanged in the various non-standard interactions is a relevant aspect in the identification of the corresponding scenarios. We discuss the identification reach at LHC on the spin-2 of the lowest-lying Randall-Sundrum resonance, predicted by gravity with one warped extra dimension, against spin-1 and spin-0 non-standard exchanges with the same mass and producing the same number of events in the cross section. We focus on the angular distributions of leptons produced in the Drell-Yan process at the LHC, in particular we use as basic observable a “normalized” integrated angular asymmetry . Our finding is that the 95% C.L. identification reach on the spin-2 of the RS resonance (equivalently, the exclusion reach on both the spin-1 and spin-0 hypotheses for the peak) is up to a resonance mass scale of the order of 1.0 or 1.6 TeV in the case of weak coupling between graviton excitations and SM particles () and 2.4 or 3.2 TeV for larger coupling constant () for a time-integrated LHC luminosity of 10 or , respectively. Also, some comments are given on the complementary rôles of the angular analysis and the eventual discovery of the predicted second graviton excitation in the identification of the RS scenario.
Department of Physics and Technology, University of Bergen, Postboks 7803, N-5020 Bergen, Norway
The Abdus Salam ICTP Affiliated Centre, Technical University of Gomel, 246746 Gomel, Belarus
University of Trieste and INFN-Trieste Section, 34100 Trieste, Italy
A common feature of the different New Physics (NP) scenarios that go beyond the Standard Model (SM) is the predicted existence of heavy new particles or “resonances”, that can be either produced or exchanged in reactions studied at high energy colliders. Such non-standard objects are expected to be in the TeV mass range, and could be revealed directly as peaks in the energy dependence of the measured cross sections.
For any model, given the expected statistics and experimental uncertainties, one can assess the corresponding discovery reach by determining the upper limit of the mass range where the resonance signal can be detected above the SM cross section to a given confidence level.
On the other hand, once a peak in the cross section is observed, further analysis is needed to distinguish the underlying non-standard dynamics against the other scenarios that potentially may cause a similar effect. In this regard, the expected identification reach is defined as the upper limit of the mass range where the model could be identified as the source of the peak or, equivalently, the other competitor models can be excluded for all values of their parameters. The determination of the spin of the resonance represents therefore an important selection among different classes of non-standard interactions.
Here, we consider the discrimination reach on the lowest-lying spin-2 Randall–Sundrum (RS) graviton resonance , that could be obtained from measurements of the Drell–Yan (DY) lepton-pair production processes () at the LHC:
The determination of the spin-2 of the lowest-lying RS graviton resonance exchange against the spin-1 hypothesis in the context of experiments at LHC has recently been discussed, e.g., in Refs. [2, 3, 4, 5], and some attention to the case of the spin-0 hypothesis has been given in Ref. . Also, the exclusion of the spin-2 hypothesis against the spin-1 Stueckelberg was discussed in Ref. . An experimental search for spin-2, spin-1 and spin-0 new particles decaying to DY dilepton pairs has recently been performed at the Fermilab Tevatron collider .
We would like to complement those analyses and assess the extent to which the domain in the RS parameters allowed by the discovery reach on the resonance is reduced by the request of simultaneous exclusion of both the hypotheses of spin-1 and spin-0 exchanges with the same mass, and mimicking the same peak in the dilepton invariant mass distribution (same number of events).
As is well known, the main tool to differentiate among the spin exchanges in the process (1) uses the different, and characteristic, dependencies on the angle between the incident quark or gluon and the final lepton in the dilepton center-of-mass frame. We shall base our discussion on the integrated center-edge asymmetry , that has the property of directly disentangling the spin-2 from vector interactions as illustrated in Refs. [8, 9]. This method represents an alternative to the use of the differential distributions , where represents the number of events.
We believe our analysis has sufficiently general features to be applicable also to the identification of other spin-2 exchange interactions, besides the RS model. We nevertheless prefer in the sequel to refer and expose in detail the procedure in the case of that mentioned scenario. Moreover, although strictly not necessary, we shall make comparisons with specific, “physically motivated” representative spin-0 and spin-1 models.
As an example of spin-0 contribution to the process (1), we can consider the sneutrino exchange envisaged by supersymmetric theories with -parity breaking () [10, 11]. In it is possible that some sparticles can be produced as -channel resonances, thus appearing as peaks in the dilepton invariant mass distribution, if kinematically allowed.
Examples of competitor spin-1 mediated interactions, that can contribute to process (1) and show up as peaks in the cross section are, besides the SM and , the heavy exchanges , and we will refer to those models for the comparison with the RS resonance exchange.
In Sec. 2 we discuss the LHC cross section and statistics for the production of a Randall–Sundrum heavy graviton. In Sec. 3 we identify the ranges in the number of events and mass that can originate from either the RS graviton or a spin-0 object, such as a sneutrino. This common range is referred to as the “signature space” of these models. A corresponding discussion is presented in Sec. 4 for the RS graviton and a spin-1 object, such as a . In Sec. 5 we review the relevant angular distributions, and in Sec. 6 we show how these can be used to identify the RS graviton by means of the asymmetry . The results are collected in Sec. 7, where regions are identified in the plane spanned by the coupling strength and the mass , where spin identification is possible. Finally, Sec. 8 presents some concluding remarks.
2 LHC cross sections and statistics for RS
Considering its great popularity in the context of models solving the gauge hierarchy problem, we here just recall that the simplest RS scenario is based on one compactified warped extra spatial dimension and two branes, such that the SM particles are confined to the so-called TeV brane while gravity can propagate in the whole 5-dimensional space. In this scenario TeV-scale, spin-2, narrow graviton resonances are predicted. The model depends on two independent parameters, that can be chosen as the dimensionless ratio , with the 5-dimensional scalar curvature and the reduced 4-dimensional Planck scale (), and , the mass of the lowest-lying graviton resonance. The masses of the higher excitations are given by , where are roots of the Bessel function (, , ,..), and are therefore unevenly spaced. The mass pattern may therefore be distinctive of the model, if higher excitations in addition to the ground state would be discovered. A correlated parameter is represented by the physical scale on the TeV brane , whose inverse controls the strength of the graviton resonance coupling to standard matter. The (theoretically) natural ranges for these parameters are and . Current discovery limits at 95% C.L. from the Fermilab Tevatron collider are for the first graviton mass: 300 GeV for and 900 GeV for .
2.1 Cross sections
In the SM, lepton pairs at hadron colliders can be produced at tree level via the following parton-level processes:
The first massive graviton mode of the RS model, in the sequel denoted simply as (and the mass ), can be produced via quark–antiquark annihilation as well as gluon–gluon fusion,
and can be observed as a peak in the dilepton invariant mass distribution. The inclusive differential cross section for production and subsequent decay into lepton pairs at the LHC can be expressed as the sum:
where and are invariant mass and rapidity of the lepton pairs, respectively, and with the lepton-proton angle in the dilepton center-of-mass frame. Explicitly:
Here, and are the even and odd parts (under ) of the partonic differential cross section . Furthermore, the -factor accounts for higher order QCD corrections and, at NLO, can be approximated by the well-known expression (see, for instance Ref. )
For simplicity, and to make our procedure more transparent, we shall use in the sequel a global, flat, factor . Although the full NLO corrections to the processes of interest here can require, as discussed in detail in Ref. , a somewhat larger -factor, especially for gluon-initiated processes, this effect would tend to cancel in the asymmetry basic to our analysis, which is determined by ratios of angular-integrated (and mass-integrated around the resonance) cross sections. It may, however, have some bearing on the statistics, rendering the event rates based on the value 1.3 a slightly conservative estimate. Finally, are parton distribution functions in the protons and , and are the parton fractional momenta:
In deriving Eq. (2.1), the relations and have been used, with the C.M. energy squared. The minus sign in the odd term in that equation allows us to interpret the angle in the parton cross section as being relative to the quark or gluon momentum (rather than the proton momentum ).
The lepton differential angular distribution, for dilepton invariant mass in an interval of size around the (narrow) resonance peak , is defined by
The cross section for the narrow state production and subsequent decay into a DY pair, , is given by:
One may notice that only terms in the partonic cross sections which are even in contribute to the right-hand side of Eq. (8), because in the case of the proton-proton collider odd terms do not contribute after the integration over the rapidity . This holds true for the SM - interference term, as well as for SM- interference, with the SM partonic cross section being pure -initiated at the considered order. Although such interference terms may appreciably contribute to the doubly-differential cross section , their contribution to the integral over needed in Eqs. (8) and (9), symmetrical around the graviton resonance mass , is negligibly small for and small resonance width (an approximation that will be assumed in the sequel), and negligible -dependence of the overlap integral within the bin. This fact is pointed out for the case of s in, e.g., Refs. [6, 15], but also holds for the graviton resonance case. Thus, in Eqs. (8) and (9), we can just retain the SM and the G pole contributions.
In Eq. (10), represents the graviton propagator, with and the mass and total width, respectively:
The total width can be written as , where is a constant depending on the number of open decay channels. Assuming the graviton decays only to the SM particles, and with partial widths explicitly given in Refs. [2, 17, 19], one finds . With in the theoretically “natural” range, this value allows to use for the graviton resonance propagator the narrow-width approximation,
The leading order angular dependencies in Eq. (10) are given by
where is the number of quark colors.
For the SM partonic cross section of Eq. (11) one has, neglecting fermion masses:
where, for fermion , , , and the propagator in the approximation is represented by
2.2 Statistical considerations
In the experimental discovery of a narrow resonance the observed width is determined by the dilepton invariant mass resolution, that we may associate to the size of the bin introduced above. Clearly, on the one hand larger would allow a larger chance of detecting the resonance and, on the other hand, for a narrow resonance falling within the bin the integral over in Eq. (9) should be practically insensitive from the size of . Conversely, such an integral should be essentially proportional to the size of for the SM background. This background is dominated by the SM Drell-Yan process, other SM background contributions turn out to amount to at most a few percent of it .
Regarding the bin size, it depends on the energy resolution. For the ATLAS detector, the bin size at invariant dilepton mass measured in TeV units, can be parameterized as :
For TeV, the term dominates in Eq. (18) and the bin size grows linearly in , so that GeV for large . Similar results, comparable to about 10%, hold for the CMS detector . Throughout the paper we will use Eq. (18) for the bin size.
At the LHC, with integrated luminosity , the number of signal (resonant) events can be computed by using and the background events are defined as (background integrated over the bin). Here, is the experimental reconstruction efficiency, taken to be 0.9 both for electrons and muons. To compute cross sections we use the CTEQ6 parton distributions . We impose angular cuts relevant to the LHC detectors. The lepton pseudorapidity cut is for both leptons (this leads to a boost-dependent cut on ), and in addition to the angular cuts, we impose on each lepton a transverse momentum cut . Analogous to previous references, in the analysis given here, we have adopted the criterion for the discovery limit that events or 10 events, whichever is larger, constitutes a signal. The number of DY background events () inside each bin, the minimum number of signal events required to detect a graviton resonance () and the resonant signal events () at various are summarized in Table 1. Only electron pairs are included in Table 1.
Fig. 1 shows the expected number of resonant (signal) events vs. resonance mass () at fb for graviton production with values of = 0.01, 0.05, 0.1 (dashed curves), and the minimum number of signal events needed to detect it above the background. With the assumption of efficiencies as stated above one finds that, with 100 of integrated luminosity, one can explore a massive graviton up to a mass of about 2.5 TeV with ( level), and this limit can be pushed to 4.5 TeV with , consistent with the results of . While the analysis above is for the specific RS model, the general features of this analysis may hold for a wider class of models which support narrow resonances and predict spin-2 intermediate states. We shall refer to a region in the space spanned by resonance mass and number of events, that can be populated by a certain model, as the “signature space” of that model. We now proceed to sketch the competing (with the spin-2 resonance) non-standard spin-0 and spin-1 interactions, and their respective signature spaces.
3 Signature spaces of RS and sneutrino in
As mentioned in Sec. 1, models based on SUSY can mimic the RS graviton in a certain part of the parameter space as far as the mass and narrowness of the resonance is concerned. At tree-level, the relevant parton process for DY lepton-pair production is in -parity breaking given by spin-0 sneutrino () formation from quark-antiquark annihilation and subsequent leptonic decay:
The corresponding partonic cross section is given by 
where the pure resonant term reads
Here, the propagator of the sneutrino is represented by
() is the mass (total decay width) of the sneutrino, and are the relevant -parity-violating couplings of and to the sneutrino, respectively. We note that the process (19), where the intermediate state is a sneutrino, requires two -parity-violating couplings to be non-zero.111A different scenario was investigated in , where only one such coupling was assumed non-zero. Then a squark would be exchanged in the - or -channel, and the angular distribution would be rather different. For the present case, the -factor has been studied for a range of sneutrino masses, and for different parton distribution functions . The value adopted for the graviton case, , remains a good approximation.
In the narrow width approximation the -exchange cross section (21) can be written as:
Here is the sneutrino leptonic branching ratio and the relevant coupling to the quarks. Indeed, due to invariance, the sneutrino, which is a -member of the doublet, can only couple to a down-type quark. This model depends therefore on two independent parameters, i.e., the sneutrino mass and . With generation indices, the -parity-violating coupling of interest is , with denoting the sneutrino generation. Among these, is rather constrained, whereas and could be as large as – for a 100 GeV sneutrino, and larger for a heavier one .
Quantitatively, the current constraints on are rather loose. The number of signal events as a function of the spin-0 mass for the case of sneutrino production with values of ranging from to in steps of 10 (dash-dotted curves), are given in Fig. 1. The calculation has been performed under the assumptions and kinematical cuts exposed in Sec. 2.2. From Fig. 1 one can easily obtain the discovery reach on sneutrino parameters ( level) and translate them into the plane (,) exhibited in Fig. 2. In this figure, the discovery region is on the left of the dashed line, and the gray ( horizontal) lines limit the “confusion” domain with the RS graviton in event rates, with and , respectively, see also Ref. .
In Fig. 1, the shaded area indicates the overlap of the LHC-discovery parameter space for the scenario via and that of the lowest RS graviton scenario via . The figure indicates that, as far as the total production cross section of DY dilepton pairs is concerned, there exists a significantly extended domain in the () plane where sneutrino production can mimic RS graviton formation in its theoretically “natural” domain (): in these respective domains, the two scenarios can lead to the same number of events under the resonance peak, . In other words, the two models are indistinguishable in the overlapping domains of their parameter spaces, indicated by the shaded area in Fig. 1. Clearly, outside the “common” shaded area, the two scenarios might be differentiated by means of event rates. For the identification, the two models must be discriminated in the “confusion” region in Fig. 1, this can be done by the spin determination of the RS resonance.
4 Signature spaces of RS and
Turning now to spin-1 resonance exchange, the differential cross section for the relevant partonic process reads at leading order
and, neglecting fermion masses:
Here, we have introduced the vector and axial-vector couplings to SM fermions, and the propagator is represented by
According to previous arguments, in the sequel we neglect interference terms in the cross section. Moreover, effects from a potential mixing are also disregarded.
As anticipated in Sec. 1, in addition to a generic spin-1 exchange, we will consider the discrimination reach on the spin-2 lowest RS resonance from the, rather popular and physically motivated, scenarios where the couplings in Eq. (27) are constrained to have fixed values. One such model is the so-called sequential model (SSM), where the couplings to fermions are the same as those of the SM .
Furthermore, we will consider: (i) the three possible scenarios originating from the exceptional group breaking; and (ii) the predicted by a left-right symmetric model that can originate from an GUT. While detailed descriptions of these models can be found, e. g., in Ref. , we just recall that the three heavy neutral gauge bosons are denoted by , and with specific coupling constants to SM matter displayed in Table 2. Regarding the case (ii), the mentioned left-right (LR) model predicts a heavy neutral gauge boson generally coupled to a linear combination of the right-handed and – currents [ and are baryon and lepton numbers, respectively]:
Here, = and are the and coupling constants with ; the parameter is restricted to the range . The upper bound corresponds to the so-called LR-symmetric model with , while the lower bound is found to coincide with the model introduced above.
All numerical values of the couplings needed in Eq. (27) are collected in Table 2, where: ; , with , and for the , and , respectively. We have introduced the notations for the and the LR models and for the ALR model. Current direct search limits on masses from the Fermilab Tevatron are of the order of 900 GeV or less .
|Left-Right model (LR)|
|Alternative Left-Right model (ALR)|
The partial decay widths into massless fermion-antifermion pairs in and LR models are functions of and , respectively. From the analysis of Refs. [12, 30] it turns out that, in the absence of “exotic” decay channels, the total width , so that the narrow width approximation to the propagator should be adequate for our numerical estimates.
The number of signal events as a function of resonance mass for the representative models summarized in Table 2, and LHC luminosity of 100 fb, are given in Fig. 3. From this figure, one can easily obtain the level discovery reaches on the corresponding masses, presented as a histogram in Fig. 4. These estimates are numerically consistent with those in Refs. [6, 12] and [31, 28, 32, 33, 34, 35].
In these cases, the signature spaces reduce to lines, and Fig. 3 shows that, at the assumed LHC luminosity, the lowest, RS spin-2, resonance can be discriminated against the ALR and SSM spin-1 scenarios already at the level of event rates in a large range of values, with no need for further analyses based on angular distributions. Only the and LR models possess a “confusion region” with the RS resonance , concentrated near the upper border of the graviton allowed signature domain. This may represent an interesting information by itself.
In the next sections, we turn to the identification of the spin-2 of the first RS resonance, vs. the spin-1 and spin-0 hypotheses.
5 Angular distributions in the dilepton channel
The normalized angular distributions of the relevant parton processes, mediated by spin-2, spin-1 and spin-0 formation and subsequent decay to DY pairs, are shown in Table 3, as summarized, e.g., also in Refs. [3, 4].
|Process||Normalized density for ,|
For simplicity, and according to the considerations made in Sec. 2, in this table only the -even terms in the parton differential cross sections are retained, -odd contributions disappear from the observables we will consider.
The correspondence between spin and angular distribution is quite sharp: a spin-0 resonance determines a flat angular distribution, spin-1 corresponds to a parabolic shape, and spin-2 yields a quartic distribution. The CDF collaboration has recently attempted angular distribution analyses using the cumulative DY data at the Tevatron collider, their results are reported in Ref. . The LHC promises tests of the spin hypotheses with significantly higher sensitivity, due to the definitely higher statistics allowed by the foreseen larger energy and luminosity.
Using Eq. (8), one can derive the angular distributions determined by spin-2 RS graviton resonance, spin-1 and spin-0 , respectively. These distributions can be conveniently written in a self-explanatory way as [ denotes the spin-2 resonance, while and denote the spin-1 and spin-0 cases, respectively]:
Detector cuts are not taken into account in the above Eqs. (30)–(33). We shall use these relations for illustration purposes, in order to better expose the most important features of the method we use. The final numerical results, as well as the relevant figures that will be presented in the sequel refer to the full calculation, with detector cuts taken into account. It turns out, however, that such results are numerically close to those derived from the application of Eqs. (30)–(33).
The angular distributions arising from the spin-2, spin-1 and spin-0 resonances are represented in Fig. 5, for the same peak masses in the three hypotheses and the same number of signal events, , under the peak. The angular distributions in this figure are somewhat distorted compared to those in Table 3, because of (i) the smearing due to the parton distributions in the protons, (ii) different partons contribute with different weight to the different channels, and (iii) detector cuts are taken into account.
6 Identification of the spin-2 of the RS graviton
6.1 Center-edge asymmetry
To assess the identification power of the LHC of distinguishing the spin-2 RS resonance from both spin-1 and spin-0 exchanges, we adopt the integrated center-edge asymmetry introduced in Refs. [8, 9]. Basically, the advantage of this observable lies in its insensitivity to spin-1 exchanges in the -channel. This property follows from the fact that such exchanges are characterized by the same -distributions as the SM - and -exchanges, see Eq. (31). Thus, deviations of from the SM predictions could be attributed to graviton exchanges and, accordingly, one could expect a particularly high sensitivity in the identification of this kind of effects. Also, being “normalized” to the cross section integrated over angles, one may hope this observable to be less sensitive to systematic uncertainties.
In the present application, we define the center-edge asymmetry, with labelling the three hypotheses we want to compare, as:
with the “center minus edge” cross section:
Here: is a, a priori free, value of that defines the separation between the “center” and the “edge” angular regions; in the approximation , are given by Eqs. (30)–(32) and the total cross sections by Eq. (33).
We assume that a deviation from the SM is discovered in the cross section for dilepton production at LHC in the form of a narrow peak in the dilepton invariant mass, and attempt the determination of the domain in the RS parameter space where such a peak can be identified as being caused by the spin-2 RS exchange, and the spin-0 and spin-1 hypotheses excluded. We also assume the integrated center-edge asymmetry evaluated within the RS model to be consistent with the measured data, and call this spin-2 model the “true” or “best-fit”model. We want to assess the level at which this “true” model is distinguishable from the other hypotheses, with spin-0 and spin-1, that can compete with it as sources of a resonance peak in dilepton production yielding in particular the same number of signal events.
Here, , s and are the fractions of resonant events for and SM background, respectively, with . They are determined by the ratios of , etc., of Eq. (30) and of Eq. (33), and shown in Fig. 6 for two values of .
Analogous definitions hold for the other cases. One should emphasize again that, for spin-1, and the SM background (predominantly from the DY continuum ) have the same form, independent of couplings and resonance mass.
As an example, in Fig. 7 (left panel) the center-edge asymmetry is depicted as a function of for resonances with different spins, same mass TeV and same number of signal events under the peak. The dot-dashed curve corresponds to the spin-2 RS graviton with . The calculation is performed using the parton distributions mentioned in Sec. 2.2, and detector cuts as well as the SM background have been accounted for. Actually, the -behavior of resulting from the full calculation is found essentially equivalent to those presented in Eqs. (36)–(38). Differences are appreciable only for close to 1, and turn out to have negligible impact on the numerical determinations of the identification reaches presented in the sequel, where the relevant chosen values of are in a range around 0.5. Indeed, since numerically the turns out to have a smooth dependence there, for definiteness we will present the results obtained from .
The deviations of the asymmetry from the prediction of the RS model, caused by the spin-0 exchange
and that caused by the spin-1 exchange
respectively, are depicted in Fig. 7 (right panel). The identification potential depends, of course, from the available statistics (as well as on systematic uncertainties). In the example of Fig. 7, the vertical bars attached to the dot-dashed curve represent the 2 statistical uncertainty on the of the RS graviton model, assumed to be the “true” model consistent with the data as stated above, with the values of and reported in the caption and integrated LHC luminosity of 100 fb. One reads from Fig. 7 that, at such (high) luminosity, the spin-2 RS graviton with mass TeV and coupling can, indeed, be discriminated from the other spin-hypotheses by means of at .
for all values of . This property is of course reproduced in Fig. 7, and allows to conclude that, in order to identify the spin-2 graviton resonance, if one is able to exclude the spin-0 hypothesis, the whole class of spin-1 models will then automatically be excluded, so that the spin-2 identification from the spin-1 hypothesis would be model-independent. Stated in a statistical language, Eq. (41) explicitly realizes the statement that discrimination of the spin-2 RS resonance from the spin-1 hypothesis requires, for a given confidence level, less events than the discrimination from the spin-0 one, as also noted in Ref. .
6.2 Numerical results for RS graviton identification
We now consider the determination of the spin-2 of the resonance, based on the assessment of the corresponding required minimal numbers of signal events under the peak, . To this purpose, we consider the deviations of the (assumed to have been measured) center-edge asymmetry from those expected from pure spin-0 exchange, , and from spin-1 exchange, , defined by Eqs. (39) and (40), respectively.
In Eq. (42), the notations are: ; ; and . We reconsider the numerical example of Fig. 7, and note that around the chosen value the gluon fusion subprocess largely dominates the deviation of Eq. (42), due to ,. Actually, it is the only contribution at , because of the vanishing at this point. This feature is found to hold more generally, also for the other values of and different from those in Fig. 7 or, in other words, this choice is optimal in the sense that shows maximal sensitivity to RS paramenters there.
To get an “estimator” that determines the spin-2 parameter space where the spin-0 hypothesis could be excluded, the deviation (42) should be compared with the statistical uncertainty on expressed in terms of the desired number () of standard deviations. We have the condition
where, taking into account that numerically at ,