The Invisible at the LHC
Frank J. Petriello, Seth Quackenbush, Kathryn M. Zurek
Physics Department, University of Wisconsin, Madison, WI 53706
, ,
We study the feasibility of observing an invisibly decaying at the LHC through the process , where is any neutral, (quasi) stable particle, whether a Standard Model (SM) neutrino or a new state. The measurement of the invisible width through this process facilitates both a model independent measurement of and potentially detection of light neutral hidden states. Such particles appear in many models where the is a messenger to a hidden sector, and also if dark matter is charged under the of the . We find that with as few as of data the invisibly decaying can be observed at over SM background for a 1 TeV with reasonable couplings. If the does not couple to leptons and therefore cannot be observed in the DrellYan channel, this process becomes a discovery mode. For reasonable hidden sector couplings, masses up to 2 TeV can be probed at the LHC. If the does couple to leptons, then the rate for this invisible decay is predicted by onpeak data and the presence of additional hidden states can be searched for. With of data, the presence of excess decays to hidden states can be excluded at 95% C.L. if they comprise 2030% of the total invisible cross section.
1 Introduction
New massive gauge bosons appear in numerous theories of physics beyond the Standard Model (SM). They appear in grand unified theories such as [1] and [2], in theories of extra spacetime dimensions as KaluzaKlein excitations of the SM gauge bosons [3], and in Little Higgs theories of the electroweak sector [4]. bosons that decay to leptons have a simple, clean experimental signature, and consequently can be searched for up to high masses at colliders. Current direct search limits from Tevatron experiments restrict the mass to be greater than about 900 GeV when its couplings to SM fermions are identical to those of the boson [5]. The LHC experiments are expected to extend the mass reach to more than 5 TeV [6].
Since the Z’ signature is clean and its QCD uncertainties are small, it is likely that the couplings of a discovered can be studied with reasonable accuracy to probe the high scale theory that gave rise to it. Many studies of how to measure properties and couplings to SM particles have been performed [7]. A recent study performed a nexttoleading order QCD analysis of properties at the LHC accounting for statistical, residual scale, and parton distribution error estimates, and concluded that four generation independent combinations of couplings could be extracted at the LHC by making full use of available onpeak differential spectra [8] (another recent study on searching for the is found in [9]). However, a degeneracy between quark and lepton couplings can not be removed by studying bosons in the DrellYan channel; all observables in this mode are unchanged if the quark couplings are scaled by a factor while the lepton couplings are scaled by . A different production mechanism must be utilized to remove this degeneracy. Possibilities are and ; however, because of SM backgrounds, all three are expected to be extremely difficult to observe at the LHC [10].
Another possible way of removing this degeneracy is by using the width. The width takes the form
(1.1) 
is the partial width for decays into invisible states such as SM neutrinos, and denote the widths for decays into quarks and leptons respectively, and represents possible other decay modes such as . This relation does not suffer from the same degeneracy as noted above. The total width can be measured by fitting the shape of the resonance peak assuming the is not too narrow. is small for large classes of models. If we make the mild theoretical assumption that invariance equates the couplings of charged leptons to those of neutrinos (satisfied in grand unified models), and note that the onpeak study of [8] showed that the combination can be measured, Eq. (1.1) becomes a quadratic equation for the unknown , that can be solved up to a twofold discrete ambiguity. The only other assumption entering this procedure is that is composed entirely of decays to neutrinos.
Besides breaking this degeneracy between quark and lepton couplings, there is an additional strong motivation for studying the invisible width of the . bosons often appear as messengers which connect the SM to hidden sectors, such as in some models of supersymmetry breaking [11] and in Hidden Valley models [12], and can decay to light particles in this hidden sector. For example, Hidden Valley models contain subTeV mass states which are electrically neutral and quasistable, with decay lengths in some cases longer than tens of meters. These exit the detector as missing energy. A sterile neutrino which is charged under the would also result in hidden decays of the . Such states may also account for the observed dark matter, as in the model of [13]. A model of millicharged dark matter from a Stueckelberg may also be found in Ref. [14].
In this paper we study whether invisible decays of the can be detected at the LHC using the channel 15]. As we will be interested in the large missing kinematic region, GeV, the experimental signature is relatively clean. Other possible channels such as . This mode has previously been used to search for invisible decays of the Higgs boson [
Our paper is organized as follows. In Section 2 we explain our choice of invisible decay channel, and discuss backgrounds. In Section 3, we subject signal and background to cuts to isolate invisible decays, and parametrize the cross section in terms of initial state radiation (ISR) and final state radiation (FSR) contributions; hidden decays, which do not couple to the Standard Model, only appear in ISR contributions. We examine typical masses and couplings that can be probed, as well as kinematic differences between ISR and FSR. In Section 4 we determine whether decays to hidden sector states can be determined apart from SM neutrinos, using predictions from onpeak data as a background. Finally, we conclude in Section 5.
2 Signal and backgrounds
We begin by explaining how we search for invisible decays. We focus on the channel . Other possible signal processes to consider are , , and . We compute the signal using MadEvent [17]; unless noted otherwise, we use MadEvent for all signal and background calculations. . These, however, are more sensitive to uncertainties such as jet energy mismeasurements and jets faking photons. They require a detailed simulation beyond the scope of our analysis. We impose the following basic acceptance cuts on the two leptons in our signal: and , where
The dominant Standard Model backgrounds to our signal fall into two categories: the production of leptons and neutrinos without a in the intermediate state, and the production of where the jets escape down the beampipe or have their energies mismeasured. We first consider SM production of leptons and neutrinos, . We compute the full SM background with all interference effects and spin correlations included. The primary subprocesses contributing to this background are and . We reduce the background using an invariant mass cut on the two leptons: . This restriction helps, but the background is still significant. Further reduction of this and the background is obtained by a separation between the two leptons in our signal, do not significantly help once the cut is imposed. cut, which we discuss in detail later. Other kinematic properties, such as the
We must also discuss the potentially large background , where the jets escape detection and fake a source of missing . The LHC hadron calorimeters have a very wide rapidity coverage, up to , but soft jets in the central region are difficult to measure. We therefore restrict ourselves to vetoing jets with GeV in the central region. Many soft jets may add up to substantial missing ; this can be a problem since the cross section is so large to begin with.
We perform a crude estimate of the two possible sources of background: jets escaping down the beampipe or soft jets in the central region. We anticipate in this analysis the missing cuts we will later impose to study 12 TeV bosons,
All signal and background processes in our study are calculated at leading order in the QCD perturbative expansion using a running scale for . The nexttoleading order corrections to the background processes are known [19], while the corrections to the signal process are easily calculable. Since we later use as our significance estimator the ratio of signal over background fluctuation, , we feel this is a conservative approach; including the factors for both and would improve our results. We note that the dependence of the nexttoleading order cross section on the renormalization and factorization scales indicates that uncertainties arising from uncalculated higher order corrections are at the few percent level or less. In most of our analysis we also neglect errors associated with imprecise knowledge of parton distribution functions. For the gauge boson production processes considered here, it is likely that LHC data can determine these to high accuracy. The analysis in Ref. [20] indicates that the parton distribution function errors for diboson process such as the backgrounds considered here may be reduced to the percent level by normalizing their rates to the LHC DrellYan data samples. Detector effects such as smearing were determined to have a small effect on lepton distributions in [16], and we neglect them in our analysis as well. We neglect other detector issues such as smearing of the cut, they are difficult to estimate with current tools. These issues should be revisited in a more complete study which makes use of LHC data, but we believe their neglect is justified in this initial analysis. distributions caused by unvetoed soft jets and the underlying event; although we expect them to be relatively unimportant due to our large missing
3 Studying the invisible
Employing the cuts and techniques described in the previous section, we determine whether invisible decays can be observed over the SM background. We map out the missing dependence in Figs. (1) and (2). The basic cuts outlined in the previous section have been implemented. Both plots show the SM background as a function of a lower cut on the missing . Two fiducial models are also shown: the sequential Standard Model and the model with an overall gauge coupling . We assume TeV for both models. The plots begin at background. We also plot the required invisible cross sections for observation at the LHC assuming , , and . Fig. (1) shows the required cross section for a statistical significance of , while Fig. (2) shows the required rate for . Two facts can be observed from these graphs. First, the optimum missing cut for TeV mass bosons is around 200 GeV, above the level where is a serious concern. Second, for realistic models a signal is observable at the LHC even with moderate integrated luminosity. is possible for both fiducial models with less than , while is possible for less than . GeV to avoid serious issues with the
We wish to do more than simply observe the invisibly decaying . We also want to measure the underlying parameters leading to these decays, and determine whether the decays are accounted for by SM neutrinos only, or whether decays to other exotic states are occuring. Although at first this appears more modeldependent, the matrix element for production actually possesses a simple structure that can be encapsulated in a few quantities. Two distinct classes of Feynman diagrams contribute to the process: finalstate radiation (FSR) graphs where the is emitted from the neutrinos, and initialstate radiation (ISR) graphs where the is emitted from the initial quark line. Examples of each type are shown in Fig. (3). We note that because of the invariant mass cut, diagrams where the leptons are emitted from the are numerically negligible. The particle labeled in the graphs can denote either a SM neutrino or a hidden sector state. If it is a hidden state, it does not couple to the boson and therefore can be produced only via ISR graphs. We have checked that the interference of ISR and FSR contributions is numerically small, indicating that only squared ISR and squared FSR graphs contribute to the signal cross section. This can be partially understood by noting that the propagator cannot be simultaneously onshell in both types of diagrams, indicating that for narrow states the interference should be suppressed.
Generically, an ISR will be softer than one from FSR, so that we can expect a corresponding preference for a softer 4), where the fraction of the total ISR or FSR crosssection surviving a given contribution. cut is shown. It is seen that the ISR contribution drops off more quickly, as expected. Also shown is the SM background, which drops off more quickly than either spectrum from ISR than FSR. This is shown in Fig. (
The relative size of the ISR and FSR contributions determines how well a decaying to hidden sector particles can be extracted. A large ISR contribution implies that nonstandard decays can be measured. The simplicity of the matrix element structure allows us to parametrize how different states decay via ISR and FSR contributions in a model independent way. To see this, we first write the cross section subject to the basic acceptance cuts and missing cut as
(3.2) 
where up and down quark contributions have been separated. Each can in turn be written as a product of two distinct terms: a piece which incorporates the matrix elements, parton distribution functions, and experimental cuts, denoted as ; a piece which depends on the charges from a given model, . We then have . The coupling structure of the various terms takes the form
(3.3) 
and
(3.4) 
where , denote the partial widths of the to SM ’s or to any invisible particle (SM ’s or hidden sector states), and is the total width. A prime on a charge indicates that it is a charge, while no prime denotes a SM charge. and subscripts denote axial and vector charges, respectively. Any model can then be constructed by dialing appropriately. The functions depend on the given model under consideration only through the mass in the narrow width approximation.
Values of the charges are given in Table (1) for the the sequential Standard Model (SSM) and model discussed previously. We also show a model with gauge coupling in which the couples to baryon number, and which also includes a hidden sector state, assumed to be a vectorlike fermion with unit charge. We present charge values for the SSM and models with the same hidden state. The increase of the when the hidden state is present can be observed in Table (1). We will see later that the values assuming only SM neutrino decays are determined once the DrellYan channel is observed. Measuring different values than predicted by DrellYan studies would indicate the presence of hidden sector decays. If leptonic decays do not occur, such as in the model, the process considered here becomes a discovery channel.
SSM  SSM  
0.274  0.212  0.292  0.197  0  
1.75  1.36  0.481  0.324  0  
0.274  0.589  0.436  1.08  1.49  
0.432  0.931  0.907  2.26  1.90  
0  0  
0  
0  
0  
0  
0  1  0  1  1  
0  0  0  0  0 
To develop some intuition, we present below several plots showing features of the cross section for different choices. For simplicity of presentation we make the simplifying assumption and . The degeneracy between and in the plots can be broken by utlizing the information fb, fb, pb, pb, evaluated for a missing cut of 150 GeV. We note that the kinematic dependences of the and quark cross sections on the missing cut are very similar. We focus on three example cases: ; and ; and . These values are roughly consistent with those present in typical models as shown in Table (1). We show in Fig. (5) the ISR fraction of the total cross section as a function of the missing cut for TeV. For , the ISR fraction of the cross section is less than 20%. The FSR matrix elements give a larger contribution to the cross section, suggesting that it will be difficult to dig out the hidden sector sector component from invisible decays. We will quantify this further later. The cross sections for the charges under consideration are shown in Fig. (6). For comparison, we overlay the curves showing the required cross sections for with from Figs. (1) and (2). We see that at least evidence is possible at the LHC for a range of values.
To study what masses can be probed, we show in Fig. (7) the cross section as a function of mass for several different example values. Since the masses are larger, the corresponding values needed for observation are larger, so we present results assuming somewhat larger charges. We show the results for missing cuts of both 150 and 200 GeV; the value that actually maximizes varies with . Included in this plot are the required cross sections for assuming . For , masses beyond 2 TeV are easily observable. If and the ISR charge is larger, the case relevant for decays to hidden sectors, only masses up to 1.25 or 1.5 TeV can be probed with .
Finally, if the does not decay into leptons but does decay to hidden sector states, will proceed by moving upward a minimum missing cut and looking for a signal to emerge. The shape of the mass. Also, if a more complicated structure of new physics than a simple isolated is discovered, we will want to determine whether the signal arises from a single new gauge boson or multiple states. spectrum should give some sensitivity to the becomes a discovery channel. The experimental search for this leptophobic
We determine the statistical measurement error for three fiducial masses, , and TeV, by performing a comparison of their missing spectra versus other masses. We set to simulate a completely leptophobic for our spectra. The cross section is divided into several bins in missing ; we take the ratio of each bin to the total rate surviving the deviation is exceeded in each case; this occurs when the total reaches . In Fig. (8) we have plotted 1 error bands for the three masses as a function of hidden cross section after the missing cut of GeV, for the SLHC luminosity of . We have not taken into account errors other than statistical, such as PDF uncertainties; we leave the inclusion of such errors for a more complete analysis. One can see, however, that given just the statistical error, there is good sensitivity to the mass given a sufficiently large cross section (reasonable for a leptophobic ) and sufficient integrated luminosity. For reference, a fb, values of , , and , for masses of TeV, TeV, and TeV, respectively. cross section corresponds to templates for many other masses and compare the ratios in each bin to the ratios for each fiducial mass, and determine for what masses a total 1 cut to normalize. We generate
4 Finding the hidden sector
We wish to study whether LHC results can determine if invisible decays occur only to SM neutrinos, or whether other states are involved. This would provide insight into possible hidden sectors to which the couples.
The crucial fact that allows this measurement to be performed is that the charges introduced in the previous section are predicted by the analysis of DrellYan production in [8] if the decays invisibly only to neutrinos. We note that
(4.5) 
and
(4.6) 
where and are the onpeak couplings determined in [8]:
(4.7) 
is fixed by and . If the decays invisibly only to neutrinos, then ; is then completely predicted by the onpeak couplings. Any deviation of from this limit indicates additional invisible decays of the .
We first determine how big an excess over the expected invisible cross section predicted by onpeak data can be observed. In addition to SM production of leptons and missing energy, the signal now becomes a background to , where the s are the hidden sector particles. In Fig. (9) we show the the size of the excess cross section over that predicted by the onpeak data which can be excluded at 95% C.L. for and of data; as this is a difficult measurement we have assumed a sizeable amount of integrated luminosity. The excess cross section for which evidence can be obtained is shown in Fig. (10). We have used a cut of from [8] must be propagated through the expressions in Eqs. (4.5) and (4.6). We present results for fractional errors on the predicted invisible cross section of 10% and 25%, which are consistent with the error propagation, as well as for the idealized limit of no error. From Figs. (6) and (7), we see that cross sections for typical values with 9), hidden sector decays leading to excess cross sections of 12 fb can be excluded at 95% confidence level. If no onpeak cross section is observed, then the left side of Figs. (9) and (10) indicate how well completely invisibly decaying bosons can be probed. Completely invisibly decaying boson cross sections can be excluded down to 0.5 fb given sufficient integrated luminosity. GeV are between 110 fb. Using the 10% error curve from Fig. ( GeV in producing these numbers. The excess cross section that can be probed depends crucially on how well the expected invisible cross section can be predicted from onpeak data. To determine this precision, the expected errors on
Several reductions of the error associated with the invisible cross section prediction are possible. With the error comes mostly from parton distribution functions; with it comes entirely from parton distribution functions. These uncertainties will be significantly improved with LHC data. In addition, one may be able to normalize the FSR contribution to onpeak data, due to the similar PDF and coupling structure. Approaching a 5% error is not inconceivable.
We now interpret this excess cross section using our effective charges. We write , where can be predicted from the onpeak data and is the portion coming from decays to hidden sector states. We plot in Fig. (11) the size of this excess cross section as a function of , where . Using this graph and keeping in mind the 12 fb cross sections, we observe that it will be difficult to significantly constrain hidden sector decays if is significantly greater than 1 TeV. For a 1 TeV state, charges in the range can be probed.
Although the 95% confidence level and reaches in the plane can be determined from Figs. (6), (7), (9), (10), and (11), since the parameter space is large and the graphs are numerous, we summarize below several canonical cases.

boson with , , : We assume a 10% error in the invisible cross section prediction when interpreting this state. Using the graphs in a similar fashion as above, the following information about can be obtained.

95% exclusion: with and with .

evidence: can probe with and with .

5 Conclusions
We have studied the feasibility of observing an invisible through the process at the LHC, where is any neutral, (quasi) stable state. We found that evidence of this process could be made with as little as of data for a standard with gauge coupling , while a discovery is possible with . With our results, using Figs. (67) in conjunction with Figs. (12), the discovery reach of LHC for observing any invisibly decaying can be computed. We parametrized our results in terms of two effective charges that completely describe production of a in conjunction with a radiated off the initial state (ISR) or the final state (FSR). We found that for a 1 TeV , any model with , can be observed at with of data. This shows that a leptophobic that cannot be observed through the usual DrellYan channel at the LHC can be discovered if it decays invisibly to hidden sector states. We showed that some sensitivity to the mass can be obtained by studying the missing spectrum.
In addition, we demonstrated that an excess invisible decay of the to hidden sector states over the predicted cross section for from onpeak data can be excluded at 95% confidence level if the size of this cross section is 2030% of the total cross section, given a 10% error on the predicted invisible cross section. The exotic states may, for example, be dark states from a “Hidden Valley” model [12]. The may be a communicator to a light hidden sector with MeV mass dark matter states as in the model of [13]; this is motivated by the INTEGRAL/SPI observation [21] of a 511 keV line toward the galactic center. To get 2030% of the invisible cross section of the from hidden decays will require in most cases a hidden sector with multiple states to compete with the SM neutrino invisible decays. In particular, when , the branching fraction to new hidden sector states must be approximately the same as the branching fraction to SM neutrinos in order to obtain a deviation in the invisible cross section predicted from onpeak data. This is on account of the hidden sector states entering only through graphs where the is radiated off the initial state quark lines; these initial state graphs usually compose a relatively small fraction of the total cross section: from Fig. (5) for the fiducial case of the effective charges for initial state and final state radiation being roughly the same. Despite this potential difficulty of observing decays to hidden sector states, we have shown that it is nonetheless feasible, given the presence of such a hidden sector. The possibility to observe such dark states through the hidden decays of a new vector gauge boson makes the accurate measurement of the invisible at the LHC an exciting and reachable goal.
Acknowledgments
The authors are supported by the DOE grant DEFG0295ER40896, Outstanding Junior Investigator Award, by the University of Wisconsin Research Committee with funds provided by the Wisconsin Alumni Research Foundation, and by the Alfred P. Sloan Foundation.
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