COHERENT constraints after the COHERENT-2020 quenching factor measurement
Dimitrios K. Papoulias
hep-ph


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        SU(2)_L ⊗ SU(2)_R ⊗ U(1)_B-L
        neutrinoless double beta decay 
        𝒞𝒫𝒞𝒫𝒯𝒫𝒯𝒞2𝒵_2Łℒet al.h.c.#1⟨ #1⟩
         lepton flavour violation lepton number violation CEνNS 
        
        
        
         
        lepton flavour violation lepton number violating SU(3)_c ⊗ SU(2)_L ⊗ U(1)_YSU(3)_c ⊗ SU(3)_L ⊗ U(1)_XSU(3)_c ⊗ SU(3)_L ⊗ U(1)_X ⊗ U(1)_N
        
        
        
        
        
        
        
        
        
        
        #1#1#1#1#1#1#1#1#1#1
        
        unsrtdipapou@ific.uv.es
        Recently an improved quenching factor (QF) measurement for low-energy
        nuclear recoils in CsI[Na] has been reported by the COHERENT Collaboration. The new energy-dependent QF is characterized by
        a reduced systematic uncertainty and leads to a
          better agreement between the experimental COHERENT data and the
          Standard Model (SM) expectation. In this work, we report updated
          constraints on parameters that describe the process of coherent
          elastic neutrino-nucleus scattering within and beyond
          the SM, and we also present how the new QF affects their
          interpretation.
          
         COHERENT constraints after the COHERENT-2020 quenching factor measurement
            D.K. Papoulias
            December 30, 2023
        ==========================================================================
        
        
        
        
        § INTRODUCTION
        
        The first observation of coherent elastic neutrino-nucleus scattering
        (CEνNS) was made at the COHERENT experiment using a CsI[Na]
        detector at the Spallation Neutron Source
        (SNS) <cit.>, providing a novel powerful
        probe for a wide range of low-energy physics searches.
        
        This motivated a large number of theoretical studies to analyze the recorded  signal for performing precision tests of the Standard Model (SM) <cit.> and for investigating possible signatures of new physics beyond the SM <cit.>. The subject became of intense interest during the latest period, and a plethora of extensive studies constantly appear covering a wide spectrum of new physics phenomena such as nonstandard interactions (NSIs) <cit.>,  neutrino  electromagnetic properties <cit.>, sterile neutrinos <cit.>, charge-parity violation <cit.>, and new mediators <cit.>. Nuclear and atomic effects were also explored in Refs. <cit.>, which may have direct implications for the neutrino-floor <cit.> and  dark matter searches <cit.>. Moreover, from the perspective of experimental physics, several experimental proposals aim to measure  at the SNS <cit.> and at reactor facilities <cit.>
        (for a review, see Ref. <cit.>).
        
        Experiments looking for  and direct dark matter signals are
        typically based on accurate measurements of the nuclear response and
        are aiming to achieve keV or sub-keV threshold capabilities depending
        on the nuclear target. In such measurements, most of the nuclear
        recoil energy is dissipated as heat and ionization, while the recorded
        energy for the case of scintillator detectors is in reality an
        electron equivalent energy whose magnitude depends on the so-called
        quenching factor (QF) <cit.>. The QF is an
        energy-dependent quantity that is different for a given isotope, and
        its calibration involves neutron scattering measurements <cit.>. Regarding the first observation of  at COHERENT with a 14.57 kg CsI[Na]
        detector, the first theoretical simulations adopted
        an energy-independent QF of 8.78 ± 1.66 % in the search region
        5–30 keV_nr <cit.>. In this
        work we employ the new energy-dependent QF resulted by the COHERENT-2020 campaign <cit.> from a refined analysis correcting systematic effects of previous measurements, i.e. Chicago-1, Chicago-2, Duke.[For the Chicago-3 analysis, see Ref. <cit.>.]
        
        We first show that the new QF measurement leads to a higher
        consistency between the SM expectation and the experimental data, a
        result that is in agreement with Ref. <cit.>. We then
        revisit various constraints on conventional and exotic parameters
        describing the  interaction and update their status. In the
        first stage, we explore the sensitivity to the weak mixing angle and
        to the average nuclear root-mean-square (rms) radius of CsI assuming purely SM
        interactions. Afterward, we reexamine the
        sensitivity of COHERENT to phenomenological parameters in the framework of new
        physics interaction channels such as vector NSIs, neutrino magnetic
        moments, and charge radii as well as in simplified scenarios with novel vector-Z^' and scalar mediators. The new  constraints are obtained on the basis of an improved 
        χ^2 fit analysis that incorporates the aforementioned quenching
        factor effects. We show that the new energy-dependent QF combined with
        the reduced uncertainty leads to stronger constraints compared to 
        previous studies.
        
        The paper is organized as follows: In Sec. <ref> we
        provide all necessary ingredients to accurately simulate the observed
         signal, in Sec. <ref> we provide the numerical
        results of our sensitivity analysis and update the constraints on the
        parameters describing the studied conventional and exotic physics
        phenomena. Finally, in Sec. <ref> we summarize the
        main outcomes of our work.
        
        
        
        § SIMULATION OF THE COHERENT  RATE
        
        
        During the  interaction, a neutrino with energy E_ν 
        scatters off a nuclear target (A,Z) with Z
        protons and N=A-Z neutrons, which in turn produces a detectable
        nuclear recoil T_A. Focusing on the COHERENT experiment, after
        summing appropriately over the nuclear isotopes x=Cs, I and
        all incident neutrino flavors ν_α= (ν_e, ν_μ,
        ν̅_μ), the number of expected  events is given by
        
        
            N_theor = ∑_ν_α∑_x = Cs, I  N_targ^x  ∫_T_th^T_A^max∫_E_ν^min^E_ν^max f_ν_α(E_ν)  𝒜  ( T_A ) (d σ_x/dT_A(E_ν, T_A) )_λ   dE_ν dT_A   ,
        
        and depends on the differential cross section (d
        σ_x/dT_A)_λ that is relevant in the framework of a
        neutrino interaction channel λ within or beyond the
         SM. The number of target nuclei contained in the CsI
        detector with mass m_det=14.57 kg is determined by Avogradro's number N_A and the stoichiometric ratio η_χ through the relation
        N_targ^x = m_detη_x/∑_x A_x
          η_x N_A. The neutrino-energy flux at the SNS consists
        of a prompt and a delayed beam that is adequately described by the
        Michel spectrum <cit.>
        
            f_ν_μ(E_ν)     = 𝒩δ(E_ν-m_π^2-m_μ^2/2 m_π)      (prompt)  , 
             
            f_ν̅_μ(E_ν)     = 𝒩64 E^2_ν/m_μ^3(3/4-E_ν/m_μ)      (delayed)  ,
             
            f_ν_e(E_ν)     = 𝒩192 E^2_ν/m_μ^3(1/2-E_ν/m_μ)      (delayed)   ,
        
        normalized to 𝒩 = r
        N_POT/4 π L^2, where L=19.3 m is the detector distance from the SNS source, and r=0.08 denotes the number of neutrinos per flavor produced for each proton on target (POT), i.e.,
        N_POT=1.76 × 10^23 for a period of 308.1 days. Assuming SM  interactions, the differential cross section with respect to the nuclear recoil energy is expressed as <cit.>
        
            (d σ/dT_A)_SM = G_F^2 m_A/π (Q_W^V)^2 (1 - m_A T_A/2 E_ν^2)  F^2(Q^2)   ,
        
        where m_A denotes the nuclear mass and G_F the Fermi coupling constant. The vector Q_W^V weak charge is given by <cit.>
        
            Q_W^V =     [  2(g_u^L + g_u^R) + (g_d^L + g_d^R) ] Z 
                + [ (g_u^L + g_u^R) +2(g_d^L + g_d^R) ] N     ,
        
        while the P-handed couplings of u and d quarks to the
        Z boson take the form
        
        
            g_u^L =   ρ_ν N^NC( 1/2-2/3κ̂_ν Nŝ^2_Z ) + λ^u,L  ,
            
            g_d^L =   ρ_ν N^NC( -1/2+1/3κ̂_ν Nŝ^2_Z ) + λ^d,L  ,
            
            g_u^R =   ρ_ν N^NC(-2/3κ̂_ν Nŝ^2_Z ) + λ^u,R  ,
            
            g_d^R =   ρ_ν N^NC(1/3κ̂_ν Nŝ^2_Z ) + λ^d,R  .
        
        Here, ŝ_Z^2 ≡sin^2 θ_W= 0.2382 is the weak mixing angle and ρ_ν N^NC = 1.0082, κ̂_ν N = 0.9972, λ^u,L = -0.0031, λ^d,L = -0.0025, and
        λ^d,R =2λ^u,R = 3.7 × 10^-5 are the radiative corrections <cit.>. Because of their tiny contributions to the  rate, axial-vector interactions, incoherent interactions, as well as contributions due to the sodium dopant of the CsI[Na] detector are neglected.
        
        
        The main source of theoretical uncertainty in the SM  process arises from the  nuclear form factor that takes into account the finite nuclear size and depends on the variation of the momentum transfer Q^2=2 m_A T_A <cit.>. Following the COHERENT Collaboration, in this work we adopt the Klein-Nystrand (KN) form factor parametrized as <cit.>
        
            F_KN = 3 j_1(Q R_A)/Q R_A[ 1 + (Q a_k )^2 ]^-1  ,
        
        where a_k = 0.7 fm is the range of the Yukawa potential (over a Woods-Saxon distribution) in the hard sphere approximation with radius R_A= 1.23 × A^1/3.  We note that regarding the old QF, slight differences from the corresponding results of Ref. <cit.> throughout the paper are due to the adoption of the KN form factor, the different neutrino-energy distribution considered, the different value of the weak mixing angle, as well as the binned χ^2 analysis performed here (see below).
        
        
        
        
        
        For a scintillation-based experiment, the  measured quantity is the number of photoelectrons (PEs) denoted here as n_PE. To account for this mechanism, the   differential rate in events vs nuclear recoil  energy gets converted to an equivalent differential rate in events vs  electron recoil energy through the application of the QF function Q_f(T_A), and that in turn gets converted to a PE spectrum via the light yield ℒ_Y= 13.348 PE/keV_ee measured for electron recoils as 
        
        
            n_PE =  Q_f(T_A) ℒ_Y T_A  .
        
        In Eq.(<ref>), the acceptance efficiency of the CsI detector is taken into account, which in terms of the photoelectron content  of the signal reads[Note that the efficiency function is instrumental and does not depend on the QF.] <cit.>
        
            𝒜(n_PE) = k_1/1 + e^- k_2 (n_PE - x_0 )Θ(n_PE)   ,
        
        with k_1= 0.6655, k_2= 0.4942, x_0= 10.8507, and the modified Heaviside function 
        
        
            Θ(n_PE) = {[            0     n_PE < 5;          0.5 5 ≤ n_PE < 6;            1  n_PE≥ 6   . ].
        
        
        
        
        Up to now, previous analyses adopted the 
          energy-independent QF of 8.78 ± 1.66 % recommended by the
          COHERENT Collaboration in Ref. <cit.> which carried a large uncertainty of 25%.[In reality, the QF uncertainty is 18.9% leading to an overall uncertainty in  rate of 25% <cit.>. We however adopt the official values reported in Ref. <cit.>.] In the present work we consider the new energy-dependent QF
        which came out of the refined COHERENT-2020 measurement
        with a reduced uncertainty by about a factor of 4 at  3.6% (for more
        details, see Ref. <cit.>). In agreement with
        Ref. <cit.>, within the SM the new calculation gives a
        theoretical value of  158 events as compared to the 174 events
        corresponding to the old QF. At this point, it is rather important to emphasize that a better agreement is now reached with the 134 events observed in
        Ref. <cit.>. The corresponding results are compared in Fig. <ref> as a function of the PE bins.
        
        
        
        § NUMERICAL RESULTS
        
        
        
        In the present study, we perform a sensitivity analysis of the parameter set 𝒮 in question (see below) that follows from a χ^2 (𝒮) fit that is relevant for the CsI detector at the COHERENT experiment and reads <cit.>
        
            χ^2 (𝒮) =  𝚊_1, 𝚊_2min [ ∑_i=4^15(N^i_meas - N^i_theor(𝒮) [1+𝚊_1] - B^i_0n [1+𝚊_2] /√(N^i_meas + B^i_0n + 2 B^i_ss))^2 
              + (𝚊_1/σ_𝚊_1)^2 + (𝚊_2/σ_𝚊_2)^2  ]   .
        
        Here, N^i_meas (N^i_theor) represents the ith bin of the observed signal (theoretical CEνNS events), and B^i_0n (B_ss^i) denotes the beam-on prompt neutron (steady-state) background events taken from the COHERENT data release <cit.>, while the analysis is restricted to the 12 energy bins corresponding to 6 ≤ n_PE≤ 30.  In Eq.(<ref>), 𝚊_1
        and 𝚊_2 are the corresponding systematic parameters with
        fractional uncertainties  σ_𝚊_1 = 12.8% (5% from
        signal acceptance determination, 5% from form factor choice, 10%
        from neutrino flux and  3.6% from the new QF) and
        σ_𝚊_2 = 25%. Note that compared to σ_𝚊_1=28%
        given in Ref. <cit.> and adopted by all similar studies
        up to now, the fractional uncertainty considered here is reduced by
        about a factor of 2.  This
        is also in agreement with estimations of previous studies addressing possible future experimental
        setups <cit.> and
        will have a direct impact on the updated constraints presented below.
        
        
        
        
         §.§ SM precision tests and nuclear physics
        
        
        
        Assuming purely SM interactions, we first extract the new sensitivity to the weak mixing angle that arises from the new QF measurement. To this end, we evaluate the χ^2(sin^2 θ_W) function and perform a sensitivity fit by varying around the central value sin^2 θ_W = 0.2382. The resultant sensitivity profiles are depicted in the left panel of Fig. <ref>. A comparison with the corresponding result assuming the old energy-independent QF is also shown. Indeed, this new calculation leads to reasonably improved results. From the fit, we find the following constraints  at 90% C.L.
        
        
            sin^2 θ_W =    0.197^+0.124_-0.080  (old QF)  ,
            sin^2 θ_W =    0.209^+0.072_-0.069  (new QF)  .
        
        Evaluating the 1σ bands δ s_W^2 according to the definition given in Ref. <cit.>, we find the values δ s_W^2=(0.057, 0.042) for the (old, new) QF case, which yield the corresponding percentage uncertainties δ s_W^2/ sin^2θ_W of (29%, 20%).
        
        We then make an effort to explore the sensitivity to the nuclear rms radius that follows from the recent COHERENT measurement. To this purpose, we employ the refined QF resulting from Ref. <cit.>, while in this case we consider the Helm form factor <cit.>
        
            F_Helm(Q^2) = 3 j_1(Q R_0)/q R_0  e^-(Q s)^2/2  ,
        
        where j_1(x) is the spherical Bessel function of the first
        kind. Here, ⟨ R^2_n ⟩^1/2 = √( 3
          / 5  R _  0  ^  2  + 3 s ^  2 ) is the nuclear rms radius,
        R_0 = 1.23   A^1/3 fm is the diffraction radius, and s=0.9 fm
        quantifies the surface thickness (for more details, see
        Refs. <cit.>). The
        resultant sensitivity profile is presented in the right panel of Fig. <ref>, showing that the constraints are 
        now stronger than  previously reported <cit.>. In
        particular at 90% C.L. we find the best fits [Note that, in this case the form factor uncertainty is neglected in Eq.(<ref>).]
        
            ⟨ R^2_n ⟩^1/2 =    5.6^+1.5_-2.1 fm  (old QF)  ,
            ⟨ R^2_n ⟩^1/2 =    5.6^+1.3_-1.25 fm  (new QF)  .
        
        In a similar manner, within 1 σ error we find the bands δ⟨ R^2_n ⟩^1/2=(1.01, 0.76) and the corresponding percentage uncertainties (18%, 14%) for the (old, new) QF measurement.
        We finally stress that the latter results remain essentially the same when considering the Klein-Nystrand form factor.
        
        
        
        
         §.§ Nonstandard interactions
        
        
        Nonstandard interactions have been a popular subject of extensive
        research during the last 15 years, with interesting applications in
        neutrino oscillations and low-energy neutrino physics (for a review, see Refs. <cit.>). For a neutrino with flavor
        α={e,μ,τ} and a quark q={u,d}, the vector-type NSI
        contributions that arise due to nonuniversal (NU) flavor-preserving and flavor-changing interactions are described in the NSI weak
        charge <cit.>
        
            Q_NSI^V =     (2 ϵ_αα^uV + ϵ_αα^dV + g^V_p) Z + (ϵ_αα^uV + 2 ϵ_αα^dV + g^V_n) N 
                + ∑_α, β[ (2 ϵ_αβ^uV + ϵ_αβ^dV) Z + (ϵ_αβ^uV + 2 ϵ_αβ^dV ) N ]   .
        
        In the context of NSI, the expected  rate is modified according
        to the substitution Q_W^V →
        Q_NSI^V in the SM differential cross section of
        Eq.(<ref>).
        
        Assuming a single nonvanishing NSI parameter at a time,
        Fig. <ref> illustrates the obtained sensitivity for
        the NU ϵ_ee^qV (ϵ_μμ^qV) couplings in
        the left (right) panel, while a useful comparison is also given for
        the case of the old QF.  The
          impact of the new QF measurement on NSI constraints becomes evident.  The left and
        right panels of Fig. <ref> show the allowed regions at
        90% C.L. in the (ϵ_ee^dV,ϵ_ee^uV) and the
        (ϵ_μμ^dV,ϵ_μμ^uV) parameter space,
        respectively.  We see that the bounds are now more
          restrictive than the corresponding results using the old QF.
          
          
        
        
         §.§ Electromagnetic neutrino interactions
        
        In this subsection, we are interested in exploring the possibility of probing nontrivial neutrino electromagnetic (EM) properties <cit.> and to revisit existing constraints  from  <cit.>. The two main phenomenological parameters that arise in the framework of EM neutrino interactions are the neutrino magnetic moment and the neutrino charge radius. For completeness, we mention that in the simplest Majorana neutrino case, the neutrino magnetic moment μ_ν is in reality expressed in terms of the neutrino transition magnetic moments Λ_i of the neutrino magnetic moment matrix <cit.>, while constraints have been recently extracted from neutrino-electron scattering <cit.> and  <cit.>.  Here, for simplicity, we consider the effective neutrino magnetic moment in the helicity-violating EM cross section <cit.>
        
            ( d σ/dT_A)_EM=π a^2_EMμ_ν^2 Z^2/m_e^2(1-T_A/E_ν/T_A) F^2(Q^2) .
        
        In Fig. <ref>, we present the updated constraint on μ_ν from our analysis with the new QF, which is also compared to the corresponding one that comes  from the old QF. The obtained upper limits at 90% C.L. read
        
        
            μ_ν  <      3.1 × 10^-9 μ_B   (old QF)  ,
            μ_ν  <      2.6 × 10^-9 μ_B   (new QF)  .
        
        From the same plot, it can be deduced that this difference is more pronounced at 99% C.L. 
        
        
        For a flavor neutrino ν_α, the associated neutrino charge
        radius ⟨ r_ν_α^2 ⟩ is another interesting
        phenomenological quantity which arises from the helicity-conserving
        charge form factor of the EM neutrino current <cit.>.
        The latter leads to a shift of the weak mixing angle as
        follows <cit.>:
        
            sin^2 θ_W →sin^2 θ_W + √(2)π a_EM/3 G_F⟨ r_ν_α^2 ⟩  .
        
        We stress  that there is not a sign flip regarding antineutrino charge radii; e.g., it holds ⟨ r_ν̅_α^2 ⟩=⟨ r_ν_α^2 ⟩ as defined in Ref. <cit.>[Reference <cit.> used a negative sign which is now corrected in Ref. <cit.>.]. In this work, we follow the definition given in Ref. <cit.> however, the shift considered here is smaller by a factor of 2.
        
        Neglecting transition charge radii and varying one parameter at a time, Fig. <ref> shows the χ^2 profiles of the neutrino charge radii ⟨ r_ν_α^2 ⟩ associated with the respective SNS neutrino flux, where the left (right) panels correspond to the old (new) QF measurement. The obtained constraints differ slightly due to the old vs new QF data. The only noticeable difference is that by employing the new QF in the case of the prompt ν̅_μ beam, the resultant constraint on ⟨ r_ν̅_μ^2 ⟩ is separated into two distinct regions at 90% C.L. It is now worthwhile to explore the simultaneous constraints that can be obtained. Figure <ref> presents the allowed regions at 90% C.L. in the (⟨ r_ν_α^2 ⟩, ⟨ r_ν_β^2 ⟩) parameter space.  As expected the allowed parameter space in all cases is more restricted using the new QF data.
        
        
        
        
         §.§ Simplified scenarios with light mediators
        
        In addition to the NSIs discussed previously in Sec. <ref>, we are now interested in simplified scenarios where the NSI is generated due to the presence of novel mediators. 
        
         
        
         
         
        In the first step, we explore the case where the  rate is
        enhanced from contributions due to a vector-Z^' mediator with
        mass M_Z^'. The relevant cross section takes the
        form <cit.>
        
            ( d σ/dT_A)_SM + Z^' = 𝒢_Z^'^2 (T_A,g_Z^',M_Z^') (d σ/d T_A)_SM  ,
        
        with the Z^' factor defined as
        
        
            𝒢_Z^' = 
            1 + 1/√(2)G_FQ_Z^'/Q_W^Vg_Z^'^ν V/2 m_A T_A + M_Z^'^2  .
        
        In the above expression, in order to reduce the number of model parameters, we consider the generalized coupling  g_Z^'^2 = g^ν V_Z^' Q_Z^'/3 A  that is expressed in terms of the vector ν_α-Z^' coupling times the respective vector charge Q_Z^'^V, under the assumption of universal quark-Z^' couplings (for more details, see Ref. <cit.>).
        
        Concentrating our attention on the case of a new scalar boson ϕ mediating the  process, the cross section takes the form <cit.>
        
            (d σ/d T_A)_scalar = G_F^2 m_A^2/ 4 π𝒢_ϕ^2 M_ϕ^4 T_A /E_ν^2 ( 2 m_A T_A + M_ϕ^2 )^2 F^2(Q^2)   ,
        
        with the corresponding scalar factor being 
        
        
            𝒢_ϕ = g^ν S_ϕ Q_ϕ/G_F M_ϕ^2  .
        
        In the same spirit of the discussion made above, for the sake of simplification our calculations involve the generalized scalar coupling g_ϕ^2 =  g^ν S_ϕ Q_ϕ/ (14 A + 1.1 Z ).[This result derives from the nuclear charge related to the scalar boson exchange; see Ref. <cit.>.]
        
        The exclusion regions in the parameter space (M_Z^',g_Z^'^2) and (M_ϕ,g_ϕ^2) for the vector and scalar scenarios, respectively, are obtained from a two parameter analysis of the COHERENT data. For both old and new QF data, the results are presented at 90% C.L. in the left (right) panel of  Fig. <ref> for vector (scalar) mediators. As in all previous cases, from this plot we conclude that the new QF data lead to generally more stringent bounds. 
        
        
        
        
        § CONCLUSIONS
        
        
        
        Focusing on the COHERENT experiment, we  reexamined the results implied from CEνNS in light of a new  QF measurement <cit.>. By using the new QF data, we came out with improved constraints regarding all the cases analyzed in this work. A full summary  is given in Table <ref>.  At first, we presented updated constraints focusing on important SM parameters namely, the weak mixing angle and the average nuclear rms radius of CsI, and we explicitly demonstrated the level of improvement. We  then concentrated on interesting phenomenological parameters beyond the SM and presented updated constraints for nonuniversal NSIs as well as for electromagnetic neutrino properties including the effective  neutrino magnetic
        moment and the neutrino charge radius. Finally, we  revisited the sensitivity of COHERENT in the framework of simplified scenarios involving massive vector and scalar mediators. We concluded that a substantial improvement on SM parameters is reached, while the improvement of beyond the SM physics constraints is also evident.  
        
        
        The author acknowledges K. Scholberg and J. Collar for useful correspondence. The author is indebted to M. Tórtola and O. Miranda for their critical comments and their help during the preparation of the manuscript.
        This work is supported by the Spanish Grants No. SEV-2014-0398 and No. FPA2017-85216-P (AEI/FEDER, UE), Grant No. PROMETEO/2018/165 (Generalitat
        Valenciana) and the Spanish Red Consolider MultiDark Grant No. FPA2017-90566-REDC.
        
        
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