New physics behind the new muon $g$-2 puzzle?

The recent measurement of the muon $g$-2 at Fermilab confirms the previous Brookhaven result. The leading hadronic vacuum polarization (HVP) contribution to the muon $g$-2 represents a crucial ingredient to establish if the Standard Model prediction differs from the experimental value. A recent lattice QCD result by the BMW collaboration shows a tension with the low-energy $e^+e^- \to \text{hadrons}$ data which are currently used to determine the HVP contribution. We refer to this tension as the new muon $g$-2 puzzle. In this Letter we consider the possibility that new physics contributes to the $e^+e^- \to \text{hadrons}$ cross-section. This scenario could, in principle, solve the new muon $g$-2 puzzle. However, we show that this solution is excluded by a number of experimental constraints.

The recent measurement of the muon g-2 at Fermilab confirms the previous Brookhaven result. The leading hadronic vacuum polarization (HVP) contribution to the muon g-2 represents a crucial ingredient to establish if the Standard Model prediction differs from the experimental value. A recent lattice QCD result by the BMW collaboration shows a tension with the low-energy e + e − → hadrons data which are currently used to determine the HVP contribution. We refer to this tension as the new muon g-2 puzzle. In this Letter we consider the possibility that new physics contributes to the e + e − → hadrons cross-section. This scenario could, in principle, solve the new muon g-2 puzzle. However, we show that this solution is excluded by a number of experimental constraints.
The expected forthcoming results of the Fermilab experiment plan to reach a sensitivity four-times better than the E821 one. Moreover, in a longer term, also the E34 collaboration at J-PARC [7] aims at measuring the muon g-2 through a new low-energy approach.
On the theory side, the only source of sizable uncertainties in a SM µ stems from the non-perturbative contributions of the hadronic sector, which have been under close scrutiny for several years. The SM prediction a SM µ in Eq. (1) has been derived using (a HVP µ ) TI e + e − , the leading hadronic vacuum polarization (HVP) contribution to the muon g-2 based on low-energy e + e − → hadrons data obtained by the Muon g-2 Theory Initiative [6] (see also ). Alternatively, the HVP contribution has been computed using a first-principle lattice QCD approach [6] (see also [29][30][31][32][33]). Recently, the BMW lattice QCD collaboration (BMWc) computed the leading HVP contribution to the muon g-2 with sub per-cent precision, finding a value, (a HVP µ ) BMW , larger than (a HVP µ ) TI e + e − [34]. If (a HVP µ ) BMW is used to obtain a SM µ instead of (a HVP µ ) TI e + e − , the discrepancy with the experimental result is reduced to 1.6σ only. The above results are respectively (a HVP µ ) BMW = 7075 (55) × 10 −11 .
The present situation regarding the leading HVP contribution to the muon g-2 can be schematically represented as in Fig. 1, where (a HVP µ ) EXP is the value of the HVP contribution required to exactly match a EXP µ assuming no NP. Hereafter, the difference between the discrepancies in Fig. 1 will be referred to as the new muon g-2 puzzle. Assuming that both (a HVP µ ) TI e + e − and (a HVP µ ) BMW are correct, we ask whether this puzzle can be solved thanks to NP effects which would bring (a HVP µ ) TI e + e − in agreement with (a HVP µ ) BMW , without spoiling the 1.6σ agreement of (a HVP µ ) BMW with (a HVP µ ) EXP . Differently from what has been usually done in the literature, here we do not assume a direct NP contribution to ∆a µ (i.e. new states that couple directly to muons). If fact, by itself this possibility could solve the longstanding discrepancy in Eq. (1), but not the new muon g-2 puzzle. Instead, in order to solve the latter, we invoke NP that modifies the e + e − → hadrons cross-section σ had .
An increase of σ had , due to an unforeseen missing contribution, has been already proposed to enhance (a HVP µ ) TI e + e − and solve ∆a µ [35][36][37][38][39]. However, the required shift in σ had is disfavoured by the electroweak fit if it occurs at √ s ≳ 1 GeV [37]. Hence, in the following, we will consider NP modifications of σ had below the GeV scale. While Refs. [35][36][37][38][39] did not specify the origin of the shift in σ had , we here assume that it is due to NP. After classifying in a model-independent way the general properties of such a NP, we investigate for the first time its non-trivial impact on e + e − and BMWc lattice results.
Crucial for our analysis is the dispersion relation used to determine (a HVP µ ) TI e + e − via σ had . This relation follows from the application of the optical theorem to the photon HVP. It will be shown that scenarios in which NP couples only to hadrons are not able to solve the new muon electrons (and hence it contributes to σ had at tree level), (a HVP µ ) TI e + e − should be subtracted of NP in the comparison with (a HVP µ ) EXP , whereas (a HVP µ ) BMW should not. In fact, in this case, the quantity that should enter the dispersion relation determining the HVP contribution should not be the experimentally measured cross-section σ had , but rather σ had − ∆σ NP had . Therefore, the tension between (a HVP µ ) TI e + e − and (a HVP µ ) BMW could be solved by invoking a negative interference between the SM and NP, that is ∆σ NP had < 0. As we will show below, the above picture selects a very specific NP scenario which entails new light particles with a mass scale ≲ 1 GeV coupling to SM fermions through a vector current.
Model-independent analysis. Hereafter we are going to examine the general properties of NP models aiming at solving the new muon g-2 puzzle via a modification of σ had . To this end, we introduce the dispersion relation where K(s) is a positive-definite kernel function with K(s) ≈ m 2 µ /3s for √ s ≫ m µ . This equation defines the HVP contribution to the muon g-2 in terms of the photon HVP, Π had , which includes possible NP effects. If the possible NP entering the photon HVP does not couple to electrons, i.e. it does not enter the hadronic cross-section at tree level, then Eq. (4) can be written as where σ had includes final-state radiation (FSR), whereas both vacuum polarization and initial-state radiation (ISR) effects are subtracted. In particular, vacuum polarization corrections can be simply accounted for by multiplying the experimental cross-section by |α/α(s)| 2 , while the correction of ISR and ISR/FSR interference effects is addressed by each experimental collaboration. In this Letter we will focus on the region where σ had is experimentally determined, i.e. √ s ≳ 0.3 GeV, since this gives the by far dominant contribution to the dispersive integral in Eq. (5).
In Fig. 2 we show a schematic classification of how NP can enter σ had . The first two diagrams are representative of FSR effects, which also unavoidably affect the photon HVP at the next-to-leading order (NLO). We can safely neglect possible NP contaminations in ISR since the bounds on NP couplings to electrons are very severe. The third diagram, where NP enters the hadronic crosssection at tree level coupling both to hadrons and electrons, is due to NP that also modifies the photon HVP at NLO. Crucially, however, its dominant contribution to the muon g-2 emerges via the tree-level shift of σ had .
Hence, when invoking NP in σ had , there are two different scenarios to be considered, depending on whether NP couples only to hadrons or both to hadrons and electrons. In the following, we analyze these two possibilities and their capability to solve the new muon g-2 puzzle.
1. NP coupled only to hadrons. This scenario is schematically represented by the first two diagrams of Fig. 2. As remarked above, real and virtual FSR must be included in σ had . However, in order to establish the impact of NP in FSR (which depends on the interplay between the mass scale of NP and the experimental cuts), it would be mandatory to perform dedicated experimental analyses imposing the various selection cuts specific of each experimental setup, which is beyond the scope of this Letter. Since the full photon FSR effect estimated in scalar QED amounts only to 50 × 10 −11 [6,40], a value well below the discrepancy between Eqs. (2)-(3), and given that light NP couplings with the SM particles are tightly constrained, the NP contributions in FSR cannot solve the new muon g-2 puzzle.
2. NP coupled both to hadrons and electrons. If NP contributes to σ had at tree level (see third diagram in Fig. 2), then only the subtracted cross-section σ had − ∆σ NP had should be included in Eq. (5). We note that the latter can be larger than σ had if ∆σ NP had < 0, thus requiring that the NP contribution is dominated by a negative interference with the SM. As (a HVP µ ) TI e + e − has been computed using σ had rather than the subtracted cross-section σ had − ∆σ NP had , the theoretical prediction of the HVP contribution in Eq. (5) is where (a HVP µ ) NP describes NP effects at LO, due to the tree-level exchange of the NP mediator (see third diagram in Fig. 2), as well as at NLO. Instead, (a HVP µ ) BMW should be shifted only by NLO NP effects. Remarkably, this scenario may allow to match Eq. (6) with (a HVP µ ) EXP , while keeping at the same time the agreement with the BMWc lattice result.
Light new physics analysis. We now explore whether the second scenario envisaged above can be quantitatively realized in an explicit NP model. Motivated by the fact that the kernel function in Eq. (5) scales like 1/s and by the fact that modifications of σ had above ∼ 1 GeV are disfavoured by electroweak precision tests [37], we focus on the sub-GeV energy range, where the dominant contribution to σ had arises from the e + e − → π + π − channel. In fact, in the SM, this channel accounts for more than 70% of the full hadronic contribution to the muon g-2.
Furthermore, the requirement of having a sizeable negative interference with the SM amplitude narrows down the general class of NP models. Indeed, the interference of scalar couplings with the SM vector current is suppressed by the electron mass, while pseudoscalar and axial couplings do not interfere. Hence, we focus on the tree-level exchange of a light Z ′ boson with the following vector couplings to electrons and first-generation quarks 1 with q = u, d and m Z ′ ≲ 1 GeV. The matrix element of the two-pion intermediate state can be expressed in terms of the pion vector form factor where q = p ′ − p and J µ em = 2 3 uγ µ u − 1 3 dγ µ d is the electromagnetic current. Exploiting the charge conjugation invariance, in the iso-spin symmetric limit, we find that ⟨π ± |J µ em |π ± ⟩ = ⟨π ± |uγ µ u|π ± ⟩ = −⟨π ± |dγ µ d|π ± ⟩ . (9) Hence, F V π (q 2 ) also describes the matrix element of the . (10) Note that if the Z ′ interactions are iso-spin symmetric, then g u V = g d V and the contribution to the π + π − amplitude vanishes. Defining σ SM+NP ππ = σ SM ππ + ∆σ NP ππ , the tree-level exchange of the Z ′ with width Γ Z ′ leads to where the pion vector form factor cancels out in the ratio. The dispersive contribution to the muon g-2 due to SM and NP can be obtained by using σ had −∆σ NP had in Eq. (5). Imposing that the current discrepancy ∆a µ is solved by NP in the hadronic cross-section, we obtain 1 Assuming only the interactions in Eq. (7), the gauge current associated to the Z ′ would be anomalous. Additional heavy fermions charged under the electroweak group can be introduced in order to make the model UV consistent.
where the lower integration limit is s exp ≈ (0.3 GeV) 2 , that is, the integral is performed in the data-driven region for the ππ channel. Approximating ∆σ NP had ≈ ∆σ NP ππ , 2 from Eq. (11) we find where we introduced the effective coupling ϵ ≡ g e V (g u V − g d V )/e 2 and the adimensional width parameter γ ≡ Γ Z ′ /m Z ′ . If both the Z ′ → ee and Z ′ → π + π − channels are kinematically open, the associated decay widths (normalized to m Z ′ ) read, respectively up to (m e /m Z ′ ) 4 corrections, and where |F V π (m 2 Z ′ )| 2 (normalized to F V π (0) = 1) can be enhanced up to a factor of 45 by the ρ resonance [11]. For m Z ′ < 2m π + ≈ 0.28 GeV, γ = γ ee , whereas for m Z ′ ∈ [0.3, 1] GeV we can approximate γ ≈ γ ππ since the e + e − channel can be safely neglected given the tight bounds on g e V . Note that possible contributions to γ stemming from non-SM final states (e.g. decays into a dark sector) yield a positive-definite shift to σ had , since they cannot interfere with the SM. Hence, they contribute with a negative shift to ∆a µ (cf. Eq. (12)), thus worsening the discrepancy. The profile of ∆σ NP had , together with its SM counterpart (taken from Ref. [14]), is shown in Fig. 3 for some representative benchmark values of the Z ′ model parameters addressing the ∆a µ discrepancy. In particular, we find that Z ′ masses below the ρ resonance require ϵ < 0, whereas Z ′ masses above it require ϵ > 0. Moreover, in order to obtain ∆a µ > 0, the interference term has to dominate over the pure resonant effect in Eq. (13). All in all, we find that the parameters of the Z ′ model that are needed to explain ∆a µ can be divided in two regions: i) m Z ′ ≳ 0.3 GeV which requires |ϵ| ≈ 10 −2 and γ ≳ 10 −3 and ii) m Z ′ ≲ 0.3 GeV which requires |ϵ| ≈ 10 −2 and basically no relevant constraint on γ (as evident from Eq. (13)). We note that in principle it could be possible to directly observe (with a dedicated scanning analysis) the new resonance in e + e − data for particular choices of the Z ′ mass and width parameters. However, since there are also configurations that can explain ∆a µ with a very narrow and light Z ′ below the experimental resolution, it still remains necessary to constrain the Z ′ couplings indirectly.
In the following, we are going to inspect whether the region of the parameter space of the Z ′ model needed to explain ∆a µ is allowed by experimental constraints. These can be divided for convenience in three classes: 1. semi-leptonic processes; 2. purely leptonic processes and 3. purely hadronic, iso-spin violating observables.
1. Semi-leptonic processes. The e + e − → qq crosssection, σ qq , has been measured with a per-cent level accuracy at LEP II for center of mass energies √ s ∈ [130, 207] GeV [41]. The leading effect to this process induced by a Z ′ exchange is given by where Q q denotes the quark charge. Requiring that the deviation from unity in Eq. (16) is less than 1% [41] leads to |g e V g q V | ≲ 4.6 · 10 −4 |Q q | which translates into ϵ ≲ 3.3 · 10 −3 . Moreover, the bound does not depend on the Z ′ mass and it acts on a coupling combination that is similar to the one entering ∆σ NP ππ , but not vanishing in the isospin symmetric limit g u V = g d V .
Hence, this bound can be considered to be conservative.
2. Leptonic processes. The Z ′ coupling to electrons is also tightly constrained. In particular, the nonobservation at BaBar of the process e + e − → γZ ′ followed by the decay Z ′ → e + e − yields g e V ≲ 2 · 10 −4 [42] if the Z ′ decays dominantly into electrons. Therefore, in our framework, this bound applies only for m Z ′ ≲ 0.3 GeV where the Z ′ → π + π − decay is not kinematically allowed. Moreover, another important bound arises from the electron g-2, yielding |g e V | ≲ 10 −2 (m Z ′ /0.5 GeV) for m Z ′ ≳ MeV.
3. Iso-spin breaking observables. The Z ′ contribution to ∆σ NP ππ is proportional to the iso-spin breaking combination g u V − g d V , which should be of O(1) in order to explain ∆a µ in the m Z ′ ≳ 0.3 GeV region. Therefore, it is natural to expect sizeable effects on other iso-spin violating hadronic observables. A relevant example is given by the charged vs. neutral pion mass squared difference, ∆m 2 = m 2 π + − m 2 π 0 . Analogously to the QED case, the quadratically divergent Z ′ loop leads to where Λ χ ≈ 1 GeV is the cut-off scale of the chiral theory and we chose m Z ′ ≪ Λ χ . Instead, for m Z ′ ≫ Λ χ , the Z ′ contribution to ∆m 2 decouples as Λ 2 χ /m 2 Z ′ ≪ 1. In practice, the NP contribution from a light Z ′ is more reliably obtained by rescaling the SM prediction from lattice QCD [43] with (g u V − g d V ) 2 /e 2 . Then, comparing the SM prediction of ∆m 2 with its experimental value, we find the 95% C.L. bound g u V − g d V ≲ 0.06. The interplay of the above constraints in the plane −g e V vs. g u V − g d V is displayed in Fig. 4 for two representative scenarios where m Z ′ = 0.1 and 0.5 GeV. The directions of the arrows indicate the excluded regions by the different experimental bounds. Instead, the red band is the region favoured by the explanation of the muon g-2 discrepancy. From Fig. 4 it is clear that, irrespectively of the Z ′ mass, there are always at least two independent bounds preventing to solve the new muon g-2 puzzle.
Conclusions. The recent lattice QCD result by the BMW collaboration shows a tension with the low-energy e + e − → hadrons data currently used to determine the HVP contribution to the muon g-2.
In this Letter we investigated the possibility to solve this tension, referred to as the new muon g-2 puzzle, invoking NP in the hadronic cross-section. A possible way to restore full consistency into the picture is to postulate a negative shift in σ had due to NP. We showed that this scenario requires the presence of a light NP mediator that modifies the experimental cross-section σ had . However, this non-trivial setup, where NP hides in e + e − → hadrons data, is excluded by a number of experimental constraints. Alternative confirmations of the e + e − determinations of the HVP contribution to the muon g-2, based on either additional lattice QCD calculations or direct experimental measurements, as proposed by the MUonE experiment [44][45][46], 3 will hence be crucial to shed light on this intriguing puzzle.
Note added. After this paper was posted on the arXiv, Ref. [49] appeared in which the authors studied the possibility of reconciling the data driven and the BMWc lattice determinations of a HVP µ by rescaling the KLOE luminosity via a NP contribution to Bhabha scattering. Hence, contrary to our analysis, in their approach NP does not directly contribute to σ had .