Neutrino Oscillation Constraints on U(1)' Models: from Non-Standard Interactions to Long-Range Forces
PPrepared for submission to JHEP
IFT-UAM/CSIC-133, YITP-SB-2020-30
Neutrino Oscillation Constraints on U (1) (cid:48) Models:from Non-Standard Interactions to Long-RangeForces
Pilar Coloma, a M. C. Gonzalez-Garcia, b,c,d
Michele Maltoni a a Instituto de Física Teórica UAM/CSIC, Calle de Nicolás Cabrera 13–15, Universidad Autónomade Madrid, Cantoblanco, E-28049 Madrid, Spain b Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos, Universitatde Barcelona, Diagonal 647, E-08028 Barcelona, Spain d Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys 23, E-08010Barcelona, Spain e C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, USA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We quantify the effect of gauge bosons from a weakly coupled lepton flavor de-pendent U (1) (cid:48) interaction on the matter background in the evolution of solar, atmospheric,reactor and long-baseline accelerator neutrinos in the global analysis of oscillation data.The analysis is performed for interaction lengths ranging from the Sun-Earth distance toeffective contact neutrino interactions. We survey ∼ set of models characterized bythe six relevant fermion U (1) (cid:48) charges and find that in all cases, constraints on the couplingand mass of the Z (cid:48) can be derived. We also find that about 5% of the U (1) (cid:48) model chargeslead to a viable LMA-D solution but this is only possible in the contact interaction limit. Weexplicitly quantify the constraints for a variety of models including U (1) B − L e , U (1) B − L µ , U (1) B − L τ , U (1) B − ( L µ + L τ ) , U (1) L e − L µ , U (1) L e − L τ , U (1) L e − ( L µ + L τ ) . We compare theconstraints imposed by our oscillation analysis with the strongest bounds from fifth forcesearches, violation of equivalence principle as well as bounds from scattering experimentsand white dwarf cooling. Our results show that generically, the oscillation analysis im-proves over the existing bounds from gravity tests for Z (cid:48) lighter than ∼ − → − eVdepending on the specific couplings. In the contact interaction limit, we find that for mostmodels listed above there are values of g (cid:48) and M Z (cid:48) for which the oscillation analysis pro-vides constraints beyond those imposed by laboratory experiments. Finally we illustratethe range of Z (cid:48) and couplings leading to a viable LMA-D solution for two sets of models. Keywords: neutrino physics, solar and atmospheric neutrinos a r X i v : . [ h e p - ph ] S e p ontents M Z (cid:48) (cid:38) − eV) M Z (cid:48) limit: the NSI regime 52.2 The finite M Z (cid:48) case: the long-range interaction regime 8 Experiments measuring the flavor composition of neutrinos produced in the Sun, in theEarth’s atmosphere, in nuclear reactors and in particle accelerators have established thatlepton flavor is not conserved in neutrino propagation, but it oscillates with a wavelengthwhich depends on distance and energy. This demonstrates beyond doubt that neutrinosare massive and that the mass states are non-trivial admixtures of flavor states [1, 2], seeRef. [3] for an overview.When traveling through matter, the flavor evolution of the neutrino ensemble is af-fected by the difference in the effective potential induced by elastic forward scattering ofneutrino with matter, the so-called Mikheev-Smirnov-Wolfenstein (MSW) mechanism [4, 5].Within the context of the Standard Model (SM) of particle interactions, this effect is fullydetermined and leads to a matter potential which, for neutral matter, is proportional tothe number density of electrons at the neutrino position, V = √ G F N e ( r ) , and which onlyaffects electron neutrinos. New flavor dependent interactions can modify the matter po-tential and consequently alter the pattern of flavor transitions, thus leaving imprints in theoscillation data involving neutrinos which have traveled through large regions of matter, asis the case for solar and atmospheric neutrinos.Forward elastic scattering takes place in the limit of zero momentum transfer, so aslong as the range of the interaction is shorter than the scale over which the matter densityextends, the effective matter potential can be obtained in the contact interaction approx-imation between the neutrinos and the matter particles. The paradigmatic example is– 1 –rovided by neutral current non-standard interactions (NSI) [4, 6, 7] between neutrinosand matter (for recent reviews, see [8–11]), which can be parametrized as L NSI = − √ G F (cid:88) f,P,α,β ε f,Pαβ (¯ ν α γ µ P L ν β )( ¯ f γ µ P f ) , (1.1)where G F is the Fermi constant, α, β are flavor indices, P ≡ P L , P R and f is a SM chargedfermion. These operators are expected to arise generically from the exchange of somemediator state heavy enough for the contact interaction approximation to hold. In thisnotation, ε f,Pαβ parametrizes the strength of the new interaction with respect to the Fermiconstant, ε f,Pαβ ∼ O ( G X /G F ) . Generically they modify the matter potential in neutrinopropagation, but – being local interactions – the resulting potential is still proportionalto the number density of particles in the medium at the neutrino position. Since suchmodifications arise from a coherent effect, oscillation bounds apply even to NSI inducedby ultra light mediators, as long as their interaction length is shorter than the neutrinooscillation length. For the experiments considered here, such condition is fulfilled as longas M Z (cid:48) (cid:38) − eV [12].Conversely, if the mediator is too light then the contact interaction approximation is nolonger valid, and the flavor dependent forces between neutrino and matter particles becomelong-range. In this case neutrino propagation can still be described in terms of a matterpotential, which however is no longer simply determined by the number density of particlesin the medium at the neutrino position, but it depends instead on the average of the matterdensity within a radius ∼ /M Z (cid:48) around it [12–16].At present, the global analysis of data from oscillation experiments provides some of thestrongest constraints on the size of the NSI affecting neutrino propagation [17–19]. Analysisof early oscillation data was also used to impose constraints on flavor dependent long-rangeforces [12–14].Straightforward constructions leading to Eq. (1.1) have an extended gauge sector withan additional U (1) (cid:48) symmetry with charge involving some of the lepton flavors and an heavyenough gauge boson. Conversely if the gauge boson is light enough a long range force willbe generated. Thus the analysis of neutrino oscillation data can shed light on the validrange of Z (cid:48) mass and coupling in both regimes. Following this approach, the marginalizedbounds on the NSI coefficients derived from the global analysis of oscillations in presenceof NSI performed in Ref. [19] were adapted to place constraints on the coupling and massof the new gauge boson both in the NSI limit [20] and in the long-range regime [16] forseveral U (1) (cid:48) flavor symmetries. However, strictly speaking, the bounds derived in Ref. [19]cannot be directly used to constraint the U (1) (cid:48) scenarios because in the latter case only flavordiagonal interactions (and only some of them depending on the U (1) (cid:48) charge) are generated,while the bounds derived in Ref. [19] were obtained in the most general parameter spacewith all relevant four-fermion interactions (flavor conserving and flavor changing) beingsimultaneously non-vanishing. In order to derive statistically consistent bounds on each U (1) (cid:48) scenario a dedicated analysis has to be performed in its reduced parameter space.With this motivation, in this work we perform such dedicated global analysis of os-cillation data in the framework of lepton flavor dependent U (1) (cid:48) interactions which affect– 2 –he neutrino evolution in matter, with interaction lengths ranging from the Sun-Earth dis-tance to effective contact neutrino interactions. In Sec. 2 we describe the models whichwill be studied and derive the matter potential generated both in the contact interactionlimit (in Sec. 2.1) and in the case of finite interaction range (in Sec. 2.2) as a function ofthe U (1) (cid:48) charges. The results of the global analysis are presented in Sec. 3. In particularthe bounds imposed by the analysis and how they compare with those from other experi-ments are presented in Sec. 3.1. An additional consideration that we take into account isthat in the presence of NSI a degeneracy exists in oscillation data, leading to the so calledLMA-Dark (LMA-D) [21] solution first observed in solar neutrinos, where for suitable NSIthe data can be explained by a mixing angle θ in the second octant. For this new so-lution to appear the new interactions must be such that the matter potential differencefor electron neutrinos reverses its sign with respect to that in the SM. It is not trivial togenerate such large effects without conflicting with bounds from other experiments, thoughmodels with light mediators ( i.e. , below the electroweak scale) have been proposed as viablecandidates [9, 10, 22–25]. Section 3.2 contains our findings on viable models for LMA-D.We summarize our conclusions in Sec. 4. We present some details of the translation of thebounds from some experiments to the models studied in an appendix. We are going to focus on U (1) (cid:48) models which can be tested in neutrino oscillation exper-iments via its contribution to the matter potential. As a start this requires that the newgauge boson couples to the fermions of the first generation.An important issue when enlarging the Standard Model with a new U (1) (cid:48) gauge groupis the possibility of mixing between the three neutral gauge bosons of the model which can,in general, be induced in either kinetic or mass terms. While kinetic mixing is fairly genericas it can be generated at the loop level with the SM particle contents, matter mixing ismodel dependent as it requires an extended scalar sector with a vacuum expectation valuecharged both under the SM and the U (1) (cid:48) . In what respects the effect of the new U (1) (cid:48) in oscillation experiments an important observation is that if the new interaction does notcouple directly to fermions of the first generation, no matter effects can be generated bykinetic mixing [20]. Thus neglecting mixing effects yields the most model independent andconservative bounds from neutrino oscillation results. So in what follows we are goingto work under the assumption that the Z (cid:48) mixing with SM gauge bosons can be safelyneglected.In addition we notice that only vector interactions contribute to the matter potential inneutrino propagation. Altogether the part of the U (1) (cid:48) Lagrangian relevant for propagationin ordinary matter has the most general form L matter Z (cid:48) = − g (cid:48) (cid:0) a u ¯ uγ α u + a d ¯ dγ α d + a e ¯ eγ α e + b e ¯ ν e γ α P L ν e + b µ ¯ ν µ γ α P L ν µ + b τ ¯ ν τ γ α P L ν τ (cid:1) Z (cid:48) α (2.1)with arbitrary charges a u,d,e and b e,µ,τ . – 3 –odel a u a d a e b e b µ b τ B − L e
13 13 − − B − L µ
13 13 − B − L τ
13 13 − B − ( L µ + L τ )
13 13 − − L e − L µ − L e − L τ − L e − ( L µ + L τ ) 0 0 1 1 − − Ref. [22]
13 13 L e + 2 L µ + 2 L τ Table 1 . Relevant charges for the matter effects in neutrino oscillation experiments correspondingto a selection of models studied in the literature.
These charges can be accommodated in generalized anomaly free UV-complete modelsincluding only the SM particles plus right-handed neutrinos [26]. If, in addition, one requiresall couplings to be vector-like and the quark couplings to be generation independent, thecondition of anomaly cancellation for models with only SM plus right handed neutrinosimposes constrains over the six charges above and one ends with a subclass of modelscharacterized by three independent charges which can be chosen to be, for example, B − L , ( L µ − L τ ) , and ( L µ − L e ) [20]: c bl ( B − L ) + c µτ ( L µ − L τ ) + c µe ( L µ − L e ) , (2.2)so a u = a d = c bl / , a e = b e = − ( c bl + c µe ) , b µ = − c bl + c µe + c µτ , and b τ = − ( c bl + c µτ ) . Forconvenience, in table 1 we list the charges corresponding to some of the models discussedin the literature.In general, the evolution of the neutrino and antineutrino flavor state during propaga-tion is governed by the Hamiltonian: H ν = H vac + H mat and H ¯ ν = ( H vac − H mat ) ∗ , (2.3)where H vac is the vacuum part which in the flavor basis ( ν e , ν µ , ν τ ) reads H vac = U vac D vac U † vac with D vac = 12 E ν diag(0 , ∆ m , ∆ m ) . (2.4)Here U vac denotes the three-lepton mixing matrix in vacuum [1, 27, 28]. Following theconvention of Ref. [29], we define U vac = R ( θ ) R ( θ ) ˜ R ( θ , δ CP ) , where R ij ( θ ij ) is arotation of angle θ ij in the ij plane and ˜ R ( θ , δ CP ) is a complex rotation by angle θ and phase δ CP .Concerning the matter part H mat of the Hamiltonian generated by the SM together withthe U (1) (cid:48) interactions in (2.1), its form depends on the new interaction length determinedby the Z (cid:48) mass as we discuss next. – 4 – .1 The large ( M Z (cid:48) (cid:38) − eV) M Z (cid:48) limit: the NSI regime In the limit of large M Z (cid:48) , the Z (cid:48) field can be integrated out from the spectrum and Eq. (2.1)generate effective dimension-six four-fermion interactions leading to Neutral Current NSIbetween neutrinos and matter which are usually parametrized in the form of Eq. (1.1). Thecoefficients ε f,Pαβ parametrizes the strength of the new interaction with respect to the Fermiconstant, with ε f,Lαβ = ε f,Rαβ = δ αβ √ G F g (cid:48) M Z (cid:48) a f b α ≡ δ αβ a f b α ε (2.5)where we have introduced the notation ε ≡ √ G F g (cid:48) M Z (cid:48) (2.6)As it is well known, only vector NSI contribute to the matter potential in neutrino oscil-lations. It is therefore convenient to define the parameters relevant for neutrino oscillationexperiments as: ε fαβ ≡ ε f,Lαβ + ε f,Rαβ = δ αβ a f b α ε . (2.7)These interactions lead to a flavour diagonal modification of the matter potential H mat = √ G F N e ( (cid:126)x ) E ee ( (cid:126)x ) 0 00 E µµ ( (cid:126)x ) 00 0 E ττ ( (cid:126)x ) (2.8)where the “ +1 ” term in the ee entry accounts for the standard contribution, and E αα ( (cid:126)x ) = (cid:88) f = e,u,d N f ( (cid:126)x ) N e ( (cid:126)x ) ε fαα (2.9)describes the non-standard part. Here N f ( (cid:126)x ) is the number density of fermion f at theposition (cid:126)x along the neutrino trajectory. In Eq. (2.9) we have limited the sum to thecharged fermions present in ordinary matter, f = e, u, d . Taking into account that N u ( (cid:126)x ) =2 N p ( (cid:126)x ) + N n ( (cid:126)x ) and N d ( (cid:126)x ) = N p ( (cid:126)x ) + 2 N n ( (cid:126)x ) , and also that matter neutrality implies N p ( (cid:126)x ) = N e ( (cid:126)x ) , Eq. (2.9) becomes: E αα ( (cid:126)x ) = (cid:0) ε eαα + ε pαα (cid:1) + Y n ( (cid:126)x ) ε nαα with Y n ( (cid:126)x ) ≡ N n ( (cid:126)x ) N e ( (cid:126)x ) (2.10)where ε pαα ≡ ε uαα + ε dαα = a p b α ε , ε nαα ≡ ε dαα + ε uαα = a n b α ε . (2.11)and we have introduced the proton and neutron Z (cid:48) couplings a p ≡ a u + a d , a n ≡ a d + a u . (2.12)As discussed in Ref. [17], in the Earth the neutron/proton ratio Y n ( (cid:126)x ) which characterizethe matter chemical composition can be taken to be constant to very good approximation.The PREM model [30] fixes Y n = 1 . in the Mantle and Y n = 1 . in the Core, with an– 5 –verage value Y ⊕ n = 1 . all over the Earth. Setting therefore Y n ( (cid:126)x ) ≡ Y ⊕ n in Eqs. (2.9)and (2.10) we get E αα ( (cid:126)x ) ≡ ε ⊕ αα with: ε ⊕ αα = ε eαα + (cid:0) Y ⊕ n (cid:1) ε uαα + (cid:0) Y ⊕ n (cid:1) ε dαα = (cid:0) ε eαα + ε pαα (cid:1) + Y ⊕ n ε nαα = (cid:2) ( a e + a p ) + Y ⊕ n a n (cid:3) b α ε . (2.13)For what concerns the study of propagation of solar and KamLAND neutrinos one canwork in the one mass dominance approximation, ∆ m → ∞ (which effectively means that G F N e ( (cid:126)x ) E αα ( (cid:126)x ) (cid:28) ∆ m /E ν ). In this approximation the survival probability P ee can bewritten as [31, 32] P ee = c P eff + s (2.14)The probability P eff can be calculated in an effective × model described by the Hamil-tonian H eff = H effvac + H effmat,SM + H effmat, Z (cid:48) , with: H effvac = ∆ m E ν (cid:32) − cos 2 θ sin 2 θ e iδ CP sin 2 θ e − iδ CP cos 2 θ (cid:33) , (2.15) H effmat,SM = √ G F N e ( (cid:126)x ) (cid:32) c
00 0 (cid:33) (2.16)and H effmat,Z’ = √ G F N e ( (cid:126)x ) ε (cid:2) a e + a p + Y n ( (cid:126)x ) a n (cid:3) (cid:32) − b D b N b N b D (cid:33) (2.17)where b D = − c b e − b µ ) + s − s c b τ − b µ ) , (2.18) b N = s c s ( b τ − b µ ) . (2.19)Following Ref. [19] we can rewrite the Z (cid:48) contribution as: H effmat,Z’ = √ G F N e ( (cid:126)x ) (cid:2) cos η + Y n ( (cid:126)x ) sin η (cid:3) (cid:32) − ε ηD ε ηN ε ηN ε ηD (cid:33) , (2.20)where the angle η parametrizes the ratio of the charges of the matter particles as: cos η = a e + a p (cid:112) ( a e + a p ) + a n , sin η = a n (cid:112) ( a e + a p ) + a n , (2.21)and ε ηD,N = (cid:113) ( a e + a p ) + a n b D,N ε . (2.22)The neutrino oscillation phenomenology in this regime reduces to a special subclassof the general NSI interactions analyzed in Ref. [19]. In particular, as a consequence of To be precise, the data analysis performed in Ref. [19] was restricted to NSI with quarks, ie a e = 0 .The formalism for matter effects can be trivially extended to NSI coupled to electrons as shown above.However, NSI coupled to electrons would affect not only neutrino propagation in matter as described, butalso the neutrino-electron (ES) scattering cross-section in experiments such as SK, SNO and Borexino. Inorder to keep the analysis manageable, in Ref. [19], and in what follows, the NSI corrections to the ESscattering cross section in SK, SNO, and Borexino are neglected. In the absence of cancellations betweenpropagation and interaction effects this renders the results of the oscillation analysis conservative. – 6 –he CPT symmetry (see also Refs. [17, 18, 29] for a discussion in the context of NSI) theneutrino evolution is invariant if the relevant Hamiltonian is transformed as H → − H ∗ . Invacuum this transformation can be realized by changing the oscillation parameters as ∆ m → − ∆ m + ∆ m = − ∆ m ,θ → π − θ ,δ CP → π − δ CP , (2.23)where δ CP is the leptonic Dirac CP phase, and we are using here the parameterizationconventions from Refs. [19, 29]. The symmetry is broken by the standard matter effect,which allows a determination of the octant of θ and (in principle) of the sign of ∆ m .However, in the presence of the Z (cid:48) -induced NSI, the symmetry can be restored if in additionto the transformation Eq. (2.23), the E αα ( (cid:126)x ) terms can be transformed as [18, 29, 33] (cid:2) E ee ( (cid:126)x ) − E µµ ( (cid:126)x ) (cid:3) → − (cid:2) E ee ( (cid:126)x ) − E µµ ( (cid:126)x ) (cid:3) − , (cid:2) E ττ ( (cid:126)x ) − E µµ ( (cid:126)x ) (cid:3) → − (cid:2) E ττ ( (cid:126)x ) − E µµ ( (cid:126)x ) (cid:3) . (2.24)Eq. (2.23) shows that this degeneracy implies a change in the octant of θ (as manifest in theLMA-D fit to solar neutrino data [21]) as well as a change in the neutrino mass ordering, i.e. ,the sign of ∆ m . For that reason it has been called “generalized mass ordering degeneracy”in Ref. [29]. Because of the position dependence of the NSI hamiltonian described by E αα ( (cid:126)x ) this degeneracy is only approximate, mostly due to the non-trivial neutron/proton ratioalong the neutrino path inside the Sun. In what follows when marginalizing over θ weconsider two distinct parts of the parameter space: one with θ < ◦ , which we denote asLIGHT, and one with θ > ◦ , which we denote by DARK.Apart from the appearance of this degenerate solution, another feature to considerin the global analysis of oscillation data in presence of NSI is the possibility to furtherimprove the quality of the fit with respect to that of standard ν oscillations in the LIGHTsector. Till recently this was indeed the case because for the last decade the value of ∆ m preferred by KamLAND was somewhat higher than the one from solar experiments. Thistension appeared due to a combination of two effects: the fact that the B measurementsperformed by SNO, SK and Borexino showed no evidence of the low energy spectrum turn-up expected in the standard LMA-MSW [4, 5] solution for the value of ∆ m favored byKamLAND, and the observation of a non-vanishing day-night asymmetry in SK, whose sizewas larger than the one predicted for the ∆ m value indicated by KamLAND. With thedata included in the analysis in Ref. [19, 34] this resulted into a tension of ∆ χ ∼ . forthe standard ν oscillations. Such tension could be alleviated in presence of a non-standardmatter potential, thus leading to a possible decrease in the minimum χ . However, withthe latest 2970-days SK4 results presented at the Neutrino2020 conference [35] in the formof total energy spectrum and updated day-night asymmetry, the tension between the bestfit ∆ m of KamLAND and that of the solar results has decreased. Currently they arecompatible within . σ in the latest global analysis [36].– 7 – .2 The finite M Z (cid:48) case: the long-range interaction regime If M Z (cid:48) is very light the four-fermion contact interaction approximation in Eq. (1.1) does nothold and the potential encountered by the neutrino in its trajectory depends on the integralof the source density within a radius ∼ /M Z (cid:48) around it. However, following Ref. [12] thegeneralized matter potential can still be written as Eq. (2.8) provided that Eq. (2.9) ismodified as: E αα ( (cid:126)x ) = (cid:88) f = e,u,d ˆ N f ( (cid:126)x, M Z (cid:48) ) N e ( (cid:126)x ) ε fαα (2.25)where ˆ N f ( (cid:126)x, M Z (cid:48) ) ≡ πM Z (cid:48) (cid:90) N f ( (cid:126)ρ ) e − M Z (cid:48) | (cid:126)ρ − (cid:126)x | | (cid:126)ρ − (cid:126)x | d (cid:126)ρ . (2.26)Taking into account that ordinary matter is neutral and only contains f = e, u, d , we canrewrite Eq. (2.25) in a way that generalizes Eq. (2.10): E αα ( (cid:126)x ) = F e ( (cid:126)x, M Z (cid:48) ) (cid:0) ε eαα + ε pαα (cid:1) + F n ( (cid:126)x, M Z (cid:48) ) Y n ( (cid:126)x ) ε nαα with F i ( (cid:126)x, M Z (cid:48) ) ≡ ˆ N i ( (cid:126)x, M Z (cid:48) ) N i ( (cid:126)x ) and i ∈ { e, n } . (2.27)For what concerns neutrinos traveling inside the Sun, the propagation effects induced by thenew interactions are completely dominated by the solar matter distribution. Denoting by (cid:126)x (cid:12) the center of the Sun and accounting for the spherical symmetry of the matter potentialwe can write: F i ( (cid:126)x, M Z (cid:48) ) (cid:39) F (cid:12) i ( | (cid:126)x − (cid:126)x (cid:12) | , M Z (cid:48) ) with F (cid:12) i ( r, M Z (cid:48) ) = 1 N (cid:12) i ( r ) · M Z (cid:48) r (cid:90) R (cid:12) ρ N (cid:12) i ( ρ ) (cid:2) e − M Z (cid:48) | ρ − r | − e − M Z (cid:48) ( ρ + r ) (cid:3) dρ . (2.28)A similar formula can be derived for neutrinos traveling inside the Earth, but in this casethe effective potential has an extra term induced by the Sun matter density. Concretely,denoting by (cid:126)x ⊕ the center of the Earth and by X (cid:9) = | (cid:126)x (cid:12) − (cid:126)x ⊕ | the Sun-Earth distance, wehave: F i ( (cid:126)x, M Z (cid:48) ) (cid:39) F ⊕ i ( | (cid:126)x − (cid:126)x ⊕ | , M Z (cid:48) ) with F ⊕ i ( r, M Z (cid:48) ) = 1 N ⊕ i ( r ) M Z (cid:48) (cid:40) r (cid:90) R ⊕ ρ N ⊕ i ( ρ ) (cid:2) e − M Z (cid:48) | ρ − r | − e − M Z (cid:48) ( ρ + r ) (cid:3) dρ + e − M Z (cid:48) X (cid:9) X (cid:9) (cid:90) R (cid:12) ρ N (cid:12) i ( ρ ) sinh( ρ M Z (cid:48) ) dρ (cid:41) . (2.29)The solar-induced contribution becomes non-negligible when the range of the interactions, (cid:14) M Z (cid:48) , is comparable or larger than the Sun-Earth distance X (cid:9) .The factors F (cid:12) i ( r, M Z (cid:48) ) and F ⊕ i ( r, M Z (cid:48) ) represent the modification of the matter po-tential due to the finite range of the interaction mediated by the Z (cid:48) with respect to thatobtained in the contact interaction limit. Such limit is recovered when the range of the new– 8 –nteractions become shorter than the typical size of the matter distribution, i.e. , R (cid:12) ( ⊕ ) .Hence: F (cid:12) ( ⊕ ) i ( r, M Z (cid:48) ) → for M Z (cid:48) (cid:29) /R (cid:12) ( ⊕ ) . (2.30)For solar neutrinos further simplification follows if one takes into account that for adiabatictransitions the dominant matter effects is generated by the potential at the neutrino pro-duction point which is is close to the Sun center. So to a very good approximation one canscale the contact interaction potential with a position independent factor F (cid:12) i (0 , M Z (cid:48) ) . Forthe Earth matter potential the position dependence of the factor F ⊕ i ( r, M Z (cid:48) ) is very weakin the current experiments, so one can also scale the contact interaction potential with anapproximate F ⊕ i (¯ r, M Z (cid:48) ) evaluated at a fix ¯ r which we take to be also ¯ r = 0 .In Fig. 1 we plot these scale factors F (cid:12) ( ⊕ ) e (0 , M Z (cid:48) ) . As seen in the figure for M Z (cid:48) (cid:46) − eV the matter potential in the Earth is more suppressed with respect to that in theSun. In principle, this opens the possibility of configurations for which the U (1) (cid:48) -inducedmatter potential in the Sun is large enough without conflicting with bounds imposed byatmospheric and long-baseline experiments. This also implies that in the combined analysisof solar and KamLAND data, for a given value of M Z (cid:48) , the effective matter potential for solarneutrinos will be suppressed by a different factor than that for KamLAND antineutrinos. Toillustrate the overall M Z (cid:48) dependence of the effect we show in Fig. 1 the effective suppressionfactor in the combined solar+KamLAND analysis calculated by scaling the results obtainedfor a specific model (concretely, for a Z (cid:48) coupled to L e − L µ , but the results are similar formodels with other couplings) for each M Z (cid:48) to those obtained in the large M Z (cid:48) regime. Asseen in the figure for M Z (cid:48) (cid:38) − eV both the effective potential in the Solar+KamLANDanalysis and the Earth matter potential relevant for atmospheric and LBL neutrinos arewell within the contact interaction regime. Conversely for M Z (cid:48) (cid:46) − eV all the matterpotentials in the analysis show deviations from the contact interaction regime.Concerning the phenomenology of neutrino oscillations in the presence of the modifiedmatter potential in this long-range interaction regime, the main difference with the NSIcontact interaction case is the impossibility of realizing the “generalized mass orderingdegeneracy” in Eqs. (2.23) and (2.24) because of the very different (cid:126)x dependence of the SMmatter potential and the one generated by the Z (cid:48) . In other words, one cannot “flip thesign of the matter hamiltonian” by adding to the standard N e ( x ) something which has acompletely different (cid:126)x profile. Hence, there is no LMA-D solution for these models. Onthe other hand, it is still possible, at least in principle, that a long-range potential leadsto an improvement on the fit to solar and KamLAND data with respect to the pure LMAsolution. We have performed a global fit to neutrino oscillation data in the framework of ν massiveneutrinos with new neutrino-matter interactions generated by U (1) (cid:48) models and character-ized by the Lagrangian in Eq. (2.1). For the detailed description of methodology and dataincluded we refer to the comprehensive global fit in Ref. [19] performed in the framework of– 9 – -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 M Z’ [eV] -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 F e ( , M Z ’ ) S un E a r t h S un - K a m e ff Figure 1 . Effective suppression factor of the matter potential due to the long range of the U (1) (cid:48) interactions as a function of M Z (cid:48) . The red (blue) curve corresponds to the potential in the Sun(Earth). The purple line corresponds to the effective combined suppression factor in the analysisof Solar+KamLAND data (see text for details). three-flavor oscillations plus NSI. In addition in the present analysis we account for the lat-est LBL data samples included in NuFIT-5.0 [36] which includes the previously cited solarneutrinos 2970-days SK4 results [35], the updated medium baseline reactor samples fromRENO [37] and Double Chooz [38], and the latest long-baseline samples from T2K [39] andNO ν A [40]. Notice that in order to keep the fit manageable we proceed as in in Ref. [19]and restrict ourselves to the CP-conserving case and set δ CP ∈ { , π } . Consequently theT2K and NO ν A appearance data (which exhibit substantial dependence on the leptonicCP phase) are not included in the fit. With these data we construct a global χ function: χ OSC+Z’ ( g (cid:48) , M Z (cid:48) | (cid:126)ω ) (3.1)for each Z (cid:48) model. In general each model belongs to a family characterized by a set of U (1) (cid:48) charges; for each family χ OSC+Z’ depends on the two variables parametrizing the new inter-action, g (cid:48) and M Z (cid:48) , plus the six oscillation parameters (cid:126)ω ≡ (∆ m , ∆ m , θ , θ , θ , δ CP ) .Following the discussion in Sec. 2, we have performed the analysis in two physicallydistinctive domains of M Z (cid:48) : • NSI Domain (DOM=NSI): In this case, the range of the induced non-standard inter-actions in both the Sun and the Earth matter is short enough for the four-fermioneffective description to hold. As seen in Fig. 1 this happens for M Z (cid:48) (cid:38) − eV.In this regime a possible conflict with the cosmological bound on ∆ N eff may appearbecause of the contribution of either the Z (cid:48) itself (if lighter than all active neutrinos)or by the extra contribution to the neutrino density produced by the Z (cid:48) decay (ifheavier than some ν mass eigenstate). These bounds can be evaded in two distinctranges of the U (1) (cid:48) interactions: – 10 – M Z (cid:48) (cid:38) MeV for which the contribution to the neutrino energy density due tothe decay of the Z (cid:48) into neutrinos is sufficiently suppressed by the Boltzmannfactor ∼ exp( − M Z (cid:48) /T ) [41]; – M Z (cid:48) (cid:46) O ( eV ) but with very weak coupling g (cid:48) < − for which the Z (cid:48) isproduced through freeze-in. In this regime, even if Z (cid:48) could decay to the lightestneutrinos this would happen after neutrino decoupling making the contributionto ∆ N eff negligible or at most within the present allowed range [42]. • Long-Range Domain (DOM=LRI): If M Z (cid:48) (cid:46) − eV interactions in the Earth andSun matter are long range, and for a given value of g (cid:48) and M Z (cid:48) the effects in the Earthare suppressed with respect to those in the Sun. As mentioned above, for g (cid:48) < − the contribution to ∆ N eff is negligible.Correspondingly we define χ OSC+Z’,DOM ( g (cid:48) , M Z (cid:48) | (cid:126)ω ) ≡ χ OSC+Z’ ( g (cid:48) , M Z (cid:48) ∈ DOM | (cid:126)ω ) (3.2)In both domains we compare the results of the fit including the new U (1) (cid:48) interaction withthose obtained in the “standard” ν -mixing scenario, which we will denote as “OSC”, andfor which the present global fit yields χ OSC,min = 718 . . (3.3)We will classify the models according to the quality of the fit in the presence of the U (1) (cid:48) interactions compared with that of OSC by defining ∆ χ LIGHT,DOM ( g (cid:48) , M Z (cid:48) ) ≡ χ OSC+Z’,DOM ( g (cid:48) , M Z (cid:48) | (cid:126)ω ) (cid:12)(cid:12) marg,LIGHT − χ OSC,min , (3.4)where by | marg,LIGHT we imply that the minimization over the oscillation parameters isdone in the LIGHT sector of parameter space. In addition the presence of a viable LMA-Dsolution can be quantified in terms of ∆ χ DARK,DOM ( g (cid:48) , M Z (cid:48) ) ≡ χ OSC+Z’,DOM ( g (cid:48) , M Z (cid:48) | (cid:126)ω ) (cid:12)(cid:12) marg, DARK − χ OSC,min , (3.5)where by | marg,DARK we imply that the minimization over the oscillation parameters is donein the DARK sector.We have surveyed the model space by performing the global oscillation analysis for agrid of U (1) (cid:48) interactions characterized by the six couplings a u,d ∈ {− , , } and a e , b e,µ,τ ∈{− , − , − , , , , } . In this way our survey covers a total of ∼ different setsof U (1) (cid:48) charges which can produce effects in matter propagation in neutrino oscillationexperiments. We first search for models for which in the LIGHT sector the new interactions lead to asignificantly better fit of the oscillation data compared to pure oscillations for some valueof g (cid:48) and M Z (cid:48) . We find that in the NSI (LRI) domain 88% (90%) of the surveyed sets of– 11 –odel (∆ χ LIGHT,LRI ) min g (cid:48) ≤ bound B − L e − . . × − B − L µ − . . × − B − L τ − . . × − B − ( L µ + L τ ) − . . × − L e − L µ − . . × − L e − L τ − . . × − L e − ( L µ + L τ ) − . . × − Ref. [22] . × − L e + 2 L µ + 2 L τ . × − Table 2 . Results for the models with charges in table 1 in the LRI regime. The second columngives minimum ∆ χ defined w.r.t. the ν oscillation (see Eq. (3.4)). The last column gives the theupper bound for the coupling of asymptotically for ultra light mediators, M Z (cid:48) (cid:46) − eV charges lead to a decrease in the χ of the global analysis when compared to the standardoscillation case. The percentages grow to 100% when restricting to the subclass of anomalyfree vector Z (cid:48) models with gauging of SM global symmetries with SM plus right-handedneutrinos matter content of Eq. (2.2). However the improvement in the quality of the fit isnever statistically significant. As an illustration we show in the second column of tables 2and 3 the minimum values of ∆ χ LIGHT,DOM for U (1) (cid:48) interactions characterized by thespecific set of charges in table 1. Comparing the two tables we notice that for some of thecases the fit can be slightly better in the LRI domain than in the NSI domain but stillbelow the σ level.Quantitatively we find that in the NSI regime none of the surveyed models yields animprovement beyond -1.9 units of χ . This is the case, for example, of a Z (cid:48) coupled to B − L e + 2 L µ + 3 L τ . Generically models in the LRI domain can provide a better fit witha reduction of up to 3.5 units of χ . For example a model with charge L e + 2 L µ − L τ and a Z (cid:48) with M Z (cid:48) ∼ × − eV provides a better fit than standard oscillations by (∆ χ LIGHT,LRI ) min = − . .But in summary, our analysis shows that none of the set of charges surveyed inboth NSI or LRI domains improved over standard oscillations at the σ level, this is min g (cid:48) ,M Z (cid:48) (∆ χ LIGHT,DOM ) was always larger than − . . Consequently for all models sur-veyed one can conclude that the analysis of neutrino oscillation experiments show no sig-nificant evidence of U (1) (cid:48) interactions. Consequently one can exclude models at a certainconfidence level – which we have chosen to be 95.45% – by verifying that their global fit isworse than in OSC by the corresponding units of χ (4 units for 95.45% CL), this is: ∆ χ LIGHT,DOM ( g (cid:48) , M Z (cid:48) ) > . (3.6) We notice that before the new results from Super-Kamiokande [35] there were models which couldimprove the mismatch between the best fit ∆ m in Solar and KamLAND. For such models, one could findvalues of g (cid:48) and M Z (cid:48) for which the fit was better than standard oscillations by more than 4 units of χ . – 12 –odel (∆ χ LIGHT,NSI ) min g (cid:48) ≤ bound (cid:18) M Z (cid:48)
100 MeV (cid:19) B − L e − . . × − B − L µ . × − B − L τ − . . × − B − ( L µ + L τ ) − . . × − L e − L µ − . . × − L e − L τ − . . × − L e − ( L µ + L τ ) − . . × − Ref. [22] . × − L e + 2 L µ + 2 L τ − . . × − Table 3 . Results for the models with charges in table 1 in the NSI regime. The second columngives minimum ∆ χ defined w.r.t. the ν oscillation (see Eq. (3.4)). The last column gives thecoefficient of the bound on the coupling over the mediator mass in units of 100 MeV. Let us stress that with the above condition we are not “deriving two-dimensional excludedregions in the parameter space”, but we are instead determining the values of g (cid:48) and M Z (cid:48) forwhich the U (1) (cid:48) model characterized by such interaction strength and interaction length,gives a fit which is worse than standard oscillations by at least 4 units of χ . As inthe LIGHT sector the standard model is recovered for either g (cid:48) → or M Z (cid:48) → ∞ , theabove condition yields also the σ excluded one-dimensional upper range of interactioncoupling g (cid:48) for each value of the interaction length (or, correspondingly, the σ excludedone-dimensional lower range of M Z (cid:48) for each value of the interaction coupling), for all modelscharacterized by a given set of charges.The corresponding excluded ranges for the Z (cid:48) coupling and mass for the models withcouplings listed in table 1 are shown in Figs. 2 and 3 for M Z (cid:48) below 1 eV and in the O ( MeV–GeV ) range, respectively – i.e. , below and above the window strongly disfavored bythe cosmological bound on ∆ N eff . In particular in Fig. 2 we observe the slope change of theoscillation exclusion ranges for masses M Z (cid:48) ∼ − – − GeV, for which the interactionlength is longer than the Earth and Sun radius and the matter potential in the Earth andin the Sun becomes saturated (see Fig. 1). Quantitatively, for the models with charges intable 1 we find that in the LRI domain the analysis of oscillation data yields the upperbound for the coupling of asymptotically ultra light mediators, M Z (cid:48) (cid:46) − eV which welist in table 2. For the sake of comparison, we also show in Fig. 2 the bounds on thesemodels imposed by gravitational fifth force searches, and by equivalence principle tests.Those are the strongest constraints in the shown range of Z (cid:48) mass and coupling. Thebounds imposed by gravitational fifth force searches were obtained by rescaling the resultsshown in Ref. [43] (which, in turn, were recasted from Ref. [44]). Limits from equivalenceprinciple tests are obtained rescaling the results from Ref. [45]. Details of the rescaling ofthe published bounds applied for the specific models can be found in the Appendix. It is Tables with the numerical values of the bounds can be provided upon request to the authors. – 13 – − − − − − − − g (cid:48) B − L e − − − − − − − B − L µ − − − − − − − B − L τ t h f o rc e E q . p r i n c . − − − − − − − g (cid:48) B − ( L µ + L τ ) − − − − − − − L e − L µ − − − − − − − L e − L τ O S C ( T h i s w o r k ) − − − − − − M Z (cid:48) ( e V ) − − − − − − − g (cid:48) L e − ( L µ + L τ ) − − − − − − M Z (cid:48) ( e V ) − − − − − − − a r X i v : . − − − − − − M Z (cid:48) ( e V ) − − − − − − − L e + L µ + L τ Figure 2 . Values of g (cid:48) and M Z (cid:48) ≤ O (eV) for which a U (1) (cid:48) model coupled to the charges labeledin each panel gives a worse fit than standard oscillation by σ , Eq. (3.6) (hatched region). Alsoshow in the figure the bounds on these models imposed by gravitational fifth force searches [43, 44]and by equivalence principle tests [45]. See text for details. – 14 –mportant to notice that these exclusion regions obtained by recasting the boundaries of thepublished regions may not correspond to the statistical condition we employed, Eq. (3.6).So the comparison has to be taken with a pinch of salt. Still, from the figures we see that,generically, for all models shown the oscillation analysis improves over the existing boundsfor Z (cid:48) lighter than ∼ − eV or ∼ − eV depending on whether the U (1) (cid:48) currentinvolves coupling to electron lepton number.Conversely the results shown in Fig. 3 for M Z (cid:48) in the O ( MeV–GeV ) range correspond to U (1) (cid:48) effects in oscillation experiments always in the NSI domain. In this case, the analysisof oscillation data results in a bound on g (cid:48) which scales as the inverse of the mediator masswith coefficients which we list in table 3. For the sake of comparison, we also show in Fig. 3a compilation of the most relevant experimental bounds on these U (1) (cid:48) models. Theseinclude constraints from electron and proton fixed target experiments, neutrino electronelastic scattering, coherent neutrino nucleus elastic scattering, white dwarf cooling andcollider constraints. Appendix A contains all relevant details on the derivation of thesebounds.As seen in Fig. 3 for most models there are values of g (cid:48) and M Z (cid:48) which are onlyconstrained by the oscillation analysis. This is particularly the case for U (1) (cid:48) couplings to B − L µ , and the model in Ref. [22] for which the only other bound applying in the shownrange of g (cid:48) and M Z (cid:48) is the one from BaBar. Finally for a Z (cid:48) coupled to B − L τ we havefound no competitive bound from other experiments in the shown window (for bounds fromother experiments relevant for larger couplings see for example Refs. [20, 46]). A subset of models can lead to a best fit in the DARK sector or at least with LMA-D within4 units of χ with respect to the standard OSC solution, ∆ χ DARK,DOM ( g (cid:48) , M Z (cid:48) ) < . (3.7)As discussed in Sec. 2.2 this can only happen in the NSI domain. In our survey we havefound that 4.8% of the set of charges studied can have a best fit in LMA-D, and 5.2%lead to LMA-D verifying Eq. (3.7). However none of these set of charges correspond to thesubclass of anomaly free vector Z (cid:48) models with gauging of SM global symmetries with SMplus right-handed neutrinos matter content (Eq. (2.2)).In particular for the first seven set of models in table 1 the LMA-D solutions lies atmore than σ from the standard oscillation fit. On the contrary the models in the last twolines yield a viable LMA-D solution The first one was proposed in Ref. [22] precisely asa viable model for LMA-D. Indeed in this case we find (∆ χ DARK,NSI ) min = 1 . , which iswithin 4 units of χ from the pure oscillation result but the best fit for this model chargesstill lies within the LIGHT sector. We also show the results for a U (1) (cid:48) model coupled to L e + 2 L µ + 2 L τ for which we find that the best fit is LMA-D with (∆ χ DARK,NSI ) min = − . .In Fig. 4 we plot as black bands the range of coupling and masses for these two setof charges verifying the condition (3.7), together with the compilation of relevant boundsfrom other experiments. – 15 – − − − − g (cid:48) B − L e C H AR M L S N D K E K - PF - W h i t e Dw a r v e s E ν CA L E E O r s a y B a B a r ( v i s . ) C O H ERE N T T E X O N O G E MM A − − − − B − L µ L H C b ( p r o m p t ) − − − − B − L τ O S C ( T h i s w o r k ) − − − − g (cid:48) B − ( L µ + L τ ) − − − − L e − L µ − − − − L e − L τ − − M Z (cid:48) ( G e V ) − − − − g (cid:48) L e − ( L µ + L τ ) − − M Z (cid:48) ( G e V ) − − − − a r X i v : . − − M Z (cid:48) ( G e V ) − − − − L e + L µ + L τ Figure 3 . Values of g (cid:48) and 5 MeV ≤ M Z (cid:48) ≤
10 GeV for which a U (1) (cid:48) model coupled to the chargeslabeled in each panel gives a worse fit than standard oscillation by σ , Eq. (3.6) (hatched region).Also show in the figure the bounds on these models imposed by a compilation of experiments aslabeled in the figure (see text for details). – 16 – − − − − − − − M Z (cid:48) (eV) − − − − − − g (cid:48) arXiv:1505.06906 viable LMA-D(This work) − − − − − − − M Z (cid:48) (eV) − − − − − − L e + 2 L µ + 2 L τ − − M Z (cid:48) (GeV) − − − − g (cid:48) arXiv:1505.06906 LHCb (prompt) − − M Z (cid:48) (GeV) − − − − L e + 2 L µ + 2 L τ White DwarvesE137E141E774KEK-PF-000OrsayBaBar (vis.)TEXONOGEMMA
Figure 4 . Range of g (cid:48) and M Z (cid:48) for which the global analysis of oscillation data can be consistentlydescribed within the LMA-D solution, Eq. (3.7), (black band) for two viable models. Also show inthe figure the bounds on these models imposed by a compilation of experiments as labeled in thefigure. See text for details. – 17 –s seen in the figure, for the model in Ref. [22], there are solutions for the Z (cid:48) couplingand mass for which LMA-D is allowed without conflict with bounds from other experimentsboth for very light mediators as well as for an O ( MeV–GeV ) Z (cid:48) . More quantitatively forultra light mediator part of the LMA-D allowed parameter space for the model in Ref. [22]is in conflict with the bounds from fifth force tests which impose the stronger constraintsfor this model in this regime. But the LMA-D is still a viable solution for − eV ≤ M Z (cid:48) ≤ . × − eV (3.8)with couplings in a very narrow band and seen in the figure, for example g (cid:48) = (9 . ± . × − for M Z (cid:48) = 10 − eV (3.9)and g (cid:48) = (1 . ± . × − for M Z (cid:48) = 2 . × − eV . (3.10)In addition LMA-D is also a viable solution for this model with MeV ≤ M Z (cid:48) ≤ MeV with g (cid:48) M Z (cid:48) = (4 . ± . × − MeV . (3.11)On the contrary, as seen in the figure, the model with charge L e + 2 L µ + 2 L τ can onlyprovide a viable LMA-D solution without conflict with bounds from other experiments forand ultra light mediator with M Z (cid:48) (cid:46) × − eV. In this work we have performed dedicated global analysis of oscillation data in the frame-work of lepton flavor dependent U (1) (cid:48) interactions which affect the neutrino evolution inmatter. The analysis is performed for interaction lengths ranging from larger than the Sunradius (covering what we label as LRI domain) to effective contact neutrino interactions(NSI domain). We survey ∼ set of models characterized by the charges of the firstgeneration charged fermions and the three flavor neutrinos. We find that • In the LIGHT sector of the oscillation parameter space the introduction of new in-teractions does not lead to a significantly better fit of the oscillation data comparedto standard oscillations, irrespective of the U (1) (cid:48) coupling in either NSI or LRI do-mains. Thus for all cases the analysis of oscillation data in the LIGHT sector resultsin excluded ranges of g (cid:48) and M Z (cid:48) . • The excluded ranges for the Z (cid:48) coupling and mass for the models with couplings listedin table 1 are shown in Figs. 2 and 3 for M Z (cid:48) below 1 eV and in the O ( MeV–GeV ) range, respectively – i.e. , below and above the window strongly disfavored by thecosmological bound on ∆ N eff . • In the regime of ultra-light mediators, for all models shown, the oscillation analysisimproves over the bounds from tests of fifth forces and of violation of equivalenceprinciple and for Z (cid:48) lighter than ∼ − eV or ∼ − eV depending on whether the U (1) (cid:48) current involves coupling to electron lepton number.– 18 – For mediators in the O ( MeV–GeV ) range we list in table 3 the derived constraintson g (cid:48) versus M Z (cid:48) . We find that for most of the considered models there are values of g (cid:48) and M Z (cid:48) for which the oscillation analysis provides constraints extending beyondthose from other experiments. • In what respects to LMA-D we find that it cannot be realized in the LRI domain.In the NSI domain we have found that 4.8% of the set of charges studied can have abest fit in LMA-D, and 5.2% lead to LMA-D as a valid solution within 4 units of ∆ χ of standard oscillations. None of these set of charges correspond to an anomaly freemodel based on gauging SM global symmetries with SM plus right-handed neutrinosmatter content (Eq. (2.2)). So, generically, Z (cid:48) models for LMA-D with gauged SMglobal symmetries require additional states for anomaly cancellation. Acknowledgement
We want to thank Renata Zukanovich for discussions and her participation in the earlystages of this work. The authors also acknowledge use of the HPC facilities at the IFT(Hydra cluster) This work was supported by the spanish grants FPA2016-76005-C2-1-P, FPA2016-78645-P, PID2019-105614GB-C21 and PID2019-110058GB-C21, by USA-NSFgrant PHY-1915093, by AGAUR (Generalitat de Catalunya) grant 2017-SGR-929. Theauthors acknowledge the support of the Spanish Agencia Estatal de Investigación throughthe grant “IFT Centro de Excelencia Severo Ochoa SEV-2016-0597”.
A Bounds from non-oscillation experiments
Here we summarize the main details of the bounds shown by the colored regions in Figs. 2and 3, as well as the procedure used to rescale them for the different models shown in eachpanel.
Bounds from searches for gravitational fifth forces
Generically gravitational fifth force experiments look for deviation from the standard New-ton potential ( ∝ /r ) between two objects.We have taken the results from Ref. [43] (which, in turn, were recasted from Fig. 10in Ref. [44]) where they present the constraints on the coupling α versus the interactionlength λ (or, equivalently, versus m = λ ) defined as the constant entering the potential ofthe fifth force V ( r ) = α N N e − r/λ r (A.1)where N , is the total charge of each object, which they take as the total number of baryons.In particular from Fig. 3 in [43] we read their boundary curve (cid:18) α max α em , m (cid:19) , (A.2)– 19 –here α em is the SM fine structure constant. In order to rescale it to the different U (1) (cid:48) models we notice that the corresponding potential for the U (1) (cid:48) interaction for the sameobjects is V (cid:48) ( r ) = C C g (cid:48) π e − r/λ (cid:48) r , (A.3)where C i is the total Z (cid:48) charge of object i and λ (cid:48) is the U (1) (cid:48) interaction length λ (cid:48) = 1 M Z (cid:48) , C i ≡ N i A i c i ≡ N i A i [ Z i ( a e + 2 a u + a d ) + ( A i − Z i )( a u + 2 a d )] , (A.4)where Z i and A i are the atomic number and mass number of the material of which theobject i is made, so c i is the charge under Z (cid:48) for each atom of the material. Assumingthat the number of protons and neutrons in the material are not very different (that is, A i /Z i ∼ ), we get V (cid:48) ( r ) = V ( r ) × g (cid:48) πα ( a e + 3 a u + 3 a d ) (A.5)with m = M Z (cid:48) . So the boundary in the g (cid:48) vs M Z (cid:48) plane will be (cid:0) g (cid:48) max , M Z (cid:48) (cid:1) = (cid:32) a e + 3 a u + 3 a d √ πα em × (cid:114) α max α em , m (cid:33) . (A.6) Bounds from searches for violation of the equivalence principle
Similarly as in the case of fifth force searches, this bound comes from precise measurementsof the gravitational potential between two objects. However, in this case one tests thedifferences of the potential for the same total mass of two different test materials, usinga pendulum which is attracted by the same mass. The bounds are taken from Ref. [45],where they define the potential due to the new force as: V G ( r ) = α G m t m s r ˆ N t ˆ N s e − r/λ . (A.7)where G is the gravitational constant, ˆ N i refers to the new interaction charge per mass unit,and the subindices t and s stand for test (or pendulum) and source masses, respectively.In Ref. [45] they assume that the new interaction couples to the number of baryons. Thus,they use beryllium and titanium as test materials, chosen to maximize the difference inbaryon number per unit mass. To be specific, we use ˆ N Be = 0 . and ˆ N Ti = 1 . ,as in Ref. [45].Noting that the total potential will be the sum of the standard gravitational potentialplus the contribution from the new interaction, the authors of Ref. [45] put a bound on η = 2 V Be − V Ti V Be + V Ti ∼ ∆ V G ( r ) Gm s m t r = α ( ˆ N Ti − ˆ N Be ) ˆ N s e − r/λ (A.8)which is used to set a constraint on α as a function of λ , (cid:0) α max , λ (cid:1) . (A.9)– 20 –his can be used to set a bound on ∆ V (cid:48) ( r ) for a general model, noting that ∆ V (cid:48) ( r ) = g (cid:48) π e − rM Z (cid:48) r C s ( C Be − C Ti )= ∆ V G ( r ) × g (cid:48) / παG · u × c Ti /A Ti − c Be /A Be ˆ N Ti − ˆ N Be × c s /A s ˆ N s (cid:39) ∆ V G ( r ) × g (cid:48) / παG · u × ( a e + a u − a d )( a e + 3 a u + 3 a d )144( ˆ N Ti − ˆ N Be ) (A.10)where u stands for the atomic mass unit in GeV ( u (cid:39) . GeV), and in the last line wehave approximated for the source material
A/Z (cid:39) and ˆ N s (cid:39) . So the boundary in the g (cid:48) vs M Z (cid:48) plane will be (cid:0) g (cid:48) max , M Z (cid:48) (cid:1) = (cid:115) N Ti − ˆ N Be )( a e + 3 a u + 3 a d )( a e + a u − a d ) 4 π G u × √ α max , λ (A.11) Bounds from white-dwarf cooling
We based the bounds shown in Figs. 3 and 4 on the study presented in Ref. [47]. There,upper bounds on new interactions are imposed on the basis that the energy losses fromplasmon decays into particles that escape the star is not larger than the energy losses dueto neutrino emission in the SM.In the U (1) (cid:48) scenarios here considered the minimum new contribution is due to Z (cid:48) mediated decays into neutrinos Γ s plasmon → ν ¯ ν,Z (cid:48) (cid:46) Γ s plasmon → ν ¯ ν, SM = C e,V G F π α em Z s π s ω s (A.12)where C e,V is the vector coupling to the electron current in the SM, Z s is the plasmonwavefunction renormalization and π s is the effective plasmon mass which enters in thedispersion relation ω s − k = π s ( ω s , k ) . Here, s = T, L refers to the plasmon polarization(transverse or longitudinal).Under the assumption that the mass of the Z (cid:48) is much larger than the frequency of theplasmon we can write its rate into neutrinos of a given flavor β due to the new interactionsas Γ s plasmon → ν β ¯ ν β ,Z (cid:48) = 13 g (cid:48) M Z (cid:48) ( a e b β ) π α em Z s π s ω s (A.13)And the upper bound obtained in Ref. [47] translates into: (cid:118)(cid:117)(cid:117)(cid:116)(cid:88) β ( a e b β ) · g (cid:48) M Z (cid:48) ≤ C e,V G F = 1 . × − GeV − . (A.14) Bounds from coherent neutrino-nucleus scattering
We have performed our own reanalysis of coherent neutrino-nucleus scattering data usingthe time and energy information from COHERENT experiment [48, 49] on CsI based on– 21 –ur recent analysis in Ref. [50] performed for NSI with a variety of nuclear form factors,quenching factors and parametrization of the background. In particular the results shownin Fig. 3 correspond to the analysis performed using the quenching factor obtained withthe fit to the calibration data of the Duke (TUNL) group [48] together with our data drivenreevaluation of the steady-state background (see Ref. [50] for details).Model predictions are obtained exactly as in [50], but replacing the cross section forcoherent scattering in the presence of NSI with that induced by the Z (cid:48) . In particular if wedefine ε (cid:48) ( Q , g (cid:48) , M Z (cid:48) ) ≡ √ G F g (cid:48) M Z (cid:48) + Q (A.15)one can use the same cross section expression for NSI simply replacing: ε q,Vαβ → δ αβ a q b α ε (cid:48) ( Q , g X , M Z (cid:48) ) (A.16)where Q = 2 M T is the momentum transfer. In this case, the differential cross section forcoherent scattering of a neutrino with flavor α reads: dσ α dT = G F π W α ( Q , g X , M Z (cid:48) ) F ( Q ) M (cid:18) − M TE ν (cid:19) (A.17)where M is the mass of the nucleus, T is the nuclear recoil energy, and E ν is the incidentneutrino energy. We have defined a modified weak charge for the nucleus as: W α ( Q , g X , M Z (cid:48) ) = Z (cid:2) g Vp + (2 a u + a d ) b α ε (cid:48) ( Q , g X , M Z (cid:48) ) (cid:3) ++ N (cid:2) g Vn + ( a u + 2 a d ) b α ε (cid:48) ( Q , g X , M Z (cid:48) ) (cid:3) . (A.18)Thus, unlike in the case of NSI, the weak charge now depends on the momentum transferredin the process.The construction of χ COH ( g (cid:48) , M Z (cid:48) ) is totally analogous to that of χ COH ( (cid:126)ε ) in Ref. [50].In consistency with the condition impose when deriving the oscillation bounds the CO-HERENT regions shown in Fig. 3 corresponds to the Z (cid:48) coupling and mass for which thefit to COHERENT data is worse than the one obtained in the SM by 4 units: χ COH ( g (cid:48) , M Z (cid:48) ) − χ COH,SM > . (A.19) Bounds from measurements of neutrino scattering on electrons
For neutrino-electron scattering experiments TEXONO and GEMMA we performed ourown analysis following the procedure in Ref. [51] which explicitly studied the bounds im-posed by those experiments in some Z (cid:48) models and provide all the relevant cross sectionexpressions.For TEXONO we use the data from Fig. 16 in Ref. [52]. This corresponds to 29882(7369) kg-day of fiducial mass exposure during Reactor ON (OFF), respectively. Theadopted analysis window is 3–8 MeV, spread out uniformly over N bin = 10 energy bins.An overall normalization constant has been manually set to reproduce the SM predictionshown in Fig 16 in [52]. Cross section has been implemented following [51]. With this weget χ SM,TEXONO = 15 . . – 22 –or GEMMA we take data from Fig. 8 in Ref. [53] and consider both phase I and phaseII data so we define χ GEMMA,TOT = χ GEMMA,I + χ GEMMA,II
This corresponds to about12000 ON-hours and 3000 OFF-hours of active time. Data was taken for a detector massof 1.5 kg. The energy window used is . MeV ≤ E ν ≤ . MeV. With this procedure weget χ SM,GEMMA = 154 . .For both TEXONO and GEMMA we draw the contours with the equivalent conditionused for the oscillation analysis so for a given set of model charges the contours are definedby χ ( g (cid:48) , M Z (cid:48) ) − χ SM = 4 . Other constraints: fixed target experiments and colliders
As mentioned in the text, the bounds from fixed target experiments and colliders are takendirectly from the literature. In particular, we consider the following set of bounds: • Electron beam dump experiments: we take these from the compilation in Ref. [54],which were obtained using data from E137 [55], E141 [56], E774 [57], Orsay [58], andKEK-PF-000 [59]. • Proton beam dump experiments: for LSND we use the results obtained in Ref. [60]which used data from Ref. [61] (see also Ref. [62]); for CHARM [63] we use the limitderived in Ref. [64]; finally, for ν -Cal [65] we use the lmit derived in Ref. [66] (see alsoRef. [67] for a similar analysis). • We also consider constraints from Z (cid:48) production in e + e − collisions in BaBaR both invisible [68] and invisible [69] final states, as well as constraints from LHCb for U (1) (cid:48) decaying into µ + µ − , the most relevant ones being from prompt decay searches [70, 71].All the bounds mentioned above were obtained for a dark photon coupled to the SMfermions via kinetic mixing. In order to recast these to bounds on the different U (1) (cid:48) modelsconsidered we have used the darkcast software [72], which takes into account the differencein production branching ratio and lifetime of the new boson, as well as its decay into a givenfinal state. References [1] B. Pontecorvo,
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