Model-Independent Bounds on Kinetic Mixing

Abstract

New Abelian vector bosons can kinetically mix with the hypercharge gauge boson of the Standard Model. This letter computes the model-independent limits on vector bosons with masses from 1 GeV to 1 TeV. The limits arise from the numerous 𝑒⁺𝑒⁻ experiments that have been performed in this energy range and bound the kinetic mixing by 𝜖≲0.03 for most of the mass range studied, regardless of any additional interactions that the new vector boson may have.

1. Introduction

The Standard Model (SM) successfully describes all known interactions of SM fermions and gauge bosons; however, there are several phenomena that motivate physics beyond the SM. Chief among these open questions is the identity of dark matter and its interactions with the SM. Recent anomalies in cosmic ray and direct detection experiments have motivated the exploration of new gauge interactions in a putative dark sector [1, 2]. New Abelian vector bosons provide one of the most robust portals for dark matter—SM interactions. The new vector boson can interact with the SM, even if no SM fermions are directly charged under the additional gauge symmetry. This interaction occurs via mixed kinetic terms between the SM's hypercharge field strength and the new Abelian field strength [3].

The Lagrangian for a kinetically mixed 𝑈(1) theory is ℒ=ℒ_SM−14𝐹^′2_𝜇𝜈−sin𝜖2𝐹^′_𝜇𝜈𝐵^𝜇𝜈+𝑚²_𝐴′2𝐴_𝜇^′2+𝑔𝐽^𝜇_𝐴′𝐴^′_𝜇, (1.1) where 𝐹^′_𝜇𝜈 is the field strength for the new vector boson, 𝐵^𝜇𝜈 is the field strength for the SM hypercharge, and 𝐽^𝜇_𝐴′ encapsulates the interactions of the 𝐴^′ with fields in the dark sector. The mixed kinetic term is interesting for several reasons. First, it is a dimension 4 operator, meaning that it can be generated at high energies without decoupling. Second, this coupling allows communication with a secluded sector that otherwise has no interactions with SM fields. These considerations have motivated a dedicated program to search for kinetically mixed vector bosons [4–7].

Most of the current program for discovering a kinetically mixed vector boson involves producing the state and searching for its subsequent decays. This method is promising, but has the drawback that it assumes that the searches can recognize the decay products of the 𝐴^′. If the dark sector has states lighter than the 𝐴^′, then the 𝐴^′ will preferentially decay to the dark sector over SM states because the kinetic mixing parameter almost always satisfies 𝜖≪𝑔. Searching for the 𝐴^′ by looking for the dark sector final states requires a wide-ranging search program because the dark sector may decay back to the SM in a variety of different ways, for example, lepton jets [8–12]. Model-independent searches are possible using completely inclusive searches, for example, 𝑒⁺𝑒⁻→𝛾+𝑋, but these are challenging and few of these searches have actually been performed.

At low masses, the best model-independent bounds arise from the (𝑔−2) measurements of the electron and muon [13]; however, the power of (𝑔−2)_𝜇 begins to weaken for 𝑚_𝐴′≳𝑚_𝜇. At masses far above collider energies, the 𝐴^′ can be integrated out and its effects can be encapsulated in higher-dimension operators, most importantly 𝑆 and 𝑇 [14–16]. 𝑒⁺𝑒⁻ colliders have probed up to √𝑠=207 GeV and therefore, the effects of the 𝐴^′ cannot be parameterized as local higher-dimension operators for masses less than this energy scale.

This letter computes the model-independent constraints on the kinetic mixing parameter, 𝜖, for masses between 1 GeV and 1 TeV by looking for the effects of virtual 𝐴^′s on precision SM observables. This approach has the benefit of not requiring any knowledge of the decay modes of the 𝐴^′ and sets an upper limit on 𝜖 regardless of the behavior of the decay modes to the dark sector.

2. Kinetic Mixing

Kinetic mixing changes the mass eigenstates and interactions of the vector bosons. What follows is a brief synopsis of the results in [17], see also [18–21]. After diagonalizing the kinetic terms and going to the mass eigenstate basis, the SM neutral current interactions are modified. Absorbing the gauge coupling constants into the definition of the currents, the neutral current interactions areℒ_int=𝑉^𝜇_Gauge𝐽_𝜇=𝑉^𝜇_Massℳ𝐽_𝜇, (2.1) where the notation for the gauge and mass eigenstates is 𝑉^𝜇_Gauge=𝐴^𝜇𝑍^0𝜇𝐴^′𝜇,𝑉^𝜇_Mass=𝐴^𝜇𝑍^𝜇𝑍^′𝜇, (2.2) and the currents are𝐽^𝜇=𝑒𝐽^𝜇_EM𝑔𝑐_w𝐽^𝜇_𝑍0𝑔𝐽^𝜇_𝐴′ (2.3) with the diagonalization matrixℳ=100−𝑐_w𝑡_𝜖𝑠_𝜉𝑠_w𝑡_𝜖𝑠_𝜉+𝑐_𝜉𝑠_𝜉𝑐_𝜖−𝑐_w𝑡_𝜖𝑐_𝜉𝑠_w𝑡_𝜖𝑐_𝜉−𝑠_𝜉𝑐_𝜉𝑐_𝜖. (2.4)𝑐,𝑠,𝑡 stand for cosine, sine, and tangent, respectively, and 𝑐_w and 𝑠_w are the cosine and sine of the weak mixing angle. The photon's interactions are unaltered due to its residual gauge invariance. The angle 𝜉 is defined as 𝑚tan2𝜉=2Δ²_𝑍′−𝑚²_𝑍0𝑚²_𝑍′−𝑚²_𝑍0²−Δ²,Δ=−𝑚²_𝑍0sin𝜃_wtan𝜖 (2.5) with 𝑚_𝑍⁰=𝑚_𝑊^±/cos𝜃_w. After changing to the mass eigenstate basis, the physical mass of the 𝑍⁰, 𝑚_𝑍, is𝑚²_𝑍=𝑚²_𝑍0−𝑚²_𝑍′sin²𝜉cos²𝜉 (2.6) and the physical mass of the new vector boson is𝑚²_𝑍′=𝑚²_𝐴′𝑐²_𝜉𝑐²_𝜖+𝑚²_𝑍0𝑠²_𝜉𝑠1+^w𝑡_𝜖𝑡_𝜉². (2.7)

These corrections to the SM neutral currents and to the mass of the 𝑍⁰ place model-independent bounds on (𝑚_𝑍′,𝜖). The next section describes the SM measurements that are sensitive to these modified neutral current interactions.

3. Precision SM Measurements

Virtual 𝑍^′ exchange modifies measured observables such as Bhabha scattering, forward-backward asymmetry measurements, 𝑚_𝑍, and the total hadronic cross sections. The mass of the 𝑍⁰ is the most powerful single measurement but the constraint is augmented by other measurements at and above the 𝑍⁰ pole. Additionally, if the 𝑍^′ has a sizeable branching ratio back to the Standard Model, resonant production of the 𝑍^′ bounds the parameter space at specific energies.

The strongest constraint on 𝜖 comes from the shift of the 𝑍⁰ mass [22]. Notice that 𝜉 in (2.6) changes sign as 𝑚_𝑍′ goes through 𝑚_𝑍⁰, meaning that the corrections to the 𝑍⁰ mass vanish at this point. Defining 𝛿𝑚=𝑚_𝑍′−𝑚_𝑍⁰, the correction to the 𝑍⁰ mass is given by𝑚_𝑍−𝑚_𝑍⁰𝑚_𝑍⁰=−𝑡²_𝜉𝛿𝑚𝑚_𝑍⁰+𝒪𝛿𝑚²𝑚²_𝑍0≤2.5×10⁻⁵, (3.1) so that as 𝛿𝑚→0, there is no bound on 𝜖 resulting from the measurement of 𝑚_𝑍. There is a reduction in the limits on 𝜖 for𝑚_𝑍′≃𝑚_𝑍⁰±0.1GeV, (3.2) where other measurements must take over for the 𝑍⁰ mass measurement.

𝑒⁺𝑒⁻ colliders measure the SM neutral current interactions and when 𝑚_𝑍′≲√𝑠≤𝑚_𝑍⁰, the 𝑍^′ couples dominantly to the electromagnetic current, which causes a kink to appear in the running of the fine structure constant. Differential Bhabha scattering measures 𝛼_EM(𝑞²) and there is a wealth of data from experiments such as OPAL [23], DELPHI [24], SLD [25], TASSO [26], CELLO [27], and TRISTAN [28]. As a result, the new vector boson changes the predictions for differential Bhabha scattering. All the experiments above have a large forward bin of cos𝜃≳0.9, so only a small range of 𝑞² is probed at each experiment. The forward bin normalizes the luminosity and cannot be used as a constraint, thereby limiting the power of these measurements. Differential Bhabha scattering for √𝑠≤𝑚_𝑍⁰ is not useful but provides additional constraints at and above the 𝑍⁰ pole, where corrections to 𝑚_𝑍 are less powerful.

In addition to differential Bhabha scattering, the forward-backward asymmetries for the bottom, charm, muon, and tau are measured at 𝑚_𝑍⁰, effectively fixing 𝛼_EM(𝑚_𝑍⁰) and sin²𝜃_w [29]. The modification to the SM neutral currents alters 𝛼_EM(𝑚_𝑍⁰) and sin²𝜃_w and leads to a conflict with other SM predictions, most notably Γ_𝑍⁰ and 𝜎_had≡𝜎(𝑒⁺𝑒⁻→hadrons), that is, Γ_𝑍⁰=Γ_𝑍⁰(𝛼_EM(𝑚_𝑍⁰),sin²𝜃_w,G_𝐹).

Resonant and on-shell production of the 𝑍^′ can be relevant even if there is a small width directly back to the SM. The 𝑍^′ has a decay width into the SM and dark sectors given by Γ_𝑍′SM≃𝜖²𝑔²𝑚_𝑍′4𝜋,Γ_𝑍′dark≃𝑔²𝑚_𝑍′.4𝜋 (3.3) The width of the 𝑍^′ into the dark sector is unknown; however, given that bounds from the 𝑍⁰ mass set 𝜖≲𝒪(10⁻²) and 𝑔 is bounded by 𝒪(1), there can be a detectable width for the 𝑍^′ back into the SM. As a way to parameterize these effects, two different dark sector widths are used in setting limitsΓ_𝑍′dark=10⁻²𝑚_𝑍′(),0().widenarrow (3.4) On-shell production of the 𝑍^′ is calculated using MadGraph 4.4.32 [30]. Only the interference between the 𝑍^′ and the SM is explicitly computed by zeroing out the |SM|² and |𝑍^′|² squared matrix elements. This results in a deviation from the SM that scales as 𝜖² and the calculations can be compared with measurements using the methods described in the next section.

The total hadronic cross sections, 𝜎_had, are measured at LEP2 with 𝑚_𝑍⁰≤√𝑠≤207 GeV [23, 24] as well as at many other experiments with 22 GeV ≤ √𝑠≤64 GeV [31]. These measurements provide additional bounds because the results from differential Bhabha scattering are not reported at every energy. While the error bars are large compared to the differential Bhabha scattering, resonant 𝑍^′ production enhances sensitivity if 𝑚_𝑍′≃√𝑠. Radiative return processes involving the 𝑍^′ could in principle constrain the theory for √𝑠 away from 𝑚_𝑍′; however, these never provide competitive measurements.

The 𝑍⁰ can have exotic decays into the hidden sector, and assuming that there are no mass thresholds in the hidden sector between 𝑚_𝑍⁰ and 𝑚_𝑍′, thenΓ_𝑍⁰_exotic≃𝑡²_𝜉𝑚_𝑍⁰𝑚_𝑍′Γ_𝑍′. (3.5) The 𝑍⁰ line shape measurement constrains Γ_𝑍⁰ in a model-independent manner giving a bound on Γ_𝑍′ [22]. The bound on the width of the Z^′ is shown in Figure 1.

Figure 1 

The model-independent upper bounds on Γ𝑍′ arising from the line shape of the 𝑍0.

In addition to precision 𝑒⁺𝑒⁻ measurements, direct searches at the Tevatron can produce on-shell 𝑍^′s. This letter finds that the Tevatron's sensitivity is just beneath precision electroweak results even assuming Γ_𝑍′dark=0 [17, 32].

4. Results

The regions in the (𝑚_𝑍′,𝜖) parameter space consistent with precision SM measurements are found by performing a global fit to the SM parameters. This letter uses a CL_𝑠 test that is a function of 𝜖, 𝑚_𝑍′ and the Standard Model parameters, 𝛼_EM(𝑚_𝑍⁰), 𝐺_𝐹 and sin²𝜃_w. 𝐺_𝐹 is fit by 𝜇 decay and does not vary in practice. Most of the data is statistics dominated so that in the case where the signal predicts an excess of events, the CL_𝑠 simplifies into CL_𝑠𝜖,𝑚_𝑍′;SMParam=_exp∑^𝑛obs_𝑛=0𝑒^−𝑁sig𝑁^𝑛_sig/𝑛!∑^𝑛obs_𝑛=0𝑒^−𝑁back𝑁^𝑛_back./𝑛! (4.1) Due to the high statistics, these Poisson summations can be approximated as gaussian integrals.

The advantage of using the CL_𝑠 method is that it is not diluted by superfluous measurements that have no a priori possibility of constraining a theory with a given (𝑚_𝑍′,𝜖), only experiments that have a significant impact.

While superfluous measurements are ignored, measurements that only slightly affect the data can have a significant influence if there are enough of them. This can be illustrated by considering 𝑁 experiments which all give the same result, only slightly different from the SM. A typical 𝜒² analysis will never exclude the SM because the 𝜒²/d.o.f. is small. The CL_𝑠 method will eventually exclude the SM because (1−𝜖)^𝑁 will be small for large 𝑁.

For 𝑚_𝑍′≫200 GeV, the effects of the new vector boson can be encapsulated in terms of local operators and coincide with the precision electroweak analyses, for example, the 𝑆, 𝑇 parameters [14–16] or more recently [33–35]. For 𝑚_𝑍′≲𝑚_𝑍⁰, the bounds are close to those from [17–21], which only use the constraint from 𝑚_𝑍.

Figure 2 shows the 95% confidence level (CL) excluded regions in the (𝑚_𝑍′,𝜖) plane obtained in this study. Wide 𝑍^′s are best constrained by the mass of the 𝑍⁰ for most of the parameter space. The exception occurs near the 𝑍⁰ mass. The forward-backward asymmetries, Γ_𝑍⁰, and 𝜎_had augment the limits when the corrections to the 𝑍⁰ mass vanish and also for 𝑚_𝑍′≃200 GeV where LEP2 forward-backward measurements are more constraining than the 𝑍⁰ mass. Limits on narrow 𝑍^′s are enhanced for 𝑚_𝑍′≃√𝑠 for the numerous 𝑒⁺𝑒⁻ experiments. The forward-backward asymmetries, hadronic cross section, and differential Bhabha scattering measurements provide the additional constraints. The peaks appearing in the exclusion region can be traced to experimental energies at which various experiments were conducted. The constraint on 𝑚_𝑍′ near the 𝑍⁰ is illustrated in the inset of Figure 2. For comparison, the bounds from (𝑔−2)_𝜇, and model-dependent 𝑒⁺𝑒⁻→𝛾𝑍^′→𝛾𝜇⁺𝜇⁻ BaBar searches from [4, 36] are shown.

Figure 2 

95% CL exclusions in the (𝑚𝑍′,𝜖). The cyan region is excluded for a “wide” 𝑍′ and the purple region is for a “narrow” 𝑍′. The blue region shows the bounds placed by CDF on direct production of 𝑍′s. The inset illustrates the constraints on 𝑚𝑍′ near the 𝑍0 pole. The bound from the (𝑔−2)𝜇 is shown in dark grey, and the light grey, dashed region shows the sensitivity from model-dependent BaBar searches.

The main improvement of this work over previous papers is the use of lower energy experiments SLD, TASSO, CELLO, and TRISTAN to measure the running of 𝛼_EM and how it changes due to the addition of a new 𝑍^′ boson. These low energy experiments augment the sensitivity to a new 𝑍^′ boson, although the single strongest constraint is still the change in mass of the 𝑍 boson. These low energy experiments are sensitive to 𝑠 channel production of the 𝑍^′ and hence the width plays an important role. The width of the 𝑍^′ depends on interactions with a hidden sector and cannot be a determined model, independently. Two different scenarios of the width of the 𝑍^′ are considered. If the 𝑍^′ is narrow, then there are many new small dips which are excluded (see Figure 2). If the 𝑍^′ has a larger width, then the effects are washed out and the results are in agreement with [21].

The model-independent limits on kinetic mixing were computed in this letter and found to be 𝜖≲0.03 for most of the mass range studied, 1 GeV < 𝑚_𝑍′<200 GeV. The possible use of radiated return to place tighter constraints on 𝑍^′ was investigated at both LEP1 and LEP2 energies; however, this channel did not help place tighter bounds on kinetic mixing. Even with the constraints found in this letter, there still is a vast parameter space available for a kinetically mixed vector boson to mediate interactions between a dark sector and the SM. The current program of searching for model-dependent decay modes at low energy experiments will augment these model-independent limits for 𝑚_𝑍′≲10 GeV. For higher energies, only the LHC will provide additional information for 200 GeV ≲𝑚_𝑍′≲ 3 TeV [17]. The relatively weak limits for 𝑚_𝑍′≳10 GeV motivate new high intensity 𝑒⁺𝑒⁻ experiments to potentially discover new interactions of this form.

Acknowledgments

The authors would like to thank D. E. Kaplan, M. Lisanti, and M. Peskin for numerous useful conversations. A. Hook, E. Izaguirre, and J. G. Wacker are supported by the US DOE under contract number DE-AC02-76SF00515. A. Hook, E. Izaguirre, and J. G. Wacker receive partial support from the Stanford Institute for Theoretical Physics. J. G. Wacker is partially supported by the US DOE's Outstanding Junior Investigator Award.