Constraints on the dark sector from electroweak precision observables

We revisit the Standard Model fit to electroweak precision observables using the latest data and the Particle Data Group value of the mass of the W boson. This analysis is repeated for the value reported by CDF. The constraints on the parameter space for dark photons arising from these electroweak precision observables are then evaluated for both values of the W boson mass. We also extend previous work by placing the first electroweak precision observable constraints on the coupling of dark photons to the fermionic dark matter sector.


Introduction
When developing extensions of the Standard Model (SM) to include potential particle candidates for dark matter (DM), the possibility that a portal exists that bridges the DM sector with SM particles is an enticing one.One interesting example of such a portal is the so-called dark photon.This couples to weak hypercharge before electroweak symmetry breaking and so mixes with both the photon and the  boson and hence couples to other SM particles.The dark photon may also couple to some or all potential dark matter particles in the dark sector.Here we consider the case of dark Dirac fermions.The model is described in the following section.A number of recent reviews have explored the theoretical implications of the presence of a dark photon along with the existing and anticipated experimental constraints on its allowed parameter space [1,2,3].
Numerous experimental searches for the dark photon have been undertaken [4,5,6,7,8].While there is no direct evidence so far, a recent study of world data on deep inelastic scattering (DIS) did report indirect evidence for its existence in the few GeV range [9].The NA64 [6] and BaBar [7] experiments have placed strong constraints on the kinetic mixing parameter,  ≤ 10 −3 , for a dark photon mass up to 8 GeV, albeit with small gaps around the  /Ψ and its excitations.The CMS Collaboration [8] has derived similarly competitive limits in the heavy mass region.The sensitivity of these limits has recently been re-examined in light of the potential coupling of the dark photon to dark matter [10].There are also several planned experiments [11,12,13] aiming to explore parts of the remaining allowed parameter space.
The dark photon framework has also been applied to ascertain whether it could explain the  boson mass anomaly reported by the CDF Collaboration [27].The favoured regions of dark photon parameter space were derived by fitting the CDF   [22,28,29,30].In this work, we will investigate how the CDF   affects the previous EWPO exclusion limits of dark photon parameters [17], by a global fit to   and the other 16 electroweak precision observables.The impact of the CDF   measurement on the fit of electroweak data in the Standard Model and beyond has been investigated recently in Refs.[31,32], including studies of new physics models with oblique corrections, the two-Higgs doublet model and dimension six Standard Model effective field theory (SMEFT).
In addition to the correction a dark photon may generate for EWPO, it is also possible that it could serve as a portal to a new DM sector that would otherwise not interact with SM particles [33,34].There are many well-motivated dark matter candidates for this sector, with masses ranging from ultra-light [35,36] to superheavy [37,38].For a DM sector consisting of dark photons coupling to light DM particles, sub-GeV Dirac fermions have been ruled out by the Planck data [39], while cosmological constraints have been placed on the variable  =  2   (  /  ′ ) 4 for scalar, pseudo-Dirac, and asymmetric DM scenarios [40], where   ′ and   are the masses of dark photon and dark matter particles, respectively, and   =  2   /4 and   is the coupling strength of the dark photon to the dark sector particles.In typical analyses,   is flexible and can vary up to the perturbativity bound.However, it is also important to attempt to place constraints on   directly, as we do here.
In this work, we revisit the electroweak constraints on the dark photon parameters.We first examine the implication of the CDF   measurement on the EWPO constraints on the dark photon kinetic mixing parameter .Then we set exclusion limits on the dark photon couplings to dark matter particles in the case of Dirac fermions with   up to   /2.
In Sec. 2, we begin by briefly reviewing the dark photon formalism, including the coupling to dark fermions.In Sec. 3 we consider the global fit to electroweak precision observables within the SM.In Sec. 4 we present the exclusion limits on dark photons and dark fermions, and finally we summarise our conclusions in Sec. 5.

Dark photon formalism
The dark photon is usually introduced as an extra  (1) gauge boson [41,42,43], interacting with SM particles through kinetic mixing with hypercharge [44] where   is the weak mixing angle,  ′  is the dark photon strength tensor and  is the mixing parameter.We use  ′ and Z to denote the unmixed versions of the dark photon and the SM neutral weak boson, respectively.Note here that we also introduced the minimal coupling of the  ′ to dark fermions .
After diagonalising the mixing term through the following field redefinitions, the dark photon and Z mass-squared matrix becomes, where By diagonalising the mass-squared matrix, one can define the physical  and   , where  is the Z −  ′ mixing angle, The masses of these physical states are [18] Due to kinetic mixing, the SM weak couplings of the  boson to both leptons and quarks will be modified, and the dark photon will also couple to SM particles.In our framework, both  and   will couple to dark fermions.The interacting Lagrangian becomes where   is the electric charge of the SM fermion  .
There are three independent parameters,  Z ,   ′ and .Alternatively, we can choose the physical mass    as a parameter instead of   ′ .From Eq. ( 7), we rewrite  2 in terms of  Z ,    and  as All the physical couplings in Eq. ( 8) depend on three parameters:  Z ,    , and .In other analyses [19,22,9,21,25] where the physical   is fixed at its experimental value, only two parameters are independent, with    and  as the usual choice.

Couplings to SM fermions
The lowest order SM couplings of the  boson to leptons and quarks,   Z = {   ,    ,    ,    } and   Z = {   ,    ,    ,    }, will be shifted to [18,21,25] where Likewise, the couplings of the physical dark photon   to SM fermions are given by

Couplings to dark fermions
Both the dark photon   and the physical  boson will couple to dark matter particles.From Eq. ( 8), we can define the effective couplings Note that in the minimal  (1)  model, where the dark photon  ′ only kinetically mixes with the physical photon, the  boson will not couple to dark matter particles.

Effective couplings
The  -pole observables can be expressed in terms of the effective couplings [45,46], which were derived in the MS renormalization scheme.The effective vector-and axialvector couplings,   and   , of the Z boson to leptons at the Z-pole  f   (  −    5 )   , can be parameterised by [46,47], where with where For the b quark, the  b couplings have non-trivial   and  ℎ dependence due to large vertex corrections [47], where In Eqs. ( 13) and ( 17), we have separated out the lowest order couplings    , , which are universal for all three respective fermion generations, Note that by introducing the dark photon, the couplings in Eq. ( 19) will be modified due to kinetic mixing (see Sec. 2.1).
In principle, the weak mixing angle sin 2   should also depend on  Z ,    , and  [48].In this paper, we will take the SM value of sin 2   in the MS scheme, sin 2   = 0.23122 [49], and check the effect of floating sin 2   when the dark photon is included.

𝑍 boson observables
The partial widths of the  into fermions can be expressed as [47] where   is the electric charge of fermion  .   and    denote corrections to the color factor in the vector and axial-vector currents, respectively.Δ   / are the mixed QED and QCD corrections, and     is the correction from the imaginary part of the loop-induced mixing of the photon and the Z boson.The values of these parameters are taken from Ref. [47].
The remaining electroweak observables can be expressed in terms of these partial widths, where  = ,  or , and  = , .Γ had = Γ  + Γ  + Γ  + Γ  + Γ  , and the total width The left-right asymmetry parameters  ℓ ,   , and   can be written at the tree level as Here we do not relate   to the effective weak mixing angle sin 2  eff , because the dark photon corrections to the couplings in Eq. ( 19) cannot be simply represented by a change in the Weinberg angle [21].

Standard Model fit
We follow the procedure performed in Ref. [17] and choose the free parameters  ℎ ,  Z ,   ,   , Δ (5)  had .
For the Standard Model fit,  Z is taken to be the physical mass of the  boson,   .These parameters are varied around their measured values to minimize the  2 SM , defined by Here,  SM = theory SM ( ℎ ,   ,   ,   , Δ (5)  had ) − exp is the difference vector between the SM predictions and the experimental values.We use the latest experimental data [49] as summarised in Tab. 1, while the  pole observables are taken from Ref. [52] with improved Bhabha cross section. PDG  refers to the world averaged value of the  boson mass from measurements at LEP, SLC, Tevatron and LHC §.Σ exp is the diagonal matrix containing the experimental errors of the corresponding observables.The matrices  characterise the correlations among these electroweak observables, which are given by Tab. 2 and Tab. 3 for the  pole [52] and quark observables [54], respectively.There is also a correlation coefficient of −0.174 between the mass and width of the  boson.
The Standard Model best fit results are given in Tab. 1, with the minimized  2 being  2 SM = 12.9.If we replace the world averaged   value by the latest CDF result [27], the minimum  2 soars to 68.2.Note that the experimental value of   ( 2  ) is not included in the fits.

Constraints on the dark photon
We start by neglecting the couplings to dark fermions by setting   = 0.For each value of    , we adjust the mixing parameter and repeat fitting to the electroweak data by allowing the parameters in Eq. ( 29) to vary.The minimum  2 one can reach depends on , and the 95% CL excluded region is defined such that The resulting upper limits on  are shown in Fig. 1.In the case of the PDG value of   (without the CDF measurement), our result (blue solid curve) is qualitatively consistent with the previous determination [17].We extend this work by performing the fit with   set to the new CDF measurement.We notice that the constraints on  will be tightened when    <   , and relaxed for    >   , as shown by the red solid curve in Fig. 1.We note that the calculation of 95% exclusion limits relies on the assumption that Wilk's theorem continues to hold even in the case of the relatively poor  2 obtained using the CDF   value.Our results should therefore be interpreted as indicative -a more detailed MC simulation is considered beyond the scope of this § We note that the ATLAS collaboration recently released an updated measurement of the  boson mass, which is more precise than their previous measurement whilst remaining compatible with it [53].
Updating the average of this and other   measurements is considered beyond the scope of this work, and we do not expect it to have a large effect on our conclusions.paper.We would expect that a new world average value of   including the CDF result should result in an exclusion curve in between.
We also search for the region in the  −    plane in which the  2 can be potentially improved.We found that, in the region of    <   , the inclusion of the dark photon will always worsen the  2 in respect to the SM fit.In Fig. 1, the blue dashed and red dashed lines represent the dark photon parameters corresponding to the maximum reduction in  2 when fitting the PDG value and the CDF value of   , respectively.In the former case, the  2 is slightly improved above the Z-pole, being  2    = 11.38 for (   , ) = (200 GeV, 0.0489).In the case with the CDF value of   , the inclusion of a heavy dark photon significantly reduces the  2 , with  2   = 33.7 for (   , ) = (200 GeV, 0.1001).The fit results are given in Tab. 4. The predicted value of   is 80.4060 GeV, reducing the discrepancy to 2.9 .
This result is consistent with Refs.[22,28,29,30], in which the dark photon was explored as a possible explanation of the anomaly in the  boson mass.Of course, while this fit represents a significant improvement over that within the SM, the  2 is still unacceptably large.
For each point of the solid lines in Fig. 1, we can also obtain the fitted   and  Z from Eq. (31).We have also checked that, by iteratively varying sin 2   according to [49] and repeating the fit in Eq. ( 31), sin 2   will converge to sin 2   | MS = 0.23117, and the changes in the resulting exclusion limits on  are very small.

Constraints on dark fermions
We then switch on the dark photon's coupling to dark fermions,   , which enters only into the total  boson decay width Γ  .It will receive extra contributions from  χ final states if   <   /2.The  boson decay width is given by: where Γ  is the  →  χ partial width, which depends on the dark photon mixing parameter  and the coupling to dark fermions   ,   ( ) −  2 SM = 3.8, for the case in which  w is taken to be the PDG (CDF) result.The blue (red) dashed curves represents the parameter space that provides the best fit, i.e the minimum  2   value that can be obtained by floating , for which  w is taken to be the PDG (CDF) result.The region in grey is not accessible due to the "eigenmass repulsion" associated with the Z mass.The EWPO limit (black dotted) is taken from Ref. [17].The best fit curve (red dashed) is consistent with the result in Ref. [22].
with  2 , χ defined in Eq. ( 12).This feature makes it possible to determine  and   separately from electroweak observables.For a given dark photon mass, we can define the 95% CL exclusion zone on these two parameters using [49]  2    (,   ) −  2 SM ≥ 5.99 .
Here, we only consider the case of  PDG

𝑊
. We first show results for heavy dark fermions with   = 10 GeV in Fig. 2 for several typical values of    .For the region in which    <   , the constraints on   become stronger as the dark photon mass increases.Conversely, for the region in which    >   , the upper bound on   becomes relaxed.
For heavier dark fermions, the invisible partial width Γ  → χ will be suppressed due to the kinematic factor in Eq. (34), Thus the resulting upper limits on   shown in Fig. 2 will be relaxed by the re-scaling factors,  = √︁  (10 GeV)/ (  ), as summarised in Tab. 5.It is not straightforward to make comparison between the existing constraints and our results in Fig. 2 and direct detection have placed constraints on the variable  =  2   (  /  ′ ) 4 for light dark matter scenarios, which are insensitive to separate factors  and   [40].For comparison, the EWPO [17] and the LHC limits [8] on  are usually multiplied by a constant   typically ranging from the QED  up to the perturbativity bound.Our results in Fig. 2 set direct upper limits on   , which are dramatically suppressed as  approaches its exclusion bound.It would be interesting to revisit the previous constraints on  by taking into account these -dependent limits on   .Table 5. Re-scaling factor to   for heavy dark fermions.

Conclusions
In this work, we revisited the constraints on the dark photon from electroweak precision observables, by investigating the impact of the latest CDF   measurement on the exclusion limits of the mixing parameter .The upper bounds of  become tightened in the region    <   , and weakened for    >   .We also searched for regions in the dark photon parameter space in which the inclusion of the dark photon could potentially improve the agreement between theory and experiment.We also introduced dark photon couplings to fermionic dark matter particles .Due to kinetic mixing, the  boson also decays invisibly to  χ in addition to the neutrino final states.Fitting to the electroweak precision observables then sets exclusion limits on   for massive dark fermions, which are rather strigent when the dark photon mass gets close to the  boson mass.These upper bounds on   become weaker as   increases.
In the future, the proposed CEPC [55], FCC-ee [56], ILC [57] and CLIC [58] are expected to measure some of the electroweak observables with significantly increased precision, which could improve the current constraints on the dark sector.Moreover, the analysis presented in this work can also be extended to pseudo-Dirac, scalar, and asymmetric dark matter scenarios.

Figure 1 .
Figure 1.The 95% CL exclusion constraints on , excluded above.The solid blue (red) curves represent the condition  2 mA D = 93 GeV mA D = 100 GeV mA D = 120 GeV mA D = 140 GeV mA D = 160 GeV

Table 1 .
[49]it results.The experimental value of   ( 2  ) is not included in the fit.PDG  is the world averaged  boson mass from measurements at LEP, SLC, Tevatron and LHC[49], while  CDF

Table 4 .
. The relic density, cosmic micro-wave background (CMB), Fit results including the dark photon.The W-boson mass is taken from the latest CDF measurement.Again, the experimental value of   ( 2  ) is not included in the fit.