Paper Title: Electromagnetic $|G_E/G_M|$ ratios of hyperons at large timelike $q^2$.







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             LFTC-24-08/91
            
            ^1Department of Physics and OMEG Institute, Soongsil University, 
            
            Seoul 06978, Republic of Korea 
            ^2LIP, Laboratório de Instrumentação e Física 
            Experimental de Partículas, 
            
            Avenida Professor Gama Pinto, 1649-003 Lisboa, Portugal
            
            ^3Departamento de Física e Departamento de Engenharia e Ciências Nucleares, 
            
            Instituto Superior Técnico (IST),
            Universidade de Lisboa, 
            Avenida Rovisco Pais, 1049-001 Lisboa, Portugal 
            ^4Laboratório de 
            Física Teórica e Computacional – LFTC,
            Universidade Cidade de  São Paulo,  
            01506-000,  São Paulo, SP, Brazil
            0§ ABSTRACT 
            In recent years, it has become possible to
            measure not only the magnitude of the electric (G_E)
            and magnetic (G_M) form factors of spin 1/2 baryons,
            but also to measure the relative phases
             of those quantities in the timelike kinematic region.
             Aiming to interpret present |G_E/G_M| data on hyperons of the baryon octet,
             as well as to predict future data, we present model calculations of
             that ratio for large invariant 4-momentum square q^2 in the timelike region (q^2>0). 
             Without any further parameter fitting, we extend to the timelike region a covariant quark model previously developed
              to describe the kinematic spacelike region (q^2 ≤ 0)
              of the baryon octet form factors.
              The model takes into account
              both the effects of valence quarks and the excitations
            of the meson cloud which dresses the baryons.
             We report here that our results are in good agreement with very recent data from BESIII,
             within the uncertainty limits of our model. We also make 
            new predictions which may be tested in the near future,
             when the range of the present measurements is expanded
             to the 20–50 GeV^2 region.
            
             
              Electromagnetic |G_E/G_M| ratios of hyperons
            at large timelike q^2
                G. Ramalho^1, M. T. Peña^2,3,  
            K. Tsushima^4 and Myung-Ki Cheoun^1
                August 5, 2024
            =======================================================================
            
            
            
            § INTRODUCTION
            
            
            In recent years it became possible to experimentally probe the structure
            of short-lived baryons through
            e^+ e^- → B B̅ and p p̅→ B B̅ experiments in
            the timelike kinematic region above the production threshold
            q^2 ≥  4 M_B^2 (M_B is the mass of the
            baryon) <cit.>.
            
            These timelike studies complement the two last decades knowledge
            on the spacelike region[In this work,  for simplicity,  we include into the spacelike region the photon point q^2 = 0,
            due to the physical continuity between the regions q^2 < 0 and q^2=0.]
            (q^2 ≤ 0) based on results from electron scattering experiments <cit.>.
            
            
            The electromagnetic structure of the baryons is represented
            by spin (J) and parity (P)
            dependent structure form factors.
            In the timelike region these form factors are complex functions of the 
            e^+ e^- or p p̅ square center-of-mass energy 
            s= q^2 <cit.>.
            
            Of particular interest are the spin 1/2 baryons (J^P =1/2^±),
            including the nucleon and hyperons 
            characterized by two form factors only, the electric and magnetic form factors.
            
            We focus here on hyperons because the experimental information
            about the nucleon is at the moment less complete.
            Also, our model uncertainty decreases with increasing q^2,
            and the convergence of the nucleon data to the asymptotic large q^2
            behavior is expected to emerge at much larger q^2 values, due
            the dominance of light quark dynamics.
            
            Given the short lifetime of hyperons, the challenges in the determination
            of their electromagnetic properties
            in the timelike region, where their weak decays can be analyzed
            in detail <cit.>
            are less significant than in the spacelike region.
            
            
            The first experiments measured the total integrated cross section
            of the e^+ e^- → B B̅ reactions for spin 1/2 baryons,
            which determine the magnitude of the  "effective"
            form factor G defined as <cit.>
            |G (q^2)|^2 = 2 τ |G_M (q^2)|^2 + |G_E (q^2)|^2/2 τ + 1,
            
            where τ = q^2/4M_B^2.
            
            These experiments provided information only about
            a combination of the electric and magnetic form factors,
            and did not reveal their relative phases.
            
            The separate determination of |G_E| and |G_M| or of the ratio
            |G_E/G_M| for spin 1/2 baryons in the timelike region 
            can be obtained by further measuring  the differential
            cross section <cit.>, while
            the knowledge of the relative phase between the two form
            factors demands in addition the determination of polarization cross sections
            (which involve complete spin separation) <cit.>.
            Today, indeed it is possible to measure the ratio |G_E/G_M|
            and the relative angle between the form factors
            for a few baryons <cit.>.
            Most measurements to date have been done
            for ground state hyperons, but in the future excited states
            may be accessed. In a first experimental period the proton and the neutron form factors
            were extracted under some assumptions about
            the relation between G_E and G_M <cit.>.
            
            
            Although the initial measurements were performed near threshold,
            it is expected that with increased accelerator power,
            the experimental energy will go higher up to the 20–50 GeV^2
            range. This extension will allow us to test the predictions described in this work.
            
            Measurements of the effective elastic form factors
            of the Λ, Σ^0,±, Ξ^0,-
            as well as the transition form factors Σ^0 →Λ
            have been performed at BaBar <cit.>, CLEO <cit.>,
            Belle <cit.> and BESIII <cit.>.
            Measurements of the ratios  |G_E/G_M| for the Λ and Σ^+
            have also been determined recently at BESIII <cit.>.
            
            Complete experiments, that measure  |G_E|, |G_M|, 
            and the relative phase between G_E and G_M have been performed
            for the Λ and Σ^+ at BESIII <cit.>, 
            and even more measurements are planned for a near future at BESIII
            and PANDA <cit.>.
            
            
            In this work, we enlarge a previous theoretical study of the effective form factors
            of J^P= 1/2^+ hyperons <cit.> to calculate the |G_E/G_M| ratios 
            for large q^2.
            
            Calculations in this momentum transfer region are important,
            given that at very large q^2, |G_M (q^2)| dominates
            the effective form factor |G (q^2)|, allowing us to probe where that regime sets in.
            
            
            The numerical calculations are based on the
            covariant spectator quark model formalism 
            for the spacelike region <cit.>
            extended to the timelike region.
            The model has been used in the study
            of electromagnetic structure of 
            nucleon resonances <cit.>
            and other baryons
            systems <cit.>.
            
            
            
            
            In the case of the nucleon is it possible to
            compare the magnitude of G_E/G_M in the spacelike
            and timelike regions <cit.>.
            
            Timelike data on the nucleon electromagnetic
            form factors can be found
            in Refs. <cit.>
            for the proton and in Refs. <cit.> for the neutron.
            
            Theoretical studies of the nucleon electromagnetic form factors
            in the timelike region can be found in Refs. <cit.>.
            
            At the moment the measurements of the nucleon form factors in
            the timelike region are incomplete, since their relative phases
            have not been measured.
            The data of the proton and neutron effective form factors
            revealed an oscillatory dependence on the center-of-mass
            energy <cit.>.
            Different explanations have been proposed <cit.>.   
            In the present work we do not discuss this subject because
            we are focused on the asymptotic region, where
            the oscillatory component is expected to be suppressed.
            
            
            Beyond the members of the baryon octet, there are also measurements
            of the effective form factors of the Ω^-, Δ
            and Λ_c^+ <cit.>.
            Theoretical works on hyperon form factors
            in the timelike region can be found in 
            Refs. <cit.>.
            
            
            
            
            
            
            § FORMALISM
            
            
            
            Within the one-photon-exchange approximation, <cit.>
            we can write the e^+ e^- → B B̅ differential
            cross section, where B stands for
            a spin 1/2 baryon B( 1/2^±), as
                  d σ_ Born/d Ω (q^2) =
            
                  α^2 β C/4 q^2[ |G_M (q^2)|^2 (1 + cos^2 θ)
                + 1/τ |G_E (q^2)|^2 sin^2 θ],
              
            where θ is the angle between the baryon
            and the initial photon in the center-of-mass frame,
            α≃ 1/137 is the fine structure constant,
            β= √(1 - 4M_B^2/q^2) is a kinematic factor
            (the baryon speed in c=1 units) and 
            C is the Sommerfeld-Gamow factor.
            This  factor takes into account the Coulomb interaction
            and can be written as C = y/1-exp(-y)
            with y= πα/β2 M_B/√(q^2)
            for charged particles, and C=1 for 
            neutral particles <cit.>.
            For charged particles the factor C converges rapidly to C ≃ 1 
            when q^2 increases.
            
            By integrating the previous relation, we obtain
            σ_ Born (q^2) =
            4 πα^2 β C/3 q^2( 1 + 1/2 τ) |G (q^2)|^2,
            
            using Eq. (<ref>).
            We can conclude then that the effective form
            factor can be calculated directly from the integrated cross section.
            However, it does not give individual information on  G_E or G_M.
            
            
            
            To untangle information on the two form factors G_E and G_M
            one uses  the differential cross section (<ref>) by
            determining the coefficients of the angular functions
            1 + cos^2 θ and sin^2 θ <cit.>.
            
            Combining the two results, one can extract the ratio |G_E/G_M|.
            
            The measurement of the phase between the form factors
            requires in addition the measurement of cross section
            with defined polarizations of the baryons <cit.>.
            
            
            
            
            
            
            
            Finally, at threshold, where q^2= 4 M_B^2, the functions
            G_E and G_M are such that G_E = G_M,
            as a consequence of the definition
            of these Sachs from factors in terms of the kinematic independent
            Dirac and Pauli form factors. Therefore one expects that G_E/G_M ≃ 1
            for measurements 
            near threshold, and the non-zero phases between the form factors are to be sought 
            above the threshold  only (q^2 > 4 M_B^2) <cit.>.
            
            
            The main conclusion of this section is that in the case
            of a spin 1/2 baryon, to obtain 
            the absolute values of the two form factors from
            experiments of the  differential cross section
            it is necessary to measure accurately angular distributions
            of the differential cross section
            besides the effective form factor |G(q^2)|.  However, as we will see in the next section the range for non-zero imaginary parts cannot extend to indefinitely large |q^2|.
            
            
            
            
            
            
            
            
            
            
            § CALCULATION OF ELECTROMAGNETIC FORM FACTORS AT LARGE Q^2
            
            
            
            
            In this work the electromagnetic form factors G_E and G_M
            in the timelike region  are obtained from extrapolating the results
            from the region q^2=-Q^2 ≤  0 to the region q^2 > 0.
            
            
            
            
            
            For that purpose we consider the large-|q^2| relations 
            
            G_ℓ (q^2)= G_ℓ (-q^2)
            
            for ℓ=E,M. 
            These asymptotic relations are a consequence of
            two general mathematical principles: 
            unitarity as well as the Phragmén-Lindelöf theorem,
            which is valid for analytic functions <cit.>.
            
            
            We notice that this theorem implies that the imaginary parts of
            the form factors in the timelike region 
            must be negligible for large q^2,
            since the form factors are real in the spacelike region.
            
            
            
            
            
            
            
            
            
            
            Due to the character of this approximation,
            we expect our estimates here to be accurate only for large q^2,
            deviating  from the condition G_E = G_M at the threshold.
            
            Since the relations (<ref>) are valid for
             the large-|q^2| mathematical limit, and we want to make predictions
            to finite q^2, we consider finite corrections to these
            asymptotic relations.
            
            We start noticing that the elastic form factors
            are divided into two regions of  q^2: (- ∞, 0] (spacelike region)
            and  [4M_B^2, + ∞),  with a gap of 4M_B^2 length
            between the photon point (q^2=0) and
            the threshold of the baryon-antibaryon production (q^2 = 4 M_B^2).  
            
            The interval (0,4M_B^2) defines an unphysical region
            where no physical baryons can be detected <cit.>.
            
            
            
            There is then an ambiguity about the center of reflection
            for the asymptotic symmetry relation of Eq. (<ref>).
            
            To take into account the ambiguity of the location of the center of symmetry 
            we replace
            q^2 → q^2 ( 1 - 2M_B^2/q^2) in (<ref>), 
            which introduces finite corrections (-2M_B^2)
            to the limit q^2= + ∞.
            
            Therefore we use <cit.>
            G_ℓ (q^2)= G_ℓ^ SL (q^2 - 2 M_B^2)
            
            for ℓ =E,M. 
            The label SL on the r.h.s. indicates that 
            theG_ℓ^ SL (Q^2) is calculated
            in the spacelike region (Q^2 = -q^2 ≥ 0).
            
            The expression (<ref>) estimates the reflection
            symmetry point underlying (<ref>)
            exactly at the center of the unphysical interval (0,4M_B^2).
            
            
            
            
            In summary, the measured form factors in the timelike region are complex functions
            characterized by their magnitudes, |G_E|, |G_M|, and their different phases.
            
            Our model calculation is based on results obtained in the q^2 ≤ 0
            region and the application of the large-q^2 relation (<ref>).
            This way, by construction, we obtain real numbers for G_E
            and G_M that carry ± signs.
            
            In principle, for very large q^2 the comparison of our results with the
            experimental values for G_E/G_M is possible
            since in that region the relative phase can only have two possibilities,
            ΔΦ =0 (when the form factors have the same sign)
            or  ΔΦ = π (if they have different signs).
            
            
            
            
            For a control of the approximation we explored the 
            band of variation of the form factors with respect to a reasonable variation
            of the symmetry point around this central value:
            when the functions G_ℓ (q^2) are positive (negative), since
            they are expected to monotonically decrease (increase)
            with increasing q^2, we take
            the upper (lower) limit values to be G_ℓ (q^2)= G_ℓ^ SL (q^2 - 4M_B^2)
            and the lower (upper) limit  G_ℓ (q^2)= G_ℓ^ SL (q^2).
            
            This procedure provides a band of variation of the
            finite corrections to the asymptotic symmetry identity.
            
            When q^2 is very large, the variation of the results becomes
            very narrow and the three estimates (central, upper and lower limits)
            converge to the same value, providing accurate predictions
            that can be tested by experimental data. Moreover, the comparison with data
            in that region provides information of the onset of the regime of
            the asymptotic symmetry (<ref>).
            
            
            
            
            
            
            
            
            § NUMERICAL RESULTS FOR |G_E/G_M|
            
            
            
            In this section we present our results of the ratio |G_E/G_M| for large q^2 values.
            The calculation method of the form factors G_E and G_M
            for the baryon octet family in the timelike region is discussed
            in detail in Ref. <cit.> which uses the
            covariant spectator quark model formalism
            for the spacelike region <cit.>. 
            In its nutshell, we use impulse approximation for the electromagnetic
            quark current, which is justified for large Q^2, since then
            the interacting photon has sufficient resolution to interact with a single quark at a time.
            As a result we can integrate over the relative internal variables
            of the spectator quarks  and therefore we end up with the
            baryon three-quark system regarded 
            as a quark-diquark configuration.
            In the model, the SU(3) symmetry quark flavor symmetry
            is broken at the level of the baryon radial wave functions
            and the constituent quark current <cit.>,
            and gluon and quark-antiquark dressing
            are effectively taken into the constituent quark structure.
            
            
            The different baryon radial wave functions are parametrized
            by two parameters, a short range scale and a long range scale,
            consistent with the expected size of the systems
            (baryons with strange quarks are more
            compact than baryons with light quarks) <cit.>.
            The quark current is parametrized by
            vector meson dominance regulated by
            light (ρ, ω) and intermediate (ϕ)
            vector meson mass poles <cit.>.
            
            
            
            
            
            
            In particular, we use the parameterization
            of the baryon octet from Refs. <cit.>.
            To the valence quark contributions
            we add also an effective description of meson cloud
            dressing processes <cit.>,    
            which are inevitably present in the low-Q^2 region.
            
            In the large-|Q^2| region, the meson cloud contributions are suppressed
            and their effects are manifest on the rescaling of
            the bare contributions to the form factors due to
            the normalization of the hyperon wave functions <cit.>.
            
            This formalism has been used to successfully describe the spacelike electromagnetic
            structure data of baryons including the low-lying nucleon resonances <cit.>.
            The free parameters of the model are fixed by the
            data for the nucleon,
            baryon octet and baryon decuplet <cit.>.
            
            
            Our calculations of the ratio |G_E/G_M| and their comparison with the data
            are shown in Figs. <ref> and <ref>,
            that depict the results for Λ, Ξ^-,0 and  the Σ^0,±, respectively.
            In the graphs for the Λ and Σ^+ we
            include the available data for |G_E/G_M|.
            How we estimate the uncertainty of the results of our model for each form factor
            was defined in the previous section.
            
            The bands of variation for the magnitude of G_E/G_M are determined
            from the independent band intervals of variation for the G_E and G_M functions.
            
            
            
            
            
            The results and their uncertainty bands near the threshold provide only 
            qualitative estimates since our results are expected to be valid only for large q^2,
            as discussed in the previous section.
            
            The deviation from the result  |G_E/G_M|=1 at threshold
            is a consequence of the approximation used.
            
            The signs of our results for the form factors in the asymptotic
            large q^2 values are indicated on the
            right side in Figs. <ref> and <ref>. 
            
            
            
            
            The separated calculations of the form factors G_E and G_M
            for Λ, Σ^0,±  and Ξ^-,0 are presented
            in the Supplementary Material <cit.>.
            
            Note that we are not attempting to predict the zeros
            of G_E/G_M, rather, we are focused on determining its magnitude at sufficiently large 
            q^2.
            In the current formalism, the zeros can only be determined within an uncertainty band of width 
             4 M_B^2 for q^2.  
            
            
            
            We discuss now how we can take into account
            the information of the relative angle
            between the form factors in the analysis of the ratio |G_E/G_M|.
            
            In complete experiments
            the ratio  G_E/G_M is determined,
            and can be written in the form
            G_E/G_M = |G_E|/|G_M| e^i ΔΦ,
            
            where |G_ℓ| is the magnitude of
            the complex form factor G_ℓ and ΔΦ
            is the phase between G_E and G_M.
            
            Once the phase is measured,
            we can extract the real part of the ratio from
            |  Re(G_E/G_M) |=
            | G_E/G_M| |cos (ΔΦ)|,
            
            which is a fraction of the measured value to |G_E/G_M|.
            This does not only show that model calculations
            of |G_E/G_M| based on real form factors 
            agree with the absolute value of the real part of G_E/G_M
            when the imaginary parts are negligible.
            
            The result (<ref>) allows also us to infer from the comparison between the
            experimental data for the r.h.s. and our model results
            for the ratio |G_E|/|G_M| (where G_E and G_M are real functions)
            if the regime where the imaginary parts of
            the form factors vanish has been attained. 
            
            The comparison of our model results for |G_E/G_M|
            with the experimental
            values for | Re(G_E/G_M)| from Refs. <cit.>
            that measure phases, are displayed in Fig. <ref>.
            
            In the graphs we notice the agreement of our results for the form factors
            (considering the model uncertainty  band) and the experimental
            data point at the largest q^2 point measured,
             q^2 ≃ 14.3 GeV^2 for the Λ
            and q^2 ≃ 8.4 GeV^2 for the Σ^+.
            
            In the graphs we ignore the uncertainties
            of the angles ΔΦ, for simplicity, and used
            their central value.
            
            The consideration of the uncertainties on ΔΦ
            only increases the error bars of the | Re(G_E/G_M)| data.
            Hopefully, future experiments will be more precise,
            which will help to determine the point were our
            model calculations provide a good description of the data
            and determine the large q^2 region where reflection symmetry sets in.
            
            
            
            
            
            
            § EFFECTIVE FORM FACTORS 
            
            
            
            In a previous work <cit.> we presented predictions
            for the effective form factors (<ref>) for
            the baryon octet members Λ, Σ^0,± and Ξ^0,-
            for q^2=10–60 GeV^2.
            At the time, only Λ, Σ^+, Σ^-, Ξ^0 and Ξ^-
            could be tested in the region of 12–15 GeV^2 by the data from CLEO <cit.>.
            
            Our estimates were on average consistent with the high q^2 data. 
            
            Since then more data for |G(q^2)| became available.
            
            In particular data for Σ^+ and Σ^- became available from BESIII <cit.>
            for the range 5.7–9.1 GeV^2.
            Surprisingly, the data are in excellent agreement with our estimates 
            (namely with the central line values, see Supplementary Material <cit.>)
            even outside the region where model validity was expected to be justified.
            
            A rough estimate of the lower limit for the application
            of our model based on the asymptotic
            relations, can be q^2 ≈ 8 M_B^2  or  q^2 ≈ 10 GeV^2.
            
            
            Recently, data from BESIII became available for
            Λ up to q^2= 21 GeV^2 <cit.>,
            and for Ξ^- up to q^2= 23 GeV^2 <cit.>.
            
            The new data are compatible with the CLEO data and extend the previous
            range covered by the experiments.
            The comparison of the new data with the model results
            from Ref. <cit.> is presented in
            Figs. <ref> and <ref> in a linear scale and in the logarithm scale for |G(q^2)|.
            
            It is clear from the figures that there is an excellent agreement between
            our predictions and the data, indicating that
            the range of the asymptotic form for the effective form factors
            may not be too far from the present day experimentally studied region.
            Future data for other hyperons may confirm or deny this tendency.
            
            
            
            
            
            
            Calculations of the effective form factors
            for all the hyperons of the baryon octet are presented in Supplementary Material <cit.>
            and compared with the world data.
            
            
            
            
            
            
            
            
            
            
            § OUTLOOK AND CONCLUSIONS 
            
            
            Experiments in new facilities (BaBar, CLEO, Belle, PANDA
            etc.) allow us to access
            the electromagnetic structure of hyperons in the elastic
            timelike region (q^2 ≥ 4 M_B^2).
            
            Till recently we had access only to
            the integrated cross section which can be used
            to determine the effective form factor |G|,
            and encode only the information
            of a certain combination of the electric and magnetic
            form factors |G_E| and |G_M|, 
            for spin 1/2 hyperons.
            
            No separated information about  |G_E| and |G_M|
            was possible till a few years ago.
            Even the separation between |G_E| and |G_M|
            for the proton and neutron was difficult
            and affected by significant errors.
            
            In the last few years it became possible
            to determine  |G_E| and |G_M| and the relative phase
            between them for Λ and Σ^+.
            
            We expect that the ratio |G_E/G_M| became
            accessible in the following years for most
            ground state hyperons (Σ^0,- and Ξ^0,-),
            as well as for charmed baryons (starting with Λ_c^+).
            
            Our calculations show a partial 
            agreement with the data in the range
            q^2 ≃ 14 GeV^2 (√(s)≃ 3.8 GeV),
            which may be an indication that our model
            calculations may be compared with future measurements
            in the range of q^2 =20–30 GeV^2
            (√(s)= 4.5–5.5 GeV)  or higher.
            
            Also the separation between the real and imaginary
            components may be tested in that range of energies.
            
            
            
            
            
              
            G.R. was supported the Basic Science Research Program 
            through the National Research Foundation of Korea (NRF)
            funded by the Ministry of Education  (Grant No. NRF–2021R1A6A1A03043957).
            M.T.P was supported by the Portuguese Science Foundation FCT
            under project CERN/FIS-PAR/0023/2021, and FCT computing project 2021.09667.
            K.T. was supported by Conselho Nacional de Desenvolvimento 
            Científico e Tecnológico (CNPq, Brazil), Processes No. 313063/2018-4 
            and No. 426150/2018-0, and FAPESP Process No. 2019/00763-0
            and No. 2023/07313-6, and his work was also part of the projects,
            Instituto Nacional de Ciência e 
            Tecnologia - Nuclear Physics and Applications 
            (INCT-FNA), Brazil, Process No. 464898/2014-5.
            The work of M.K.C. was supported by the National
            Research Foundation of Korea
            (Grant Nos. NRF–2021R1A6A1A03043957 and NRF–2020R1A2C3006177).
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            SuppM
              Supplementary material.
            
            
            
              
            
             Supplemental Material
            
            14cmWe present here calculations of the form factors
            G_E and G_M and the effective form factor G
            in the timelike region for the hyperons
            Λ, Ξ^-,0 and  the Σ^0,±.
            
            
            § FORM FACTORS G_E AND G_M
            
            
            The absolute values of the form factors G_E and G_M
            are presented in Fig. <ref>.
            The signs of G_E and G_M can be inferred from
            the labels included in the figures.
            The representation -G_E or -G_M indicate
            that the functions are negative in that region.
            The cusps indicate the zero of G_E.
            
            
            
            § EFFECTIVE FORM FACTORS G
            
            
            
            The effective form factors are presented in Fig. <ref>.
            The limits of the theoretical calculations
            are represented by the dashed lines.
            
             DataΣ^+:  Belle <cit.>,
            BESIII <cit.>,
            and CLEO <cit.>;
             DataΣ^0: BaBar <cit.>, BESIII <cit.>, 
            CLEO <cit.> and Belle <cit.>;
             DataΣ^-: BESIII <cit.>; 
             DataΛ: BaBar <cit.>, BESIII <cit.>
            and CLEO <cit.>;
             DataΞ^0: BESIII <cit.> and CLEO <cit.>;
             DataΞ^-: BESIII <cit.> and
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 Paper Title: An alternative way to decipher the nature of the doubly charmed tetraquark $T_{cc}(3875)^+$: its antiparticle $T_{{\bar c}{\bar c}}(3875)^-$ photoproduction off nuclei near threshold.









An alternative way to decipher the nature of the doubly charmed tetraquark T_cc(3875)^+: its antiparticle
T_c̅c̅(3875)^- photoproduction off nuclei near threshold
    E. Ya. Paryev

Institute for Nuclear Research of the Russian Academy of Sciences

Moscow, Russia
    August 5, 2024
==================================================================================================================================================================



§ ABSTRACT 
We study the inclusive photoproduction of T_c̅c̅(3875)^- mesons (which are the antiparticles of the doubly charmed tetraquarks T_cc(3875)^+ discovered recently by the LHCb Collaboration) from nuclei in the near-threshold energy region within the nuclear spectral function approach by considering incoherent direct (γp(n) → D^+(D^0)T_c̅c̅(3875)^-Λ^+_c) photon–nucleon T_c̅c̅(3875)^- creation processes as well as five possible different scenarios for their internal structure with the main goal of clarifying the possibility to decipher this structure (and, hence, that of T_cc(3875)^+) in photoproduction via integral and differential observables. We calculate the absolute and relative excitation functions for T_c̅c̅(3875)^- production off ^12C and ^184W target nuclei at near-threshold photon beam energies of 30–38 GeV, the absolute differential cross sections for their production off these target nuclei at laboratory polar angles of 0^∘–10^∘ as well as the A and momentum dependences of the relative (transparency ratios) cross sections for T_c̅c̅(3875)^- production at photon energy of 35 GeV within the adopted scenarios for the T_c̅c̅(3875)^- meson intrinsic structure. We demonstrate that the absolute and relative observables considered show a certain sensitivity to these scenarios. Therefore, the measurement of such observables in future experiments at the proposed high-luminosity electron-ion colliders EIC and EicC in the US and China in the near-threshold energy region might shed light on the T_cc(3875)^+ internal structure.






§ 1. INTRODUCTION


 The study of exotic heavy hadrons, which exhibit properties incompatible with the
predictions of the conventional quark model for quark-antiquark mesons and three-quark baryons and which
are composed of four or five quarks (and antiquarks) and are named as tetraquark and pentaquark
states, respectively, has received considerable interest in recent years and is one of the most exciting
topics of the nuclear and hadronic physics nowadays owing to the expectation to better understand perturbative
and nonperturbative aspects of QCD and to extend our knowledge about the world of "elementary" particles.
(for a recent overview of the multiquark states see Refs. [1–5]). This interest was triggered by
the discovery of X(3872) resonance (also known as χ_c1(3872)) by the Belle Collaboration in
2003 [6] as a narrow peak in the vicinity of the D^0D̅^*0 (D̅^0D^*0) mass threshold in the J/ψπ^+π^- invariant mass distribution in exclusive B^±→ K^±X(3872) → K^±(J/ψπ^+π^-) decays.

Recently (in 2021), the LHCb Collaboration announced the first observation of the isoscalar doubly charmed tetraquark state T_cc(3875)^+ in the D^0D^0π^+ mass spectrum in the prompt production of the proton-proton
collisions at LHC [7, 8]. This state has a quark content ccu̅d̅, a mass of approximately 3875 MeV and
manifests itself as a very narrow peak in this spectrum just below the D^0D^*+ mass threshold
[^)Thus, its mass with respect to this threshold (m_D^0+m_D^*+=3875.1 MeV) and width are
δm_ exp=-360±40^+4_-0 keV, Γ=48±2^+0_-14 keV when the T_cc(3875)^+ is fitted
with a unitarized Breit-Wigner shape [8].]^)
.
Its spin-parity quantum numbers were estimated as J^P=1^+ by the experiment. The T_cc(3875)^+ is the second
doubly charmed hadron that has been experimentally observed up to now. In 2017, 2018, the LHCb Collaboration reported the observation of the doubly charmed baryon Ξ^++_cc with quark content ccu through the non-leptonic decay modes
Ξ^++_cc→Λ^+_cK^-π^+π^+ [9] and Ξ^+_cπ^+ [10].
After the experimental discovery of the T_cc(3875)^+ meson, its internal structure has been intensively debated.
Since the mass of T_cc(3875)^+ lies just below the nominal D^0D^*+ and D^+D^*0 mass thresholds
[^)The latter one lies at a slightly higher energy of m_D^++m_D^*0=3876.51 MeV than the former and difference between them is only 1.41 MeV.]^), it is natural to treat it as an
extremely narrow D^+D^*0-D ^0D^*+ weakly bound molecular state in a relative S-wave with a very small width and binding energy, dynamically generated from the interaction of the coupled channels D ^0D^*+ and D^+D^*0 in the isospin I=0 sector (see, for example, Refs. [11–14] and those cited below). On the other hand, the T_cc(3875)^+ has possibilities to be either a compact multiquark state where the quarks and antiquarks form compact configurations with the size of a typical hadron [15–17] or a mixture of a molecular and compact tetraquark states [18, 19]
[^)It is worth noting that the large value of the ratio of the partial radiative decay widths of the
χ_c1(3872) into ψ(2S)γ and J/ψγ final states measured very recently by the
LHCb Collaboration makes the interpretation of the χ_c1(3872) state as a pure
D^0D̅^*0+D̅^0D^*0 molecule questionable and provides a strong argument in favor of a compact
component also in its structure [20].]^).
All these plausible interpretations of the T_cc(3875)^+ exotic state were supported by many current theoretical studies (see references herein below). But, in spite of large efforts the full understanding of the T_cc(3875)^+ structure remains a challenging open problem.

Most of the above various interpretations are based on the analysis of the three-body hadronic and radiative decays of
T_cc(3875)^+ via intermediate states D^0D^*+ and D^+D^*0 into the channels D^0D^0π^+, D^0D^+π^0
and D^0D^+γ, using the experimental values of its mass and total decay width as inputs [21].
However, the internal structure of T_cc(3875)^+ could also be studied in high-energy proton–proton and heavy–ion collisions at hadron colliders where a large number of charm quarks is produced, since its abundance in these collisions could reflect its structure [22]. In these collisions the T_cc(3875)^+ mesons can be produced as a compact tetraquarks from the four-quark coalescence in the QGP phase at the critical temperature or as a hadronic molecules from the coalescence of charm mesons D and D^* formed at the end of the mixed phase [22] (cf. Ref. [23] containing the similar statement for the exotic hadron X(3872)) and in them their such observables as the transverse momentum distributions and total production yields are expected to be completely different for both these configurations [24].
Alternatively, the structure of T_cc(3875)^+ can also be investigated through its antiparticle
T_c̅c̅(3875)^- photoproduction off nuclei at near-threshold energies in the same manner as it was done for the X(3872) resonance in Ref. [25].
The advantage of photoproduction compared to the hadronic collisions is that the interpretation of data from such future experiments is clearer due to insignificant strength of initial-state photon interaction and since in it a
fewer individual exclusive elementary T_c̅c̅(3875)^- meson production channels need to be accounted for.

In this work, we present the detailed predictions for the absolute and relative excitation functions for T_c̅c̅(3875)^- production off ^12C and ^184W target nuclei as well as for the A and momentum dependences of the relative (transparency ratios) cross sections for T_c̅c̅(3875)^- production from γA reactions at threshold energies obtained in the framework of the first collision model within the possible different scenarios for the T_c̅c̅(3875)^- (and, hence, for the T_cc(3875)^+) meson intrinsic structure. The predictions can be tested by future high-precision measurements at electron-ion colliders EIC [26] and EicC [27] with the aim of discriminating between these scenarios.



§ 2. FRAMEWORK: DIRECT T_C̅C̅(3875)^- PHOTOPRODUCTION MECHANISM


 Direct T_c̅c̅(3875)^- photoproduction off nuclei in the near-threshold photon beam
energy region E_γ≤ 38 GeV of interest
[^)Which corresponds to the center-of-mass energies W of the photon-proton system W ≤ 8.5 GeV,
or to the relatively "low" excess energies ϵ above the D^+T_c̅c̅(3875)^-Λ^+_c production threshold W_ th=m_D^++m_T_c̅c̅(3875)^-+m_Λ^+_c (m_D^+, m_T_c̅c̅(3875)^- and m_Λ^+_c are the D^+, T_c̅c̅(3875)^- mesons and Λ^+_c hyperon free space masses, respectively ) 0 ≤ϵ≤ 0.47 GeV and in which the doubly anticharmed tetraquark T_c̅c̅(3875)^- can be observed in γp and γA reactions at the proposed electron-ion colliders EIC [26] and EicC [27] in the US and China.]^)
can occur in the following elementary processes with the lowest free production threshold (≈ 33.9 GeV) [28, 29]:


    γ+p → D^++T_c̅c̅(3875)^-+Λ^+_c,



    γ+n → D^0+T_c̅c̅(3875)^-+Λ^+_c.

The D^+, D^0, T_c̅c̅(3875)^- mesons and Λ^+_c hyperons, produced in these processes, are sufficiently energetic.
Thus, for example, the kinematically allowed D^+, T_c̅c̅(3875)^- mesons and final Λ^+_c
hyperon laboratory momenta in the direct process (1), proceeding on the free target proton at rest, vary within the momentum ranges of 5.749–11.125 GeV/c, 13.748–20.066 GeV/c and 7.335–13.061 GeV/c, respectively, at incident photon beam energy of E_γ=35 GeV.
Since the neutron and D^0 meson masses are approximately equal to the proton and D^+ meson masses, respectively,
the kinematical characteristics of final particles, produced in the reaction (2), are close to those of
final particles in the process (1). The binding of target nucleons and their Fermi motion will distort the
distributions of the outgoing high-momentum D^+, D^0, T_c̅c̅(3875)^- mesons and Λ^+_c
hyperons as well as lead to a wider accessible for them momentum intervals compared to those given above
(cf. Figs. 8 and 9 given below). Since the medium effects are expected to be reduced for high momenta [30, 31],
we will ignore the medium modifications of these final particles in the case when the reactions (1), (2)
proceed on a nucleons embedded in a nuclear target
[^)It should be pointed out that the properties of the D, T_cc(3875)^+ and T_c̅c̅(3875)^- mesons and Λ^+_c hyperons in a strongly interacting nuclear environment at low energies have been theoretically investigated, respectively, in the works [32–42], [43, 44] and [45, 46].]^)
.

  Then, neglecting the distortion of the incident photon at energies of interest
and describing the T_c̅c̅(3875)^- meson absorption by intranuclear nucleons by the absorption cross section σ_T_c̅c̅(3875)^-N, we represent the total cross section for the production of
T_c̅c̅(3875)^- mesons off nuclei from the direct processes (1), (2) as follows [25]:


    σ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ)=I_V[A,σ_T_c̅c̅(3875)^-N]
    <σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A,

where


    I_V[A,σ]=2π∫_0^Rr_dr_∫_-√(R^2-r_^2)^√(R^2-r_^2)dz
    ρ(√(r_^2+z^2))
    exp[-σ∫_z^√(R^2-r_^2)ρ(√(r_^2+x^2))dx],



    ρ(r)=ρ_p(r)+ρ_n(r),   r=√(r_^2+z^2)    or r=√(r_^2+x^2);



    <σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A=
    ∫∫
    P_A( p_t,E)d p_tdE
    σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*))

and


    s^*=(E_γ+E_t)^2-( p_γ+ p_t)^2,



    E_t=M_A-√((- p_t)^2+(M_A-m_p+E)^2).

Here, σ_γp→ D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*)) is the "in-medium"
total cross section for the production of T_c̅c̅(3875)^- mesons in process (1)
[^)In Eq. (3) we assume that the T_c̅c̅(3875)^- production cross sections
in γp and γn reactions are the same.]^)
at the "in-medium" γp center-of-mass energy √(s^*);
ρ_p(r),  ρ_n(r) (r is the distance from the nucleus center)
and P_A( p_t,E) are normalized to the numbers of protons Z, neutrons
N and to unity the local proton, neutron densities and the
spectral function of target nucleus with mass number A (A=Z+N), having mass M_A and radius R
[^)The information about the spectral function of target nucleus, used in the subsequent calculations,
is given in Refs. [47, 48] and the local nucleon densities, employed in them, will be defined below.]^);
p_t  and E are the internal momentum and binding energy of the struck target proton
just before the collision; p_γ and E_γ are the momentum and energy of the incident photons;
m_p is the proton bare mass.

As in Ref. [25], we suppose that the "in-medium" cross section
σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*)) for T_c̅c̅(3875)^- production in process (1)
is equivalent to the free space cross section
 σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))), in which
the free space center-of-mass energy squared s(E_γ) for given photon energy E_γ, presented by the formula


    s(E_γ)=W^2=(E_γ+m_p)^2- p_γ^2=m_p^2+2m_pE_γ,

is replaced by the in-medium expression (7).
For the free total cross section σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))) no data are available up to now. Therefore,
in the photon energy range W ≤ 8.5 GeV of interest we have adopted for it the following parametrization of the results of the calculations of this cross section here (for a cutoff parameter α=1.7) within the approach [28], in which the γp → D^+T_c̅c̅(3875)^-Λ^+_c reaction is studied by considering the central diffractive mechanism and T_c̅c̅(3875)^- as a D̅D̅^* molecule:


    σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ)))=2.409(1-s_ th/s(E_γ))^1.745 [ pb],

where


    s_ th=W_ th^2=(m_D^++m_T_c̅c̅(3875)^-+m_Λ^+_c)^2.

In the energy regime W ≤ 8.5 GeV, the cross section of the γp → D^+T_c̅c̅(3875)^-Λ^+_c reaction can reach the values of the order of 0.05 pb [28]. Although, this cross section
is quite small here, one may hope that its measurements cold be performed in the future at the electron-ion colliders in the US [26] and China [27] (cf. also [28]). The empirical formulas (10), (11) will be used in our subsequent calculations
as a guideline for a reasonable evaluation of the T_c̅c̅(3875)^- yield in γA reactions.

We discuss now the local nucleon densities, employed in our calculations of the T_c̅c̅(3875)^- photoproduction on the target nuclei ^12_6C, ^27_13Al, ^40_20Ca, ^63_29Cu, ^93_41Nb, ^112_50Sn, ^184_74W, ^208_82Pb and ^238_92U considered in the present study.
For lightest nucleus ^12_6C we use the same proton and neutron density profiles of the harmonic oscillator
model [49, 50]. For nuclei ^27_13Al, ^40_20Ca and ^63_29Cu for the proton and neutron densities, ρ_p(r) and ρ_n(r), we have employed in our present calculations the Woods-Saxon distributions with the same radial parameters for protons and neutrons given in Ref. [25].
For medium-weight ^93_41Nb, ^112_50Sn and heavy ^184_74W, ^208_82Pb, ^238_92U target nuclei, we adopted for protons and neutrons the two-parameter Fermi density distributions with proton and neutron density parameters also reported in Ref. [25]. As in Ref. [25], we use here the neutron density ρ_n(r) in the
'skin' form.

To go further, we have to determine the T_c̅c̅(3875)^-–nucleon absorption cross section
σ_T_c̅c̅(3875)^-N. It is a crucial to estimate this cross section in different theoretical pictures describing  the possible underlying structure of T_c̅c̅(3875)^- for its interpretation
through the T_c̅c̅(3875)^- photoproduction on nuclei since there is a direct dependence of the rate of this production on the absorption cross section σ_T_c̅c̅(3875)^-N
(cf. Eq.(3)). This rate is expected to be different for different options for the inner structure of the
T_c̅c̅(3875)^-. That is, in our approach the rate of the T_c̅c̅(3875)^- meson
photoproduction off nuclei reflects the sensitivity to this structure. This remarkable feature can be used to better
determine/constrain the inner structure of this exotic state in the future dedicated experiments. In our present study we consider three scenarios for the hypothetical T_c̅c̅(3875)^- meson based on the proposed in literature three popular configurations for the discovered doubly charmed tetraquark T_cc(3875)^+:
i) a molecular state - a loosely bound, with large spatial size characterized by the r.m.s. radius of the order of 5–7 fm or more,  S-wave state in the D̅^0D^*-(D^-D̅^*0) system consisting of an anticharmed D̅^0 and D^*- or D^- and D̅^*0 mesons having the quarks contents c̅u and c̅d or c̅d and c̅u, respectively, [51–65]
[^)Loosely bound S-wave charm-meson molecules like X(3872) and T_cc(3875)^+ have universal
properties  determined by their binding energies [66, 67]. Accordingly, within the molecular hypothesis, the r.m.s. size r_T of the separation of the D̅^0 and D^*- constituents in the T_c̅c̅(3875)^- molecule
can be estimated from the relation r_T=1/√(4μ_0δ_T_c̅c̅(3875)^-)
by knowing the binding energy δ_T_c̅c̅(3875)^-=m_D̅^0+m_D^*--m_T_c̅c̅(3875)^- of the T_c̅c̅(3875)^- molecule [66, 67]. Here, μ_0 is the D̅^0D^*- reduced mass. Assuming that this binding energy is equal to that δ_T_cc(3875)^+ of the T_cc(3875)^+
state relative to the D^0D^*+ mass threshold and taking δ_T_cc(3875)^+=360±40^+4_-0 keV as input
in the case when an unitarized Breit-Wigner shape is used to fit the T_cc(3875)^+
[8], we find that the anticharmed mesons in the T_c̅c̅(3875)^- have a sufficiently large r.m.s. separation: r_T=5.3±0.3 fm. The typical scale for the relative momentum between the D̅^0 and
D^*- mesons [66, 68], √(2μ_0δ_T_c̅c̅(3875)^-)≃25–30 MeV/c, is much smaller than their average laboratory momenta ∼ 8.1 and 8.8 GeV/c, respectively (see above). With these, we expect that the application of the considered below quasi-free approximation to evaluate the T_c̅c̅(3875)^-–nucleon absorption cross section in molecular scenario is sufficiently well justified.]^),
ii) compact, ∼ 1 fm, diquark [ud] and antidiquark [c̅c̅] tetraquark state [24, 69–72],
and iii) a hybrid state - a state in which the T_c̅c̅(3875)^- wave function contains a molecular and
compact four-quark components [18, 73].

In the "pure" molecular picture of the T_c̅c̅(3875)^-, whose wave function takes the
following charge-conjugation form


    |T_c̅c̅(3875)^->_ mol=1/√(2)(|D^-D̅^*0>-|D̅^0D^*->)

compared to that


    |T_cc(3875)^+>_ mol=1/√(2)(|D^+D^*0>-|D^0D^*+>)

for the T_cc(3875)^+ in the isoscalar channel [52, 64, 74, 75]
[^)It should be pointed out that in a molecular picture (13) the T_cc(3875)^+ couples equally
to the D^+D^*0 and D^0D^*+ channels. However, in some works (see, for example, Refs. [53, 64, 65]) it was predicted that the T_cc(3875)^+ flavor wave function contains about 70% of the D^0D^*+ and 30% of the D^+D^*0
molecular components. Since in our model the specific flavor content of the T_cc(3875)^+ (and, respectively, of the T_c̅c̅(3875)^-) has no influence on the determination of the T_c̅c̅(3875)^-–nucleon absorption cross section in a molecular scenario, we will treat the T_c̅c̅(3875)^- below in this
scenario as a D^-D̅^*0-D̅^0D^*- molecule.]^)
and in which the constituents (anticharmed mesons) are weakly bound and have large spatial separation in coordinate space (see above)
the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N^ mol can be naturally estimated in the quasi-free approximation [22, 76, 77]. In this approximation, the binding energy of
the anticharmed mesons is ignored. 
They are taken to be on-shell and they are considered to fly together with an average laboratory momentum of the order of 8 GeV/c each. In this approximation the
T_c̅c̅(3875)^- is absorbed when a nuclear nucleon (proton or neutron) interacts (elastically or inelastically) with the D̅^0 or with the D^*- (or with the D^- or with the D̅^*0).
In each of these interactions the other anticharmed meson is a spectator.
Then, assuming that the total cross sections of the free D^*-N and D̅^*0N interactions are the
same as those for the D^-N and D̅^0N ones [78, 79], we can evaluate the cross section for T_c̅c̅(3875)^- absorption, σ_T_c̅c̅(3875)^-N^ mol, as [25, 77]:


    σ_T_c̅c̅(3875)^-N^ mol=σ_T_c̅c̅(3875)^-p^ mol=σ_T_c̅c̅(3875)^-n^ mol,

where the T_c̅c̅(3875)^-p and T_c̅c̅(3875)^-n absorption cross sections, σ_T_c̅c̅(3875)^-p^ mol and σ_T_c̅c̅(3875)^-n^ mol,
are given by


    σ_T_c̅c̅(3875)^-p(n)^ mol≈σ_D̅^0p(n)^ el+σ_D̅^0p(n)^ in+σ_D^-p(n)^ el+σ_D^-p(n)^ in+σ_D^-p →D̅^0n(σ_D̅^0n → D^-p).

Here, σ_D̅^0p(n)^ el(in) and σ_D^-p(n)^ el(in) are the elastic (inelastic) cross
sections of the free D̅^0p(D̅^0n) and D^-p(D^-n) interactions, respectively. And,
σ_D^-p →D̅^0n and σ_D̅^0n → D^-p are the total cross sections of the free charge-exchange reactions D^-p →D̅^0n and D̅^0n → D^-p, correspondingly. In the calculations we employ for them the following constants, which are relevant to the momentum regime above of 1 GeV/c of our interest:
σ_D̅^0p(n)^ el=σ_D^-p(n)^ el=10 mb,
σ_D̅^0p(n)^ in=σ_D^-p(n)^ in=0,
σ_D^-p →D̅^0n=σ_D̅^0n → D^-p=12 mb [78, 79]. Adopting these values, we get that
σ_T_c̅c̅(3875)^-p^ mol=σ_T_c̅c̅(3875)^-n^ mol=32 mb and, in line with Eq. (14), σ_T_c̅c̅(3875)^-N^ mol=32 mb.

In the second scenario, in which the T_c̅c̅(3875)^- meson is treated as a compact c̅c̅ud
tetraquark state, its breakup cross section σ_T_c̅c̅(3875)^-N^ 4q can be approximated
by the following relation [77]:


    σ_T_c̅c̅(3875)^-N^ 4q≈1/3σ_T_c̅c̅(3875)^-N^ mol.

For σ_T_c̅c̅(3875)^-N^ mol=32 mb we obtain that σ_T_c̅c̅(3875)^-N^ 4q=10.7 mb. It is interesting to note that this value for the quantity
σ_T_c̅c̅(3875)^-N^ 4q is similar to that of 13.3 mb, adopted in Ref. [25]
for the high-momentum X(3872)–nucleon absorption cross section σ_X(3872)N^ 4q considering
the X(3872) meson as a compact tetraquark with radius r_4q=0.65 fm
and approximating the latter cross section by a geometrical cross section σ_4q^ geo=πr_4q^2.

In the hybrid scenario, it is assumed that the T_c̅c̅(3875)^- wave function is a linear superposition of the compact c̅c̅ud (or 4q) and the molecular D̅D̅^* components [18, 73]:


    |T_c̅c̅(3875)^->_ hyb=α|c̅c̅ud>+β/√(2)(|D^-D̅^*0>-|D̅^0D^*->).

Here, α^2 and β^2 represent the probabilities to find a compact and non-compact hadronic configurations,
respectively, for the normalization


    α^2+β^2=1.

The limiting case of α^2=0, β^2=1 corresponds to the pure molecular interpretation of the T_c̅c̅(3875)^- state, while the case of α^2=1, β^2=0 refers to its pure compact four-quark treatment.
Following Refs. [18, 43, 44, 73], we assume that the T_c̅c̅(3875)^- wave function (17) contains 80% of the non-molecular 4q component and 20% of the molecular D̅D̅^* component ((α^2,β^2)=(0.8,0.2)), 50% of the compact component and 50% of the non-compact component
((α^2,β^2)=(0.5,0.5)) as well as 20% of the non-molecular component and 80% of the molecular component
((α^2,β^2)=(0.2,0.8)).
These five options for the non-molecular and molecular probabilities of the T_c̅c̅(3875)^- cover the bulk of theoretical and experimental information presently available in this field and they will allow us to see the sensitivity of the T_c̅c̅(3875)^- meson photoproduction cross sections from the direct processes (1), (2) on nuclei to its internal structure. In the hybrid picture (17) of the T_c̅c̅(3875)^-, we can represent  the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N^ hyb in the following incoherent probability-weighted sum [25]:


    σ_T_c̅c̅(3875)^-N^ hyb=α^2σ_T_c̅c̅(3875)^-N^ 4q+β^2σ_T_c̅c̅(3875)^-N^ mol.

In view of the above, we set σ_T_c̅c̅(3875)^-N^ 4q=10.7 mb and σ_T_c̅c̅(3875)^-N^ mol=32 mb.
With these values, the T_c̅c̅(3875)^- absorption cross section (19) is σ_T_c̅c̅(3875)^-N^ hyb=15.0, 21.35 as well as 27.7 mb for the non-molecular and molecular probabilities of the T_c̅c̅(3875)^- 80% and 20%, 50% and 50% as well as 20% and 80%, respectively. We summarize below the results obtained above for the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N in the considered schemes for the T_c̅c̅(3875)^- tetraquark inner configuration:


    σ_T_c̅c̅(3875)^-N={[  10.7  mb         ,;          ;  15.0  mb         ,;          ; 21.35  mb         ,;          ;  27.7  mb         ,;          ;    32  mb         . ].

Therefore, the T_c̅c̅(3875)^- as a D̅D̅^* molecule has the largest absorption cross section and, hence, it is expected to be more easily absorbed than its other configurations in a nuclear medium.
We will use these values for the quantity σ_T_c̅c̅(3875)^-N in our calculations.

Neglecting the nuclear effects considered: the target nucleon binding and Fermi motion, from Eq. (3)
we obtain the following simple expression for the total cross section
σ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ):


    σ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ)=I_V[A,σ_T_c̅c̅(3875)^-N]
    σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))),

where the elementary cross section σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))) is given above by Eq. (10).
This expression is valid only at photon energies well above threshold
[^)In our case, the difference between the expressions (3) and (21) is small at
incident photon energies around 37 GeV and more, and it becomes substantial at photon energies close to the
threshold energy of 33.9 GeV as our calculations have shown
(compare magenta short-dashed and cyan short-dashed-dotted curves in figures 1–4 given below).]^)
.
As a measure for the
T_c̅c̅(3875)^- absorption cross section σ_T_c̅c̅(3875)^-N in nuclei and, hence, for the T_c̅c̅(3875)^- intrinsic configuration we will use the so-called T_c̅c̅(3875)^- transparency ratio defined as [80, 81]:


    S_A=σ_γA →T_c̅c̅(3875)^-X(E_γ)/A σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))),

i.e. the ratio of the inclusive nuclear T_c̅c̅(3875)^- photoproduction cross section divided by A times the same quantity on a free proton. It should be noted that this relative observable is more
favorable compared to those based on the absolute cross sections for the aim of obtaining the information
on the T_c̅c̅(3875)^- nuclear absorption [25] (and, hence, on the T_c̅c̅(3875)^- internal configuration).
It is expected that the direct processes (1), (2) will be dominant in the T_c̅c̅(3875)^- production in γA interactions close to threshold
[^)At least, at incident photon energies ≤ 35 GeV, where
the elementary processes γN → DT_c̅c̅(3875)^-Λ^+_cπ
with one pion in the final states are expected to be suppressed in T_c̅c̅(3875)^- production in γA reactions compared to those of (1), (2) due to their larger production threshold energy (≈ 35.1 GeV) in free γN collisions. Moreover, since the main inelastic channel in γN
collisions at these photon energies is the multiplicity production of pions with rather
low energies the secondary pion-induced T_c̅c̅(3875)^- production processes are energetically suppressed at them as well.
We expect that the full inclusive nuclear T_c̅c̅(3875)^- photoproduction cross section σ_γA →T_c̅c̅(3875)^-X(E_γ), entering into Eq. (22), is entirely exhausted by that of Eq. (3) from the direct T_c̅c̅(3875)^- production processes (1), (2) not only at photon energies E_γ≤ 35 GeV, but also at higher energies considered in the present study.]^).
Then, according to Eqs. (3) and (4), we have:


    S_A=σ_γA →T_c̅c̅(3875)^-X^( dir)(E_γ)/A σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ)))=
    I_V[A,σ_T_c̅c̅(3875)^-N]/A<σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A/σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))).

Usually, the transparency ratio of the meson is normalized to a light nucleus like ^12C [80, 82].
With this and Eq. (23), for T_c̅c̅(3875)^- we have:


    T_A=S_A/S_C=12 σ_γA →T_c̅c̅(3875)^-X^( dir)(E_γ)/A σ_γ C→T_c̅c̅(3875)^-X^( dir)(E_γ)=
    12 I_V[A,σ_T_c̅c̅(3875)^-N]/A I_V[ C,σ_T_c̅c̅(3875)^-N]<σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A/<σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_ C.

In the case when direct T_c̅c̅(3875)^- production processes (1), (2) proceed on a free target
nucleons being at rest the expression (24) for the quantity T_A is reduced to the following simpler form:


    T_A ≈12 I_V[A,σ_T_c̅c̅(3875)^-N]/A I_V[ C,σ_T_c̅c̅(3875)^-N].

The formulas (21)–(25) will be used in calculations of the integral observables considered at photon energies of our interest.







Before going further, we consider now the momentum-dependent inclusive differential cross section for production
of T_c̅c̅(3875)^- mesons with momentum p_T_c̅c̅(3875)^- and total energy
E_T_c̅c̅(3875)^- (or with momentum p_T_c̅c̅^- and total energy E_T_c̅c̅^- for brevity) on nuclei in the direct processes (1) and (2) in γA reactions. According to Eqs. (3), (6) and Ref. [83], we represent this cross section as follows:


    dσ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^-=
    I_V[A,σ_T_c̅c̅(3875)^-N]
    <dσ_γp → D^+T_c̅c̅(3875)^-Λ^+_c( p_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^->_A,

where the off-shell elementary differential cross section
dσ_γp → D^+T_c̅c̅(3875)^-Λ^+_c( p_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^- for T_c̅c̅(3875)^- production in process (1)
is given by (cf. Ref. [83])


    dσ_γp → D^+T_c̅c̅(3875)^-Λ^+_c( p_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^-=π/4σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*))/I_3[s^*,m_D^+,m_T_c̅c̅(3875)^-,m_Λ^+_c]E_T_c̅c̅^-λ[s^*_D^+Λ^+_c,m_D^+^2,m_Λ^+_c^2]/s^*_D^+Λ^+_c,

assuming for simplicity that the T_c̅c̅(3875)^- angular distribution in reaction (1) is
isotropic in the γp c.m.s. at photon beam energies of interest.
Here,


    I_3[s^*,a,b,c]=(π/2)^2∫_(b+c)^2^(√(s^*)-a)^2λ[x,b^2,c^2]/xλ[s^*,x,a^2]/s^*dx,



    λ(x,y,z)=√([x-(√(y)+√(z))^2][x-
    (√(y)-√(z))^2]),



    s^*_D^+Λ^+_c=s^*+m_T_c̅c̅(3875)^-^2-2(E_γ+E_t)E_T_c̅c̅^-+2( p_γ+ p_t)
     p_T_c̅c̅^-

and the quantity s^* is defined above by Eq. (7).
At the initial photon energies of interest the T_c̅c̅(3875)^- mesons are produced at very small
laboratory polar angles
[^)Thus, for example, the maximum angle of T_c̅c̅(3875)^- meson production off a free proton at rest in reaction (1) is about 2.5^∘ at photon energy of 35 GeV.]^)
. Therefore, we will calculate their momentum distribution from target nuclei considered
for the laboratory solid angle ΔΩ_T_c̅c̅^- = 0^∘≤θ_T_c̅c̅^-≤ 10^∘ and 0 ≤φ_T_c̅c̅^-≤ 2π.
Then, in line with Eq. (26), we can get the following expression for this distribution:


    dσ_γA→T_c̅c̅(3875)^-X^( dir)
    (E_γ,p_T_c̅c̅^-)/dp_T_c̅c̅^-=
    ∫_ΔΩ_T_c̅c̅^-^dΩ_T_c̅c̅^-dσ_γA→T_c̅c̅(3875)^-X^( dir)
    (E_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^-p_T_c̅c̅^-^2


    =2πI_V[A,σ_T_c̅c̅(3875)^-N]
    ∫_cos10^∘^1dcosθ_T_c̅c̅^-<dσ_γp→ D^+T_c̅c̅(3875)^-Λ^+_c(p_γ,
    p_T_c̅c̅^-,θ_T_c̅c̅^-)/dp_T_c̅c̅^-dΩ_T_c̅c̅^->_A.




























§ 3. RESULTS AND DISCUSSION


 We discuss now the results of our calculations in the framework of the approach outlined above.
First of all, we consider the excitation functions for production of T_c̅c̅(3875)^- mesons off
^12C and ^184W nuclei. They were calculated in line with Eq. (3)
for five adopted options for the T_c̅c̅(3875)^- absorption cross section σ_T_c̅c̅(3875)^-N in nuclear medium (cf. Eq. (20)) as well as in line with formula (21) for the free target nucleon
at rest with the value of σ_T_c̅c̅(3875)^-N=32 mb corresponding to the molecular picture for the T_c̅c̅(3875)^-, and are given in Figs. 1 and 2, respectively.
One can see that the difference between calculations with and without
accounting for the target nucleon Fermi motion (between magenta short-dashed and cyan short-dashed-dotted curves)
is sufficiently small at well above threshold photon energies ∼ 37–38 GeV, while at lower incident energies its impact on the T_c̅c̅(3875)^- yield is substantial.
It can be seen yet that the T_c̅c̅(3875)^- meson total cross sections show a definite sensitivity to the
absorption cross section σ_T_c̅c̅(3875)^-N and, hence, to the T_c̅c̅(3875)^- internal structure, for both target nuclei and for all photon energies considered.
But their absolute values are quite small even at above threshold photon energies of ∼ 35–38 GeV.
At these energies, they are ∼ 0.1–0.5 and 1–5 pb for carbon and tungsten target nuclei, correspondingly,
which is accessible at the proposed electron-ion colliders [26, 27] (see below).
For the heavy target nucleus ^184W, there are experimentally distinguishable
differences between the considered options:
σ_T_c̅c̅(3875)^-N=32 mb and σ_T_c̅c̅(3875)^-N=27.7 mb, σ_T_c̅c̅(3875)^-N=27.7 mb and σ_T_c̅c̅(3875)^-N=21.35 mb, σ_T_c̅c̅(3875)^-N=21.35 mb and σ_T_c̅c̅(3875)^-N=15.0 mb, σ_T_c̅c̅(3875)^-N=15.0 mb and σ_T_c̅c̅(3875)^-N=10.7 mb
or between the considered scenarios:
pure molecular state and hybrid state with the non-molecular and molecular probabilities of 20% and 80%,
hybrid state with the non-molecular and molecular probabilities of 20% and 80% and
hybrid state with the non-molecular and molecular probabilities of 50% and 50%,
hybrid state with the non-molecular and molecular probabilities of 50% and 50% and hybrid state with the non-molecular and molecular probabilities of 80% and 20%, hybrid state with the non-molecular and molecular probabilities of 80% and 20% and pure 4q compact state for the T_c̅c̅(3875)^-–nucleon absorption cross section or for the T_c̅c̅(3875)^- intrinsic configuration. They are ∼ 12, 21, 25, 20% for all considered photon energies. For the light target nucleus ^12C, the sensitivity of the cross sections to these quantities becomes somewhat lower and, respectively, the same differences as above become somewhat smaller. They are ∼ 6, 11, 11, 8%. Measurement of such small effects is a difficult experimental challenge.
On the other hand, the differences between calculations corresponding to the molecular configuration and hybrid configuration with the non-molecular and molecular probabilities of 50% and 50%, hybrid configuration with the non-molecular and molecular probabilities of 50% and 50% and pure 4q compact configuration for the T_c̅c̅(3875)^- are yet not so small. They are ∼ 18, 20%. This means that, at least, these configurations could be tested in the future T_c̅c̅(3875)^- measurements on carbon and light target nuclei like carbon as well.
To motivate such measurements at the future colliders EicC and EIC, it is desirable to estimate the T_c̅c̅(3875)^- production rates (the event numbers) in the γ^12C and γ^184W reactions.
For this purpose, we translate the T_c̅c̅(3875)^- photoproduction total cross sections, reported above, into the expected event numbers for the T_c̅c̅(3875)^- from the reactions
γ^12 C(^184 W) →T_c̅c̅(3875)^-X, T_c̅c̅(3875)^- →D̅^0D̅^0π^-, D̅^0 → K^+π^-, D̅^0→ K^+π^-.
To estimate the total numbers of the T_c̅c̅(3875)^- events in a one-year run at EicC, one needs
to multiply the above T_c̅c̅(3875)^- photoproduction total cross sections of 0.1–0.5 and 1–5 pb
on the carbon and tungsten target nuclei, respectively, by the integrated luminosity of 60 fb^-1 [27, 84]
as well as by the detection efficiency and by the appropriate branching ratios
Br[T_c̅c̅(3875)^- →D̅^0D̅^0π^-]≈ 60% and
Br[D̅^0 → K^+π^-]≈ 4%
[^)We suppose here that Br[T_c̅c̅(3875)^- →D̅^0D̅^0π^-]=Br[T_c c(3875)^+ → D^0D^0π^+]≈ 60% [18, 55, 58, 85] and Br[D̅^0 → K^+π^-]=Br[D^0 → K^-π^+]≈ 4% [86].]^)
.
With a realistic 50% detection efficiency [27],
we estimate about of 3–15 and 30–150 events per year for the T_c̅c̅(3875)^- signal in the cases of the ^12C and ^184W target nuclei, respectively. The event numbers for
T_c̅c̅(3875)^- →D̅^0(→ K^+π^-)D̅^0(→ K^+π^-)π^- at EIC
are about of 15–75 and 150–750 with an integrated luminosity of 300 fb^-1 [26, 84].
We see that a sufficiently large number of T_c̅c̅(3875)^- events could be observed (especially at
EIC collider) at well above threshold photon energies of 35–38 GeV.
Thus, the electron-ion colliders EicC and EIC provide a good platform to study the nature of the T_c̅c̅(3875)^- (and, respectively, of the T_cc(3875)^+) in near-threshold photonuclear reactions.

To gain further insight, we show in Figs. 3 and 4 the energy dependences of the
ratios of the cross sections, presented in Figs. 1 and 2, to the cross section, calculated in the molecular picture for the T_c̅c̅(3875)^- for an off-shell target nucleons, on the same linear scale for ^12C and ^184W target nuclei, respectively. It is clearly seen that for the heavy ^184W target nucleus
the sensitivity of the T_c̅c̅(3875)^- meson yield to variations in its intrinsic structure is
significantly stronger than that in the case of light ^12C target nucleus and it would definitely be possible for ^184W nucleus to distinguish between the results corresponding to all considered choices for this structure at all initial photon energies. At the same time, for light ^12C target nucleus we see a big enough distinctions between
the curves corresponding to the pure molecular configuration, hybrid configuration with the compact and non-compact probabilities of 50% and 50% and pure compact tetraquark configuration for the T_c̅c̅(3875)^-. On the basis of the foregoing, we can conclude that the structure of T_c̅c̅(3875)^- mesons could be studied at the future electron-ion colliders via the energy dependence of their absolute (and relative) total production cross sections in inclusive near-threshold photonuclear reactions. Figures 3 and 4 also show clearly that the difference between the absolute T_c̅c̅(3875)^- photoproduction cross sections, calculated in the pure molecular picture for the T_c̅c̅(3875)^- with allowance for the impact of the binding of intranuclear nucleons and their Fermi motion on the direct processes (1), (2) and without it (cf. Eqs. (3) and (21)) becomes indeed small only for above-threshold photon energies ∼ 37–38 GeV.

Additional information about the sensitivity of the cross sections to the internal structure of the
T_c̅c̅(3875)^- meson is contained in Figs. 5 and 6, where the A–dependences of the transparency ratios S_A and T_A from the direct reaction channels (1), (2) in γA
(A=^12C, ^27Al, ^40Ca, ^63Cu, ^93Nb, ^112Sn, ^184W, ^208Pb, and ^238U)
collisions, calculated for the photon beam energy of 35 GeV in line with Eqs. (23) and (24), respectively,
and for the values of the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N given above by Eq. (20).
It is seen that the transparency ratio S_A depends more strongly on this cross section
or on the adopted interpretation of T_c̅c̅(3875)^- than that for the quantity T_A.
Thus, there are a sizeable changes ∼ 8, 13, 17, 11% in the ratio S_A
between calculations corresponding to the cases when the loss of T_c̅c̅(3875)^- mesons in nuclear
matter is determined by their absorption cross sections of 32 and 27.7 mb, 27.7 and 21.35 mb, 21.35 and 15 mb, 15 and 10.7 mb, respectively, (cf. Eq. (20)) already for comparably "light" nuclei (^27Al, ^40Ca).
For the medium-mass (^93Nb, ^112Sn) and heavy (^184W, ^238U) target nuclei these changes are
larger and already are measurable. They are about 10, 18, 22, 18% and 13, 21, 27, 21%, respectively.
We see that the highest sensitivity of the quantity S_A to the T_c̅c̅(3875)^-
structure is observed, as is expected, for heavy target nuclei.
For the quantity T_A the analogous changes are smaller and are almost insignificant. They are about
2, 3, 5, 3%, 4, 7, 10, 10% and 6, 10, 15, 12%, correspondingly, in the cases of "light", medium-mass
and heavy target nuclei indicated above.
Therefore, we conclude that the observation, at least, of the A dependence of the transparency ratio S_A, especially for large mass numbers A, in the future high-precision photoproduction experiments at electron-ion colliders
opens the possibility to discriminate between all considered inner configurations of the T_c̅c̅(3875)^- tetraquark state.
At the same time, the precise T_c̅c̅(3875)^- photoproduction data on the A dependence of the transparency ratio T_A in the range of large A, obtained in such experiments, could also additionally help to distinguish, at least, between a pure compact tetraquark state and compact tetraquark-molecule mixture with the compact and molecular probabilities ∼ 50% and 50%, compact tetraquark-molecule mixture with the compact and molecular probabilities ∼ 50% and 50% and a pure molecular configuration of the T_c̅c̅(3875)^- exotic state.

In addition to the A dependence of the transparency ratio T_A at photon energy of 35 GeV presented in Fig. 6, we have investigated the initial photon energy dependence of this ratio for the ^184W/^12C combination. This dependence is shown in Fig. 7. It was calculated in line with Eq. (24) for five adopted options for the T_c̅c̅(3875)^- meson intrinsic structure and for the off-shell target nucleons as well as in line with simple formula (25) for its pure molecular interpretation and for the free target nucleons at rest. One can see that there is a certain
sensitivity of the transparency ratio T_A to various interpretations of T_c̅c̅(3875)^- considered
at all studied photon energies. At these energies, the sensitivity is similar to that available in Fig. 6 for heavy nuclei. Inspection of this figure tells us yet that the simple formula (25) describes well the quantity T_A at above
threshold photon energies (at energies E_γ > 33.9 GeV), where (and at subthreshold energies E_γ < 33.9 GeV) it exhibits practically a flat behavior. This behavior of the transparency ratio T_A can also be used for discriminating between possible scenarios for the T_c̅c̅(3875)^- internal structure.
Thus, it is optimistic that this relative (also integral) observable can, additionally, be used to help
determine this structure.

We discuss now the T_c̅c̅(3875)^- differential observables.
Figs. 8 and 9 show the absolute T_c̅c̅(3875)^- meson momentum distributions from the direct
processes (1), (2), respectively, in γ^12C and γ^184W reactions, determined in line with
Eq. (31) for laboratory polar angles of 0^∘–10^∘, for photon beam energy of 35 GeV and
for five adopted values of the T_c̅c̅(3875)^-–nucleon absorption cross section
in the considered interpretations of T_c̅c̅(3875)^-.
Our results indicate that the differential cross sections for producing the T_c̅c̅(3875)^- mesons
on ^12C nucleus are roughly one order of magnitude smaller than those on ^184W nucleus and, therefore,
they have a lesser chance to be observed (at EIC).
The latter ones have, according to estimates given above, a rather measurable (at EIC) strength ∼ 0.1–0.3 pb/(GeV/c) in the central momentum region of 13–19 GeV/c and also show a rather sizeable variations
[^)Which are similar to those shown in Fig. 2.]^)
, when going from the T_c̅c̅(3875)^-–nucleon absorption cross section value of 10.7 to 32.0 mb.
Such behavior of the T_c̅c̅(3875)^- production differential cross sections on tungsten target nucleus
can also be used to decipher the T_c̅c̅(3875)^- inner structure from comparison our model
calculations with future experimental data.

 Finally, in Fig. 10 we show the "differential" transparency ratio T_A for target combination W/C for
T_c̅c̅(3875)^- mesons produced in the direct reaction channels (1), (2) at laboratory angles
of 0^∘–10^∘ by 35 GeV photons as a function of the T_c̅c̅(3875)^- laboratory
momentum. This ratio is calculated in line with Eq. (24) by using in it the differential cross sections presented
in Figs. 8 and 9 instead of the total cross sections. One can see that the sensitivity of the transparency ratio
T_A to the adopted configurations of T_c̅c̅(3875)^- is similar to that available in Fig. 7 for
the "integral" quantity T_A considered as a function of photon energy. The transparency ratio T_A exhibits dips
at momenta ∼ 9–10 GeV/c. This can be explained by the fact that the differential cross sections for
T_c̅c̅(3875)^- production on ^184W nucleus "are bent down" at these momenta (see Fig. 9)
due to the off-shell kinematics of the direct photon-nucleon collision and the role played by the nucleus-related
effects such as the target nucleon binding and Fermi motion, encoded in the nuclear spectral function
P_A( p_t,E). The spectral functions for ^12C and ^184W, employed in the calculations,
are different [48]. This behavior of the "differential" transparency ratio T_A can additionally be adopted for discriminating between possible schemes for the T_c̅c̅(3875)^- internal structure.

In conclusion, the absolute total and differential cross sections for production of T_c̅c̅(3875)^-
mesons in near-threshold photonuclear reactions as well as their relative (transparency ratios) yields might allow one to shed light on the T_c̅c̅(3875)^- (and, respectively, on the T_cc(3875)^+) intrinsic configuration.



§ 4. CONCLUSIONS


 Motivated by the experimental discovery of a doubly charmed tetraquark state T_cc(3875)^+, in this work we have studied theoretically the inclusive photoproduction of T_c̅c̅(3875)^- mesons (which are the antiparticles of the T_cc(3875)^+) from nuclei in the near-threshold energy region within the nuclear spectral function approach by considering incoherent direct (γp(n) → D^+(D^0)T_c̅c̅(3875)^-Λ^+_c) photon–nucleon T_c̅c̅(3875)^- creation processes as well as five possible different scenarios for their internal structure with the main goal of clarifying the possibility to decipher this structure (and, hence, that of T_cc(3875)^+) in photoproduction via integral and differential observables. We have calculated the absolute and relative excitation functions for T_c̅c̅(3875)^- production off ^12C and ^184W target nuclei at near-threshold photon beam energies of 30–38 GeV, the absolute differential cross sections for their production off these target nuclei at laboratory polar angles of 0^∘–10^∘ as well as the A and momentum dependences of the relative (transparency ratios) cross sections for T_c̅c̅(3875)^- production at photon energy of 35 GeV within the adopted scenarios for the T_c̅c̅(3875)^- meson intrinsic structure. We show that the absolute and relative observables considered reveal a certain sensitivity to these scenarios. Hence, they may be an important tool to clarify it and that of the T_cc(3875)^+. The measurements of these observables could be performed in the future experiments at the proposed high-luminosity electron-ion colliders EIC and EicC in the US and China.



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 Paper Title: Dark Matter Search Results from 1.54 Tonne$\cdot$Year Exposure of PandaX-4T.








School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for Particle Astrophysics and Cosmology (MoE), Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240, China
Jinping Deep Underground Frontier Science and Dark Matter Key Laboratory of Sichuan Province
School of Physics, Beihang University, Beijing 102206, China
Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China
Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing, 102206, China
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, China
State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, China
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
International Research Center for Nuclei and Particles in the Cosmos & Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing 100191, China
School of Physics, Peking University, Beijing 100871, China
Yalong River Hydropower Development Company, Ltd., 288 Shuanglin Road, Chengdu 610051, China
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, 201800 Shanghai, China
Center for High Energy Physics, Peking University, Beijing 100871, China
Research Center for Particle Science and Technology, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao 266237, Shandong, China
Key Laboratory of Particle Physics and Particle Irradiation of Ministry of Education, Shandong University, Qingdao 266237, Shandong, China
Department of Physics, University of Maryland, College Park, Maryland 20742, USA
New Cornerstone Science Laboratory, Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 201210, China
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai, 519082, China
School of Physics, Nankai University, Tianjin 300071, China
Department of Physics, Yantai University, Yantai 264005, China
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
School of Medical Instrument and Food Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Shanghai Jiao Tong University Sichuan Research Institute, Chengdu 610213, China
SJTU Paris Elite Institute of Technology, Shanghai Jiao Tong University, Shanghai, 200240, China
School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China
School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China
College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu 610059, China


















 



















[Spokesperson: ]jianglai.liu@sjtu.edu.cn





















[Corresponding author: ]taoyi92@sjtu.edu.cn




























 









 









PandaX Collaboration




             

§ ABSTRACT 
In this letter, we report the dark matter search results from the commissioning run and the first science run of the experiment.
A blind analysis is carried out on the entire data set. The data processing is improved compared to previous work, unifying the low-level signal reconstruction in a wide energy range up to 120 keV.
With a total exposure of 1.54 tonne·year, no significant excess of nuclear recoil events is found.
The lowest 90% confidence level exclusion on the spin-independent cross section is 1.6 × 10^-47 cm^2 at a dark matter mass of 40 .
Our results represent the most stringent constraint for a dark matter mass above 100 .



                              

Dark Matter Search Results from 1.54 Tonne·Year Exposure of 
    Zhizhen Zhou
    August 5, 2024
============================================================



Overwhelming astrophysical and cosmological evidence points to the existence of dark matter (DM) in our Universe <cit.>.

Direct detection of dark matter particles through their potential interaction with ordinary matter is pivotal to our understanding of the fundamental nature of DM. 
Among many experimental endeavors, liquid xenon time projection chambers (TPCs) have spearheaded the detection sensitivities <cit.>, with their sensitivity steadily approaching the irreducible neutrino-nucleus scattering background <cit.>.


The experiment <cit.> is located in the B2 hall of the China Jinping Underground Laboratory (CJPL-II) <cit.>.
The xenon TPC is inside a dual-vessel stainless steel cryostat, which holds 5.6 tonnes of total liquid xenon. The cryostat is housed in a 10-meter diameter by 13-meter height stainless steel tank, which is filled with ultrapure water to provide radioactivity shielding.

The xenon cooling and purification system comprises three cooling heads and two independent re-circulation loops, continuously removing the electronegative impurities via hot getters <cit.>. 

The detector features a cylindrical, dual-phase xenon TPC delineated by 24 reflective polytetrafluoroethylene (PTFE) panels, with an 1185 mm separation across opposing panels at room temperature.
Its internal electrical fields are established through a cathode grid at the bottom, and a gate and anode mesh positioned immediately below and above the liquid level, with a vertical spacing of 1185 mm and 10 mm, respectively.

Energy depositions produce prompt scintillation (S1) and delayed electroluminescence (S2) photons, which are detected by a total of 368 Hamamatsu R11410-23 3-inch photomultiplier tubes (PMTs) positioned at the top and bottom of the TPC. Each PMT's waveform is read out by a digitizer channel running at a sampling rate of 250 MS/s, with a self-trigger threshold set at around 1/3 of a photoelectron <cit.>. The S1-S2 detection allows precise energy and position reconstruction of the events.


The ratio S2/S1 also provides excellent discrimination between electron recoil (ER) and nuclear recoil (NR) events.



More detailed information of configurations and operation can be found in Ref. <cit.> and the associated references therein.



The commissioning run contains 95 days of stable data taking operation from November 28, 2020, to April 16, 2021 (Run0), separated into five sets according to different detector conditions.
Multiple analyses of the Run0 data have been conducted to search for various DM model particles <cit.>.
To reduce the tritium background identified in Run0, a dedicated offline xenon distillation <cit.> was carried out after Run0.
resumed operations in the summer of 2021, collecting over 164 days of scientific data from November 16, 2021, to May 15, 2022 (Run1). Afterwards, the operation was halted until the end of 2023 due to ongoing construction at CJPL-II. In this paper, we present a combined DM analysis using data from both Run0 and Run1.


Although most of the detector configurations remained the same as in Run0, several experimental issues occurred during Run1. In total, an additional fifteen PMT channels were malfunctioned: eight due to a short in a shared negative high-voltage supply <cit.>, and the remaining seven were turned off due to high afterpulses and dark rates. 
The shorted eight channels are adjacent to each other on the top array, which are also in the vicinity of three dead channels happened earlier in Run0. Events in this region suffer from non-negligible charge loss, therefore special measures were taken in the data selection. 

Additionally, liquid level control failed throughout Run1. As a result, each time the re-circulation conditions changed, the induced liquid level variations led to a (correctable) difference in the amplification of S2 signals.



Accordingly, Run1 data is divided into six subsets.




In this analysis, we re-blinded the Run0 data to maintain consistent treatment between Run0 and Run1. The combined analysis includes the following major improvements:



  
Unified signal reconstruction procedure
Signal reconstruction refers to the offline identification and reconstruction of S1 and S2 signals from raw data, and the building of physical “events” to become basic units of the analysis. 
We adopt the “rolling gain” method instead of using the PMT gains obtained from weekly calibration with light-emitting diodes <cit.>.
As previously, after hit clustering, the S1 and S2 signals are identified based on the number of hits, the charge ratio between the top and bottom arrays, and the signal width.
To discriminate S1s from fragments of single electron ionization S2 signals, we impose a series of requirements on S1 when paired with S2, including criteria on the S1 pulse shape profile, light pattern topology, and noise levels in PMT waveforms immediately before and after.
The event building is now performed with a 2-ms fixed window length (maximum electron drift time is about 900 μs).
This allows more consistent event level parameters, e.g. waveform noise parameter, in comparison to the dynamic event window used in Ref. <cit.>, and further eases the selection efficiency evaluation.
The improved signal reconstruction procedure is consistently applied to data from the low energy threshold up to 120 keV in electron equivalent energy, to facilitate future analyses on solar pp neutrinos, etc.



  
Blinded data selection 
Data selection criteria are set “blindly”.
We conduct dedicated optimizations on data selection cuts based solely on all low-energy ER and NR calibration data <cit.>.
We eliminate periods with high S1 rates and veto the “afterglow” time window after large signals (varying with the signal size), resulting in a reduction of approximately 12% of the exposure.
At the signal level, the data quality cuts include those related to the S1 and S2 pulse shapes, as well as the charge distribution on PMTs.
Furthermore, the S1 top-bottom charge partition is required to be consistent with the vertical position of the event, and the width of S2 should correlate with the drift time according to diffusion effect.
For Run 1, these cuts are relaxed if S2 is reconstructed close to the malfunctioned PMTs (green dashed regions in later Fig. <ref>).
At the event level, events with excessive noise in the waveform are further removed.
To identify genuine single-site events, instead of simply counting the number of S2s, multiple waveform cuts are developed and applied. 
For the DM search, we set the range of corrected S1 to be [2, 135] PE (with at least two PMTs in coincidence) and that for S2 to be [120, 20 000] PE, respectively.
This corresponds to an average ER (NR) energy window from 1 (4) to 25 (94) keV for Run0 and 1 (3) to 26 (103) keV for Run1. 

Candidate events are also required to be above the 99.5% NR quantile to further suppress surface background and multi-site events depositing their energy partially below the cathode plane.

The efficiencies and associated uncertainties in data selection are established using calibration and simulation data. <cit.>.












  
Improved spatial and temporal signal corrections
The event position reconstruction mostly follows the methods developed in Refs. <cit.>.
An extra azimuth-angle-dependent scaling correction is applied based on the uniformly distributed  Kr-83 ER events <cit.>. 
Within the fiducial volume (FV), the uniformity of  Kr-83 events in four quadrants is within ±8%, reflecting the impact of malfunctioned PMTs on the position reconstruction.
A series of S1 and S2 charge corrections vs. event position are performed, using the in situ
( Rn-222 5.6 MeVαs) and injected ( Kr-83 41.5 keV internal conversion electrons) radioactive sources, as described in Ref. <cit.>. 
To avoid systematic effects due to dead channels and saturation of the top PMTs, bottom-only S2 (i.e. S2) is used for energy reconstruction.
The electron-equivalent energy E_ee of a single-site event in is reconstructed as:

    E_ee = W_q (Q^c_S1/g_1+Q^c_S2_b/g_2_b),

where W_q = 13.7 eV, and Q^c_S1 and Q^c_S2_b are the corrected S1 and bottom S2 charges in photoelectrons, respectively.
The detector parameters (g_1, g_2_b) are pre-fitted using the mono-energy ER peaks of  
 Kr-83 (41.5 keV), Xe-131 (163.9 keV) and Xe-129 (236.2 keV). Due to variations in the operation conditions such as the liquid levels, the values of (g_1, g_2_b) in subsets of Run0 and Run1 are related by constant factors, which can be tracked and boot-strapped using in situ α peaks from radon decay. The final values of (g_1, g_2_b) 
used for the DM analysis is derived from the best fit to the continuous ER 

and NR 

calibration data <cit.>.
The mean values in Run0 and Run1 are (0.100±0.005, 4.1±0.4) and (0.091±0.004, 5.0±0.5).



  
Comprehensive signal response model
To model the ER and NR signal responses in the detector, as well as to optimize selection criteria, several low-energy calibrations are carried out with radioactive sources. The ER response is calibrated using the β-decay progenies of Rn-220 and Rn-222, which are injected through one of the circulation loops <cit.>. Neutrons produced by a deuterium-deuterium (D-D) neutron generator or an Am-241Be source are employed to calibrate the NR response. The signal response model of the detector is derived from a simultaneous fit of all the calibration data from Run0 and Run1 <cit.>.
















The following major background compositions are considered, also summarized in Table <ref>. All background components are re-estimated using modified data processing and selection efficiencies, and our updated Run0 estimates generally agree with Ref. <cit.> within uncertainties.

One of the top ER background contributions comes from β-decays of  Pb-214 and Pb-212, the decay progenies of Rn-222 and Rn-220, respectively.  
The average decay rate of Rn-222 is measured with its decay αs at a level of 7.1 ± 0.2 μBq/kg in Run0, and increases to an average of 8.7 ± 0.3 μBq/kg in Run1.
The level of Pb-214 is depleted from Rn-222 due to the re-circulation flow and electric field in the TPC <cit.>.
Since Pb-214 β-decay has a characteristic shape at high energy, we take the best-fit Pb-214 level from Ref. <cit.> for Run0, then scale it to Run1 according to the measured Rn-222 α rate.

Similar procedures are adopted for βs from  Pb-212.
Due to the short half-life of Rn-220 (55 s) and the long half-life of Pb-212 (10.6 hr), Rn-220 α rate does not reflect the internal activity of  Pb-212.
Instead, the level of the daughter Po-212 α decays is taken as an estimate, which is 0.1 μBq/kg for both Run0 and Run1.

The background from Kr-85 β-decay is estimated through the correlated emission of β-γ via the metastable state Rb-85 (514 keV) with a branching ratio of 0.44%. 
Average Kr/Xe ratios in Run0 and Run1 are 0.52 ± 0.27 ppt and 0.94 ± 0.28 ppt respectively, under the assumption of a 2×10^-11 isotropic concentration of Kr-85 <cit.>, leading to an expected background of 80 ± 40 and 289 ± 88 events within the DM detection window in Run0 and Run1, respectively.


The estimation of solar neutrino-electron elastic scattering is adopted from Ref. <cit.>, which incorporates neutrino flux from the standard solar model, three-flavor neutrino oscillation, and the Standard Model neutrino-electron scattering with xenon atomic effects <cit.>.
The two-neutrino double-β decay of Xe-136 is calculated based on the half-life measurement from our recent study <cit.>. Additionally, the double electron capture from the LL-shell of Xe-124 from XENON <cit.> is incorporated as a single Gaussian peak at 10 keV. The contributions of LM- and LN-shell double electron capture are minor and thus not considered in this analysis.

The radioactivity of detector materials, mainly stemming from the PMTs and stainless steel vessels, is assessed using a Monte Carlo (MC) simulation <cit.>.
The data match the simulations in the energy and position distributions for high-energy gamma rays. The total contribution to background is 147 ± 12 events.

The unstable Xe-127, produced through the capture of cosmogenically generated neutrons, decays via electron capture. The L-shell X-ray with an energy of 5.2 keV contributes to the ER background. In this analysis, the rate in Run1 is estimated from that obtained in Run0 <cit.> using the decay half-life of 36.3 days. The contribution in Run1 is almost negligible. The total background is 7.7 ± 0.8 events.

To allow an approximate online assessment of the tritium background, we perform one-dimensional spectral fits only based on S1 using a set of relaxed cuts, while keeping S2 data blinded. In the final DM data fit, tritium levels from both runs are again left floating. One observes a significant decrease of tritium level in Run1, from 2.04 ± 0.25/tonne/day at the end of Run0 to 0.26 ± 0.07/tonne/day in Run1. These findings are consistent with an independent analysis with one-dimensional energy spectrum fit [X. Zeng et al. (PandaX), in preparation.].

The low-energy neutron background originating from detector materials is evaluated as the weighted average of three different estimates <cit.>, by scaling the measured high-energy xenon-capture γ rays, by scaling the measured multi-site neutron events, and by direct MC prediction. The contributions of multi-site neutrons that satisfy the single-site cuts are also taken into account in all three estimates. The total number of neutron background is 1.7 ± 0.3 events.

Solar  B-8 neutrinos can produce coherent elastic scattering (CEνNS) <cit.>, thereby contributing to the NR background in the low-energy region.
Our new estimate of 0.31 events for Run0 is about half of that in Ref. <cit.>, due to the update of the signal response model and a smaller FV, yet still consistent within the previously assigned uncertainty.
The total contribution of  B-8 CEνNS is 1.0 ± 0.3 events in Run0 and Run1 combined.

The surface background comes from the low-energy β-decay events due to radon plate-out on the PTFE panels. The S2 of these events are significantly suppressed due to distorted electric field lines close to the PTFE surface. A data-driven background model is established using Po-210 α surface events, whose S2s are similarly suppressed. Within the optimized FV selection, the total surface background is estimated to be 0.26 ± 0.12 events.

Unphysical accidental coincidence (AC) backgrounds are estimated similarly as in Ref. <cit.>.
The S1 (S2) identified in a fixed 2-ms event window without any pairable S2 (S1) is defined as an isolated-S1 (isolated-S2).
In Run0 and Run1, the rates of isolated-S1 (isolated-S2) are 13.3 Hz (0.125 Hz) and 14.4 Hz (0.146 Hz), respectively, under a set of loose selection cuts.
The AC rate is estimated by manually pairing randomly scrambled isolated-S1 and isolated-S2 pulses, and then applying the full data selection procedure. 

The pre-unblinding AC backgrounds were estimated as 5.6 ± 1.7 and 7.9 ± 2.4 events for Run0 and Run1, respectively [These values are updated after unblinding, see later.]. In the two runs, numbers of AC background do not quite scale with exposures primarily due to different cut efficiencies.

















The FV in Run0 and Run1 is optimized separately, utilizing a figure-of-merit based on signal efficiency and background levels <cit.>.
The final fiducial masses are 2.38 ± 0.04 tonne (Run0) and 2.48 ± 0.05 tonne (Run1).
The uncertainties are derived from the non-uniformity of Kr-83 and  Rn-220/222 calibration events, therefore encompassing systematics arising from malfunctioned top PMTs.
The total accumulated exposure is 1.54 tonne·year.


The unblinding of science data was performed with a three-step procedure, by sequentially unblinding about 10%, 50%, then the entire data set from both runs, and performing sanity checks on the data selections.
No adjustments were made to the data selection. 
After fully unblinding the data, we realized that our AC background was estimated too low by about a factor of 2, due to an inconsistent acceptance cut on the scrambled data.
Instead of making post-unblinding tightening to the cuts, we updated the nominal AC predictions (Table <ref>), which were then fed into the likelihood fits.
As a result, the overall AC contribution in this analysis is increased from Ref. <cit.>, but the impact is minor to our DM sensitivity for a DM mass above 40 . 











After unblinding, a total of 2490 final candidate events are identified in Run0 and Run1 within the FV and DM ROI selection window.
As presented in Fig. <ref>, twelve (Run0) and twelve (Run1) events are found below the NR median curves in log_10(S2_ b/S1) vs. S1.
These events are uniformly distributed inside the TPC (Fig. <ref>) and in time. 
Compared with the six events found below the NR median in the previous Run0 analysis <cit.>, five of them (#2, #3, #5, #9, #11) are the same, 
and one event is removed by the newly optimized FV and the afterglow cut. All seven newly observed events are near the boundaries of one or two selection cuts: events #4, #6, #7, #8, #10, and #12 do not satisfy previous afterglow or several previous quality cuts, and event #1 was above the previous NR median curve.


An unbinned likelihood function <cit.> is constructed, with an update of nuisance parameters from a combined-fit signal response model utilizing all available low-energy ER and NR calibration data <cit.>.
The likelihood fit takes background and nuisance parameters from Table <ref> as inputs.
Due to their similar distributions in S1 and S2, we combine the continuous ER components ( Rn-220/222, Kr-85, solar neutrino, Xe-136, and detector materials) with their nominal contribution in Table <ref> into a single “Other ER” component.
The overall normalization is independently determined from the data within 20 to 30 keV, which agrees with the sum of individual components but with better statistical uncertainty.
The tritium levels in both runs are set floating. 
In general, background-only fit results agree with pre-unblinding estimates (Table <ref>). 
For those events below the NR median curves, based on the likelihood composition of each event in Fig. <ref>, events with small S1 are dominated by AC background and large S1 events are ER leaks from tritium in Run0 and “Other ER” in Run1.

The DM search is carried out for a mass range between 5 and 10 . The standard conventions for direct dark matter searches <cit.> are adopted, using a profile likelihood ratio method to derive statistical interpretation <cit.>. A local best fit is observed for a DM mass at 6 with a strength of 4.3 events, corresponding to the slight excess of events observed at low S1 and S2 (Fig. <ref>).

Testing against a background-only hypothesis, the local significance is 1.8σ, but reduced to 1.2σ (global p-value of 0.11) after the look-elsewhere effect correction <cit.>. Alternatively, if the central value of the B-8 flux from our recent measurement <cit.> is used, the local significance of the excess at 6 is reduced to 1.4σ. On the other hand, a downward fluctuation is observed for DM masses above 15 due to fewer events appearing within the core of DM signal region.
The derived 90% CL upper limits (without power constraint <cit.>) on the spin-independent (SI) DM-nucleon cross section is shown in Fig. <ref>. The limit is slightly outside +1σ sensitivity band at the low mass end, and approaches -1σ when the DM mass increases to above 15 , consistent with the data features discussed above. 





The lowest excluded cross section of 1.6 × 10^-47 cm^2 occurs at a DM mass of 40 . In comparison to earlier results from XENONnT <cit.> and LZ <cit.>, we obtain the most stringent upper limit for DM mass larger than 100 .




In summary, we present the results of a DM search from a blind analysis of data collected during Run0 and Run1 at PandaX-4T, amounting to a cumulative live exposure of 1.54 tonne·year.
No significant event excess is observed above the expected background. Our analysis derives the 90% CL upper limits on the spin-independent DM-nucleon scattering cross section, with the lowest excluded value of 1.6 × 10^-47 cm^2 at 40 DM mass.
These findings establish a new, stringent constraint at a DM mass exceeding 100 . During the CJPL construction shutdown, completed upgrades to its PMT, electronics, DAQ, and external water veto systems.
The second science run has now commenced, with ongoing efforts to further suppress background.
We expect that the sensitivity of the DM search will improve by a factor of 2-3 with a complete 6-tonne·year exposure.






This project is supported in part by grants from National Science Foundation of China (Nos. 12090060, 12090061, 12205181, 12222505, 12325505, U23B2070), a grant from the Ministry of Science and Technology of China (Nos. 2023YFA1606200, 2023YFA1606201, and by Office of Science and Technology, Shanghai Municipal Government (grant No. 22JC1410100, 21TQ1400218). We thank for the support by the Fundamental Research Funds for the Central Universities. We also thank the sponsorship from the Chinese Academy of Sciences Center for Excellence in Particle Physics (CCEPP), Hongwen Foundation in Hong Kong, New Cornerstone Science Foundation, Tencent Foundation in China, and Yangyang Development Fund. Finally, we thank the CJPL administration and the Yalong River Hydropower Development Company Ltd. for indispensable logistical support and other help. 








 Paper Title: TrackSorter: A Transformer-based sorting algorithm for track finding in High Energy Physics.






^1University of California-Berkeley
^2Lawrence Berkeley National Laboratory
yashmelkani02@gmail.com, xju@lbl.gov

§ ABSTRACT 
Track finding in particle data is a challenging pattern recognition problem in High Energy Physics. It takes as inputs a point cloud of space points and labels them so that space points created by the same particle have the same label. The list of space points with the same label is a track candidate. We argue that this pattern recognition problem can be formulated as a sorting problem of which the inputs are a list of space points sorted by their distances away from the collision points and the outputs are the  space points sorted by their labels. In this paper, we propose the algorithm: a Transformer-based algorithm for  pattern recognition in particle data. uses a simple tokenization scheme to convert space points into discrete tokens. It then uses the tokenized space points as inputs and sorts the input tokens into track candidates. is a novel end-to-end track finding algorithm that leverages Transformer-based models to solve pattern recognition problems. It is evaluated on the TrackML dataset and has good track finding performance.





§ INTRODUCTION


The High Luminosity Large Hadronic Collider (HL-LHC) plans to collide two proton beams at the unprecedented center of mass energy of 14 TeV at an instantaneous luminosity of up to 7.5×10^34cm^-2s^-1. That corresponds to an average number of proton-proton collisions per beam crossing (i.e. pileup), ⟨μ⟩, of up to 200. HL-LHC brings opportunities and challenges. To cope with the challenges, the ATLAS <cit.> and CMS <cit.> experiments decided to build a new fully silicon-based inner tracker detector <cit.>. The new inner trackers will have better raditation tolerance, increased granularity, reduced material, and large readout bandwith to fulfill the requirement of the HL-LHC Runs. Take the ATLAS's new innder detector, ITk, as an example. ITk consists of a Pixel detector at a small radius and a large area Strip detector surrounding it. The Pixel detector consists of about five billion finely segmented silicon sensors, most of which have a pitch of 50×50 μm^2 and the rest 25×100 μm^2. The Strip detector comprises 23,000 long and skinny silicon sensors (75.5 μm× 24.1 or 48.2 mm). Each event with ⟨μ⟩=200 produces about 300,000 space points, out of them only about 10,000 space points come from particles of interest. Finding the tracks of interests from a point cloud of space points is a challenging pattern recognition problem. 

Our work is inspired by the remarkable capabilities of Large Language Models (LLMs) such as BERT <cit.>, GPT <cit.>, Llama <cit.>, and grok-1 <cit.>. The foundation of our work is to convert space points into discrete token ids (i.e. tokenization). Tokenization is a critical step in efficient learning and handling out-of-vocabulary words for LLMs learning natural language. For example, one can use letters as tokens for the English language. Doing so would result in a small vocabulary and can construct all out-of-vocabulary words. However, it is inefficient for learning because semantic relationships among letters are lost during tokenization. 
Tokenizing a point cloud of measurements in High Energy Physics (HEP) presents a unique challenge: converting variables from a continuous, multi-dimensional space into discrete spaces. Although some information from the continuous space will inevitably be lost during tokenization, this loss may be acceptable as long as it does not compromise the accuracy of the underlying physics. After all, all physics measurements inherently contain some level of uncertainties.
In the context of jet physics, the Omnijet framework  <cit.> explored three schemes for tokenizing jet constituents: physics-inspired binning of contitunents' kinematic variables (binning in short), conditional and unconditional tokenization via the vector-quantization variational autoencoder (i.e. VQ-VAE) <cit.> technique, which is also used in Ref <cit.> to build the codebook index. The binning scheme adjusts the bin width to match measurement uncertainties and the study shows a small bias in the reconstructed jet mass and poor resolution (Fig. 4 in Ref. <cit.>). The VQ-VAE schemes enjoy better performances with a larger number of tokens. In our previous research <cit.> on particle tracking data, we utilized the unique detector module IDs as the token identifiers for space points created on that detector module. While this tokenization process results in the loss of precise positional and other details of the space points, it enables us to directly train LLMs with the tracking data.

We argue that the track finding problem can be formulated as a sequence-to-sequence (seq2seq) problem, illustrated in Fig. <ref>. The input sequence is a list of space points ordered by their distances away from the collision point. And the output sequence is a list of the same space points ordered by their labels and their distances away from the collision point. Language models like Bart <cit.> are very good at solving seq2seq problems like machine translation, text summary, and question-answering. The Bart model consists of a bidirectional encoder (like BERT) and a left-to-right decoder (like GPT). A similar model is used in our work.






§ DATA


This study is based on the TrackML dataset <cit.>, which simulates the top quark pair production from proton-proton collisions at the HL-LHC. To simulate the effect of event pileup and produce realistic detector occupancy, a Poisson random number (with μ=200) of Quantum Chromodynamics "minimum bias" events are overlaid on top of the tt̅ collisions. The TrackML detector is a set of concentric cylindrical layers of pixelated sensors (the barrel) complemented by a set of circular disks (the endcaps) to ensure nearly 4π coverage in solid angle, as pictured in Fig. <ref>.

The detector is divided into nine volumes, each consisting of 2 to 7 layers. Each layer contains multiple silicon modules. There are 18,737 detector modules in the TrackML dataset. We use data from volume 8, 13, and 17, summing up to 14,000 modules. We introduce two custom tokens to indicate the start of the output sequence [SOS] and the end of each track . Therefore, as detector module IDs are treated as token identifiers, the vocabulary size in our work is the sum of the number of detector modules and the two special tokens; that's 14,002.  

 

Our study utilizes particles that have space points from at least 6 unique layers. Our training dataset uses tracks from 100 events that meet this condition, totaling 349k tracks. The validation dataset is similarly constructed from 10 events (35k tracks). A testing dataset for performance analysis is curated with an additional condition that each track has an average < 5 GeV, containing 67k tracks.

In natural language processing workflows, discrete tokens are first embedded into a continuous, dense vector representation, i.e. Word2Vec <cit.>. We used the continuous bag of words framework <cit.> to train a Word2Vec model by asking the model to predict a target word using all the words in a context window. To train the model, we randomly pair each track with another track and construct a target sentence following the ordering scheme shown in Fig. <ref> for each track pair. In total, our training data contains 349k sentences and 8M tokens. The “Generate Similar` (Gensim) library <cit.> is utilized for training. To achieve a reasonable embedding performance, the model uses an embedding dimension of 64, a context window of 20 tokens, and is trained for 100 epochs. In the future, we can use the TrackingBERT <cit.> method to embed detector modules.



§ MODEL AND TRAINING


The model utilizes the encoder-decoder structure of transformers <cit.>. Both encoder and decoder networks are composed of a stack of identical transformer modules, each having a single-head self-attention mechanism and a position-wise fully connected feed-forward network. We only use a single attention head because our embedding dimension is 64, which is relatively small. And the feed forward layers have a dimension of 256. The output of the decoder network is fed into a linear layer that spans the dimension of the vocabulary. Our model contains six bi-directional encoder layers followed by six left-to-right decoder layers, totaling 1.6M trainable parameters. We inject positional encoding into the input sequence to provide information on the order of the detector modules. 

The model is trained to autoregressively predict the correct sequence of tokens for 371 epochs using the Adam optimizer <cit.> in conjunction with the CosineAnnealingLR scheduler. Model weights corresponding to the lowest validation loss were saved.  




§ RESULTS AND DISCUSSIONS


During model inference, we utilize the greedy search algorithm to construct model predictions: given an input sequence, a count mask is created to store the number of instances of each token in the input sequence. The count value for the [SOS] and tokens are set to 0 and 100, respectively. The model is first fed with the [SOS] token and predicts the next token by calculating logits for each token in the vocabulary. The logits of tokens that have a value of zero in the count mask are set to zero. The token with the highest model logit is considered as the next token; thus, it is appended to the output sequence. Its corresponding count value is decremented by one so that it would not appear again in the output sequence. The updated output sequence is then fed back to the model for the next token prediction. In the case that the greedy algorithm predicts the token, the model logit of the token will be set to zero in the next step. 
This is to prevent the algorithm from predicting the token in consecutive steps. 
Note that after predicting the token, the model will decide what the next token will be. That means the model may predict track candidates in an arbitrary order.
Such predictions are repeated until all input tokens are in the output sequence and the last predicted token is the token. This termination condition is not ideal when noise space points [Noise space points are those created either from electronic noises or low-particles (i.e. < 200 MeV).] are presented as in real data. 

The model performance is evaluated by the tracking reconstruction efficiency, defined as the fraction of particles that are matched to at least one reconstructed track. Reconstructed track candidates are  matched to particles if (1) 75% of module hits in the reconstructed track are in the true particle track and (2) 75% of module hits in the true particle track are in the reconstructed track. To assess our model, each track in our testing dataset was randomly paired with another track before being passed to the model. Fig. <ref> shows the model's performance with respect to track length and track . 
The model performance is fairly stable against the track length, indicating the model can catch long-distance information. 
We note that tracking efficiency as a function of resembles the distribution of particle in our testing dataset, see figures on the top panel in Fig. <ref>. This suggests that an uniform sampling of particle during training may make model performant in all regions.  




We observed that a larger model size leads to better performance. We expect a larger dataset size to be beneficial to improve model performance. In our training dataset, we find several tokens represented in only a handful of samples. This adversely affects Word2Vec training of our initial embedding vectors as well as training of the model itself.

In smaller scale experiments focusing on the inner barrel detector region, we came up with physics-inspired embedding vectors for detector modules, such as the global coordinates, rotation matrix and pitch components, and geometric features of each detector module. The model performance resulting from this embedding scheme does not perform well compared to the more traditional Word2Vec implementation. 


The model works fairly well with two tracks per input. but it remains to be studied whether the can scale effectively in a dense environment like the HL-LHC, where each event contains 10k particles resulting in 100k detector hits. 
This would form the event-level context window for the language model. 
This large context window may not pose a problem, as the leading LLM models already have substantial capacities. 
For instance, GPT-4o supports a 128k context window <cit.>, 
Claude 3 Sonnet extends up to 200k tokens <cit.>, 
and the Gemini-1.5 model can handle up to 1 million tokens in production <cit.>. 



§ CONCLUSIONS


We reformulated the particle tracking problem as a sequence to sequence problem and proposed a Tranformer-based track sorting algorithm to address it. This algorithm achieves good tracking reconstruction efficiency, even for low-particles (< 1 GeV). Our work leverages a language model-style architecture to tackle high-energy problems. Our study trained the language model from scratch. It remains to be studied open large language models can be fine-tuned for HEP problem-solving.



§ DATA AND SOFTWARE AVAILABILITY

Data can be found at Kaggle [https://www.kaggle.com/c/trackml-particle-identification] and code is avaible at Github [https://github.com/YashMelkani/Track-Sorting-Tutorial].



§ ACKNOWLEDGEMENTS

This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231.



jhep



 Paper Title: Final state interactions for high energy scattering off atomic electrons.










Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA



CALT-TH/2024-026



§ ABSTRACT 
    We consider the scattering of high energy leptons off bound atomic electrons focusing primarily on final state interactions 
    between the outgoing energetic electron, and the heavy residual charged “debris” in the final state. These effects are inherently absent from calculations for a free electron at rest. Coulomb exchanges are enhanced by the large number of electrons in the atomic debris, and are unsuppressed by non-relativistic velocities in the debris. We find that these exchanges can be resummed using operator methods, and cancel at the level of the cross section until at least O(α^3). Furthermore, we argue that both final and initial state Coulomb exchanges (enhanced by the number of electrons in the atom) do not affect the cross section until at least O(α^3). Transverse photon couplings  to non relativistic electrons are proportional to their small velocities, and rotational invariance suppresses their contribution to O(α^3).   Our results are relevant for precision experiments involving  neutrinos, electrons, positrons, and muons scattering off of atomic electrons in a fixed target. 



    



 Final state interactions for high energy scattering off atomic electrons
    Mark B. Wise
    August 5, 2024
========================================================================






§ INTRODUCTION  



Fixed target experiments play an important role in high energy physics at the intensity frontier. Examples include neutrino experiments <cit.>, searches for feebly interacting particles <cit.>, rare processes <cit.>, and precision measurements <cit.>. It is often implicitly assumed that atomic electrons can be treated as free and at rest when they participate in a scattering interaction with an incident high energy beam. This is a good leading approximation, however for high precision experiments sub-leading corrections must be understood theoretically. 

In recent work <cit.> we have studied the corrections due to atomic binding arising both from phase space, and the initial state (many-body) wavefunction of the atom. These corrections stem from the finite three-momentum of electrons in bound atomic orbitals, and the shifted kinematics from binding energies. Both of these effects enter at the same order and they must be treated simultaneously. They account for the non-perturbative bound-state dynamics i.e., from iterated Coulomb exchange in the initial state. 

Further corrections arise from perturbative photon exchange between “hard” leptons, and “soft” (i.e., non-relativistic) electrons and nuclei. These effects are inherently absent from calculations for a free electron at rest. For example, for neutrino scattering off an atomic electron, the outgoing ∼ GeV electron can exchange photons with the ionized “atomic debris”. Incident charged lepton scattering (e.g., μ e →μ e) involves both initial and final state photon exchange with atomic electrons or nuclei.

In this work, we focus mostly on perturbative final-state interactions involving the outgoing energetic electron (in <ref> we also comment on the “shake-up” of the initial state from Coulomb exchange with an incoming charged lepton). We specifically focus on ladder-graphs involving photon exchanges with the final state debris. If we track the charge of the debris, Q_B, and the charge of the electron, Q_e, separately then the (gauge invariant) class of corrections we consider are of order[We explicitly neglect “mixed” corrections such as those of order O(Q_BQ_e^3).] (Q_B Q_e)^n.  Although Q_B=+e, matrix elements involving the debris are enhanced for multi-electron atoms. This enhancement occurs for momentum transfers on the order of the atomic scale (i.e., the inverse Bohr radius) when the struck electron is able to see a partially (but not fully) screened atomic nucleus.  To match the order studied in Ref. <cit.>, requires working to order (Q_B Q_e)^2. 


These corrections are a necessary input for an extraction of hadronic vacuum polarization from μ e →μ e scattering <cit.> as is being pursued by the MuonE collaboration <cit.>. For neutrino electron scattering (relevant for neutrino flux normalization measurements <cit.>), we find these corrections are much smaller than the uncertainty from the 1-loop radiative corrections computed in Ref. <cit.> (roughly ∼ 0.5%).


We use a non-relativistic power counting for photon interactions with the atomic system, and standard perturbative counting for the highly relativistic electron in the final state. We find a number of interesting results. First, Coulomb corrections cancel in the differential cross section upon integration over the final state phase space of the atomic debris; this result holds to all orders in perturbation theory [i.e., resumming graphs of order (Q_B Q_e)^n] in the limit of small energy splittings of the debris. For hydrogen, the cancellation can be understood in terms of a final-state wavefunction, whereas for multi-electron atoms it stems from an operator-level identity. Second, sub-leading corrections due to a single transverse photon exchange in are also shown to cancel. Finally, a class of corrections which are sensitive to the energy splittings of the atomic debris and appear at order e^2 can be shown to have vanishing interference with the leading order matrix element. Taken together these results imply that final state interactions (as defined above) do not alter the predictions of Ref. <cit.>  up to and including O(Q_e^2Q_B^2) in our power counting. A similar result holds for interactions that involve the initial atomic state.


In everything that follows, we will omit factors of Q_e and Q_B and simply count powers of α. This includes factors of α that stem from bound-state dynamics i.e., that the typical velocity of an electron in an atom is O(α). We aim to work to an accuracy of O(α^2). 



The rest of the paper is organized as follows: In <ref> we discuss the case of scattering neutrinos from hydrogen. We find that Coulomb distortion of the final state cancels at the level of the cross section after integrating over the final-state proton's phase space. Next in <ref> we show how to generalize our analysis to multi-electron atoms where the corrections similarly cancel at the level of the cross section. We find an operator level expression for Coulomb photons, and also discuss sub-leading corrections such as transverse photon exchange with non-relativistic electrons in the final state debris. Transverse photons are discussed in <ref>. We find the non-relativistic nature of the electrons combined with rotational invariance of the initial atom implies that transverse photons first enter at O(α^3). In <ref> we describe how to apply our results at the level of a measurable cross section. We show that the relevant sum-rules that control cross sections (inclusive with respect to the debris)  are unaffected by Coulomb exchange through O(α^2). We discuss how this result applies both to neutrino electron scattering and muon electron scattering.  
Finally in <ref> we provide a summary and outlook.





§ HYDROGEN  


Let us first focus on neutrino scattering off hydrogen as the simplest example where only final state interaction are present, 


    ν(k) + (*0) →ν(k') + e^-(p') + p^+(h') .


The outgoing proton is heavy, and couples dominantly to Coulomb modes with couplings to transverse photons suppressed by |h'|/m_p∼α m_e/m_p. While discussing hydrogen, we will therefore focus exclusively on the exchange of Coulomb photons in the final state. 

Consider an outgoing scattering state |f⟩=|e^-(p') p^+(h')⟩ labeled by the asymptotic momentum of the electron and proton. Including the iterated Coulomb potential, the state |f⟩ satisfies a Lippmann-Schwinger equation 


    |f⟩ = |e^-(p')⟩_0 |p^+(h')⟩_0 + 1/E_f - Ĥ_0 - 0V̂|f⟩ ,


where |e^-(p')⟩_0|p^+(h')⟩_0 is the free-particle state. 
Here V̂ is the Coulomb interactions, Ĥ_0 is the free-Hamiltonian and E_f = √(p'^2+m_e^2) + √(h'^2+m_p^2) is the exact energy of |f⟩; the causal regulator (- 0) is fixed by demanding that |f⟩ be an out-state. 
The final state can be written in terms of its valence free-particle Fock states, as[Strictly speaking the Dirac wavefunction should be spin-dependent, but we suppress spin indices for simplicity. The answer is unaffected in the kinematic regime of interest <cit.> and <ref> is correct as written.]


    |f(p',h')⟩ = √(2E_e(p'))√(2 E_p(h'))   ∫^3 q/(2π)^3ϕ̃_p',h'(q)/√(2E_e(p'+q))√(2E_p(h'-q))|e^-(p'+q)⟩_0|p^+(h'-q)⟩_0 .



Note that when V̂=0, the scattering wavefunction is ϕ̃_p',h'(q)= (2π)^3 δ^(3)(q). The two-particle scattering state is normalized according to 


    ⟨f(p',h')|f(p,h)⟩ =  2   E_e(p) (2π)^3 δ^(3)(p-p') 
       × 2 E_p(h)(2π)^3 δ^(3)(h-h') .


The scattering state's wavefunction depends only on the asymptotic relative momentum f'≡p'-h' so that ϕ̃_p',h'(q) = ϕ̃_f'(q).  The scattering state wavefunction is normalized according to 


    ∫   ^3 q/(2π)^3ϕ̃_f'^*(q+f') ϕ̃_f(q+f)= (2π)^3 δ^(3)(f'-f) .


It is readily checked that <ref> can be substituted into <ref> to obtain <ref>. In coordinate space <ref> becomes, 


    ∫   ^3 x  ^ (f-f')·x ϕ_f'^*(x) ϕ_f(x)= (2π)^3 δ^(3)(f'-f) .




We similarly write the initial hydrogen atom in terms of free-particle valence states, 


    |(*0)⟩= ∫^3 p/(2π)^3   √(2M_H)/√(2E_e(p))√(2 E_p(p))
       ×ψ̃(p)|e^-(p)⟩_0 |p^+(-p)⟩_0 .


The bound state's wavefunction is normalized according to 


    ∫^3q/(2π)^3 |ψ̃(q)|^2=1 .


The relevant matrix element is 


    ℳ = f(p',h') ν(k')ℋ_W(0)ν(k) ,


where ℋ_W is the weak Hamiltonian density evaluated at the origin of spacetime. Using the above definitions, and _0⟨p^+(h'-q)|p^+(-p)⟩_0= 2 E_p (2π)^3 δ^(3)(q-(h'+p)), we arrive at 


    ℳ   = ∫^3 p/(2π)^3√(2M_H 2 E_p/2E_e)ϕ̃_p'-h'^*(p+h') ψ̃(p) 
       × M(p'+p+h',k';p,k) 
       ≃∫^3 p/(2π)^3√(2M_H 2 E_p/2E_e)ϕ̃_p'-h'^*(p+h') ψ̃(p) 
       × M(p',k';p,k) ,


where in the second line we have neglected the p+h' (both small) relative to p' (large); the error due to this approximation is very small ∼ |p|/|p'| <cit.>. The free-electron matrix element is defined as


    M(p',k';p,k)=  _0⟨ e^-(p')ν(k')| ℋ_W| e^-(p)ν(k)⟩_0 .


As mentioned above, in the absence of Coulomb exchanges ϕ̃_p'-h'(q) → (2π)^3 δ^(3)(q), and <ref> reduces to Eq. (14) of Ref. <cit.>.



 §.§ Perturbation theory  





It is instructive to compute the matrix element perturbatively. This both serves to illustrate the size of Coulomb corrections at the level of the amplitude and the connection between diagramatics and wavefunctions. Furthermore, by analyzing different regions of loop momenta we learn which characteristic scales dominate the Coulomb corrections. 

The O(α) correction involves an integral of the form (including a photon mass, μ_γ, to regulate IR divergences)


    I^(1)=∫^3 q/(2π)^3ψ̃(h'-q)  γ_0(p'+q+m) /2p'· q+q^2+ 0e^2/q^2-μ_γ^2+ 0 ,


where contraction against u̅(p') is implicit. 
The energy transfer is small, q_0 ≃ (2h'·q + q^2)/(2 m_p). We will suppress the causal regulators (+ 0) in what follows. 

Let us focus on a 1s orbital of hydrogen for concreteness,


    ψ̃_ 1s(h'-q) ≃8√(π)Λ^5/2/(|h'-q|^2 + Λ^2)^2 ,


where we have used the non-relativistic approximation and introduced Λ=1/a_0=α m_e with a_0 the Bohr radius. 

It is interesting to first understand the limit of large momentum transfers. These may occur in two qualitatively different regions of phase space. First consider the final state proton's momentum to be small |h'|∼Λ. Then, if the Coulomb photon mediates a large momentum transfer we must have a large initial momentum for the proton which is suppressed by the hydrogenic wavefunction; in this case we find a power suppression of ∼Λ^4/|q|^' 4. Alternatively, one can consider a large final state proton momentum with h'∼q. In this case the integral factorizes into a tree-level matrix element (for a proton at rest) multiplied by the wavefunction at the origin ψ_ 1s(x=0) (which is suppressed by powers of α). 

Hence the integral in <ref> is dominated by |q|∼Λ where one finds


    I^(1)∼αΛ^-3/2∼αψ̃_ 1s ,


which is O(α) relative to the leading expression.  We may therefore approximate I^(1) by expanding the integrand counting |h'|∼ |q|  ≪ E'_e, 


    I^(1)≃ e^2 ∫^3 q/(2π)^3 ψ̃_1s(h'-q)   1 /v'·q1/q^2+μ_γ^2 ,


where v'=p'/E'; this corresponds to the eikonal approximation. To get a sense of the size of the one-Coulomb-exchange correction, we may evaluate at the kinematic point h'=0. Then the integral is 



    I^(1)|_h'=0   ≃ e^2 ∫^3 q/(2π)^38√(π)Λ^5/2/(q^2 + Λ^2)^21 /v'·q1/q^2+μ_γ^2
        = α/|v'|[log( μ_γ^2 /Λ^2) + 1] ×ψ̃(h'=0)  .


As expected the correction is controlled by α/|v'|∼α and is naively much larger than the binding corrections computed in <cit.>. At the special kinematic point  h'=0 the correction is purely imaginary, but is in general complex[<Ref> has the form of a convolution and can be written in coordinate space ∫^3 x  ^-h'·x f(x) ψ_ 1s(x). If for h'=0 this is purely imaginary, then for non-zero h' it has both real and imaginary parts.] (with non-zero real and imaginary parts) for kinematics such that h'≠ 0. 


If we keep the next corrections from the electron's propagator we find terms suppressed by Λ/E' or Λ/m_p (coming from the energy transfer q_0=q^2/2m_p). As a concrete example, consider the correction proportional to q^2/2m_p. Expanding  1/(v'·q + q^2/2m_p)  leads to the correction, 


    δ I^(1)|_h'=0   ≃e^2 /2m_p∫^3 q/(2π)^38√(π)Λ^5/2/(q^2 + Λ^2)^2(1 /v'·q- 0)^2
       =α/|v'|^2(Λ/4m_p) ×ψ̃(h'=0) ,


which is suppressed by (Λ/m_p) relative to the leading result.[In the case of a multi-electron atom, the final state debris can contain electrons with kinetic energy T∼q^2/m_e. In this case the suppression is Λ/m_e ∼ O(α).] 


<Ref> is suppressed by the explicit perturbative factor of α, but is not suppressed by any hierarchy of scales. Coulomb corrections must therefore be controlled to at least two-loop order. As we discuss below, the Coulomb corrections actually exponentiate and cancel after integrating over the phase space of the outgoing proton. The exponentiation of Coulomb corrections can be seen transparently in coordinate space and we therefore return to the description in terms of final state wavefunctions. 




 §.§ Coordinate space representation  


For non-zero recoil momentum, h'≠ 0, we can obtain a useful representation of the integrals by noticing they have the form of a convolution. At n^ th-order in (e^2) we encounter integrals of the form,


    I^(n) =   ∫^3 x   E_f^*(n)(x) ^-h'·xψ(x) .


The function E_f^*(n) (x) is a series of nested convolutions. For example E_f^*(1) is given by, 


    E_f^*(1)(x)= e^2∫^3q_1/(2π)^3^q_1·x1/v'·q_11/|q_1|^2 ,


and E_f^*(2) by 


    E_f^*(2)(x)= e^4∫   ^3q_2/(2π)^3^q_2·x∫^3q_1/(2π)^31/v'·q_2
       ×1/|q_1-q_2|^21/v'·q_11/|q_1|^2 .


Inserting <ref> and <ref> into <ref>, writing ψ(x) = ∫^3p/(2π)^3 ^p·xψ̃(p), and carrying out the integral over ^3x one easily arrives at the momentum space representation of I^(1) and I^(2) [cf. <ref>]. 


Next, summing over n we find 


    ∑_n I^(n) =   ∫^3 x  ψ(x) ^-h'·x E_f^*(x)  ,


with 


    E_f^*(x)  = ∑_n E_f^*(n) (x) .


This resummation of diagrams is equivalent to solving the Lippmann-Schwinger equation and E_f(x). The function E_f(x) is the eikonal approximation to ϕ_f(x). For both relativistic and non-relativistic electrons, the eikonal wavefunction satisfies <cit.>


    [ v'·*∇ + V(x)] E_f(x) = 0 ,


with V(x) the Coulomb potential. When approximating ϕ_f with E_f,  <ref> hold up to corrections suppressed by[In the rest frame of the atom the momentum f≃p', and therefore p' labels the state.] Λ/|p'| as can be checked usign the method of stationary phase. The solution to <ref> exponentiates in coordinate space,


    E_f(x) = exp[-/|v'|∫_z^∞ s  V(|x_⊥|,s)] ≡^φ(x) ,


where |x_⊥| is the the spatial coordinate perpendicular to v', and z labels the coordinate direction parallel to v'.

<Ref> can be written in coordinate space rather than momentum space. One then finds, 


    ℳ=√(2M_H 2 E_p/2E_e)∫^3 x  ^-h'·x^-φ(x) M(-*∇) ψ(x) ,


where we have abbreviated M(p',k';p,k)= M(p) and transformed to coordinate space where the matrix element should be interpreted as an operator M(-*∇). 

Let us now consider the matrix element squared integrated over phase space. Importantly, in the m_p→∞ limit, the energy conserving delta function is independent of the proton's recoil momentum. Furthermore, corrections to the phase space due to a finite h' are suppressed by h'/E_e' and are negligible. We therefore have an unconstrained integral over h', 


    ∫^3 h'/(2π)^3 |ℳ|^2  =    ∫^3 h'/(2π)^3∫^3x ∫^3 y  ^h'·(y-x)
         ×^φ(y)^-φ(x)
       ×[ψ^*(y)  M^*(-←*∇_y)  M(-→*∇_x)ψ(x) ]  .
 

The integral over h' produces δ^(3)(x-y) and we find 


    ∫^3 h'/(2π)^3 |ℳ|^2     = ∫^3x 
          M(-*∇)ψ(x)^2
       = ∫^3p/(2π)^3 ψ̃(p)^2 × M(p)^2 .


This leads to a result which is identical to that obtained neglecting final state interactions. We see that the Coulomb corrections cancel only after integrating over final-state phase space. This result is independent of the details of the initial state wavefunction. 





§ MULTI-ELECTRON ATOMS  
 

The analysis for hydrogen is greatly simplified by the presence of a single recoiling proton in the final state. Inelastic excitations of the final state proton are gapped-out by large hadronic energy scales ∼ 300  MeV and can be safely neglected. By way of contrast when considering helium and other multi-electron atoms inelastic excitations must be included. The remnant system now contains a complex spectra of excitations with energy splittings on the order of ∼ 1  Ry= 12α^2 m_e and these are always energetically accessible. Furthermore, while the coupling of a proton to transverse photons is suppressed by 1/m_p, the coupling of electrons is only suppressed by 1/m_e, which necessitates the inclusion of both transverse and Coulomb modes. 

The addition of a single electron (i.e., going from hydrogen to helium)  therefore introduces a substantial increase in theoretical complexity. Nevertheless, as we will show below, the non-relativistic nature of the atomic debris allows us to simplify the analysis and retain theoretical control. Many of the properties derived above for hydrogen generalize readily to multi-electron atoms provided they are re-formulated at an operator level; for instance we will show that the exponentiation of Coulomb corrections persists.  In what follows we will first motivate our power counting, then analyze Coulomb photons neglecting energy splittings in the debris, and finally discuss sub-dominant corrections. 




 §.§ Power counting  



In this section we sketch our power counting, motivating the scaling of loop momentum, and the organizing principles behind our calculation. Following the analysis for hydrogen we take regions where |q|∼Λ∼α m_e to dominate all loop integrals. This guarantees that the atomic debris remains non-relativistic in both intermediate and final states. When considering photon exchange in the non-relativistic limit it is useful to separate different modes from one another. We will phrase our power counting in Coulomb gauge which simplifies the analysis and work in the rest frame of the atom. 

Non-relativistic particles have an unsurpressed coupling to Coulomb photons. The interaction strength between two particles with charges Q_1 and Q_2 with relative velocity β can be parameterized as Q_1 Q_2/β. For interactions among particles in the initial atom and/or atomic debris this coupling is non-perturbative, and the Coulomb interaction must be retained at leading order. We therefore include the intra-Coulomb interaction in the energies of atomic states which we will sum using completeness relations.  For Coulomb exchange with the “struck” electron we count Coulomb modes as O(α).  

The coupling of a particle to transverse photons are velocity suppressed in the non-relativistic limit. The atomic nucleus is sufficiently heavy that its velocity can be neglected entirely, while atomic electrons have velocities on the order of v_e∼α. We therefore count the exchange of transverse photons between the struck electron and the atomic debris as O(α v_e) ∼ O(α^2). 

Finally, in what follows we will often expand propagators making use of non-relativistic kinematics. For on-shell states |X⟩ coupled to a photon with momentum q,  we count energy splittings as Δ E_X ∼q^2/m_e ∼α |q|. Therefore we count powers of Δ E_X/|q| as O(α). 








 §.§ Analysis of Coulomb photons  


Let us first analyze the dominant final state interactions mediated by the exchange of Coulomb modes. In a multi-electron atom the wavefunction is replaced by the matrix element of the field operator with a state |B⟩. Let us begin from the in-out matrix element of the weak interaction Hamiltonian density (evaluated at x=0), 


    ν' e' B'ℋ_WAν =  u̅_k'γ_μ P_L u_k e' B'ψ̅Γ^μψA .


Next we insert a complete set of  in-states with electric charge Q=+e,


    e' B'ψ̅Γ^μψA = _B e' B'ψ̅BΓ^μBψA


The matrix element, e' B'ψ̅B, in the above equation involves in- and out-states and can be computed using standard techniques for scattering amplitudes. 


 
Notice that the matrix element involving ψ̅ is an “in-out” matrix element and so standard Feynman rules immediately apply. We now define 


    ψ_BA(p) ≡ _ inB(p)ψA(0)_ in ,


which is the analog of the hydrogenic wavefunction. 
 

A diagrammatic interpretation of e' B'ψ̅B is now clear. Drawing the highly energetic electron with a single line, and the atomic debris with a double line, we have 


    10pte' B'ψ̅B   =diagram_soft(20,6.5) 
        i1,i2o1,o2doublei1,b1,b2,o1plaini2,t1,t2,o2decor.shape=square,decor.size=10,decor.filled=emptyi2square,tension=0,shadedb2,t2,t1,b110pt   ,


where the white square is the insertion of ψ̅ and the shaded box includes all possible photon exchanges. In what follows we will expand in Δ E_X/|q|, but resumming an arbitrary number of Coulomb exchanges. 



  §.§.§ Leading order in Δ E_X/|q|


As shown in <ref>, for Coulomb exchanges <ref> can be reduced to an operator level expression.  When Δ E_X is neglected relative to |q| the only operator that appears[If energies of intermediate states are retained then the Hamiltonian also appears in an operator-level identity .]  is the Fourier transform of the the charge density at t=0, ρ̂(q). The atomic debris is non-relativistic, the energetic electron is ballistic (i.e., eikonal), and Coulomb modes are instantaneous; taken together this implies that only one time-ordering contributes to the matrix element.   Since ρ̂(q), commutes with itself for different arguments, i.e., [ρ̂(q), ρ̂(q')]=0, the sum over orders in perturbation theory ultimately exponentiates. 

The result is that the in-out matrix element in <ref> can be written as


    e'(p') B'(h')ψ̅B(h') = u̅(p') B'(h')Ŵ_v'B(h),


where, up to O(e^4), and evaluating operators at t=0, 


    Ŵ_v'=1 + e^2∫^3 q_1/(2π)^31/v'·q_1ρ̂(q_1)/q_1^2
       + e^4∫^3 q_1/(2π)^3^3 q_2/(2π)^31/v'·(q_1+q_2)ρ̂(q_2)/q_2^21/v'·q_1ρ̂(q_1)/q_1^2 
        +  …


The operator Ŵ_v' resembles a Wilson line, but is defined in terms of the operator ρ̂(q), as opposed to the photon field Â_μ. Matrix elements of Ŵ_v' may be thought of as resumming operator insertions of ρ̂, 


    6pt⟨Ŵ_v⟩ =  ∑_nwilson-line(30,7.5) 
        
        
        i1,b1,b2,b3,b4,b5,b6,o1i2,t1,t2,t3,t4,t5,t6,o2phantom,tension=4i1,b1,b2,b3,b4,b5,b6,o1dbl_dots,tension=0b1,t1dbl_dots,tension=0b2,t2dbl_dots,tension=0b5,t5dbl_dots,tension=0b6,t6phantomt3,v,b4label=…vplain,tension=4i2,t1,t2,t3,t4,t5,t6,o2label=⟨ρ̂(q_1) ρ̂(q_2)   … ρ̂(q_n-1) ρ̂(q_n)⟩b36pt    .



Here, the dotted lines refer to Coulomb modes (with propagators /q^2), and the solid line to the eikonalized electron.  See Ref. <cit.> for a similar construction in the context of Glauber gluons in QCD. 

The second integral in <ref> can be symmetrized between q_1 and q_2. Then, using standard eikonal identities, 


    1/2(1/v'·q_1 + 1/v'·q_2)= 1/2v'· (q_1+q_2)/(v'·q_1)(v'·q_2) ,


it is straightforward to show [in complete analogy with the eikonal wavefunction in <ref>] that <ref> exponentiates 


    Ŵ_v'   = exp[ e^2∫^3 q/(2π)^31/v'·q- 0ρ̂(q)/q^2+μ_γ^2]
       = exp[ - e^2∫_0^∞ s V̂(ŝv')  ] ,


where we have made the infrared regulator μ_γ explicit [cf. <ref>].  <Ref> is the operator level generalization of <ref> evaluated at x=0. We have introduced the Hermitian operator V̂ 


    V̂(x) = ∫^3 q/(2π)^3^q·xρ̂(q)/q^2+μ_γ^2 ,


which manifests that Ŵ_v' is unitary, Ŵ_v'^†Ŵ_v'=1. For a point-like particle (such as the proton), the matrix element of Ŵ_v' reduces to a standard Wilson line, expanded in the limit of small energy-transfer.


Upon summing over all states |B⟩⟨B| (which is the analog of folding with the wavefunction for hydrogen), we obtain the compact formula


    e' B'ψ̅Γ^μψA = u̅(p') B'(h')Ŵ_v'ψ̂A(0) .


Therefore, at the level of the cross section, we have the simple replacement <cit.>,


    Aψ̂^†δ(…) ψ̂A→Aψ̂^†Ŵ_v'^†δ(…) Ŵ_v'ψ̂A .


where the argument of the delta function includes the Hamiltonian, which does not commute with Ŵ_v'. This operator level replacement holds for arbitrary atoms. It relies only the non-relativistic dynamics of the atomic debris and on the scale separation between the energy of the outgoing electron, E'_e and the binding energies of the atoms,  ϵ_A∼ϵ_B. 

The results of this section apply to all orders in the Coulomb interaction in the limit of negligible energy transfer i.e., neglecting corrections of order Δ E_X/Λ. Phenomenologically (e.g., for an extraction of space-like hadronic vacuum polarization) we  expect working to O(α^2) to be sufficient; this corresponds to two Coulomb modes. At this order, however,  sub-leading corrections not captured in <ref> must also be considered. 




  §.§.§ Higher order in Δ E_X/|q| 


The leading effect arrises from corrections to the struck electron's propagator due to finite energy-level splittings, Δ E_X, in the debris. The contribution from the atomic-energy Δ E_X appears at O(αΔ E_X/|q|) ∼ O(α^2) and must therefore also be included. Using the expansion


    u'_0u'· q_1 +  0= -1 v' ·q_1 - 0+q^0  (v' ·q_1-0)^2 +…


where u_μ=γ(1,  v') is the four-velocity of the outgoing electron. 


For a single Coulomb exchange (i.e., at order e^2) the energy transfer is given by q_1^0=h^'0-h^0. The first correction [coming from the second term on the right-hand side of <ref>] gives rise to shift in <ref> that is proportional to 


    ⟨ B'(h')|[H,∫^3 q_1  (2 π)^3V̂(q_1)  (v' ·q_1- 0)^2]|B(h) ⟩ .


Note that q_1 in the integral of <ref> is actually fixed by momentum conservation. The operator in the matrix element above 


    O= [H,∫^3 q_1  (2 π)^3V̂( q_1)  (v'·q_1- 0)^2] ,


is anti-Hermitian [recall that V̂(q_1)^†=V̂(-q_1)] and therefore at linear order it's contribution to the cross section, which is proportional to

    Δσ∝∫^3 k_2  (2 π)^3∫^3 k_1  (2 π)^3⟨ A| â^†_k_2( O+ O^†)â_k_1 |A ⟩ ,

vanishes. This correction (at the amplitude level) was discussed in <ref> for the case h'=0 when the incoming atom is hydrogen so that  B and B' are a single proton.







 §.§ Transverse photons  


Having demonstrated the exponentiation of Coulomb modes, we now turn to transverse photons which enter first at O(α^2) in our power counting. Therefore, to match the size of the binding corrections considered in Ref. <cit.>, we will only study the exchange of a single transverse photon.  In Coulomb gauge a crucial difference between Coulomb and transverse modes appears in their propagators:


    D_00(q)= /q^2    vs.   D_ij(q)=-/q^2+ 0 [ δ_ij-q_i q_j/q^2] .


Transverse modes have poles in the complex q_0 plane and are propagating degrees of freedom. Coulomb modes are instantaneous, and non-propagating. 

When considering contributions from transverse photons it is therefore not possible to completely separate final states from initial states. We therefore organize our discussion along separate lines for final state interactions and interactions involving the initial state. 




  §.§.§ Exchange in the final state


Let us consider a contribution to <ref> from the exchange of a single transverse photon. The spatial photon propagator in Coulomb gauge we have


    D_ij= -/q_0^2 - q^2+ 0 [ δ_ij-q_i q_j/q^2] →/q^2- 0[ δ_ij] .


Where we have made use of current conservation, and an expansion in q_0/|q| in the second step. At this order (single transverse photon exchange) q_0 is explicitly fixed to be small by on-shell conditions from the atomic-debris side of the diagram.  

It is then clear from the structure of the loop amplitude that to this order in the expansion, transverse photons can be included by making the operator level replacement ρ̂→ρ̂ - v'·Ĵ. Counting non-relativistic matrix elements of the current operator ⟨Ĵ⟩∼ O(α)  then guarantees that transverse photons do not spoil the exponentiation at O(α^2). 



  §.§.§ Exchange with the initial state


There are also corrections that arise from the exchange of photons between the outgoing struck electron and the initial atom. Hard photons, with virtuality on the order of m_e^2 or larger, will factorize from bound state effects and are already included in existing calculations of QED corrections to ν e →ν e scattering with free electrons. 

Corrections involving soft photons may, however, be sensitive to atomic structure. Any transverse photon coupling from the atom to the debris is doubly suppressed by non-relativistic velocities and enters at O(α^3). We therefore focus on diagrams (shown below) in which the transverse photon couples to the outgoing ballistic electron. It is instructive to compare the one-loop diagrams involving the exchange of a Coulomb (dotted line) vs. a transverse (wavy line) photon,  

    coulomb-graph(16,16)
        tib,i1,i2,i3,i4,i5,itob,o1,o2,o3,o4,o5,otbdashesb,tdouble,tension=2i3,v1,v_hardphantomv_hard,o3doublev_hard,o1plain,tension=1.1v_hard,v2,otdbl_dots,tension=0,left=0.5,label=qv1,v2decor.shape=square,decor.size=10,decor.filled=emptyv_hardlabel=Ai3label=Bo1label=eottransverse-graph(16,16)
        tib,i1,i2,i3,i4,i5,itob,o1,o2,o3,o4,o5,otbdashesb,tdouble,tension=2i3,v1,v_hardphantomv_hard,o3doublev_hard,o1plain,tension=1.1v_hard,v2,otphoton,tension=0,left=0.5,label=qv1,v2decor.shape=square,decor.size=10,decor.filled=emptyv_hardlabel=Ai3label=Bo1label=eot20pt   .

We have drawn a dashed line where we insert a complete set of states in <ref>. 
The Coulomb graph vanishes, because its propagator supplies no poles in the complex q_0 plane, and the poles from the atom and the debris both lie on either the upper- or lower-half of the complex plane (see Appendix B of <cit.> for a more detailed discussion). A similar argument applies to Coulomb exchanges between the initial state atom and final state debris. Physically these results follow from the non-relativistic nature of the bound-state constituents, and the fact that the Coulomb interaction is instantaneous. 

The graph involving a transverse photon has additional poles from the photon propagator that invalidate the argument given above. Performing the integral over q_0 by closing  the contour in the upper half plane picks up only the photon's pole. This corresponds to an intermediate state that contains a photon [and contributes to the sum in <ref>]. 
However transverse photons couple proportional to a three momentum (i.e., Ĵ = ∑_i p̂_e,i/m_e)  and in the scattering amplitude this is integrated against the initial state multiparticle wave function.  For rotationally invariant targets this integral gives a contribution to the cross section that vanishes,[Strictly speaking, the expectation value of momentum can be proportional to the spin of the atom. For unpolarized targets this cancels upon spin averaging. ] because ⟨p̂_e ⟩≃0. Therefore, the contribution from transverse photons to the cross section can occur first at order e^2 ⟨p̂^2 ⟩ /m_e^2 ∼ O(α^3).









§ APPLICATIONS 
 


We now return to the phenomenological examples discussed in the introduction. We begin with neutrinos scattering from atomic electrons. This is convenient since in the previous sections we have already analyzed ν e →ν e scattering for atomic electrons in detail. In the absence of final-state interactions, the cross section[The same formula applies for the differential cross section σ /Π_e where Π_e is the phase space of the struck electron <cit.>.] for neutrino-atom scattering can be written as <cit.>, 


    σ   =  ∫ϵ∫^3 p/(2π)^3  S_A(ϵ,p)
       1/2 ω1/2E∫Π_e 1/2ω'δ( E-ϵ)1/2∑_ spins | M|^2  .


 Here M(k',p';k,p), is the free-particle matrix element for  ν(k) e(p) →ν(k') e(p') scattering and S_A(ϵ,p) is the non-relativistic spectral function of the target atom A, 


    S_A=∫^3 q/(2π)^3Aâ^†_qδ(ϵ_A+Ĥ-ϵ)  â_pA .


Binding corrections can be efficiently incorporated by expanding | M|^2 in ϵ and p <cit.>. 

To include final-state Coulomb exchange one simply makes the replacement S_A →S̃_A where 



    S̃_A=∫^3 q/(2π)^3Aâ^†_qŴ_v'^†δ(ϵ_A+Ĥ-ϵ) Ŵ_v'â_pA .


The same procedure (mentioned above) of expanding | M| in terms of p and ϵ still applies. We note that <ref> is reminiscent of the opperatorial definition of a parton distribution function. 


The unitarity of Ŵ_v' guarantees that corrections disappear after integrating over ϵ (which removes the delta function). The first term for which this is not true is the energy weighted sum ∫ϵ (-ϵ) × (…), which introduces the insertion of the Hamiltonian. The non-commutativity of Ĥ and Ŵ_v' then gives


    ∫ϵS̃_A(ϵ,p)    =  ∫^3 q/(2π)^3Aâ^†_qâ_pA ,
    ∫ϵ (-ϵ) S̃_A(ϵ,p)    =  ∫^3 q/(2π)^3Aâ^†_qŴ_v'^†  [Ŵ_v'â_p,Ĥ] A .


<Ref> generalizes the Koltun sum rule <cit.> to include final-state Coulomb interactions in the leading non-relativistic approximation. 

In our counting ϵ∼ O(α^2), and so when computing O(α^2) corrections to the cross section we can set Ŵ_v'=1 in <ref>. This demonstrates that Coulomb corrections do not alter the predictions of the cross section until at least O(α^3). 

For the same reason, when considering the sub-leading O(α^2) corrections we only need to interfere these contributions with the tree-level matrix element evaluated at p=0 and ϵ=0 (hereafter referred to as “1”). 
As argued in <ref> the contributions from final state transverse photon exchange, and corrections due to finite energy splitting in the debris do not contribute to the cross section at O(α^2).  We therefore conclude that all final-state interaction effects can be neglected at O(α^2) for observables that are inclusive with respect to the atomic debris (i.e., that have integrated over the debris' phase space). We note however, that corrections involving transverse photon exchange with the initial atom must still be evaluated for a complete O(α^2) analysis. 




 §.§ Neutrino flux normalization


One application of our results is for neutrino flux normalizations at long-baseline experiments such as DUNE and T2HK.  These experiments require the ν e →ν e cross section to be known with a precision on the order of (ideally at least a factor of ten smaller than) one percent. In reality these experiments measure hard outgoing electrons from atomic scattering. For example in the case of DUNE the relevant process is ν Ar→ν e B with B all the soft atomic debris. 

As discussed in <cit.>, atomic binding corrections are much smaller than power-counting would suggest due to accidental cancellations that come from the structure of the tree-level matrix element. Coulomb corrections vanish upon integrating over the final state debris, and transverse photon exchange vanishes at least until O(α^2).   We therefore conclude that all possible sources of QED corrections to ν A →ν e B scattering are under sufficient control (i.e., better than 1%) for neutrino flux normalization. 





 §.§ Extraction of hadronic vacuum polarization


Another process for which the results of this paper are relevant is the t-channel scattering of charged leptons off atomic electrons. We have in mind (in particular) muons scattering on electrons bound to carbon 


    μ(k)  ^12 C(0) →μ(k') e(p') B(h') .


This is relevant for the MuonE experiment <cit.> which aims to extract hadronic vacuum polarization at space-like kinematics. MuonE's data can potentially give an independent determination of hadronic vacuum polarization's contribution to the anomalous magnetic moment of the muon <cit.>. The theory demands of the experiment are a relative precision of 10^-5 <cit.>, which necessitates the inclusion of bound state effects. 

In Ref. <cit.> we have computed the shift in the shape of σ / t due to atomic binding effects. The results of this paper imply that final-state photon exchange between the struck electron and the atomic debris do not alter the conclusion of our previous work at O(α^2) (the working-order of Ref. <cit.>). 

Muon electron scattering, however, involves a charged muon which may also exchange photons with both the initial atom and the outgoing debris. The results of this paper are therefore not sufficient for a complete account of all of these effects, however we can comment immediately on a sub-class of diagrams. Let us first match the hard-photon mediated μ-e scattering vertex onto an effective four-Fermi contact operator, 𝒪̂. The operator's Wilson coefficient includes all short-distance radiative corrections.  

We are then interested in matrix elements of the form B'e'μ'𝒪̂Aμ. If we insert a complete set of states before the operator insertion, we can re-sum Coulomb exchange into an operator level expression involving Ŵ_v_μ^† where v_μ is the 3-velocity of the initial muon. We can resum Coulomb exchange between the final state leptons and the atomic debris as described above using the product of Ŵ_v_μ' and   Ŵ_v_e'. These operators commute with one-another since they only involve ρ̂(x) or equivalently ρ̂(q). Our experience with final state Coulomb exchanges suggests that for high energy muon atomic electron scattering, all the effects of soft Coulomb exchanges can be captured by making the replacement  


    â_p→Ŵ_v_μ'Ŵ_v_e'â_pŴ_v_μ^† ,


in the definition of the spectral function. The physical interpretation being that the Wilson-line before â_p “shakes up” the initial state. This leads to different sum rules than in the case of neutrino electron scattering, 


    ∫ϵS̃_A(ϵ,p) =  ∫^3 q/(2π)^3AŴ_v_μâ^†_qâ_pŴ_v_μ^†A .


We are interested in whether or not the operators Ŵ_v_μ and Ŵ_v_μ^† modify the sum rules that are relevant for our analysis. Corrections to the Koltun sum rule, and to the average kinetic energy are necessarily O(α^3) and can therefore be neglected in our O(α^2) analysis. 

 
The leading-order sum rule, ∫^3p/(2π)^3 ∫ϵS̃_A(ϵ,p) = Z, could in principle receive an O(α) or O(α^2) correction. Integrating over ^3 p, using ψ̂(0)= ∫^3 p/ (2π)^3 â_p and carrying out both integration over p and q we find


    ∫^3 p/(2π)^3∫ϵS̃_A(ϵ,p) = AŴ_v_μρ̂_e Ŵ_v_μ^†A .


where ρ̂_e=ψ̂^†ψ and the fields are evaluated at x=0. Then, since the electron charge density ρ̂_e commutes with Ŵ_v_μ we conclude that (provided <ref> holds) the modified spectral function satisfies the same sum rule as when Coulomb exchanges are ignored, 


    ∫^3 p/(2π)^3∫ϵS̃_A(ϵ,p) = ∫^3 p/(2π)^3∫ϵ  S_A(ϵ,p) = Z  ,


where Z is the atomic number of the state A. This is not surprising since Coulomb exchanges do not change the number of electrons in the atom. We therefore conclude that Coulomb exchanges in both the initial and final state do not interfere with the leading order amplitude for μ ^12 C→μ e B^+ scattering until at least O(α^3). 





§ CONCLUSIONS  
 

We have considered corrections to scattering cross sections that arise when a target electron is bound in an atom. The focus of the present paper is on final-state photon exchange and compliments our previous work on corrections due to the initial bound state <cit.>. 

We have studied which regions of loop momentum dominate virtual corrections, and find that photons with virtuality on the order of the atomic scale Λ∼α m_e dominate the integrals. We have organized these virtual corrections using a power counting scheme that makes use of the non-relativistic features of the atomic bound state, A, and final state debris, B^+. 

Within this power-counting, Coulomb exchanges provide the dominant leading correction (before summing over final states). We have studied these graphs in detail and find that they can be resummed in the leading non-relativistic approximation at an operator level. This includes arbitrary multi-particle intermediate states (e.g., ^12 C^3+ e^-e^-) with charge Q=+1.  Our work then provides an operator level definition of Coulomb corrections in the eikonal approximation for a system with small excitation energies. We find that the unitary nature of the resummed Coulomb corrections (they resemble a Wilson line) demonstrates that these corrections do not enter until at least O(α^3) for neutrino electron scattering. Our experience suggests that they likely do not enter for μ e →μ e scattering either (this second result relies on <ref> holding). 

We have also considered sub-leading corrections in the final state, such as transverse photon exchange (suppressed by non-relativistic velocities in the atom) and corrections to Coulomb exchanges from finite-energy splittings among different final states in the atomic debris. We find that both of these corrections (already down by α^2) do not interfere with the tree-level cross section after integrating over the final states of the debris, and therefore do not influence the cross section at O(α^2). We conclude that all effects necessary for a sub-percent prediction of the neutrino (atomic) electron cross section are under control. 


In the context of μ e →μ e scattering, the results of Ref. <cit.> and of the current work account for all of the leading “Z-enhanced” corrections to the cross section. These correspond to the class of diagrams that are O(Q_e^2 Q_B^2), with Z-enhancements in medium-heavy atoms; the correction to the cross section per electron scale parametrically as ∼ Z^4/3α^2. 



At O(e^4), there can be further corrections from final state interactions that stem from “mixed” corrections involving both Coulomb and transverse photons e.g., 


    diagram_mixed(20,6.5) 
        i1,i2o1,o2doublei1,b1,o1plaini2,t1,t2,t3,o2photon,tension=0,left=0.5t1,t3dbl_dots,tension=0t2,b1decor.shape=square,decor.size=10,decor.filled=emptyi210pt   .


These only receive Z-enhancements for Coulomb mode, but not for the transverse photon. Furthermore, loops with a transverse photon scale like α/(4π), whereas Coulomb modes scale like α/|v'| ∼α. We therefore expect that these “mixed” corrections scale as ∼ Z^2/3α^2/(4π) which is smaller than the purely kinematic binding corrections by[This estimate does not account for any large logarithms (or accidental numerical enhancements) that may appear in the mixed radiative corrections. ] Z^-2/3/(4π) ≈ 0.02. We therefore defer their evaluation to future work.





In summary, we have made further progress on understanding radiative corrections to leptonic interactions with atomic electrons. We find that photon exchanges (in both the initial and final state) between the atomic system and any high energy leptons cancel after integrating over the phase space of the atomic debris. The formalism for Coulomb corrections developed in this work is appropriate for systems with many low-lying energy levels that should be summed over. This is different from conventional treatments in terms of a static background potential. The results in this paper account for all maximally Z-enhanced corrections at O(α^2). The standard non-relativistic power-counting applied in this work should serve as a useful guide for future studies of atomic corrections to muon electron scattering for the MuonE experiment. 




§ ACKNOWLEDGEMENTS

Some of this work was done at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.  RP is supported by the Neutrino Theory Network under Award Number DEAC02-07CH11359. RP and MW are supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award Number DE-SC0011632, and by the Walter Burke Institute for Theoretical Physics. 









§ EXPONENTIATION OF COULOMB CORRECTIONS  


In this section we derive <ref> by considering the perturbative series that arises from Coulomb exchange. We show that within a non-relativistic approximation, the Coulomb exchange leads to an expression involving a unitary operator that is closely related to a Wilson line . The main novelty of our derivation is the inclusion of arbitrary states of the atomic debris between subsequent insertions of the charge-density operator. In particular, we do not model Coulomb exchanges using a (classical) background potential, and instead include all possible non-relativistic excited states of the debris as intermediate states in the perturbative expansion. 

We find that only one time-ordering contributes (the uncrossed ladder graphs). After expanding the ballistic electron propagator to take advantage of the small energy transfers (fixed by the non-relativistic atomic debris) the expressions are independent of intermediate state energies at leading order. This allows the intermediate states to be removed via completeness, and a simple operator-level identity remains. We now discuss two different derivations, on in the time-domain and one in the frequency-domain. 



 §.§ Marks Way  


It is convenient to define,


    ∫^4x ^ Q· x⟨ B'(h')e(p')|ψ̅(x)|B(h) ⟩ = u̅I ,


where Q is a residual electron four momentum and our convention is that the final state electron has zero residual momentum. Factoring out the energy momentum conserving delta function we write, I(q_1)=(2π)^4δ(Q+(h'-h))J(q_1). Next, I (or J) can be expanded in a series I=I^(0)+I^(1)+ …  where the superscript is the number of Coulomb exchanges. The leading order piece is I^(0)=(2π)^4δ(Q+(h'-h)), or equivalently J^(0)=1 and for the one photon exchange in Coulomb gauge I^(1) we have, treating the electron as ballistic and using the appropriate vertex and propagator,


    I^(1)=e^2 u^'0 u'· Q∫^4x_1 ^ Q· x_1⟨ B'(h')|J_0(x_1)|B(h)⟩ .


Here u^μ is the electron four velocity (u^μ=γ(1,  v') in the rest frame of the atom) and we have used the fact that the constituents of the states are non-relativistic so matrix elements of the electromagnetic 3-vector current are suppressed. Translating the charge density operator J_0(x)→ J_0(x=0) to the origin of space-time and performing the integration over x_1 the above is equal to 


    I^(1)= u^'0 u'· Qe^2  Q^2 (2π)^4   δ(Q+(h'-h))
       ⟨ B'(h')|J_0(0)|B(h)⟩ .


Next we use 


    J_0(0)=∫^3q_1  (2π)^3J̃_0( q_1)  ,


and  ⟨ B'(h')|J̃_0( q)|B(h)⟩∝ (2π)^3δ( Q- q). Note that J̃_0( q_1) is also evaluated at t=0. It can be combined with the factor 1/ q_1^2 into the Fourier transform of the potential energy operator Ṽ. Finally since non relativistic energies  are small compared to momenta we can write u'· Q≃ -u'^0 v'· Q yielding


    J^(1)= e^2 ⟨ B'(h')|∫^3q_1  (2π)^3Ṽ( q_1) v'· q_1|B(h)⟩ .


At second order we have


    I^(2)= ∫^4 q_2  (2 π)^4 u^'0 u'· q_2+ 0 u^'0 u'· Q +0 e^2  q_2^2 e^2  | Q- q_2|^2      
       ∫^4 x∫^4 y ^- (Q-q_2)· x^- q_2· y
       ⟨ B'(h')|T{J_0(y)J_0(x)}|B(h)  ⟩ .
 

First imagine doing the q_2^0 integration (see <ref> for an explicit discussion of the method by contours). The ballistic electron propagator only has a pole in the lower half q_2^0 complex plane. Consequently in the time ordered product of the currents only the time ordering where y^0-x^0 >0 contributes and we can so we can replace[Since 1=θ(x^0-y^0)+θ(y^0-x^0) and the second theta function gives no contribution.] 


    T{J_0(x)J_0(y)}→ J_0(y)J_0(x)θ(y^0-x^0)→ J_0(y)J_0(x) ,

in the integral. Next insert a complete set of states |X_1(p_X_1)⟩ between the two currents and translate the currents to the origin of space-time.
Then perform the coordinate integrations. This gives  a product of two delta functions, (2π)^4δ(p_X_1-h'+q_2)(2π)^4δ(h-p_X_1-q_2+Q). They  can be rewritten as a product of  the overall four momentum conservation delta function and one that performs the q_2 integration setting q_2=h'-p_X_1.  This yields


    J^(2)≃_X_11 v'·( h'- h)1  v'·( h'- p_X_1)   e^2  | h'- p_X_1|^2e^2  | h- p_X_1|^2⟨ B'(h') | J_0(0)|X_1(p_X_1)⟩⟨ X_1(p_X_1)|J_0(0)|B(h)⟩ .


Next express the charge density operator at the origin and an integral over the three dimensional Fourier transform, and use the fact that matrix elements conserve momentum to reassemble terms into the Fourier transform of the potential operator Ṽ. At this point the sum over intermediate states can be done yielding

    J^(2)≃ e^4 1  v'· ( h'-  h)⟨ B'(h')|∫^3 q_1  (2π)^3 Ṽ( q_2)1  v'·q_2∫^3 q_1  (2π)^3 Ṽ(q_1) |B(h)⟩ .



Since the matrix element under the integral is proportional to δ^(3)([h-h']-[q_1+q_2] ) this can be rewritten as, cf. <ref>, 

    J^(2)=  e^4 ∫^3 q_1  (2π)^3    ∫^3 q_2  (2π)^3 1  v' · (q_1 +q_2)1  v'· q_2
       ⟨ B'(h')|Ṽ( q_2) V( q_1) |B(h)⟩ .

Obviously this procedure can be repeated to deduce expressions for all the J^(n).





 §.§ Ryan's way  


Let us repeat the above analysis with a different emphasis and method. Consider the n^ th-order in perturbation theory. Photon exchange between the electron line and the charged-debris |B⟩ leads to a conventional diagrammatic expansion (with fermion and photon propagators) and a “blob” on the bottom of the diagram with an incoming ionic system B and an outgoing system B'. This “blob” corresponds to a multi-current correlator, G(q_1,…, q_n) of time-ordered electromagnetic currents (specifically Ĵ_0=ρ̂ for Coulomb exchange), 


    G^(n)=∫{ x}^+∑ q_i · x_iB'T{Ĵ_0(x_1) …Ĵ_0(x_n)}B.


In analogy with the transition from I to J discussed above, it is convenient to factor out an overall momentum conserving delta function and introduce 𝒢. 


    G^(n)(q_1,…, q_n) =    (2π)^4 δ(Σ q- Q) 
         ×𝒢^(n)(Q; q_2,…, q_N)  .


where Q = h'-h  is enforced by the delta function. Notice that 𝒢^(n) is a function of n-1 variables, while Q is a parameter fixed by the states |B(h)⟩ and ⟨B'(h'|. 

Making use of the eikonal approximation on the highly energetic electron leg, and working in Coulomb gauge, we then find



    e' B'ψ̅B = u̅(p') [ 
    ⟨B'|B⟩ + e^21/q_1^2u'_0/u'·Q𝒢^(1)(Q) + e^4∫^4 q_2/(2π)^41/q_2^2u'_0/u'· q_21/|Q-q_2|^2u'_0/u'·  Q𝒢^(2)(Q; q_2) + …]  .


Extension to higher orders is straightforward: one appends an extra integral, an extra fermion and photon propagator, and uses 𝒢^(n)(Q; q_2,…, q_n). The term in square brackets in <ref> corresponds to J as defined beneath <ref>. 

It is useful to write Ĵ_0(0) in terms of its components which supply definite momentum transfer k. For example,


    𝒢^(1)(Q)= B'(h')Ĵ_0(0)B(h) = ∫^3 q_1/(2π)^3B'ρ̂(q_1)B ,


where ρ̂(q_1) = ∫^3x  ^-q_1·xĴ_0(x). 

We now proceed with a direct evaluation of the loop integrals by contour integration. Let us study 𝒢^(2)(Q; q_2) as an explicit example. Begin by explicitly time ordering the currents in 𝒢^(2). Inserting a complete set of states, re-writing Ĵ_0(0) in terms of ρ̂(q_1), and carrying out the remaining integral over ^4x we obtain (defining ω_2=q_2^0), 


    𝒢^(2)(Q; q_2)= ∫^3q_1/(2π)^3_X B'ρ̂(q_1)X-/ω_2-Δ E_X + 0Xρ̂(q_2)B+B'ρ̂(q_2)X-/ω_2 -Δ E_X'- 0 Xρ̂(q_1)B,




where Δ E_X = E_X - E_B and Δ E_X'=E_X-E_B'. This result is obtained by translating the currents to t=0 and integrating the exponentials. It essentially amounts to a derivation of time-ordered perturbation theory, and the two terms in <ref> can be interpreted as a crossed and uncrossed Feynman diagram.

Next notice that because 1/u'· q_2 ≡ 1/(u'· q_2 + 0) only has a pole in the lower-half plane. We may then close our contour in the upper-half plane which selects only the second term in <ref>. At higher orders in perturbation theory all time orderings with poles in the lower-half plane vanish after contour integration; these correspond to crossed ladders. It follows that only the uncrossed ladders contribute to the amplitude to any order in perturbation theory, in agreement with the time-domain analysis presented in <ref>. 



Expanding the eikonal propagators and dropping the (now on-shell) energy transfer Δ E_X gives, 


    [u'_0/u'· q_2 + 0]_ω_2=Δ E_X'≃1/-v'·q_2 + 0(1+O(Δ E_X'/|q|)) .


The propagator structure is then independent of Δ E_X, and the sums over intermediate states reduce to the identity by completeness. We are then left with <ref>, with the n^ th order contribution being given by 



    e^2n∫[^3q_1/(2π)^3…^3q_n/(2π)^3 ]1/v'·q_11/v'·(q_1+q_2)…1/v'· (∑_i q_i)∏_i ρ̂_i(q_i)/q_i^2 = e^2n/n![∫^3 q/(2π)^31/v'·q- 0ρ̂(q)/q^2]^n ,


where the equality follows from symmetrizing the eikonal propagators inside the curly braces (introducing the factor of 1/n!), and using the soft-photon identity <cit.>, 


    ∑_ permutations1/x_π(1)1/x_π(1) + x_π(2)…1/x_π(1) + x_π(2)+ … + x_π(n) = 1/x_11/x_2…1/x_n .











apsrev4-1




 Paper Title: Waveform resampling with LMN method.


Minimal Quantum Circuits for Simulating Fibonacci Anyons
    Layla Hormozi
    August 5, 2024
========================================================





§ INTRODUCTION



Resampling is the process of transforming a signal in the form of samples collected at discrete points in some domain to a new set of samples collected at novel points in that domain.
The quality of a resampling is typically judged 
by how accurately these novel samples represent the original, underlying signal.
One central category of resampling methods is interpolation.
It asserts some model of the underlying signal and is normalized to preserve the local signal amplitude.
Other normalization policies may be asserted.
For example, an application may require the resampling to leave the integral of the signal invariant.

The computational expense of a particular software implementation of a resampling method is another important consideration.
Various software packages that are used in the analyses of data from particle physics experiments provide resampling method implementations including Boost <cit.>, Eigen <cit.>, SciPy <cit.> and ROOT <cit.>.


Methods can be placed in two broad categories.
The first category includes those methods that operate in the sampled domain (e.g. interval or time domain)
and include
Cardinal B-spline interpolation <cit.>, Barycentric Rational interpolation <cit.>, Hermite interpolation <cit.>, and Catmull Rom spline interpolation <cit.>.
Interval domain methods can be fast but tend to not preserve spectral features.

The second category includes methods that consider the signal spectrum or otherwise operate in the frequency domain.
These include Whittaker-Shannon interpolation <cit.> and trigonometric interpolation <cit.>.
Typically, these methods apply operations in the frequency domain that are efficient while the discrete Fourier transform (DFT) as implemented with the Fast Fourier Transform (FFT) algorithm <cit.>[Gauss developed an algorithm akin to FFT in 1805 for a similar purpose of interpolating orbital measurements <cit.>.]
provide an efficient means to transform a signal between interval and frequency domain representations.
While the FFT is indeed fast, its complexity still grows as 𝒪(Nlog N) in the size N of the input signal.
Naive resampling methods can require inflating the original signal size by a large factor.
This inflation subjects the frequency domain method to ruinously large N.

Scaling problems can be exacerbated when the signal is large and multi-dimensional.
Interval-domain resampling can be extended to this case.
For example the bilinear interpolation <cit.> is a commonly employed method to interpolate images in the two-dimensional interval or pixel domain.
As with the one-dimensional case, these extensions to two-dimensional intervals tend to distort spectral features which can be problematic in some applications.
One dimensional methods that resample in the frequency domain may also be extended to multiple dimensions in a simple manner when those dimensions are independent.  They require application of a series of DFT along each dimension to reach their frequency domain.  An FFT applied to a two-dimensional signal of size N × M then costs 𝒪(MNlog N + NMlog M).
The inflation required by naive resampling methods then become yet more costly.

In this paper, we describe a variant of trigonometric interpolation, the LMN method, that implements an optimized resampling of discrete waveforms in the frequency domain by avoiding the inflation that comes with naive algorithms consisting of an integer upsample factor followed by an integer downsample factor.
The quality and computational expense of this method are compared to others that are used in the field.


This work is motivated by a technical task required in the processing of data from detectors used by the Deep Underground Neutrino Experiment (DUNE) <cit.>, a next-generation, long-baseline neutrino oscillation experiment.
DUNE’s primary physics objectives include determining the neutrino mass hierarchy, search for leptonic CP violation, and search for physics beyond the standard model of particle physics.
DUNE's primary detector technology is the liquid argon time projection chamber <cit.> (LArTPC).
To understand the signals from these detectors, their operation is summarized.
When a charged particle traverses the liquid argon in a LArTPC detector, ionization electrons are produced along the particle's trajectory.
Under the influence of an external electric field, the ionization electrons drift at a constant velocity toward a stack of three parallel “anode planes”.
Each anode plane is comprised of an array of parallel electrodes spanning the width of their plane.
The ionized electrons induce a negative electric current as they approach an electrode and a positive current as they drift past.
This induced current is amplified, filtered, digitized and sampled and the resulting discrete waveform is recorded.
Given the location of each electrode, these sampled waveforms represent a tomographic measurement of the distribution of drifting electrons in two dimensions: transversely across the electrode array and longitudinally along the electron drift direction.
No electron position information is available in the dimension along the length of the electrodes.
The anode planes are oriented so that their arrays of electrodes are at mutually unique angles.
Given a collection of these two-dimensional tomographic views from all anode planes, a three-dimensional image of the ionized electrons can be constructed.
From that image, it is possible to reconstruct information about the neutrino-argon interaction.
Finally, neutrino properties can be deduced from the ensemble of many such reconstructed interactions.
Lengthy and complex software developed by DUNE and the larger LArTPC community is applied to theses recorded waveforms to perform this extraction of physics information.

DUNE's far detector underground cavern provides for four independent cryostats, each containing approximately 17.5 kt of liquid argon. 
Currently, two far detector module designs are in the process of being implemented.
Each is based on a variant of common LArTPC design principles.
The modules are known by the direction of their electron drift: the first far-detector module FD1-HD <cit.>  employs a horizontal drift and the second FD2-VD <cit.> drifts in the vertical direction.
They also utilize different anode plane and readout technologies.
In particular, the FD2-VD design divides its liquid argon volume into two regions and 
supplies anode plane electrodes as traces on large printed circuit boards
installed near the top and bottom of the cryostat.
Supported by the international DUNE collaboration, the design specifies different types of readout electronics for top and bottom volumes. 
The top electronics are based on the design developed for the dual-phase liquid argon detector technology <cit.>.
The bottom electronics design <cit.> is in common with that used for the FD1-HD <cit.> module.
It is chosen to minimize the overall detector capacitance, which in turn increases the signal-to-noise ratio. 
This choice is driven by the necessarily long cable paths from the bottom anode planes to the cryostat roof. 

One consequence of employing different readout electronics technologies in FD2-VD is that a variety of sampling periods are employed. 
Specifically, the top electronics employs analog-digital converters (ADC) with a sampling period of 500ns while the bottom ADCs sample at 512ns. 
In addition, the FD1-HD will sample at 512ns.
Prior LArTPC experiments (MicroBooNE <cit.>, ProtoDUNE-SP <cit.>, SBND) for which most of the existing data processing software was developed sample at 500ns.  

The readout electronics apply low-pass filters to their input analog signals in order to satisfy the Nyquist-Shannon <cit.> condition in the subsequent digitization stage.
Any sampling period that places the Nyquist frequency comfortably above the signal bandwidth may be chosen so that the information in the signal will be faithfully captured.
While the exact choice of sampling period is not critical, employing multiple sampling periods across the detectors leads to a technical complication for the data-processing software.
It is desirable to resample the ensemble of waveforms across the variety of detectors to produce waveforms with a common sampling period prior to applying the large software suite.
If the resampling assures the Nyquist frequency remains above the signal bandwidth and it does not otherwise distort the waveforms then this resampling will preserve all information relevant to the underlying physics.


This paper is organized as the following.
In <ref>, the signal processing procedure applied to LArTPC data is briefly reviewed as it comprises the first stage of data processing and it places strong constraints on the quality of any resampling method.
The LMN resampling method is then described in <ref>. 
In <ref>, the performance of the LMN method in terms of accuracy and computational cost is compared with some popular interpolation methods.
The application of the LMN method to realistic LArTPC data is shown in <ref> using simulated data. 
Finally, a summary and discussions can be found in <ref>.




§ LARTPC SIGNAL PROCESSING















When charged particles traverse the liquid argon medium, electrons are ionized along the particle trajectory and drift toward the anode planes under the influence of the external electric field. 
While drifting, the distribution of ionization electrons undergoes diffusion and absorption effects <cit.>.
The TPC signal is then formed with i) the distribution of ionized electrons arriving at the anode plane, ii) the field response describing the resulting induced current on the anode electrodes <cit.> (see Ref. <cit.> for a recent example of calculating field response in three dimensions), iii) the electronics response, which amplifies, shapes and converts current to a voltage signal that is band limited, and iv) the ADC that samples the continuous voltage signal to produce discrete-time waveforms.

The goal of the signal processing procedure is to effectively invert this signal formation process.
It reconstructs a measure of the distribution of ionization electrons arriving at the anode electrodes given the recorded waveforms. 
The kernel of the procedure deconvolves a model of the overall detector response, which includes field and electronics responses, from the waveforms.
A technique to perform this deconvolution as a per-waveform filter that is applied in the frequency domain via discrete Fourier transform (DFT) was introduced to the LArTPC field in Ref. <cit.>.
It relied on a one-dimensional (1D) model that described how the drifting electrons induce current in an electrode as a function of time.
This was later improved by a new deconvolution that extending the model two dimensions (2D) by adding variation as a function of distance transverse to an electrode.
This new 2D method was first applied in the MicroBooNE experiment <cit.> where it demonstrated improvements compared to the 1D method.
The 2D method has since been utilized in the ProtoDUNE detectors, (DUNE prototype detectors), where it also achieved good performance <cit.>. 

For simplicity and to share the notation of <ref>, we describe the 1D technique that performs a deconvolution over the time dimension.  The 2D deconvolution is conceptually a straight-forward extension.
The technique extracts a finite, sampled sequence ŝ[n] that estimates the number of electrons drifting past a given electrode  s[n] ≡ s(t_n) ≡∫_t=t_n^t_n+Δ t q(t)dt, n∈[0,...,N-1] over a sample time period [t_n,t_n+Δ t] to produce a sequence of N samples.
This signal s[n] is not directly observable as it represents the true distribution of ionization electrons.
Instead, a detector provides a measure m[n] of the signal after the signal has been altered by a detector response modeled in general as r[n',n].  In addition, noise is inescapable and is modeled as η[n].  In terms of these sampled sequences, the measure is modeled as


    m[n] = ∑_n' r[n',n] · s[n'] + η[n].

The first term is a convolution of the detector response and the signal.  
For time-invariant detectors, the response may be replaced with r[n',n] → r[n'-n].  This form allows the discrete Fourier transform (DFT, defined below in <ref>) to be applied to rewrite <ref> as M[k] = R[k]S[k] + 𝒩[k] were k enumerates frequency domain samples. 
Neglecting noise for the moment allows for an estimate of the signal to be formed that may be expressed in the frequency domain as,

    S[k] ≈Ŝ[k] = M[k]/R[k].


Naively, the inverse DFT (defined below in <ref>) may be applied to recover a time domain estimate ŝ[i].
However, several aspects of real detectors coincide to make that final step impractical.
First, the low-pass filter that is applied by the detector electronics to the analog input is designed to vanish at high frequency in order to satisfy the Nyquist-Shannon condition for the subsequent sampling.
Next, the electrodes comprising the initial anode planes do not collect drifting electrons.
This means their net induced current is zero and their response is bipolar in time resulting in R[k→ 0] → 0.
Finally, real-world noise can not be neglected.
The noise to which LArTPC detectors are subjected fills a broad-band spectrum 𝒩[k] that can extend up to and beyond the Nyquist frequency while some sources will introduce noise in the circuit downstream of the band-limiting low-pass filters. 
Thus, the presence of noise in the measure M[k] and a response R[k] that is necessarily small or vanishing leads to  <ref> producing a naive signal estimate that is artificially amplified at high frequency and, for the initial anode planes, will diverge at low frequency.

To combat this practical problem, digital filtering is applied to the measure in both the frequency and time domains.
The time domain filter[The 2D technique also filters in the transverse or “channel” domain.] is introduced by rewriting <ref> as

    S[k] ≈Ŝ[k] = M[k]/R[k] F[k].

From this result, an estimate ŝ[n] in the time domain may be constructed by applying the inverse DFT.
In fact two such estimates are found, each with their own filter.
The first is constructed with a “Wiener-inspired” filter that maximizes signal-to-noise.
The second uses a (truncated) Gaussian-shaped filter to give an unbiased measure of the signal.

The time-domain digital filtering consists of identifying “signal regions-of-interest” (ROIs).  Each ROI selects a fragment of the signal estimate after the Wiener-inspired filtered has been applied and that is expected to have a high signal-to-noise ratio.
The algorithms that identify the signal-ROIs are complex heuristics created by a development process that considered data from real LArTPC detectors such as MicroBooNE <cit.>.
These ROIs are then applied to the Gaussian-filtered estimate and a new signal baseline is calculated for each region based on values at either ends of the ROI. 
This process effectively combats the problematic combination of noise with the small or vanishing portions of the response spectrum. 






§ LMN RESAMPLING METHOD
 




We review established resampling methods on which the LMN method is based. 
We then describe the LMN method, which is essentially identical to the basic method but adds a constructed, special-case optimization.
Finally, a discussion of normalization, information loss and mitigation of artifacts is given.



 §.§ Resampling basics



A continuous signal g(t) may be sampled at times t_n = n· T_s, n∈ [0,...,N_s-1] with period T_s to produce a time-domain sequence g⃗_s ≡{g_s[n]}, g_s[n] ≡ g(t_n).
From this sequence, another sequence
g⃗_r ≡{g_r[n]}, n∈ [0,...,N_r-1] may be constructed that corresponds to a resampling of g(t) with a sampling period T_r  T_s.
This correspondence may be exact or aproximate 
depending on the resampling method, the bandwidth of the signal and whether an overall upsampling (T_r < T_s) or downsampling (T_r > T_s) is performed.

One general resampling method known as fractional resampling <cit.> decomposes the resampling procedure into two stages.  First, the original sequence g⃗_s is upsampled by an integer factor L to produce an intermediate sequence g⃗_u with a larger number of samples N_u = LN_s.
Second, this intermediate sequence is downsampled by an integer factor M to produce the final sequence g⃗_r with a number of samples N_r = N_u/M.
The simple fraction ≡L/M = N_r/N_s characterizes the fractional resampling.


If the duration of the original signal is unchanged in the resampling then N_s T_s = N_r T_r and thus = T_s/T_r.  However, if T_s/T_r is irrational then fractional resampling can not be applied exactly.  In such cases, a simple fraction ≈T_s/T_r may be chosen to approximate the desired resampling ratio.  The choice can be arbitrarily precise at the cost of potentially large values of L and M.




 §.§ Discrete Fourier transform



The resampling used here is based on a subclass of these integer fractional methods that operate on the Fourier representation of the sampled signal <cit.>.
The discrete Fourier transform (DFT) is applied to the time-domain sequence g⃗ of size N to produce the frequency-domain sequence G⃗≡{G[k]} = DFT(g⃗) with


    G[k] = ∑_n=0^N-1g[n] · e^-i2πnk/N, k ∈𝕀.

This sequence is infinite and periodic, G[k] = G[k+N] over size N, and has two points of Hermitian symmetry which are important for the resampling.
The first symmetry is around the “zero frequency” sample G[0] so that G[k] = G^*[-k].
The second is around the Nyquist frequency and is expressed differently if the number of samples is even or odd as


    G[H-k] =
    
    G^*[H+k+2],      N ∈𝒩_even
    
    G^*[H+k+1],      N ∈𝒩_odd

Here, H gives the size the “half spectrum” that consists of the sub-sequence of frequency-domain samples that are not Hermitian-symmetric with themselves.  It is given by

    H =
    N/2-1 ,    N ∈𝒩_even
    N-1/2 ,    N ∈𝒩_odd

For N ∈ℕ_even, the sample G[H+1] is real and called the “Nyquist bin” as the Nyquist frequency lands at the center of its frequency band.  For N∈ℕ_odd, the Nyquist frequency is at the lower edge of the sample G[H+1].

There are two common ways to represent a finite subsequence of G⃗ of size N.
First is the “Nyquist-centered” range where the subset of samples {G[1],...,G[H]} are Hermitian-reflected about the Nyquist frequency starting with G[H+1] = G^*[H] for N ∈ℕ_odd or with G[H+2] = G^*[H] for N∈ℕ_even.
The second is the “zero-centered” range where
the same subsequence of size H is instead Hermitian-reflected about G[0].
Influenced by this second choice, the reflected subsequence is often said to consist of the “negative frequency” samples.

The inverse DFT may be applied to G⃗ to recover the original time-domain sequence g⃗ as

    g[n] = 1/N∑_k=0^N-1G[k] · e^i2πnk/N, n ∈𝕀.

The result of the inverse DFT is also an infinite, periodic sequence.  Note, the form of the normalization term 1/N shown here is the convention followed by most software providing the DFT.  Additional discussion of normalization is in Sec. <ref>.



 §.§ Resampling in the Fourier domain



An upsampling from N_s to N_u = LN_s can be accomplished in the frequency-domain by 
inserting (L-1)N samples at both ends of the zero-centered range (or equivalently  in the middle of the Nyquist-centered range) of G⃗_s to produce an intermediate spectrum sequence G⃗_u with samples giving,


    G_u[k] =
    
    G_s[k],     k ∈ [-H_s, ..., 0, ..., H_s] 
    
    0     ,    otherwise

Here, H_s represents the “half spectrum” size for the original sampling as defined in Eq. (<ref>) and it is assumed that N_s ∈ℕ_odd.
When N_s ∈ℕ_even, Eq. (<ref>) applies with the addition that the Nyquist bin at G_s[H+1] requires special treatment which is described in Sec. <ref>.

If N_u/M∉ℕ then the final M-wise downsampling 
must be performed in the time domain after applying the inverse DFT of (large) size N_u = LN_s to G⃗_u.
This allows any remainder sequence of interval domain samples of size N_u mod M to be ignored and corresponds to an effective change in the duration of the resampled signal compared to the original.

In the special case that N_u/M ∈ℕ, the downsampling may be performed in the Fourier domain by removing  N_u/M(M-1) samples from the middle of the Nyquist-centered range to produce G⃗_r of size N_r with terms (expressed zero-centered),

    G_r[k] = G_u[k], k ∈ [-H_r, ..., 0, ..., H_r].

Here again, H_r gives the “half spectrum” size as defined in Eq. (<ref>) for the resampled sequence. Eq. (<ref>) is strictly for N_r ∈ℕ_odd and treatment of the Nyquist bin for the case N_r ∈ℕ_even is put aside until Sec. <ref>.

Obviously, in this special case, the upsampling and downsampling stages may be combined so that the net number of frequency-domain samples is changed by Δ N = N_r - N_s.  When Δ N > 0 a net upsampling is performed by inserting this number of zero-valued samples similar to Eq. (<ref>).  
Conversely, Δ N < 0 indicates a net downsampling is performed with (the absolute value of) this number of samples removed as in Eq. (<ref>).
To finish, the intermediate spectrum G⃗_r of size N_r is formed and the inverse DFT is applied to produce a resampled time-domain sequence g⃗_r.  In this special case, the size of the forward DFT is N_s and the size of the inverse DFT is N_r, both of which are typically many factors smaller than the inflated N_u.  It is seeking this special case that leads to the LMN optimization.



 §.§ Special-case optimization



To give it a name, creating this special case or simply exploiting it when it is accidentally met is called here the LMN method.
The special case NL/M∈ℕ  can be met for any given T_s and T_r that meet a rationality condition described below and when there is freedom to choose a value of N_s even if it may differ from the size of a given input signal.
It is possible to generate a series of values for N_s that satisfy N_sL/M ∈ℕ.  
Constructing the special case begins by considering that the resampling will shift the sampling frequency by an amount


    Δ F = 1/T_r - 1/T_s = T_s-T_r/T_sT_r.

Next, this shift must span an integer number δ N ∈ℕ of original frequency-domain samples and thus must cover a frequency range


    δ F = δ N/N_sT_s.

Equating δ F = Δ F results in the relation


    N_s = δ N · T_r/T_s - T_r∈ℕ.

The smallest value of |δ N| that satisfies Eq. (<ref>) results in a minimal sequence size N_s = N_min for which the special case applies.  Any multiple of N_min will also satisfy the special case.
If found, we may select the multiple of N_min that best approximates the size of a given input signal.

The satisfactory value for δ N may be found as

    δ N = T_s - T_r/(T_r, T_s - T_r).

Here, (·,·) returns the greatest common divisor of its arguments.
As the arguments are in general real valued, this function may apply the Euclidean algorithm to calculate its result.
Computer floating-point round-off error will require the algorithm to test for convergence against a small, non-zero error parameter.
Note, a lack of convergence within this error indicates that the ratio of its arguments is not rational.
When this LMN rationality condition is not met, this form of fractional resampling can not be applied exactly.  
This is similar to the rationality condition of the integer fraction method on which LMN is based.

The choice of the special case size N_s of course determines the size of the forward DFT and (along with the chosen resampling ratio ) the size of the inverse DFT. 
The freedom to select a multiple of δ N may be exploited for a secondary optimization.
The speed of the DFT strongly depends on the size of prime factors that make up the length of the sequence being transformed <cit.>.
A DFT size with large prime factors requires substantially more processing than nearby larger sizes.
Thus, all else being equal, an application of the LMN method may find that an otherwise unnecessary extension  of the original interval domain sequence by an additional δ N may lead to an overall faster resampling.
The effects of this dependence on the sizes of the prime factors of a sequence length can be seen in the performance results presented in <ref>.



 §.§ Treatment of Nyquist bins



In general, when N ∈ℕ_even the sequence G⃗ possesses a Nyquist bin sample.  When resampling in the Fourier domain, this sample requires special treatment in two cases. The first case <cit.> is when a net upsampling from size N_s ∈ℕ_even is performed.
Half of the Nyquist bin sample value G_s,nyquist = G_s[H_s+1] is transferred to each of the two equivalent samples in the resampled spectrum,
G_r[H_s + 1] = 1/2G_s,nyquist  and G_r[-H_s - 1] = 1/2G_s,nyquist.  Note, neither of these two samples in the upsampled spectrum G⃗_r are themselves a Nyquist bin.
The second and opposite case is when a net downsampling to size N_r ∈ℕ_even is performed.
The Nyquist bin in the resampled spectrum must be set to the absolute value of either of the two corresponding samples from the original spectrum, for example as G_r[H_r + 1] = |G_s[H_r + 1]|.



 §.§ Normalization



A “round trip” of a sequence through the forward and inverse DFT requires an overall normalization of 1/N shown in Eq. (<ref>).
We follow here the dominant convention to apply this normalization fully as part of the inverse DFT.
Other conventions are possible as long as a total of 1/N normalization is applied in the round trip.
For example the symmetric normalization places 1/√(N) on both forward and inverse DFT.
All such conventions produce the same result as long as N is unchanged throughout the round trip.

However, resampling in the Fourier domain necessarily performs the forward DFT with size N_s and the inverse DFT with size N_r  N_s.
Care is needed to select a normalization convention that preserves a desired invariant.
The choice of invariant may depend on how an application interprets the quantity measured by signal samples.

For each of three interpretations considered below, the nominal 1/N applied as part of the inverse DFT is taken for granted and an additional normalization of the form g_r[n] → g'_r[n] = Ag_r[n] is provided.  It is also assumed that only zero-value frequency samples are inserted or removed.  Later we consider the case where this assumption is not held.



  * Interpolation:


  An interval-domain sample gives a measure of the instantaneous value of the underlying continuous signal.
  This interpretation requires resampling to preserve the local amplitude of the interval-domain samples and thus act as an interpolation.

  The nominal DFT convention applies a normalization of 1/N_r and so the signal amplitude will be reduced in an upsampling and increased in a downsampling compared the a direct resampling of the continuous signal.
  Thus, an additional normalization of A_0 = N_r/N_s must be applied to the resampled sequence for correct interpolation.
  Alternatively, applying this normalization is equivalent to applying the basic DFT normalization fully as part of the forward DFT instead of following the accepted convention to apply it fully as part of the inverse DFT.


  * Integral:  An interval-domain sample represents the integral of the underlying continuous signal over the sample period.
This interpretation must preserve the sum of the time-domain sequence samples through the resampling process as expressed by

    ∑_n=0^N_s-1 g_s[n] = A_1A_0∑_n=0^N_r-1 g_r[n].

If we were to apply only the interpolation normalization A_0 it would be equivalent to integrating each of N_r points over the original sample period T_s while integration over the resampled period T_r is proper to preserve the equality between sampled and resampled sums.  Thus an additional normalization of A_1 = T_r/T_s = N_s/N_r must be applied in addition to the interpolation normalization.
These two normalization terms cancel A_1A_0 = 1 and thus the nominal DFT normalization convention is directly appropriate for the integral interpretation.


  * Energy:  An interval-domain sample gives a measure of a probability amplitude, field or other quantity where the sum-of-squares must be conserved through the resampling.
  This sum is conserved through the forward DFT as 
expressed by Parseval's theorem which equates an energy measure in the time domain E_t to an energy measure in the frequency domain E_f.  In terms of the original sampling, this may be written as

    E_t,s≡∑_n=0^N_s-1|g_s[n]|^2 = 1/N_s∑_k=0^N_s-1|G_s[k]|^2 ≡ E_f,s.

The energy after resampling may be written as

    E_f,r≡1/N_r∑_k=0^N_r-1|G_r[k]|^2 = 1/N_r∑_k=0^N_s-1|G_s[k]|^2 = N_s/N_rE_f,s.

The effective replacement r → s in the sum of right hand term is allowed as we have assumed only zero-amplitude frequency samples have been added or removed in the resampling.  Thus, in order for Parseval energy to be conserved through the resampling, an additional normalization A_2 = √(%s/%s)N_rN_s must be multiplied to the resampled time-domain sequence, g_r[n] → g'_r[n] = A_2g_r[n].


In the above discussion, it is assumed that only zero-amplitude frequency-domain samples are inserted or removed in the resampling.  In the case of upsampling, this may assured by construction.  In the case of downsampling it is only assured if the original signal is band-limited to be below the new, smaller Nyquist frequency.  When this band limit is not assured, a downsampling effectively applies a perfect low-pass filter with a cut-off at the resampled Nyquist frequency 1/2T_r.  The removal of the non-zero samples represents actual loss of signal information.




 §.§ Potential artifacts


The method just described, and indeed any based on a round-trip via DFT, assumes the signals are periodic and band-limited.  Real-world signals are typically neither.  Resampling such signals can induce artifacts.



The original sampling produces a sequence over a finite interval window.  It is typically inescapable that the end of that sequence will not “wrap around” to form a smooth connection to the beginning of the sequence.  Even in a special case where an underlying signal is constructed to be periodic it typically includes aperiodic noise.

In the frequency domain, another source of artifacts can arise.  While it is typical to band-limit signals with a low-pass filter prior to the original sampling, real world low-pass filters merely attenuate and do not fully remove power above their “cut-off” frequency.  Another source of high-frequency content is noise that is picked up by  electronics circuit components that exist after the low-pass filter.  Insertion or removal of frequency samples is effectively equivalent to the application of a sharp low-pass filter.  This results in some amount of ringing in the interval domain after the inverse DFT is applied.



The LMN method provides ways to mitigate the endpoint discontinuities of non-periodic signals. 
To achieve the special case N_s size, the original signal must either be truncated or extended in the interval domain. 
When extended, there is freedom to choose the values applied when padding the signal to size N_s. 
Simple schemes can be employed to allow the endpoints of the signal to be smoothly, periodically connected. 
For example, padding can be filled with an interval-domain linear interpolation between last and first sampled values of the original signal. 
Depending on the degree of discontinuity, the sharp “kinks” this can create may still produce ringing. 
If significant, more sophisticated padding models can be applied to form a smoother connection.
If the application allows, these artifacts can also be mitigated by exploiting the intermediate frequency-domain representation. 
There, a low-pass filter spectrum can be multiplied.
Both of these forms of mitigation require minimal additional computational expense.




§ PERFORMANCE COMPARISON WITH SELECTED METHODS

The performance of a resampling algorithm can be evaluated in different ways and it will vary subject to the shape of the considered signal as well as the resampling ratio. 
Therefore, it is important to evaluate various performance metrics and to do so for a range of signal frequencies, noise levels, and sampling ratios. 
In this section, the performance of the LMN method in a waveform resampling task will be compared with selected interpolation methods including: i) linear interpolation, ii) polyphase filtering, and iii) cardinal cubic B-spline.



 §.§ Considered signals


The measure introduced in <ref> is idealized by a signal generated from an analytic function -1≤ f ≤ 1 representing the convolution s * r and a white-noise term c_nη formed by scaling samples drawn from the normal distribution.




This idealized measure is then sampled at period T_s ≡1/N_s to produce an initial sampled sequence g[n] = f[n] + c_n ·η[n], n∈[0,...,N_s-1].





The following analytic functions were used to generate sampled sequences that will be input to ressampling.







         f(t)    = sin(c_f ·t)         single freq  

         f(t)    = 1/2 (cos(c_f ·t) + cos(1.4 ·c_f ·t + π/3))         multi freq 

         f(t)    = exp(-t^2) ·cos(c_f ·t^2)         gauss pulse


Here, c_f is a frequency parameter that allows to generate a similar signal with a different frequency. Examples of such signals can be seen in <ref>. 



 §.§ Performance metrics


The initial sequences are resampled g[n]→ g_r[n'], n∈[0,...,N_s-1], n'∈[0,...,N_r-1].
The top of <ref> shows an example of a signal of the gauss pulse type (<ref>) with one choice of frequency and noise parameters.
In the middle, the results of resampling this signal from N_s=500 to N_r=512 points while conserving total signal duration are shown for the considered resampling methods.
This is representative of upsampling a signal with sample period 512ns to one with 500ns.
The bottom panel shows the difference between the resampled discrete noisy signal (“Model”) and the “pure” noise-free signal function f(t) evaluated at the resampled times (“Truth”).










 §.§ Scaling with signal frequency

In order to evaluate the quality of a resampling across a wide spectrum of signal frequencies, sequences were generated with the single freq function (<ref>) for different values of the frequency parameter, c_f. Each sequence was then resampled from N_s=500 to N_r=512 points using the considered resampling methods.
To quantify how well the resampled signal matches the underlying true signal, the mean-squared error (MSE) statistic of <ref> is calculated.


    MSE = 1/N_r∑_i=0^N_r-1(f[n]-g_r[n])^2.



<ref> shows the MSE as a function of the signal frequency parameter c_f for the case of no noise (top panel) and a chosen non-zero value of c_n (bottom panel). 
For the noiseless case, it can be seen that the LMN method results in mean squared errors very close to or identical to zero across all considered values for c_f. 


The other considered resampling methods exhibit higher MSE with increasing signal frequency parameter. 
In the case of a noisy signal, the MSE for signals with a small frequency parameter is lowest for the linear interpolation. 
This is not necessarily desired, as the noise level should in principle remain untouched by resampling.
This reduction of noise is the consequence of an implicit spectral distortion.



In order to understand the inaccuracy of some of the resampling methods at high signal frequencies,
frequency-domain spectra of original and resampled sequences are show in in <ref> for the case c_f = 200.
In the figure, the original signal spectrum is nearly identical to the resampled spectra from the LMN and polyphase filtering methods.
Resampling with the other methods, however, introduces artifacts below and above the signal frequency and show reduced power at the sole frequency of the true signal.





 §.§ Processing time scaling with sampling rate



Besides being lossless, a good resampling method should also be computationally efficient.
As a way to evaluate this efficiency, the various resampling algorithms are deployed to resample the same noisy signal 100 times from which average processing times are determined.
The original signal with N_s=500 data points is resampled to sizes between 501 and 565 points.
<ref> shows the average processing time as a function of the resampled size.
In the figure, the Naive Resampler case gives an expectation for run times when LMN optimization is not used but rather the intermediate signals are inflated to large sizes in order to enable independent integer upsampling followed by integer downsampling.
All methods utilizing DFT exhibit large variance in average processing times across the different resampled sizes, N_r. 
Comments on this origin of this speed variance may be found in <ref>.
Depending on the precise number of resampled data points, either the LMN method or linear interpolation requires the least computation, whereas the naive approach is typically two orders of magnitude slower.






















§ LARTPC WAVEFORM RESAMPLING


The motivation for developing the LMN resampling method was to solve the practical and technical challenges of processing large amounts of data with differing sampling periods from the different DUNE LArTPC detectors.
Thus it is imperative to demonstrate the method on waveforms that are representative of what the LArTPC detectors produce.
This section illustrates the resampling of waveforms with period 512ns to those with period 500ns.  
The waveforms are produced by the current state-of-the-art detailed LArTPC signal and noise simulation algorithms from the Wire-Cell Toolkit <cit.>.

This simulation models the physical processes involved in electron drift<cit.> through liquid argon, induced current in anode electrodes via field and electronics response, noise effects and ADC digitization as introduced in <ref>.
The kernel of the simulation involves a convolution of the distribution of ionization electrons after they have drifted through the volume with the field and electronics responses. 
The field responses are calculated with a 2D model of the anode electrodes, the various applied electrical potentials and a model of electron mobility in liquid argon. 
An example of the resulting field responses for one anode plane, calculated with GARFIELD <cit.>, is shown in <ref>. 
It is represented as a 2D array giving the amount of induced electric current in an electrode of interest at different electron drift path starting locations transverse to (“Pitch”) the electrode and along (“Time”) the nominal drift path. 
The longitudinal dimension is sampled with a period of 100ns. 
As part of the convolution kernel operation, the simulation will apply an integer downsampling. 
With 100ns responses, a downsampling factor of five achieves the target ADC sample period of 500ns.




The 100ns sampling can not be downsampled by an integer factor to the target ADC sample period of 512ns and so a new set of field responses are needed. 
As the field calculations are computationally costly, we elect here to apply LMN resampling to the results at the nominal 100ns to produce results sampled at 64ns. 
This allows the simulation to downsample by an integer factor of eight and achieve the desired 512ns ADC sampling period. 
This resampling of the field responses also provides an auxiliary test of LMN method.

The induced current in an anode plane electrode that is due to a nearby electron drift path (e.g. the central row from <ref>) is shown in <ref>).
Such a path contributes a large fraction of the total induced current compared to more distant paths.
The I_s shows the original response while I_r shows the LMN-resampling of this sequence.  Note, this resampling is normalized following the interpolation interpretation described in <ref>.  



High frequency ringing can be seen in the resampled field response.  This is due to the fact that the original field response is not fully band limited at 100ns.  This can be seen in the small step down at the original 5MHz Nyquist frequency.  After convolution with the slower electronics response, which is a low-pass filter, in order to form the overall detector response this ringing is naturally abated as illustrated in <ref>, 
The overall detector response is in units of induced electric charge and not induced electric current.
That is, the integrated normalization introduced in <ref> is taken by multiplying the current by the sample period. 



Both sets of responses, each at a given sampling, are first input to the simulation in order to output signal-only, voltage-level waveforms. 
Such measures are not observable in a real detector but by comparing the native 500ns, 512ns and the 512ns→500ns LMN-resampling waveforms we may observe deviations or artifacts that may otherwise be hidden with realistic noise and ADC integer truncation.
Examples from one representative channel of each anode plane show very good alignment in the time domain as seen in <ref>.



Waveforms measured by real detectors includes signal and noise added at the voltage level and then scaled and truncated to integer by the ADC.
These waveforms for native 500ns and 512ns simulation as well as the 512ns→500ns LMN-resampling at ADC level are shown in  <ref>. 
The detailed noise components in the 512ns waveform are closely followed in the 512ns→500ns resampling.
It can also be seen that the main signal body is faithfully resampled, within the variation due to differently generated noise between the native 500ns simulation and the noise shared by the native 512ns simulation and the 512ns→500ns resampling.



The 2D signal processing described in <ref> is sensitive to spectral distortions.  As a final comparison, signal processing using the responses corresponding to the 500ns sampling is applied to the native 500ns ADC-level signal and noise simulation and the 512ns→500ns resampling.  The results from the same channels selected above are shown in <ref>.  The small variations are due to the noise components that differ between the two waveforms.











§ SUMMARY

In this paper we describe the LMN method which achieves a fast and robust waveform resampling based on the Fast Fourier Transformation algorithm.
The principle of the LMN method is introduced.
It exploits freedom to resize the original signal to enable a special case optimization that in turn allows for upsampling and downsampling by integer factors to be performed simultaneously in the frequency domain.
Intermediate signals are kept to modest sizes and thus ruinous computational costs are avoided.
Normalization for a number of different signal interpretations is provided.
The potential for a resampling to produce artifacts is raised and mitigation procedures enabled by the method are outlined.
The performance in terms of quality and computational cost of the LMN method is compared with some popular methods.
The LMN method is generally among the fastest and most accurate.
By exercising the simulation and signal processing procedures used for Liquid Argon Time Projection Chambers the effectiveness of the LMN method is demonstrated in a particle physics experiment. 

We would like to thank Laura Zambelli for her insightful comments and valuable suggestions during the DUNE collaboration review of this paper.





 Paper Title: Characterization of the RD50-MPW4 HV-CMOS pixel sensor.









1]B. Pilslcor1
Bernhard.Pilsl@oeaw.ac.at

1]T. Bergauer
5]R. Casanova
1]H. Handerkas
1]C. Irmler
3]U. Kraemer
6]R. Marco-Hernandez
6]J. Mazorra de Cos
7]F. R. Palomo
2]S. Powell
1]P. Siebererfn2
3]J. Sonneveld
1]H. Steininger
2]E. Vilella
2]B. Wade
2]C. Zhang
4]S. Zhang

 [cor1]Corresponding author
 [fn2]Now at Paul Scherrer Institut (PSI).

[1]organization=Austrian Academy of Sciences, Institute of High Energy Physics, 
                 addressline=Nikolsdorfergasse 18,
                 postcode=1050, 
                 city=Vienna, 
                 country=Austria                 
[2]organization=Department of Physics, University of Liverpool,
                 addressline=Oliver Lodge Building, Oxford Street,
                 postcode=L69 7ZE, 
                 city=Liverpool,
                 country=UK
[3]organization=NIKHEF,
                 addressline=Science Park 105,
                 postcode=1098 XG,
                 city=Amsterdam,
                 country=Netherlands
[4]organization=Physikalisches Institut, Rheinische Friedrich-Wilhelms-Universitaet Bonn,
                 addressline=Nussallee 12,
                 postcode=53115, 
                 city=Bonn,
                 country=Germany
[5]organization=Institute for High Energy Physics (IFAE), 
Autonomous University of Barcelona (UAB),
                 addressline=Bellaterra,
                 postcode=08193, 
                 city=Barcelona,
                 country=Spain
[7]organization=Department of Electronic Engineering, University of Sevilla,
                 addressline=Calle San Fernando 4,
                 postcode=41092, 
                 city=Sevilla,
                 country=Spain
[6]organization=Instituto de Fisica Corpuscular (IFIC), CSIC-UV,
                 addressline=Parque Cientifico, Catedratico Jose Beltran 2,
                 postcode=46980, 
                 city=Paterna (Valencia),
                 country=Spain

§ ABSTRACT 
The RD50-MPW4 is the latest HV-CMOS pixel sensor from the CERN-RD50-CMOS working group, designed to evaluate the HV-CMOS technology in terms of spatial resolution, radiation hardness and timing performance. Fabricated by LFoundry using a 150nm process, it features an improved architecture to mitigate crosstalk, which has been an issue with the predecessor RD50-MPW3 <cit.>, allowing more sensitive threshold settings and full matrix operation. Enhancements include separated power domains for peripheral and in-pixel digital readout, a new backside-biasing step, and an improved guard ring structure supporting biasing up to 500V, significantly boosting radiation hardness. Laboratory measurements and test beam results presented in this paper show significant improvements over its predecessor regarding noise behavior, spatial resolution, and efficiency.



RD50-MPW, HV-CMOS, DMAPS, CERN-RD50






§ CHARACTERISTICS


The RD50-MPW4 comprises a 64 × 64 pixel matrix with a pitch of 62 × 62 μ m^2. The threshold of each pixel can be tuned by a 4bit trimDAC. A subset of sensors was backside processed to allow backside biasing and thinned from 300μ m→280μ m. 




§ LABORATORY ASSESSMENTS





To assess the leakage current and breakdown behavior, IV measurements were performed. Fig. <ref> shows breakdown at V ≈270V for the topside biased sensor and V ≈490V for the backside processed sample. Furthermore, a current increase at 200V, which most likely is an effect of the sensor reaching full depletion, is observed for the topside biasing case. The two sudden current increases at 300V and 400V at the backside biased sample can't be explained yet. Edge TCT measurements for both biasing approaches are planned to clarify these effects.



To qualify the trimDACs capabilities to harmonize the pixel response, injection measurements with the in-pixel injection capacitance of C ≈2.8fF have been performed. The injection voltage V_inj, 50 at which 50% of the injected hits get detected was measured in fig. <ref> and shows a standard deviation of σ≈ 50e^-. 




§ TEST-BEAM RESULTS



To evaluate the sensor's performance in an actual particle beam, a test campaign was conducted at DESY <cit.>. In this campaign 4.2GeV e^- and the Adenium telescope <cit.> were utilized. The Corryvreckan <cit.> framework was used for data analysis. The standard used parameters for bias voltage and threshold settings for both biasing flavors are V_Bias = -190V and V_Thr = 30mV→ Q_Thr≈2500e^-.



The evaluation of the standard deviation of the biased and the unbiased spatial residuals, as shown in fig. 
<ref> for the X axis,  allows the depiction of the spatial resolution in X as σ_X ≈17.8μ m and σ_Y ≈16.49μ m in Y direction. The discrepancy between the two dimensions can be explained by the rectangular shape of the Alpide pixels used within the telescope. These residuals are dominated by single hit clusters, as an average of 1.3 pixels per cluster is measured, which is the reason for only a slight improvement of the resolution compared to the binary resolution of 62μ m / √(12)≈17.9μ m.



Efficiency measurements in fig. <ref> show full efficiencies >99% up to thresholds of 200mV which corresponds to 𝒪(5000e^-).




§ CONCLUSION


Laboratory and test beam measurements have demonstrated that the RD50-MPW4 sensor has successfully addressed several issues present in its predecessor, the RD50-MPW3. A comparison between the topside and backside biasing schemes reveals that backside biasing is superior, as evidenced by IV measurements and efficiency at high threshold values.




§ ACKNOWLEDGEMENTS


This work has been partly performed in the framework of the CERN-RD50 collaboration. 
The measurements leading to these results have partially been performed at the Test Beam Facility at DESY Hamburg (Germany), a member of the  Helmholtz Association (HGF). The research leading to these results has received funding from the European Union's Horizon Europe research and innovation program under grant agreement no. 101057511.

elsarticle-num-names 




 Paper Title: Fully strange tetraquark resonant states as the cousins of $X(6900)$.













yaoma@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China

wlwu@pku.edu.cn
School of Physics, Peking University, Beijing 100871, China

lu.meng@rub.de
Institut für Theoretische Physik II, Ruhr-Universität Bochum,  D-44780 Bochum, Germany 

chenyanke@stu.pku.edu.cn
School of Physics, Peking University, Beijing 100871, China

zhusl@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China


§ ABSTRACT 

We conduct systematic calculations of the S-wave fully strange systems with “normal" (J^P C=0^++, 1^+-, 2^++) and “exotic" (J^P C=0^+-, 1^++, 2^+-) C-parities, which are the 
strange analogue of the fully charmed tetraquark state X(6900).
Within a constituent quark potential model, we employ the Gaussian expansion method to solve the four-body Schrödinger equation and the complex scaling method to identify resonant states. We obtain a series of resonant states and zero-width states in the mass range of 2.7 to 3.3 GeV, with their widths ranging from less than 1 MeV to about 50 MeV. Their rms radii strongly indicate that they are compact tetraquark states. Among these states, the T_4s,2^++(2714) may be the most likely one to be observed experimentally. We urge the experimental exploration of the 2^++ s s s̅s̅ state around 2.7 GeV in the ϕϕ channel. Since the lowest S-wave s s s̅s̅ state is around 2.7 GeV, the compact P-wave s s s̅s̅ states are expected to be heavier. Hence, ϕ(2170) and X(2370) are unlikely to be compact tetraquark states.





Fully strange tetraquark resonant states as the cousins of X(6900)
    Shi-Lin Zhu 0000-0002-4055-6906
    August 5, 2024
==================================================================






§ INTRODUCTION
 

Since 2003, the interest in multiquark states has been reignited by the experimental discovery of a series of candidates (for recent reviews, see Refs. <cit.>). These discoveries have significantly enriched the hadron spectrum. The study of their properties and structures enhances our understanding of the non-perturbative nature of the strong interaction.

Inspired by their many strangeness decays, there are several promising candidates for the fully strange tetraquark states. 


 
  * The state ϕ(2170) (was Y(2175)) was first reported in 2006 by the BaBar Collaboration in the process of e^+ e^-→ϕ(1020) f_0(980) <cit.>. Later it was confirmed by the BES and BESIII Collaborations in the J / ψ→ηϕ f_0(980) process <cit.>, and the Belle Collaboration in the e^+ e^-→ϕπ^+π^-and e^+ e^-→ϕ f_0(980) processes <cit.>. 
 
 
  * The state X(2370) was first observed in 2010 by the BESIII Collaboration in the π^+π^-η^' invariant-mass spectrum <cit.>. Later it was confirmed in 2019 by the BESIII Collaboration in the K K̅η^' invariant-mass distribution <cit.>, which suggests that the X(2370) likely contains four valence s or s̅ quarks.

 
  * The pseudoscalar state X(2500) was first observed in 2016 by the BES Collaboration in the process J / ψ→γϕϕ <cit.>, together with the scalar state f_0(2100), and three tensor states, the f_2(2010), f_2(2300) and f_2(2340).

 
  * The X(2239) was seen in 2018 by the BESIII Collaboration in the cross section of e^+ e^-→ K^+ K^- <cit.>.
 
 
  * An evidence for a structure X(2100) in the ϕη^' mass spectrum of the J / ψ→ϕηη^' decay was reported in 2019 by the BESIII Collaboration <cit.>.

 
  * Another structure in the ϕ f_0 cross section around 2.4 GeV, labelled as X(2436) <cit.>, was reported in 2007 by the BaBar Collaboration <cit.>. It is also seen by the BES Collaboration <cit.> and Belle Collaboration <cit.>, and evidenced by the BESIII Collaboration <cit.>. However, the latest BESIII  results indicate that its statistical significance is no more than 2σ <cit.>.


Among the aforementioned possible T_s s s̅s̅ candidates, all the established ones have negative parity. The existence of the positive parity states remains uncertain. Additionally, the LHCb Collaboration recently discovered a fully charmed tetraquark candidate X(6900) <cit.>, which was later confirmed by the CMS and ATLAS <cit.>. 
The QCD Lagrangian leads to the same quark gluon interaction if one omits the quark mass difference. The discovery of X(6900) implies the possible existence of the strange analogue T_s s s̅s̅. 

Inspired by these experimental observations, various theoretical investigations have been conducted on the fully strange tetraquark systems. Regarding the ϕ(2170) state with J^P C=1^–, some authors used the QCD sum rule <cit.>, quark model <cit.> and diquark-antidiquark scheme <cit.> to interpret it as an s s s̅s̅ tetraquark state, although other interpretations were not  excluded <cit.>. Similarly, the X(2370) state was also explained as an s s s̅s̅ tetraquark state with 0^-+ <cit.>. The pseudoscalar state X(2500) was interpreted as a P-wave s s s̅s̅ tetraquark state of J^P C=0^-+ <cit.>. In Ref. <cit.>, the X(2500), X(2239), f_2(2300) were assigned as 0^-+, 1^–, 2^++ s s s̅s̅ tetraquark states, respectively. Moreover, the structure X(2100) was interpreted as a 1^+- s s s̅s̅ state <cit.>. In Ref. <cit.>, the resonant states X(2100) and X(2239) were treated as the axial-vector and vector tetraquarks with the quark content s s s̅s̅, respectively. The structure X(2436) was explained as a P-wave s s s̅s̅ tetraquark state of J^P C=1^– <cit.>. The s s s̅s̅ tetraquark states with other quantum numbers were also expolred  <cit.>.

So far, there is no experimental conclusion regarding the existence of positive parity s s s̅s̅ states, and theoretical investigations on these states are also limited. Although some theoretical calculations on the fully strange tetraquark system in quark models have been conducted <cit.>, they are relatively sparse and less refined compared to those on tetraquark systems containing heavy quarks. The studies in Refs. <cit.> focus exclusively on the diquark-antidiquark picture, obtaining only the states above the dimeson thresholds. These states would become scattering states or resonant states once the dimeson configuration is incorporated. Only the resonant states have physical significance and should be identified using nontrivial approaches. In Refs. <cit.>, the s s s̅s̅ tetraquark resonant states were investigated via the real scaling method. This method detects resonant states by avoiding the cross pattern of energy levels across a parameter controlling the spatial extension of the wave function, which is usually quite rough. 

To address these issues, we will conduct systematic calculations of the positive parity S-wave fully strange system. We will employ the Gaussian expansion method (GEM) <cit.> to solve the four-body Schrödinger equation within a nonrelativistic constituent quark potential model. Additionally, we will adopt the complex scaling method (CSM) <cit.> to identify genuine resonant states from meson-meson scattering states. In our recent works, we have incorporated both the dimeson and diquark-antidiquark configurations to carry out benchmark calculations for tetraquark bound states <cit.> and resonant states <cit.>. We will continue to include both configurations for the fully strange tetraquark system. In this study, we will investigate both the “normal" C-parity states (J^PC=0^++, 1^+-, 2^++) and the “exotic" C-parity states (J^PC=0^+-, 1^++, 2^+-), with the “exotic” C-parity systems referring to those that cannot be composed of two S-wave ground quarkonia.

This paper is arranged as follows. 
In Sec. <ref>, we introduce the theoretical framework, including the Hamiltonian in the constituent quark potential, the construction of the wave function, the complex scaling method, and the approach to analyzing the spatial structures.
In Sec. <ref>, we present the numerical results for the properties of the fully strange resonant states.
Finally, we give a brief summary and discussion in Sec. <ref>.




§ THEORETICAL FRAMEWORK
 



 §.§ Hamiltonian
 

The nonrelativistic Hamiltonian of a four-quark system reads

    H=∑_i^4(m_i+p_i^2/2m_i)+∑_i<j=1^4 V_i j ,

where m_i and p_i are the mass and momentum of quark i.

In this study, we will employ the AL1 constituent quark potential model <cit.> containing the one-gluon-exchange (OGE) interaction and a linear confinement term.
The AL1 potential reads 

    V_i j=-3/16   λ_i^c ·λ_j^c(-κ/r_ij+λ r_i j-Λ
       +8πκ'/3m_im_jexp(-r_ij^2/r_0^2)/π^3/2r_0^3s_i·s_j),






where r_0=A(2m_im_j/m_i+m_j)^-B. λ^c represents the SU(3) color Gell-Mann matrix and s_i is the spin operator of quark i. The parameters of the potential have been determined by fitting the meson spectra in Ref. <cit.> and are listed in Table <ref>. The calculated meson masses in AL1 model are shown in Table <ref>. 





















 §.§ Wave function construction
 

The wave function bases of the sss̅s̅ tetraquark states can be expressed as the direct product of spatial wave function ϕ, spin wave function χ_s, and color wave function χ_c.

    ψ=𝒜(ϕ⊗χ_s⊗χ_c),

where 𝒜 is the antisymmetrization operator, representing the exchange of identical quarks. For the sss̅s̅ system, 𝒜=(1-P_12)(1-P_34), where P_i j permutes the ith and jth (anti)quarks. 


For the spatial wave function, the Gaussian expansion method (GEM) <cit.> is employed. Namely, the spatial wave function is expanded using the following basis:

    ϕ_n l m(r)=√(2^l+5 / 2/Γ(l+3/2) r_n^3)(r/r_n)^l e^-r^2/r_n^2 Y_l m(r̂),

where the r_n is taken in geometric progression, r_n=r_0 a^n-1. Y_l m is the spherical harmonics.

For a four-body system, one can work on the Jacobi coordinates to exclude the center-of-mass motion. In principle, one could construct a complete basis functions using any type of Jacobi coordinates. However, to get rid of the complexity from the angular momentum, we use only S-wave bases constructed in different types of Jacobi coordinates to partially compensate for higher partial waves (see Ref. <cit.> for details). In this work, we include two types of Jacobi coordinates, the diquark-antidiquark structure and the dimeson structures, as shown in Fig. <ref>.


In our calculation, we choose n_max=12 bases for each coodinate, and the settings of the basis parameters are 

    {[ r_0=0.4 fm, r_n_max=2.0 fm              s-s  or s̅-s̅; r_0=0.4 fm, r_n_max=2.0 fm               (s s)-(s̅s̅); r_0=0.4 fm, r_n_max=1.3 fm                       s-s̅; r_0=0.5 fm, r_n_max=4.5 fm              (s s̅)-(s s̅) ],.

where we use relatively large r_n_max for the coordinate between two s s̅ clusters to better describe the meson-meson scattering states and possible molecular states.

For the spin wave function, the bases for total spin S=0,1,2 are

    S=0:{[ [(s_1 s_2)_0(s̅_3 s̅_4)_0]_0; [(s_1 s_2)_1(s̅_3 s̅_4)_1]_0 ],. 
        S=1:{[ [(s_1 s_2)_0(s̅_3 s̅_4)_1]_1; [(s_1 s_2)_1(s̅_3 s̅_4)_0]_1; [(s_1 s_2)_1(s̅_3 s̅_4)_1]_1 ],. 
        S=2:[ [(s_1 s_2)_1(s̅_3 s̅_4)_1]_2 ].


Two optional color wave function bases sets are

    color-I: {[ [(s_1 s_2)_3̅(s̅_3 s̅_4)_3]_1;  [(s_1 s_2)_6(s̅_3 s̅_4)_6]_1 ],.


    color-II: {[ [(s_1 s̅_3)_1(s_2 s̅_4)_1]_1; [(s_1 s̅_3)_8(s_2 s̅_4)_8]_1 ]..

Since both sets are orthogonal and complete, using either set will yield exactly the same results <cit.>.

Besides the antisymmetrization operation in Eq. (<ref>), since the s s s̅s̅ system carries definite C-parity, it is necessary to add (or subtract) the C-transformation of each term in Eq. (<ref>) to ensure that the overall wave function has the desired positive (or negative) C-parity. The specific C-transformation is

    [(q_1 q_2^')^s_1_c̅(q̅_3 q̅_4^')_c^s_2]^S_1_cϕ(r_1, r_2, r_3, r_4)
       .(-1)^S-s_1-s_2[(q_1 q_2^'))_c̅^s_2(q̅_3 q̅_4^')_c^s_1]^S_1_cϕ(r_3,r_4,r_1,r_2)

for the bases with color-I set, and

    [(q_1 q̅_3)_c^s_1(q_2^'q̅_4^')_c^s_2]^S_1_cϕ(r_1, r_2, r_3, r_4)
       (-1)^s_1+s_2[(q_1 q̅_3)_c^s_1(q_2^'q̅_4^')_c^s_2]^S_1_cϕ(r_3,r_4,r_1,r_2)

for the bases with color-II set. The particle q and q' can be different quarks, although they both represent the same s quark in the s s s̅s̅ system.




 §.§ Complex scaling method
 

The complex scaling method (CSM) is a direct approach to obtain the energies and the decay widths of resonant states in a many-body system by performing an analytical continuation of the Schrödinger equation <cit.>. This is achieved by carrying out a complex rotation on the coordinate r and momentum p, given by

    U(θ) r=r e^i θ,    U(θ) p=p e^-i θ.

Under the rotation, the Hamiltonian in Eq. (<ref>) becomes

    H(θ)=∑_i=1^4(m_i+p_i^2 e^-2 i θ/2 m_i)+∑_i<j=1^4 V_i j(r_i j e^i θ).

Meanwhile, for the resonant states with pole positions within the range of the rotated angle, their wave functions become normalizable by integration, thereby solvable through localized Gaussian bases in the same way as bound states. As a result, solving the complex-scaled Schrödinger equation will simultaneously yield the eigenenergies of bound states and resonant states within the rotated angle.

A typical pattern of the solved eigenenergies in the complex energy plane is: The bound states lie on the negative real axis of the energy plane. The continuum states align along beams originating from thresholds with Arg(E)=-2 θ. The resonant states with mass M_R and width Γ_R are located at E_R=M_R-i Γ_R / 2, and only those within |Arg(E_R)|<2 θ can be solved. The positions of the bound and resonant states remain unchanged with the variation of the rotation angle.  One can find more details in Refs. <cit.>.



 §.§ Spatial structure


The root-mean-square (rms) radius is a good physical quantity for reflecting the spatial structure of the tetraquark states, and a widely used criterion for distinguishing between compact and molecular tetraquark states.

The conventional definition of the rms radius under CSM is 

    r_i j^rms,C≡Re[√((Ψ(θ)|r_i j^2 e^2 i θ| Ψ(θ))/(Ψ(θ) |Ψ(θ)))],

where the Ψ(θ) is the obtained complex wave function of the tetraquark state. The round bra-ket represents the so-called c-product <cit.> defined as

    (ϕ_n |ϕ_m) ≡∫ϕ_n(r) ϕ_m(r) d^3r,

without taking complex conjugate of the bra-state. This procedure ensures the function inside the integral is analytic, thereby the expectation value of the physical quantity remains stable as the rotation angle changes. The rms radius calculated from the c-product is generally not real; however, its real part can still reflect the internal quark clustering behavior if the resonant state is not too broad, as discussed in Ref. <cit.>.

Nevertheless, for tetraquark states containing identical quarks, this definition may fail to identify the molecular nature of their spatial structure due to the antisymmetrization of identical quarks, as discussed in Ref. <cit.>. To eliminate this ambiguity, we have proposed a new approach to calculate the rms radii of the fully heavy Q Q Q̅Q̅ system in Ref. <cit.>, which can also be applied to the s s s̅s̅ system. In this new approach, the total wave function is non-orthogonally decomposed into

    Ψ(θ) =     ∑_s_1 ≤ s_2[[(q_1 q̅_3^')_1_c^s_1(q_2 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_3, r_4;θ).
        - [(q_2 q̅_3^')_1_c^s_1(q_1 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_2, r_1, r_3, r_4;θ) 
        - [(q_1 q̅_4^')_1_c^s_1(q_2 q̅_3^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_4, r_3;θ)
        + .[(q_2 q̅_4^')_1_c^s_1(q_1 q̅_3^')_1_c^s_2]_1_c^S ϕ(r_2, r_1, r_4, r_3;θ)]
    
    =    𝒜[ ∑_s_1 ≤ s_2[(q_1 q̅_3^')_1_c^s_1(q_2 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_3, r_4;θ)]
    ≡   𝒜Ψ_13,24(θ),

where s_1, s_2 sum over all spin configurations with total spin S. The particle q and q' can be different quarks, although they both represent s quark in the s s s̅s̅ system. We use the decomposed wave function Ψ_13,24(θ) to define the new rms radius

    r_ij^rms,M≡Re[√((Ψ_13,24(θ)|r_i j^2 e^2 i θ| Ψ_13,24(θ))/(Ψ_13,24(θ) |Ψ_13,24(θ)))] .

This newly defined rms radius can better describe the spatial structure of the molecular-type tetraquark states containing identical quarks. We will provide the results for both definitions of the rms radius in our calculation.




§ NUMERICAL RESULTS
 



 §.§ States with “normal" C-parity


The “normal” C-parity s s s̅s̅ systems refer to those that can be composed of two S-wave ground quarkonia, namely, the  systems with J^PC=0^++,1^+-,2^++. The complex eigenenergies yielded are shown in the upper panel of Fig. <ref>. We choose different complex scaling angles θ=9^∘,12^∘,15^∘ to distinguish bound states and resonant states, which remain stationary as the angle changes, from the continuum states. The markers that align along beams originating from meson-meson thresholds with Arg(E) = -2 θ are the continuum state eigenenergies. The results show that no bound state is found below any of the thresholds, while a series of resonant states are obtained and marked out by black circles. Their complex energies, proportions of different color configurations, and two kinds of rms radii are summarized in Table <ref>. For convenience, we label the tetraquark resonant states obtained in our calculations as T_4s, J^PC(M), where M is the mass of the state.










The masses of the resonant states are in the region (2.7,3.3) GeV, and their widths range from a few MeV to several tens of MeV. For the resonant states above the 1S2S di-meson threshold, our calculation actually underestimates their widths because the decay width of the 2S meson has not been accounted for. The rms radii of all the obtained states approximately match the sizes of the 2S mesons. Furthermore, the newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states.



In tetraquark systems, there are abundant dimeson thresholds. Each of these thresholds represents a branch cut connecting two different Riemann sheets. In our calculations, we must choose appropriate rotation angle to make the pole of interest emerge. Meanwhile, due to numerical reasons, the pole close to the continuum solutions will be affected by them, as illustrated in Refs. <cit.>. Therefore, to reduce uncertainties, the rotation angle should be chosen such that the continuum states are not too close to the resonant state. In practices, we adjust the angle to make each pole and its rms radii stable. 






Comparing the fully strange systems with ccc̅c̅ using the AL1 model in our previous work <cit.>, we observe a similar overall pattern. However, the fully strange systems exhibit an additional quasi-bound state with 2^++ below the ϕϕ(2S) threshold. This difference may stem from the mass-dependent color-magnetic term in the potential.

For the 0^++ states, the five resonant states can all decay into η'η', ϕϕ, η'η'(2S), ϕϕ(2S) final states, although there may be some effects if the mixing between the I=0 η(nn̅) and η'(ss̅) is taken into account. The T_4s,0^++(2917) and T_4s,0^++(3133) have an additional decay channel into η'η'(3S). The higher T_4s,0^++(3175) and T_4s,0^++(3248) have two additional channel into η'η'(3S) and η'(2S)η'(2S). For the 1^+- states, the five resonant states can all decay into η'ϕ, η'ϕ(2S) and η'(2S)ϕ. The  T_4s,1^+-(3142) has an additional decay channel into η'ϕ(3S). And the higher T_4s,1^+-(3228) and T_4s,1^+-(3237) have two extra η'ϕ(3S) and η'(3S)ϕ decay channels. For the 2^++ states, the lowest T_4s,2^++(2714) below the ϕϕ(2S) threshold have only one decay channel ϕϕ. Other two 2^++ resonant states T_4s,2^++(2993) and T_4s,2^++(3164) have an additional decay channel into ϕϕ(2S). The T_4s,2^++(3266) has an extra decay channel to ϕϕ(3S).

Among these states, the state T_4s,2^++(2714) is the most likely one to be observed experimentally. Its mass is the lowest, and its decay channel ϕϕ predominantly decays into K^+K^-K^+K^-, which are all charged particles that can be detected directly by detectors, leading to high detection efficiency and resolution.




 §.§ States with “exotic" C-parity


The “exotic” C-parity systems refer to those that cannot be composed of two S-wave ground quarkonia, namely, the system with J^PC=0^+-,1^++,2^+-. The complex eigenenergies yielded are shown in the lower panel of Fig. <ref>. We obtain a series of resonant states and zero-width states, whose energies, proportions of different color configurations, and two kinds of rms radii are summarized in Table <ref>. The rms radii approximately match the sizes of the 2S mesons. The newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states. 

For the J^PC=0^+- and 2^+- systems, there is no S-wave di-quarkonium threshold. So the calculated states appear as bound states on the complex energy plane. However, there may actually be higher-wave thresholds below these states, which are not included in the calculation. If the higher-wave mesons are included, the properties of these states may change and their widths may no longer be zero.

For the J^PC=1^++ system, the masses of the resonant states range in (2.7,3.3) GeV, and their widths are small, with the largest being only a few MeV. Generally, one would expect the widths of these resonant states to arise from their decays into lower-energy meson-meson channels. However, their narrow widths suggest that such decays are suppressed. The reason for the suppression has been discussed in detail in Ref. <cit.>. Simply put, when the rms radii between different i,j quarks are roughly equal, the transition operator of the color-electric term becomes approximately proportional to the identity operator. Therefore, the color mixing transition is suppressed. In the s s s̅s̅ system, however, the contribution from the color-magnetic term is not as significantly suppressed by the quark mass as it is in the heavy quark sector. Therefore, the transition can still occur via spin mixing, which, although limited, can lead to slightly larger widths of these resonant states. All three J^PC=1^++ resonant states can decay via ϕϕ(2S), though this process is suppressed.








§ SUMMARY AND DISCUSSION
 

We calculate the mass spectrum of the S-wave fully strange tetraquark systems with “normal" C-parity J^PC=0^++,1^+-,2^++ and “exoitic" C-parity J^PC=0^+-,1^++,2^+- using the AL1 quark potential model. We employ the GEM to solve the four-body Schrödinger equation, and the CSM to identify genuine resonant states from meson-meson scattering states.

We obtain a series of resonant and zero-width states in the mass region (2.7,3,3) GeV, with widths ranging from less than 1 MeV to around 50 MeV. Apart from the state T_4s,2^++(2714), all the others lie above the M(1 S) M^'(2 S) dimeson thresholds.

We calculate two kinds of rms radius for all states, the conventional rms radius r_ij^rms,C and the newly defined rms radius r_ij^rms,M based on the decomposed wave function. For all states, both kinds of rms radii approximately match the sizes of the 2S mesons, and the newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states. 

Among these states, the state T_4s,2^++(2714) is the most likely to be observed experimentally due to its lowest mass and its decay into ϕϕ, which predominantly produces directly detectable charged K^+K^- pairs, ensuring high detection efficiency and resolution.

Recalling the experimental candidates for s s s̅s̅ tetraquark states, their masses are all in range of 2.1 to 2.5 GeV, whereas the lowest S-wave resonant state in our calculations is around 2.7 GeV. This suggests the states like ϕ(2170) and X(2370) are unlikely to be compact s s s̅s̅ tetraquark states. Since the lowest S-wave state is already as high as 2.7 GeV, it's unlikely that a P-wave compact state would be lower than the S-wave ones. However, they could still be P-wave molecular-type resonant states. In this work, a state T_4s,2^++(2714) is found near the threshold of ground ϕ combined with a radially excited ϕ(2S), although it does not exhibit a molecular spatial structure. Introducing P-wave thresholds might yield molecular-type resonant states near the threshold of an S-wave meson combined with a P-wave meson, which could be lower in mass. This possibility awaits careful investigation in the future.

We emphasize that the fully charmed and fully strange tetraquark resonant states share many common characteristics. Their internal structures and underlying dynamics are nearly the same. All of them are compact states. The discovery of X(6900) implies the existence of its fully strange cousins, which can be searched for in the ϕϕ or ϕϕ(2S) channels. While X(6900) can only be produced at LHC now, the fully strange tetraquark resonances may be observed at BESIII, Belle II and LHC. The future discovery of these states shall shed light on both the charmed and strange compact tetraquark states.     












We are grateful to Zi-Yang Lin, Jun-Zhang Wang and Liang-Zhen Wen for the helpful discussions. This project was supported by the National Natural Science Foundation of China (11975033 and 12070131001). This project was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Project ID 196253076-TRR 110). The computational resources were supported by High-performance Computing Platform of Peking University.










 Paper Title: Test-beam measurements of instrumented sensor planes for a highly compact and granular electromagnetic calorimeter.











IFIC, CSIC and Universitat de València, C/ Catedràtic José Beltrán Martínez 2,
46980 Paterna, Spain




The LUXE experiment is designed to explore the strong-field QED regime in interactions of high-energy electrons from the European XFEL in a powerful laser field. One of the crucial aims of this experiment is to measure the production of electron-positron pairs as a function of the laser field strength where non-perturbative effects are expected to kick in above the Schwinger limit.
For the measurements of positron energy and multiplicity spectra, a tracker and an electromagnetic calorimeter are foreseen. 
The expected number of positrons varies over ten orders of magnitude and has to be measured over a widely spread low-energy background. To overcome these challenges, a compact and finely segmented calorimeter is proposed. The concept of a sandwich calorimeter made of tungsten absorber plates interspersed with thin sensor planes is developed.
The sensor planes comprise a silicon pad sensor, flexible Kapton printed circuit planes for bias voltage supply and signal transport to the sensor edge, all embedded in a carbon fiber support. The thickness of a sensor plane is less than 1 mm. A dedicated readout is developed comprising front-end ASICs in 130 nm technology and FPGAs to orchestrate the ASICs and perform data pre-processing. As an alternative, GaAs are considered with integrated readout strips on the sensor. Prototypes of both sensor planes are studied in an electron beam of 5 GeV. Results will be presented on the homogeneity of the response and edge effects.



Test-beam measurements of instrumented sensor planes for a highly compact and granular electromagnetic calorimeter
    Melissa Almanza Soto1melissa.almanza@ific.uv.es on behalf of the LUXE Collaboration

    August 5, 2024
==================================================================================================================




§ INTRODUCTION


The Laser Und XFEL Experiment (LUXE) is an experiment proposal with the mission of studying the behavior of Quantum Electrodynamics (QED) in the strong-field regime. In particular, LUXE will take precision measurements of electron-photon and photon-photon interactions in the transition from perturbative to non-perturbative QED. LUXE also opens the possibility to perform a sensitive search for new particles beyond the Standard Model that couple to photons <cit.>. 

The experiment will use the 16.5 GeV electron beam from the European X-Ray Free-Electron Laser and collide it with a high-intensity laser to reach and surpass the critical field strength ℰ_cr=m_e^2c^3/(eħ)=1.32× 10^18 V/m, known as the Schwinger limit. 

In its initial phase, the experiment will use an existing 40 TW laser. Later on, the laser power will be upgraded to 350 TW for the second phase. LUXE will use the 10 Hz electron bunch rate of XFEL, with collisions occurring only at 1 Hz to match the laser pulse rate. The remaining electron bunches will be used for background measurements <cit.>.   

The experiment is planned to have two modes which involve the production of electrons, positrons, and photons. In the first mode, beam electrons will collide directly with the high-power tightly-focused laser pulse. For the second mode, the high energy electrons will first be converted into a photon beam and then this photons will interact with the laser pulse.
The multiplicities and energies of the particles produced during the interactions will be measured by a set of detectors. An electromagnetic calorimeter and a tracker compose the system to measure the number of positrons and their energy spectrum. 



§ THE POSITRON ELECTROMAGNETIC CALORIMETER FOR LUXE


For both modes of the experiment, the same electromagnetic calorimeter is foreseen for the positron side of LUXE, this detector is known as ECAL-P. 

The main challenge in the design of ECAL-P is the vast range of multiplicities for positrons between the two modes of the experiment. The expected positron rates vary from 10^-4 to more than 10^6 per bunch crossing <cit.>.
In the low multiplicities expected for the γ-laser mode, the calorimeter must be able to perform measurements with an excellent background rejection. In contrast, for the high positron multiplicities of the e-laser mode the detector must achieve a good linearity. 

The proposed solution consists of a compact sampling calorimeter with a small Molière radius and a high granularity. 

The ECAL-P design consists of tungsten absorber plates and thin active layers.  Tungsten was chosen as the absorber material, as it has a small Molière radius of 9.3 mm. The detector will have 21 tungsten absorber layers with a thickness of 3.5 mm, equivalent to one radiation length. The active layers are to be kept to less than 1 mm in thickness.

Two sensor materials are proposed for the sensitive layers: Gallium Arsenide (GaAs) and Silicon.

The GaAs sensors were produced by the National Tomsk State University. The sensor is made out of a single GaAs crystal compensated with chromium. Each pad has an area of 4.7 × 4.7  mm^2 with a separation of 0.3 mm between pads. A distinctive feature of the GaAs sensors is the presence of aluminum traces embedded in the gaps between the pads. The aluminum traces are connected to the pads and take the signal to the top edge of the sensor plane. The metal traces favor the compact design of the calorimeter by keeping all the electronics on top of the detector and outside of the absorber and the active volume of ECAL-P. 
An advantage of GaAs over silicon is that the former has a higher radiation tolerance, this being the reason why it has been proposed for other calorimeters such as BeamCal for the ILD <cit.>, detector in which the ECAL-P design has been based on.



The silicon sensors are produced by Hamamatsu. The sensor pads have a 5.5 × 5.5  mm^2 area and a 320 μm thickness. The space between pads is 0.01 mm. A flexible Kapton PBC with copper pads and traces connected to the sensor pads with conductive glue will guide the signal to the edge of the sensor. The sub-layer components of a silicon sensor instrumented plane are shown in Fig. <ref>. Similar signal transport techniques have been tested in various geometries of the LumiCAL detector prototypes <cit.>,<cit.>. 

The front-end ASICs for the readout of the sensor planes of ECAL-P are called Fcal ASIC for XFEL Experiment (FLAXE). These ASICs are based on the FCAL ASIC for Multiplane Readout (FLAME), designed for the silicon sensors of the LumiCAL detector. The FLAME ASICs have been used in several beam-tests of the FCAL collaboration and were used by the LUXE ECAL-P group in the beam-test described in the following section. 




§ TEST-BEAM MEASUREMENTS OF ACTIVE PLANES

A beam-test of four prototypes of the ECAL-P sensor planes was performed at the DESY-II test-beam facility in September of 2022. Two 16 × 8 pad arrays of silicon sensors and two 15 × 10 pad arrays of GaAs sensor planes were put to test in a 5 GeV electron beam. The silicon sensors tested were 500 μm thick, differing from the 320 μm thick sensors proposed for the ECAL-P. 

The FLAME ASICs were used for the readout of the sensor planes in the beam-test. 
These ASICs are 32-channel ASICs in CMOS 130 nm with analog front-end and 10-bit ADC in each channel, followed by two fast (5.2 Gbps) serializers and data transmitters, more specifications for each component can be found in table <ref>.



A beam telescope composed of 6 Alpide sensors was used to measure the tracks of the beam electrons and predict their entry point on the sensor plane. All 6 telescope planes were set in between a beam collimator and the sensor plane under test.  



 §.§ Performance of the sensor planes


The readout channels used for the beam-test were calibrated for differences in the pre-amplification, these calibration factors were applied to the test-beam data. 

The response of the silicon sensor plane to 5 GeV electrons was simulated using Geant4. The simulation included the full material of the sensor planes (carbon fiber frame, flex PCB, sensor, high-voltage flex PCB). As shown in Fig. <ref>, the simulation shows a good agreement with the test-beam measurements around the Most Probable Value (MPV). 



Using the telescope data, the area of each pad was subdivided into pixels. The signal distribution of electrons hitting each section was plotted and the MPV was calculated using a Landau function convoluted with a Gaussian fit. 
The MPV distribution as a function of the pad's local X and Y coordinates shows a drop in the MPV when approaching the edges of the GaAs sensor pad, as seen in Fig. <ref>. As for the silicon sensors, Fig. <ref> shows an L-shaped area with a 1% rise in amplitude with respect to the average. The same effect was observed for all the pads studied.




To quantify the edge effects, the pad area was separated into thin strips. The MPV of the signal distribution of electrons hitting each strip area was calculated. At the very edge of the GaAs pad, the drop amounts to 50% of the MPV of the center of the pad. The drop in signal is shown for a typical pad in Fig. <ref>. A drop of 2 to 3% at the edges of the pad can be observed in Fig. <ref> for the silicon sensor.







A study was performed in which the signals from two adjacent pads were added for events in which electrons are predicted to enter the detector along the gap between the two pads. Scanning along the horizontal coordinate for the GaAs sensor, one observes a maximum drop in signal of 42% with respect to the average, as shown in Fig. <ref>.

Along the vertical coordinate, the signal decreases 13% with respect to the average. This effect was not observed for the silicon sensor along either coordinate. 








Lastly, the MPV of the signal distribution was calculated for every pad in a sensor plane. To discard the edge effects of the GaAs pads and make a comparison between the two types of sensors, only the signals generated from electrons predicted to hit the center area of a pad were used. The distributions of the MPV are shown in Figs. <ref> and <ref> for the silicon and the GaAs sensor, respectively.
The results show that the GaAs sensor has an overall larger MPV than that of silicon, this is due to the higher density of the material. 

 




The homogeneity of the response of the GaAs sensor was found to be comparable to that of silicon after discarding electrons hitting the edges of pads. The homogeneity of the response in all pads is within 3% for both sensors. 



§ CONCLUSION

 Individual pad-response studies were possible using the beam telescope. The GaAs sensors with the aluminium traces present edge-effects involving a drop in the amplitude of signals. The same effect is found for the silicon sensor in a smaller scale. Gain calibration factors were applied to the test-beam data. After removing the edges from the GaAs sensor pads, the homogeneity of the response was found to be comparable to that of the silicon sensor. 
A Monte Carlo simulation of the instrumented silicon plane was performed and was found to be a good modeling of the test-beam data.



§ ACKNOWLEDGEMENTS

We thank the DESY technical staff for continuous assistance and the DESY directorate for their strong support and the hospitality they extend to the non-DESY members of the collaboration. This work has benefited from computing services provided by the German National Analysis Facility (NAF) and the Swedish National Infrastructure for Computing (SNIC). The measurements leading to these results have been performed at the Test Beam Facility at DESY Hamburg (Germany), a member of the Helmholtz Association (HGF). M.A.S. acknowledges the financial support by Spanish MICIU/ AEI and European Union / FEDER via the grant PID2021-122134NB-C21. EM and by Generalitat Valenciana (GV) via the Excellence Grant CIPROM/2021/073. M.A.S also acknowledges the support from the MCIN with funding from the European Union NextGenerationEU and Generalitat Valenciana in the call Programa de Planes Complementarios de I+D+i (PRTR 2022) through the project with reference ASFAE/2022/015.



 Paper Title: Considerations on measuring spin-spin correlation of hyeprons in heavy-ion experiments.






dyshen@fudan.edu.cn
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China

Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China

Brookhaven National Laboratory, Upton, New York 11973


§ ABSTRACT 
     The significant global spin alignment observed for ϕ mesons in heavy-ion collisions has sparked intense discussions about its origin and implications. One explanation suggests that fluctuations in the strong force field may introduce strong spin correlations between strange (s) and anti-strange (s̅) quarks, leading to the global spin alignment of ϕ mesons. Extending this line of research, the theoretical community has proposed studying the spin correlation between Λ and Λ̅ hyperons. In this paper,  we construct experimental observables and make connections between them and the theoretical proposed quantities.
    

Considerations on measuring spin-spin correlation of hyeprons in heavy-ion experiments
    Aihong Tang
    July 2024
======================================================================================

The concept of global polarization in heavy-ion collisions was first proposed in Ref.<cit.>. In off-central heavy-ion collisions, quarks are predicted to be globally polarized along the direction of angular momentum due to spin-orbital coupling in strong interactions. This effect has recently been confirmed experimentally through measurements of Λ global polarization <cit.>. However, another related phenomenon, vector meson spin alignment, cannot be explained solely by vorticity or other  mechanisms, most of which are conventional <cit.>. The measured ρ_00, the 00-component of the spin density matrix (with 1/3 indicating no spin alignment), of ϕ mesons is significantly above 1/3 <cit.>, while predictions from the quark coalescence model, considering Λ data, suggest <cit.>  ρ_00 below 1/3 with a much smaller deviation from 1/3. This discrepancy between hyperon polarization and vector meson spin alignment implies richer spin physics beyond global quark polarization. It was later realized that spin correlation between quarks and anti-quarks plays a crucial role in determining vector meson spin <cit.>. For instance, short-range correlations between s and s̅ quarks due to strong-force fields successfully explain the observed ϕ meson spin alignment <cit.>.

The spin-spin correlation of spin 1/2 particles was usually studied in helicity-frame in e^+e^- collisions or p+p collisons <cit.>, and the correlation function is defined as 

    c_ij = f_↑↑ + f_↓↓ - f_↓↑ - f_↑↓/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓,

where f_m_im_j=⟨ m_im_j | ρ̂ | m_im_j ⟩ denotes the fraction of fermion pairs in the spin state of | m_im_j  ⟩ along spin quantization direction, with ρ̂ being the spin density matrix. In general, the quantization directions of particle i and j differ in helicity-frame. However, in heavy-ion collisions, we are interested in global spin-spin correlations, meaning the quantization directions of particle i and j are fixed. In this case, Eq. <ref> is modified as <cit.>:

    c'_ij = c_ij - P_i P_j,

where P_i and P_j are the polarization of particle i and j respectively. It is clear that the c'_ij represents ⟨σ̂_m_iσ̂_m_j⟩ - ⟨σ̂_m_i⟩⟨σ̂_m_j⟩, where σ̂_m is the Pauli operator along spin quantization direction. 

The left-hand side of Eq. <ref> represents the global spin correlations in heavy-ion collisions and can be calculated by measuring the c_ij and global polarization. In the following, we demonstrate how to measure c_ij using the example of Λ-Λ spin correlation. This method can be easily extended to other hyperons. 

We begin with the proton distribution from Λ weak decay,

    dN/dcosθ^*=1/2(1+α_Λ P_Λcosθ^*),

where θ^* is the angle, in Λ's rest frame, between proton momentum and Λ spin quantization direction. P_Λ defines Λ polarization and α_Λ is a decay constant. For Λ pairs in the spin state | ↑↑⟩ (both are fully polarized thus P_Λ=1), the distribution of proton-pairs can be written as

    dN_↑↑/dcosθ_i^*dcosθ_j^*=1/4 [1+α_Λ (cosθ_i^*+cosθ_j^*) + α_Λ^2cosθ_i^*cosθ_j^*  ],

where N_↑↑ denotes proton pairs from Λ pairs in | ↑↑⟩ state. The footnote i and j denote the first proton and the second proton, respectively. Similarly, for Λ pairs in the spin state of | ↓↓⟩, | ↑↓⟩, and | ↓↑⟩, the proton pairs are described by 

    dN_↓↓/dcosθ_i^*dcosθ_j^*   =   1/4 [1-α_Λ (cosθ_i^*+cosθ_j^*) + α_Λ^2cosθ_i^*cosθ_j^*  ], 
    dN_↑↓/dcosθ_i^*dcosθ_j^*   =   1/4 [1+α_Λ (cosθ_i^*-cosθ_j^*) - α_Λ^2cosθ_i^*cosθ_j^*  ], 
    dN_↓↑/dcosθ_i^*dcosθ_j^*   =   1/4 [1-α_Λ (cosθ_i^*-cosθ_j^*) - α_Λ^2cosθ_i^*cosθ_j^*  ].

In reality, Λ pairs are mixture of the above spin states, with the fractions given by f_m_im_j. Therefore, the general distribution of daughter proton pairs can be expressed as follows :

    dN/dcosθ_i^*dcosθ_j^*   =    f_↑↑dN_↑↑/dcosθ_i^*dcosθ_j^* + f_↓↓dN_↓↓/dcosθ_i^*dcosθ_j^* + f_↑↓dN_↑↓/dcosθ_i^*dcosθ_j^* + f_↓↑dN_↓↑/dcosθ_i^*dcosθ_j^*
       =   1/4 [ 1 + Aα_Λcosθ_i^* 
    + Bα_Λcosθ_j^* +Cα_Λ^2cosθ_i^*cosθ_j^*  ],

where 

    A=f_↑↑-f_↓↓+f_↑↓-f_↓↑/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓, 
    
    B=f_↑↑-f_↓↓-f_↑↓+f_↓↑/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓, 
    
    C=f_↑↑+f_↓↓-f_↑↓-f_↓↑/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓.

The parameter C is the c_ij defined in Eq. <ref>. Indeed, the parameter A could be reduced to (f_↑-f_↓)/(f_↑+f_↓) which represents the polarization of the first Λ by definition, where f_↑=f_↑↑+f_↑↓ (f_↓=f_↓↑+f_↓↓) is the fraction of it in the spin state | ↑⟩ (| ↓⟩). Similarly, the parameter B represents the polarization of the second Λ. It is reasonable to expect that parameter C reflects the spin correlation between the first and second Λ. 
Mathematically, the c_ij can be obtained by the integral of cosθ^*_icosθ^*_j, it is equivalent to calculate the average ⟨cosθ^*_icosθ^*_j ⟩ in the experiment. 

Based on Eq. <ref>, we establish the connection between experimental observable and the Λ spin correlation as used in the context of theoretical work: 

    c'_ΛΛ=9/α^2_Λ⟨cosθ^*_icosθ^*_j ⟩ - P_Λ^2.

In experiment, the spin quantization direction (orbital angular momentum direction) is estimated from the event plane, which has a finite resolution. To correct for event plane resolution, similar to Ref. <cit.>, we project Eq. <ref> onto transverse plane, resulting in :

    dN/dΔϕ_i^*dΔϕ_j^* = 1/16π(1+Aα_ΛsinΔϕ_i^* + Bα_ΛsinΔϕ_j^*+Cα_Λ^2π/4sinΔϕ_i^*sinΔϕ_j^*),

and c'_ΛΛ can be calculated by

    c'_ΛΛ=64/π^2α_Λ^2⟨sinΔϕ_i^*sinΔϕ_j^* ⟩ - P_Λ^2,

where Δϕ_i^* (Δϕ_j^*) is the azimuthual angle relative to the reaction plane for proton i (j) in first (second) Λ rest frame. Considering a finite event plane resolution, Eq. <ref> can be expressed as

    c'_ΛΛ=64/π^2α_Λ^2⟨sin(ϕ_i^*-Ψ_ EP)sin(ϕ_j^*-Ψ_ EP) ⟩/⟨cos^2(Ψ_ EP - Ψ_ RP) ⟩ - P_Λ^2,

where Ψ_ EP and Ψ_ RP denote the event plane and the reaction plane, respectively. Note that

    ⟨cos^2(Ψ_ EP - Ψ_ RP) ⟩ =   1 + ⟨cos(2(Ψ_ EP - Ψ_ RP)) ⟩/2,

and ⟨cos(2(Ψ_ EP - Ψ_ RP)) ⟩ can be determined from sub-event method in experiment <cit.>.

Building on the structure of balance function <cit.>, we propose a spin balance function to detect the difference of spin correlation between “same sign” pairs (Λ-Λ or Λ̅-Λ̅) and “opposite sign” pairs (Λ-Λ̅),

    B(c'_OS,c'_SS) = 1/2(c'_ΛΛ̅ - c'_ΛΛ/c'_ΛΛ̅ + c'_ΛΛ + c'_Λ̅Λ - c'_Λ̅Λ̅/c'_Λ̅Λ + c'_Λ̅Λ̅ ).

The advantage of the above equation is that the detector effect and flavor independent correlation will be canceled, only additional correlation between s and s̅ (in Λ-Λ̅ pairs) remains.

To summarize, in this paper, we investigated the correlation function used in a theoretical context, identified its connection to experimental observables, and outlined the procedure to measure it in experiments.  Additionally, we proposed a method for measuring  the  correlation using a spin balance function, which minimize the impact of detector and other trivial effects. This work will be instrumental in studying hyperon spin correlations and facilitating the understanding of spin-spin correlations and their implications for fluctuations in the strong force field. 




§ ACKNOWLEDGMENT

This work was inspired by the theoretical study <cit.> and benefited from discussions with X.-N. Wang, Z.-T. Liang, Q. Wang, X.-L. Sheng., and Z. Tu.  






 Paper Title: Exploring atmospheric neutrino oscillations at ESSnuSB.







Consorcio ESS-bilbao, Parque Científico y Tecnológico de Bizkaia, Laida Bidea, Edificio 207-B, 48160 Derio, Bizkaia, Spain

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden
The Oskar Klein Centre, AlbaNova University Center, Roslagstullsbacken 21, 106 91 Stockholm, Sweden

Sofia University St. Kliment Ohridski, Faculty of Physics, 1164 Sofia, Bulgaria

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

University of Milano-Bicocca and INFN Sez. di Milano-Bicocca, 20126 Milano, Italy

University of Milano-Bicocca and INFN Sez. di Milano-Bicocca, 20126 Milano, Italy

University of Milano-Bicocca and INFN Sez. di Milano-Bicocca, 20126 Milano, Italy

Consorcio ESS-bilbao, Parque Científico y Tecnológico de Bizkaia, Laida Bidea, Edificio 207-B, 48160 Derio, Bizkaia, Spain

Department of Physics, Lund University, P.O Box 118, 221 00 Lund, Sweden

Department of Physics, Lund University, P.O Box 118, 221 00 Lund, Sweden

Department of Physics and Astronomy, FREIA Division, Uppsala University, P.O. Box 516, 751 20 Uppsala, Sweden

Corresponding authors: S. Choubey, T. Ohlsson and S. Vihonen
Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden
The Oskar Klein Centre, AlbaNova University Center, Roslagstullsbacken 21, 106 91 Stockholm, Sweden

Department of Physics, Lund University, P.O Box 118, 221 00 Lund, Sweden

Faculty of Engineering, Lund University, P.O Box 118, 221 00 Lund, Sweden
European Spallation Source, Box 176, SE-221 00 Lund, Sweden

University of Milano-Bicocca and INFN Sez. di Milano-Bicocca, 20126 Milano, Italy

AGH University of Krakow, al. A. Mickiewicza 30, 30-059 Krakow, Poland 

European Spallation Source, Box 176, SE-221 00 Lund, Sweden




IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

CERN, 1211 Geneva 23, Switzerland

Department of Physics and Astronomy, FREIA Division, Uppsala University, P.O. Box 516, 751 20 Uppsala, Sweden

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

Institute of Nuclear and Particle Physics, NCSR Demokritos, Neapoleos 27, 15341 Agia Paraskevi, Greece

Cockroft Institute (A36), Liverpool University, Warrington WA4 4AD, UK

Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
Center for Interdisciplinary Research and Innovation (CIRI-AUTH), Thessaloniki, Greece

Institute for Advanced Research, Nagoya University, Nagoya 464–8601, Japan

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

Institute of Nuclear and Particle Physics, NCSR Demokritos, Neapoleos 27, 15341 Agia Paraskevi, Greece

Center of Excellence for Advanced Materials and Sensing Devices, Ruđer Bošković Institute, 10000 Zagreb, Croatia

Dipartimento di Matematica e Fisica, Universitá di Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy

University of Cukurova, Faculty of Science and Letters, Department of Physics, 01330 Adana, Turkey
Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86, B-9000 Ghent, Belgium

Institute for Experimental Physics, Hamburg University, 22761 Hamburg, Germany

Center of Excellence for Advanced Materials and Sensing Devices, Ruđer Bošković Institute, 10000 Zagreb, Croatia



Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86, B-9000 Ghent, Belgium

Department of Physics, Lund University, P.O Box 118, 221 00 Lund, Sweden

Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86, B-9000 Ghent, Belgium

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
Center for Interdisciplinary Research and Innovation (CIRI-AUTH), Thessaloniki, Greece

University of Cukurova, Faculty of Science and Letters, Department of Physics, 01330 Adana, Turkey

European Spallation Source, Box 176, SE-221 00 Lund, Sweden



Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
Center for Interdisciplinary Research and Innovation (CIRI-AUTH), Thessaloniki, Greece

Physics Laboratory, School of Science and Technology, Hellenic Open University, 26335, Patras, Greece 

Deceased
European Spallation Source, Box 176, SE-221 00 Lund, Sweden
Department of Physics, Lund University, P.O Box 118, 221 00 Lund, Sweden

Department of Physics and Astronomy "G. Galilei", University of Padova and INFN Sezione di Padova, Italy

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

University of Milano-Bicocca and INFN Sez. di Milano-Bicocca, 20126 Milano, Italy

CERN, 1211 Geneva 23, Switzerland

Dipartimento di Matematica e Fisica, Universitá di Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy

INFN Sez. di Padova, Padova, Italy

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

Consorcio ESS-bilbao, Parque Científico y Tecnológico de Bizkaia, Laida Bidea, Edificio 207-B, 48160 Derio, Bizkaia, Spain

Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86, B-9000 Ghent, Belgium

University of Cukurova, Faculty of Science and Letters, Department of Physics, 01330 Adana, Turkey

Corresponding authors: S. Choubey, T. Ohlsson and S. Vihonen
Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden
The Oskar Klein Centre, AlbaNova University Center, Roslagstullsbacken 21, 106 91 Stockholm, Sweden

Department of Physics and Astronomy, FREIA Division, Uppsala University, P.O. Box 516, 751 20 Uppsala, Sweden

Department of Physics and Astronomy "G. Galilei", University of Padova and INFN Sezione di Padova, Italy

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

Sofia University St. Kliment Ohridski, Faculty of Physics, 1164 Sofia, Bulgaria

Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
Center for Interdisciplinary Research and Innovation (CIRI-AUTH), Thessaloniki, Greece

IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

Institute of Nuclear and Particle Physics, NCSR Demokritos, Neapoleos 27, 15341 Agia Paraskevi, Greece

INFN Sez. di Padova, Padova, Italy

Department of Civil, Environmental and Natural Resources Engineering Luleå University of Technology, SE-971 87 Lulea, Sweden

Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
Center for Interdisciplinary Research and Innovation (CIRI-AUTH), Thessaloniki, Greece

IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

Consorcio ESS-bilbao, Parque Científico y Tecnológico de Bizkaia, Laida Bidea, Edificio 207-B, 48160 Derio, Bizkaia, Spain

CERN, 1211 Geneva 23, Switzerland

Institute of Nuclear and Particle Physics, NCSR Demokritos, Neapoleos 27, 15341 Agia Paraskevi, Greece



European Spallation Source, Box 176, SE-221 00 Lund, Sweden

University of Milano-Bicocca and INFN Sez. di Milano-Bicocca, 20126 Milano, Italy

Institute for Experimental Physics, Hamburg University, 22761 Hamburg, Germany

European Spallation Source, Box 176, SE-221 00 Lund, Sweden

Sofia University St. Kliment Ohridski, Faculty of Physics, 1164 Sofia, Bulgaria

Physics Laboratory, School of Science and Technology, Hellenic Open University, 26335, Patras, Greece 

Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
Center for Interdisciplinary Research and Innovation (CIRI-AUTH), Thessaloniki, Greece

Sofia University St. Kliment Ohridski, Faculty of Physics, 1164 Sofia, Bulgaria

Institute of High Energy Physics (IHEP) Dongguan Campus, Chinese Academy of Sciences (CAS), Guangdong 523803, China

Corresponding authors: S. Choubey, T. Ohlsson and S. Vihonen
Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden
The Oskar Klein Centre, AlbaNova University Center, Roslagstullsbacken 21, 106 91 Stockholm, Sweden

IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

Institute of Nuclear and Particle Physics, NCSR Demokritos, Neapoleos 27, 15341 Agia Paraskevi, Greece



ESSnuSB Collaboration



§ ABSTRACT 
This study provides an analysis of atmospheric neutrino oscillations at the ESSnuSB far detector facility. The prospects of the two cylindrical Water Cherenkov detectors with a total fiducial mass of 540 kt are investigated over 10 years of data taking in the standard three-flavor oscillation scenario. We present the confidence intervals for the determination of mass ordering, θ_23 octant as well as for the precisions on sin^2θ_23 and |Δ m_31^2|. It is shown that mass ordering can be resolved by 3σ CL (5σ CL) after 4 years (10 years) regardless of the true neutrino mass ordering. Correspondingly, the wrong θ_23 octant could be excluded by 3σ CL after 4 years (7 years) in the case where the true neutrino mass ordering is normal ordering (inverted ordering). The results presented in this work are complementary to the accelerator neutrino program in the ESSnuSB project.


Exploring atmospheric neutrino oscillations at ESSnuSB
    O. Zormpa
    August 5, 2024
======================================================





§ INTRODUCTION


Atmospheric neutrinos are one of the most formidable neutrino sources in the Nature. Cosmic-ray interactions in the atmosphere very often result in hadronic showers that produce neutrinos as a side product. The neutrinos and antineutrinos produced in this way can have a wide range of energies and directions as they gain their origin from cosmic rays, spanning over neutrino energies from hundreds of MeV up to the PeV scale. As most of the atmospheric neutrinos traverse very long distances inside the Earth before they can be observed in any neutrino detector, atmospheric neutrinos are sensitive to effects that arise from neutrino interactions with matter.

The standard theory of three-flavour neutrino oscillations states that the mixing of three active neutrinos can be parameterized with three mixing angles θ_12, θ_13 and θ_23, two mass-squared differences Δ m_21^2 ≡ m_2^2 - m_1^2 and Δ m_31^2 ≡ m_3^2 - m_1^2, and one charge-parity (CP) phase δ_CP. Experimental efforts to study neutrino oscillations with neutrinos of accelerator, reactor, atmospheric and solar origin have determined the values of the mixing angles and the mass-squared differences within 1%–5% precision at 1σ confidence level (CL) and hinted that δ_CP may be CP-violating <cit.>. Next-generation neutrino oscillation experiments aim to determine whether the neutrino masses m_1, m_2 and m_3 follow the normal ordering, Δ m_31^2 > 0, or the inverted ordering, Δ m_31^2 < 0, by studying neutrino oscillations with both neutrinos and antineutrinos. It is also to be discovered whether the CP phase is CP-violating, sinδ_CP≠ 0, or CP-conserving, sinδ_CP = 0, and whether the mixing angle θ_23 resides in the low octant, θ_23 < 45^∘, or the high octant, θ_23 > 45^∘. Future neutrino oscillation experiments will also test the precision of the Standard Model by looking for non-standard interactions and additional neutrino families.

The European Spallation Source neutrino SuperBeam (ESSnuSB) project <cit.> aims to study leptonic CP violation by sending high-power neutrino and antineutrino beams over a baseline length that is 360 km long. The main source of neutrinos in this project would be the ESS linear accelerator, which is capable of creating ultra-pure muon neutrino beams with 5 MW output. The advantage of ESSnuSB would be its access to neutrino oscillations at the second oscillation maximum, which is expected to have significant potential to accurately measure the value of δ_CP. The second oscillation maximum is also expected to enable measurements on the standard neutrino oscillation parameters with high precision <cit.>. The prospects of ESSnuSB also include other physics cases related to neutrinos, such as the coherent elastic neutrino-nucleus scattering <cit.> and searches for physics beyond the Standard Model <cit.>.

In the present work, we examine the prospects of measuring atmospheric neutrino oscillations at the ESSnuSB far detector facility. The ESSnuSB far detector utilizes the Water Cherenkov technology, where neutrino properties are reconstructed by observing the Cherenkov light that is emitted from neutrino interactions with water. The far detector facility is planned to consist of two identical water cylinders that would be placed inside the mine in Zinkgruvan in central Sweden at the depth of 1 km. The combined fiducial mass of the proposed far detector facility is 540 kt, which would make the ESSnuSB far detectors approximately 2.9 times larger than the Hyper-Kamiokande detector <cit.>. The geographical location of Zinkgruvan has a relatively high flux of atmospheric neutrinos thanks to its proximity to the North Pole, making the conditions at ESSnuSB far detectors promising for the study of atmospheric neutrino oscillations. The experimental program of ESSnuSB would complement the prospects of the currently planned neutrino experiments such as DUNE <cit.>, Hyper-Kamiokande <cit.>, IceCube-Gen2 <cit.> and KM3NeT <cit.>.

The numerical study of atmospheric neutrino oscillations is carried out as follows. We first generate a large set of Monte Carlo (MC) samples for atmospheric neutrino interactions at ESSnuSB far detectors using the neutrino event generator GENIE <cit.>. An in-house written analysis software based on Python is then used to emulate detector response in the ESSnuSB far detectors and compute sensitivities to neutrino mass ordering, θ_23 octancy and precisions on sin^2θ_23 and Δ m_31^2 assuming a 5.4 Mt·year total exposure. The neutrino oscillation probabilities used in this analysis are calculated numerically with the General Long-Baseline Experiment Simulator (GLoBES) <cit.>. The results obtained in this work are complementary to the accelerator physics program of the ESSnuSB project[The complementarity between atmospheric and accelerator neutrinos at ESSnuSB was previously studied in Ref. <cit.>, where the authors assumed a MEMPHYS-like detector with 1 Mt fiducial mass and the atmospheric neutrino fluxes of Gran Sasso.].

This article is divided into the following sections. Section <ref> provides a brief review of atmospheric neutrino oscillations in vacuum and matter. The ESSnuSB far detector complex as well the atmospheric neutrino flux are described in section <ref>. Section <ref> presents the analysis techniques used on the MC samples, while the major numerical results are shown in section <ref>. We finally provide concluding remarks in section <ref>.



§ NEUTRINO OSCILLATIONS


The concept of neutrino oscillations is a quantum phenomenon that derives from the non-conventional nature of neutrino mass. It is known that neutrino mass states and flavour states do not coincide, leading to the possibility that a neutrino born in flavour state ν_α may be found in a different flavour state ν_β after propagating distance L with energy E_ν. This mixing between neutrino flavour and mass eigenstates can be described with a complex unitary matrix,

    |ν_α⟩ = ∑_i=1^3 U_α i^* |ν_i⟩,

where α = e, μ or τ. Here |ν_i⟩ are eigenstates in the mass basis and |ν_α⟩ are the states in the flavour basis, respectively. In the standard parametrization of leptonic mixing, the mixing matrix U is the so-called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix and it is given by

    U =
        [     1     0     0;     0  c_23  s_23;     0 -s_23  c_23 ][           c_13              0  s_13 e^-iδ_CP;              0              1              0; -s_13 e^-iδ_CP              0           c_13 ][  c_12  s_12     0; -s_12  c_12     0;     0     0     1 ],

where c_ij and s_ij are defined as cosθ_ij and sinθ_ij, respectively. The parametrization in the PMNS matrix (<ref>) involves three leptonic mixing angles θ_12, θ_13 and θ_23 and one CP phase δ_CP. When neutrinos propagate in vacuum, the neutrino oscillation probabilities can be computed from the time-evolution operator as P_ν_α→ν_β (E_ν, L) = |S|^2 ≡ |e^-i H_0 L|^2, where the vacuum Hamiltonian H_0 is given by

    H_0 = 1/2E_νU
        [        0        0        0;        0 Δ m_21^2        0;        0        0 Δ m_31^2;          ]
        U^†,

and the full probability formula can be written as

    P_ν_α→ν_β (E_ν, L) = δ_αβ - 4 ∑_i>jℛ(U_α i^* U_β i U_α j U_β j^*) sin^2Δ_ij±∑_i>jℐ(U_α i^* U_β i U_α j U_β j^*)sin2Δ_ij.

The quantity Δ_ij≡ L Δ m_ij^2/(4E_ν) in equation (<ref>) defines the oscillation mode and the sign of the second term is positive for neutrinos and negative for antineutrinos. Neutrino oscillations in vacuum therefore depend on six independent parameters, now including the two mass-squared differences Δ m_21^2 and Δ m_31^2 in addition to the mixing angles and the CP phase.

In atmospheric neutrino oscillations, the relevant oscillation channels are ν_e →ν_e, ν_μ→ν_μ, ν_e →ν_μ and ν_μ→ν_e. When L / E_ν≪ 1, the oscillation probabilities are driven by the Δ_31 mode, since Δ_21≪ 1. In this case, the neutrino oscillation probabilities can be approximated with the analytical formulas <cit.>

    P_ν_e →ν_e (E_ν, L)    ≃ 1 - sin^2 2θ_13sin^2 ( LΔ m_31^2/4 E_ν),
    
        P_ν_μ→ν_μ (E_ν, L)    ≃ 1 - 4cos^2θ_13sin^2θ_23 (1 - cos^2θ_13sin^2θ_23) sin^2 ( L Δ m_31^2/4 E_ν),
    
        P_ν_μ↔ν_e (E_ν, L)    ≃sin^2 θ_23sin^2 2θ_13sin^2 ( L Δ m_31^2/4 E_ν),

where the neutrino oscillation probabilities are given at zeroth-order in the small parameter Δ m_21^2 / Δ m_31^2. Equation (<ref>) gives the parameter dependence of the neutrino oscillation probabilities for both the ν_μ→ν_e and ν_e →ν_μ channels. As expected, the oscillations between muon and electron neutrino states are driven by the leptonic mixing parameters θ_23 and Δ m_31^2, and also by the mixing angle θ_13. On the other hand, the sensitivity to the CP phase δ_CP arises from the sub-leading terms that are not present in formulas (<ref>)–(<ref>).

The majority of atmospheric neutrinos are created about 15 km above the Earth's surface. Atmospheric neutrinos may therefore undergo distances between 15 km and 12 742 km, the latter of which is equivalent to neutrinos passing through the full diameter of the Earth. Taking the matter effects into account, the effective Hamiltonian can be written in the mass basis as

    H_m = 1/2E_ν[ m_1^2     0     0;     0 m_2^2     0;     0     0 m_3^2;       ]
        + U^†[ a 0 0; 0 0 0; 0 0 0;   ]
        U,

where a = ±√(2) G_F N_e is positive for neutrinos and negative for antineutrinos, G_F is the Fermi coupling constant and N_e is the number density of electrons in the Earth. The mixing matrix U is the PMNS matrix we defined in equation (<ref>). For constant matter density, the neutrino interactions with matter can be described with effective mixing parameters which reduce the oscillation probabilities back to the vacuum formulas we provided in equations (<ref>)–(<ref>). This is achieved through the following transformations:

    Δ m_31^2    →Δ m_31^2 √(sin^2 2θ_13 + (Γ - cos 2θ_13)^2),
    sin^2 2θ_13   →sin^2 2θ_13/sin^2 2θ_13 + (Γ - cos 2θ_13)^2,

where we define Γ = a E_ν / Δ m_31^2. One can readily see from equations (<ref>) and (<ref>) that the effective mixing is maximal when Γ = cos 2θ_13 and the neutrino oscillation probabilities are significantly enhanced. Neutrinos undergoing these conditions are therefore said to go though resonant transition.

As the distances that atmospheric neutrinos can traverse inside the Earth vary significantly, the constant matter density approach is not applicable and a detailed matter density profile must be used. The effective operator describing the propagation of neutrino mass eigenstates in N layers of constant matter density can be written as <cit.>

    X = ∑_k [ ∏_j ≠ k2 E_ν H_m - m_j^2 I/m_k^2 - m_j^2] e^-i m_k^2 L / (2 E_ν),

where k = 1, 2, …, N and I is the identity matrix. Equation (<ref>) presents the propagated eigenvalues m_i^2 / (2E_ν) of the constant matter density Hamiltonian H_m. The rows of the matrix X then represent the propagated eigenvectors of the neutrino mass matrix. The neutrino oscillation probabilities that take into account neutrino interactions with matter can therefore be obtained from the formula

    P_ν_α→ν_β(E_ν, L) = | (U X U^†)|^2,

where U is the PMNS matrix defined in equation (<ref>) and α, β = e, μ and τ. A convenient way to approximate the varying matter density is to implement the Preliminary Reference Earth Model (PREM) <cit.>, which treats the internal structure of the Earth as a finite number of layers with constant matter density. To calculate the matter density for a given propagation distance inside the Earth, one must determine the zenith angle θ_z of the incoming neutrino. The full three-flavour neutrino oscillation probabilities can then be calculated as 

    P_ν_α→ν_β(E_ν, h, θ_z) = | (U ∏_i^N X(L_i, ρ_i, E_ν) U^†)_αβ|^2,

where h is the production height at which the atmospheric neutrino is produced. Note that the neutrino trajectory is now sliced into N layers such that L = ∑_i L_i. Each layer is assumed to have a constant matter density ρ_i.



§ THE ESSNUSB FAR DETECTORS


The ESSnuSB far detector facility is planned to consist of two identical Water Cherenkov detectors. The far detectors have the shape of standing cylinders with a height of 76 m and a diameter of 76 m. Each cylinder is expected to hold ultra-pure water of about 270 kt fiducial mass, giving the total fiducial mass of the far detector complex as 540 kt. The cylinders are to be instrumented with photo-multiplier tubes (PMTs), which would reduce the fiducial volume by 2 m from the cylindrical surface. The PMT structure is planned to feature inward-pointing 20-inch PMTs with the purpose of detecting Cherenkov light from charged particles that are produced in neutrino interactions. There would also be outward-pointing PMTs to be used as a veto. The area covered by the inward-pointing PMTs would form 30% coverage. After the fiducial cuts, the active volume of the detectors has the dimensions of 70 m height and 70 m diameter. 

The chosen site for the ESSnuSB far detector complex is the mine in Zinkgruvan in central Sweden. The 1100 m rock overburden provides a sufficient protection from the main backgrounds to the experiment's accelerator neutrino program. The atmospheric neutrino flux at this location is expected to be similar to that of the Pyhäsalmi mine in Finland and several percent higher than the atmospheric neutrino flux in Kamioka in Japan <cit.>. The atmospheric neutrino flux is the highest in the tangential plane on the Earth's surface, cosθ_z = 0, and decreases towards the directions where the neutrino flux is perpendicular, cosθ_z = ± 1. The muon-to-electron flavour ratio of the atmospheric neutrinos is about 2:1, with a large number of the neutrinos carrying sub-GeV energies. We compute the atmospheric neutrino fluxes from the average of the respective fluxes that correspond to the solar minimum and the solar maximum periods[The flux files are available in .].

In this work, the performance of the ESSnuSB far detector complex is studied in the context of atmospheric neutrino detection. The neutrino events emerging from atmospheric neutrino interactions are generated with the GENIE <cit.> event generator, whereas the detector geometry is created using the ROOT geometry package <cit.>. An illustration of the detector geometry can be found in Figure <ref>. The total fiducial mass of the far detector complex is taken to be 540 kt, which can be achieved by creating a ROOT geometry of two cylindrical volumes with 70 m diameter and 70 m height each. The detector material is taken to be ultra-pure water, where 88.89% of active mass is formed by ^16O atoms and 11.11% by ^1H atoms. The PMT structure and the cylindrical containers are not taken into account in the MC event generation and are therefore neglected in the ROOT geometry.




§ ATMOSPHERIC NEUTRINO ANALYSIS


The statistical analysis of the simulated neutrino events is given as follows. The GENIE event generator <cit.> is used to generate MC samples which are processed to obtain neutrino events for the test and the true hypotheses. The processed MC events are then analysed with the likelihood function:

    χ^2 = 2 ∑_n=1^2000( E_n - O_n + O_n logO_n/E_n) + ∑_i=1^5( ζ_i/σ_i)^2.

Here E_n and O_n correspond to the expected and observed neutrino events in the n^ th analysis bin and ζ_i is the nuisance parameter modeling the i^th systematic uncertainty with standard deviation σ_i. The systematic uncertainties are treated with the pull method <cit.>, where

    E_n = E_n, 0( 1 + ∑_i=1^5 f_n, iζ_i ).

In this equation, E_n, 0 represents the original MC expectation and f_n, i is the coefficient that determines the weight of the pull parameter ζ_i. The analysis bins n runs through 100 energy and 20 cosine zenith angle bins, which are distributed evenly over neutrino energies E_ν∈[0.1,100] GeV and neutrino cosine zenith angles cosθ_z∈[-1,1]. In order to assess the effect of detector response, the MC events undergo a folding process, where Gaussian smearing is applied to the neutrino energies E_ν and also to the neutrino cosine zenith angles cosθ_z in each analysis bin. The number of atmospheric neutrino events in each analysis bin is furthermore multiplied by the appropriate detector efficiency. Similar techniques have been used for Water Cherenkov detectors in e.g. Ref. <cit.>. As the Water Cherenkov technology planned for the ESSnuSB far detectors is not sensitive to the sign of the electric charge of primary leptons, the neutrino and antineutrino events of the same lepton flavor are analyzed without distinction between CP charges[Some sensitivity to the CP charge of neutrinos and antineutrinos could be accomplished by via gadolinium-doping, which has been successfully implemented in Super-Kamiokande, see Ref. <cit.>. Investigations are currently underway to include gadolinium-doping in ESSnuSB far detectors, which could benefit from the treatment.]. We adopt the detector efficiencies from Ref. <cit.>, taking into account leptonic flavor and CP charge of each atmospheric neutrino. We furthermore use 30% resolution for sub-GeV neutrino energies and 10% resolution for multi-GeV neutrino energies, respectively, and a constant 10^∘ resolution for the neutrino cosine zenith angles. We have checked that these resolution functions reproduce rather well the mass ordering results obtained by the Hyper-Kamiokande collaboration with the atmospheric neutrino fluxes at Kamioka in Japan and the detector size normalized to the Hyper-Kamiokande detector <cit.>.

The neutrino oscillations are taken into account by applying the so-called reweighing method. For each MC event, the oscillation probabilities are computed according to the initial energy and cosine zenith angle of the associated neutrino. The reweighing is carried out by assigning each MC event a random number S ∈ [0, 1], which is then compared to the relevant neutrino oscillation probability. For instance, if a muon neutrino event is assigned a random number S that satisfies S < P_ν_μ→ν_e, the event is classified as an oscillated electron neutrino event. Correspondingly, the event is classified as an unoscillated muon neutrino event if P_ν_μ→ν_e < S < P_ν_μ→ν_e + P_ν_μ→ν_μ and an oscillated tau neutrino event if S > P_ν_μ→ν_e + P_ν_μ→ν_μ, respectively. Electron neutrino events as well as electron antineutrino and muon antineutrino events are treated analogously. The matter densities are interpolated from PREM, which is evaluated with 81 layers to ensure sufficient detail in the matter effects. We resolve the neutrino propagation distances from their corresponding zenith angles with the following equation

    L = √((R + h)^2 - (R - d)^2 sin^2 θ_z) - (R - d)cosθ_z,

where R is the radius of the Earth, h is the height where atmospheric neutrinos are born and d is the depth of the neutrino detector. In our analysis, we assume the detector to be located in the mine at d = 1 km. We furthermore assume the neutrino production height h to be 15 km and the diameter of the Earth R to be 6371 km, respectively. The probability calculation is executed with GLoBES, while the reweighing method is implemented in the main analysis code.

The evaluation of the systematic uncertainties is based on the approach described in Ref. <cit.>. The systematic uncertainties are evaluated with the pull parameters ζ_1, ζ_2, ζ_3, ζ_4 and ζ_5 in equation (<ref>). Each analysis bin is assigned weights f_1, n, f_2, n, f_3, n, f_4, n and f_5, n that determine the impact from individual pull parameters in the n^th analysis bin.  In the analysis of the MC events generated for ESSnuSB far detectors, there are five different uncertainties that influence the analysis of atmospheric neutrinos in ESSnuSB: (i) flux normalization error, (ii) cross-section normalization error, (iii) zenith angle dependence error, (iv) energy tilt error and (v) detector error. The summary of the systematic uncertainties and the values used in this work is given in Table <ref>. Systematic errors (i) and (ii) are ordinary normalization errors derived from uncertainty in the atmospheric neutrino fluxes and cross-sections, respectively. The zenith angle uncertainty (iii) arises from the uncertainty on the zenith angle bins and it depends on the value of neutrino cosθ_z. The energy tilt error (iv) is calculated directly from the ratio of atmospheric neutrino fluxes. The detector uncertainty (v) is considered to be a normalization error.


We now discuss the systematic uncertainties related to zenith angle dependence and energy tilt errors. The zenith angle error arises from the uncertainty in the zenith angle binning. It is independent of the neutrino energy and it can be calculated directly from the value of the neutrino cosine zenith angle cosθ_z. In this work, we calculate the weights for zenith angle dependence as 5% of the neutrino cosθ_z value. The error weights associated with the zenith angle dependence therefore belong to the interval f_3, n∈ [-5%, 5%]. On the other hand, the energy tilt error takes into account potential deviations from the power law dependence of the atmospheric neutrino fluxes. We treat the energy tilt error using the method discussed in Ref. <cit.>. In this approach, the MC events are generated using atmospheric neutrino fluxes that have been perturbed by a small deviation δ (called the tilt error) from the standard atmospheric neutrino flux spectrum
. The distorted fluxes Φ_δ are computed from the original fluxes Φ_0 as

    Φ_δ (E) = Φ_0 (E) ( E/E_0)^δ≃Φ_0 (E) ( 1 + δlogE/E_0),

where E is the neutrino energy in the distorted spectrum and E_0 is a reference upon which the power-law deviation is imposed. The error weights f_4, n are then extracted for every analysis bin n by comparing the generated MC samples at the original and distorted scales. Following the example in Ref. <cit.>, we obtained the distorted fluxes at δ = 5% and reference energy E_0 = 2 GeV. The weights were determined for the tilt error by generating MC samples for 100 years of ESSnuSB far detector exposure using both the nominal and the distorted atmospheric neutrino fluxes and calculating the relative difference in the MC samples. We estimate the resulting weights to fall mostly within the range f_4, n∈ [-5%, 5%] for all analysis bins.

We perform the analysis of the generated MC samples with a grid scan. The parameters θ_12 and Δ m_21^2 are fixed at sin^2 θ_12 = 0.303, Δ m_21^2 = 7.41×10^-5 eV^2 in the scan, whereas the parameters θ_23 and Δ m_31^2 are varied. The values are adopted from  <cit.> assuming normal ordering without atmospheric neutrino data from Super-Kamiokande. The mixing angle θ_13 is fixed according to the reactor neutrino measurements at sin^2θ_13 = 0.02225. We additionally scan the CP phase δ_CP over range [0, 2π). The scan ranges are listed in Table <ref>.





§ NUMERICAL RESULTS


The analysis of the MC samples is carried out with pre-computed neutrino oscillation probabilities. Figure <ref> presents neutrino oscillation probabilities in the oscillation channels ν_μ→ν_e, ν_μ→ν_μ, ν̅_μ→ν̅_e and ν̅_μ→ν̅_μ. The neutrino oscillation probabilities are provided as functions of neutrino energy E_ν and neutrino cosine zenith angle cosθ_z. The probabilities were calculated using the neutrino oscillation parameter values that are given in Table <ref>. The matter effects were included in the probability calculation by using 81 constant matter density layers which have been obtained from PREM. The color coding in Figure <ref> indicates the values of neutrino oscillation probabilities, where the darker shades represent high neutrino oscillation probabilities and lighter areas low probabilities. The differences between the neutrino and antineutrino oscillation probabilities can be observed in the probabilities computed for the conversion channels ν_μ→ν_e and ν̅_μ→ν̅_e, which are presented in the top-left and the bottom-left panels, respectively. The most interesting features in these two panels are found in the segment cosθ_z ∈ [-1, -0.8], which corresponds to atmospheric neutrinos that traverse near the full diameter of the Earth. The neutrino oscillation probabilities belonging to this segment reach the first local maximum for ν_μ→ν_e at the neutrino energies E_ν∼ 2 GeV, while the antineutrino channel ν̅_μ→ν̅_e shows no significant neutrino oscillation probabilities in the same region. This is an artifact of matter effects enhancing oscillations in the neutrino channel for the normal neutrino mass ordering. Moreover, neutrino energies E_ν∼ 6 GeV display notable distortions in the neutrino disappearance channel ν_μ→ν_μ but not in the antineutrino disappearance channel ν̅_μ→ν̅_μ. This distortion is due to the matter effects in the neutrino channel for the normal neutrino mass ordering. Changing the sign of Δ m_31^2 would switch the role of matter effects in the oscillation probabilities in the neutrino channels ν_μ→ν_e and ν_μ→ν_μ and the antineutrino channels ν̅_μ→ν̅_e and ν̅_μ→ν̅_μ, enabling the determination of the neutrino mass ordering.



The analysis of the generated MC samples is carried out in the following physics scenarios. We first examine the physics potential to exclude the wrong neutrino mass ordering at the ESSnuSB far detectors. The MC samples are then used to compute the sensitivity to the θ_23 octant. We finally estimate the precisions at which the ESSnuSB setup can determine the leptonic mixing parameters θ_23 and Δ m_31^2 and provide the evolution of sensitivities to the neutrino mass ordering and the θ_23 octant as functions of time. Both normal and inverted orderings are taken into account throughout this section.

The expected numbers of atmospheric neutrino events in the ESSnuSB far detectors are presented in Figure <ref>. The total number of atmospheric neutrino events corresponds to 5.4 Mt·year exposure. The top-left and bottom-left panels of the figure show the total numbers of electron-like and muon-like events, respectively. The events have been binned for neutrino energies E_ν∈ [0, 100] GeV and neutrino cosine zenith angles cosθ_z ∈ [-1,1] with bin sizes Δ E_ν = 1 GeV and Δcosθ_z = 0.1. The top-right panel depicts the relative difference of the atmospheric neutrino events Δ N_NO / N_NO = |N_ NO - N_ IO| / N_NO for the electron-like events, where N_ NO represents the number of electron-like events for normal ordering and N_ IO for inverted ordering. The bottom-right panel shows the same quantity for the muon-like events. The relative differences are shown for neutrino energies E_ν∈ [0, 10] GeV. The determination of the θ_23 octant and the precision measurements on sin^2 θ_23 and Δ m_31^2 can be done by observing neutrino oscillations in any of the neutrino channels ν_μ→ν_μ, ν_e →ν_μ, ν_μ→ν_e, as has been shown in equations (<ref>)–(<ref>). The sensitivities to these quantities are therefore proportional to the number of electron-like and muon-like events in the generated MC samples. On the other hand, the sensitivity to the neutrino mass ordering mainly arises from the difference in the number of neutrino events that are observed with the normal and inverted ordering hypotheses. The bottom-right panels of Figure <ref> shows that the relative difference between the two orderings is the most significant for the muon-like events in the neutrino energy bin E_ν∈ [1, 2] GeV and neutrino cosine zenith angle bin cosθ_z ∈ [-1, -0.9]. The relative difference in this analysis bin is about 75%. The next most significant contribution is found in the neutrino energy bin E_ν∈ [0,1] and the neutrino cosine zenith angle bin cosθ_z ∈ [-0.4,-0.3], where the relative difference in the muon-like events is about 20%. The sensitivities furthermore depend on the systematic uncertainties[The sensitivity to the neutrino mass ordering arises mainly from the neutrino energy bin E_ν∈ [5, 6] GeV, where the average difference between the charged lepton and neutrino zenith angles is about 9^∘. We have explicitly checked that the contributions from lower neutrino energies are sub-dominant due to the effect of the systematic uncertainties.].


The sensitivities to probe the neutrino mass ordering and the θ_23 octant with the ESSnuSB setup are shown in Figure <ref>. The left panel shows the statistical significance at which the wrong neutrino mass ordering can be ruled out as a function of sin^2 θ_23 in the case when the true mass ordering is normal ordering (blue bands) and inverted ordering (red hashes), respectively. The right panel shows the corresponding sensitivity to rule out the wrong θ_23 octant. The mass ordering sensitivities are given as the number of standard deviations N_σ = √(χ^2), which are computed by minimizing the χ^2 function over the neutrino oscillation parameter values that are consistent with the wrong mass ordering. In other words, the true data is obtained under normal ordering and fitted data under inverted ordering when the true mass ordering is normal, and the other way round when the true mass ordering is inverted. Similarly, the θ_23 octant sensitivities are obtained by minimizing over the parameter values that conform with the wrong θ_23 octant. The band widths correspond to the dependence on δ_CP, which is varied over the values δ_CP = 0, π/2, π and 3π/2. ESSnuSB can be expected to determine the neutrino mass ordering by 4.8σ–10.9σ CL for normal ordering and 4.3σ–8.9σ CL for inverted ordering. The sensitivity to the neutrino mass ordering generally increases as a function of sin^2 θ_23, with a local minimum at sin^2θ_23 = 0.55 for inverted ordering. The CP phase δ_CP has a non-negligible role in the determination of the neutrino mass ordering. If the true neutrino mass ordering is the normal ordering, the variation in the neutrino mass ordering sensitivity is the largest at sin^2 θ_23≃ 0.45 and lowest at sin^2 θ_23≃ 0.42, where varying δ_CP causes the sensitivities to change by 2.0σ CL and 0.7σ CL, respectively. If the true neutrino mass ordering is the inverted ordering, the variation is about 1σ CL regardless of the sin^2 θ_23 value. The CP phase δ_CP has conversely smaller effect on the determination of the octant of θ_23, where the variation is less than 0.5σ CL for normal ordering and less than 1.1σ CL for inverted ordering. Increasing the number of scan points for δ_CP may affect the band widths, however, we do not expect the changes to be significant.



Atmospheric neutrino oscillations can provide competitive sensitivities for both the neutrino mass ordering and the θ_23 octant determinations. The sensitivity to reject the inverted ordering for sin^2 θ_23 = 0.451 with atmospheric neutrino oscillation data for ESSnuSB is about 4.8σ–6.8σ CL and normal ordering 4.9σ–6.0σ CL, respectively. The sensitivity to reject the high octant solution sin^2 θ_23 > 0.50 is approximately 4.4σ–4.5σ CL for normal ordering and 3.3σ–3.5σ CL for inverted ordering.

Atmospheric neutrino oscillations can be used to constrain the values of the leptonic mixing parameters θ_23 and Δ m_31^2. The precisions on these mixing parameters are illustrated in Figure <ref>. The one-dimensional χ^2 distributions are presented as functions of sin^2 θ_23 (left panel) and |Δ m_31^2| (right panel) assuming the neutrino mass ordering to follow either normal ordering (blue bands) or inverted ordering (red hashes). In both panels, the true values of the relevant mixing parameters are assumed to be sin^2θ_23 = 0.451 and |Δ m_31^2| = 2.507×10^-3 eV^2, whereas δ_CP is varied over its allowed values δ_CP∈ [0, 2π). For convenience, the allowed 3σ CL ranges for ESSnuSB far detectors are shown with grey bands for normal ordering and light grey bands for inverted ordering, respectively. As before, the results correspond to 5.4 Mt·year total exposure. The mixing angle θ_23 can be constrained to 0.424 < sin^2θ_23 < 0.484 for normal ordering and 0.419 < sin^2θ_23 < 0.498 for inverted ordering, whereas the magnitude of the mass-squared difference Δ m_31^2 can be restricted to 2.502 × 10^-3 eV^2 < |Δ m_31^2| < 2.510 × 10^-3 eV^2 for normal ordering and 2.498 × 10^-3 eV^2 < |Δ m_31^2| < 2.518 × 10^-3 eV^2 for inverted ordering, respectively. As can be observed from the results, the effect of the δ_CP variation is relatively small both in the sin^2 θ_23 and the Δ m_31^2 resolutions.


Figure <ref> presents the sensitivities to the neutrino mass ordering and the θ_23 octant as functions of time. As before, the true values for the oscillation parameters are provided in Table <ref> and the sign of Δ m_31^2 is fixed according to the selected true mass ordering. For this choice of the neutrino oscillation parameter values, the sensitivities to the neutrino mass ordering overlap for normal ordering and inverted ordering. The corresponding sensitivities for the θ_23 octant determination on the other hand are higher for normal ordering than for inverted ordering. The effect of the δ_CP variation is also lower in the θ_23 octant determination. Figure <ref> shows that the wrong mass ordering can be ruled out at 3σ CL after about 4 years of data taking regardless of the true ordering, whereas the 5σ CL milestone can be reached for the majority of the considered δ_CP values. As before, the widths of both sensitivity bands correspond to the uncertainty on δ_CP. The wrong θ_23 octant on the other hand can be ruled out by 3σ CL after 4 years for normal ordering and 8 years for inverted ordering. 





§ SUMMARY


This work presents a study on atmospheric neutrino oscillations at the ESSnuSB far detector facility. The ESSnuSB project proposes a megaton-scale Water Cherenkov neutrino detector to observe neutrinos from the European Spallation Source. In addition to detecting neutrinos from the accelerator facility, the ESSnuSB far detectors would be capable of observing neutrinos from non-beam sources. Atmospheric neutrinos present an excellent opportunity to study neutrino oscillations with long-baseline lengths and strong matter effects, therefore complementing the physics program of the ESSnuSB project.

In the present work, we investigated the physics prospects of observing atmospheric neutrino oscillations at ESSnuSB in the standard three-flavor oscillation paradigm. The expected experimental sensitivities were examined for the determination of the neutrino mass ordering, the discovery of the θ_23 octant and the precision measurements on θ_23 and Δ m_31^2. It is found that ESSnuSB is able to determine the correct neutrino mass ordering at 3σ CL after 4 years and 5σ CL after 10 years of data taking when the value of δ_CP is not known, regardless of the mass ordering. It is also shown that ESSnuSB would be able to determine the θ_23 octant at 3σ CL after 4 years if the neutrino mass ordering is normal ordering and 7 years if it is inverted ordering. The atmospheric neutrino data collected by the ESSnuSB far detectors could also provide individual constraints on the values of θ_23 and Δ m_31^2. The sensitivities derived in this work are complementary to the beam-based long-baseline neutrino oscillation program for ESSnuSB.




§ ACKNOWLEDGEMENTS

Funded by the European Union, Project 101094628. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them. The numerical calculations presented in this work were carried out in part with resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS), which is partially funded by the Swedish Research Council through grant agreement no. 2022-06725. The authors are also grateful for the computing resources that are provided by the Division of Condensed Matter Theory at KTH Royal Institute of Technology.




 Paper Title: First Measurement of the Total Inelastic Cross-Section of Positively-Charged Kaons on Argon at Energies Between 5.0 and 7.5 GeV.

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p#1The DUNE CollaborationThe DUNE Collaboration§ ABSTRACT 
    ProtoDUNE Single-Phase (ProtoDUNE-SP) is a 770-ton liquid argon time projection chamber that operated in a hadron test beam at the CERN Neutrino Platform in 2018. We present a measurement of the total inelastic cross section of charged kaons on argon as a function of kaon energy using 6 and 7 GeV/c beam momentum settings. The flux-weighted average of the extracted inelastic cross section at each beam momentum setting was measured to be 380±26 mbarns for the 6 GeV/c setting and 379±35 mbarns for the 7 GeV/c setting. 
    First Measurement of the Total Inelastic Cross-Section of Positively-Charged Kaons on Argon at Energies Between 5.0 and 7.5 GeV
    R. Zwaska
    August 1st, 2024
===============================================================================================================================



§ INTRODUCTION


Liquid argon time projection chambers (LArTPCs) may be used to measure the trajectories of charged particles with millimeter resolution. This capability makes the detectors, like those of the Deep Underground Neutrino Experiment (DUNE) far detector modules, sensitive to studying GeV-scale and MeV-scale neutrinos and searching for physics beyond the Standard Model <cit.>. An example of important physics that can be done using the DUNE far detector modules is a search for proton decay to a final state with a neutrino and a charged kaon (p→ν+K^+), which is predicted to be dominant in a broad class of supersymmetric Grand Unified Theories <cit.>. Unlike searches in Water Cherenkov detectors <cit.>, DUNE can detect the final-state kaon, which has a momentum of 330 MeV/c absent final-state interactions. The efficiency of observing this signature is sensitive to modeling kaon transport in the LAr medium, which is limited by the dearth of kaon-argon scattering data. This search for nucleon decay requires a representative model of kaon transport and interactions in liquid argon to ensure an accurate simulation of signal events. Without reliable data and simulations, the relevant uncertainties for the kaon cross section on argon cannot be constrained. This can lead to large systematic uncertainties in nucleon decay searches with a potentially biased cross-section model.

As a first step toward collecting high-quality kaon-argon interaction data, the ProtoDUNE Single-Phase (ProtoDUNE-SP) large-scale prototype of a DUNE far detector module was exposed to a test beam from the H4-VLE beamline at CERN that included kaons at 6 and 7 GeV/c <cit.>. ProtoDUNE-SP is a 770-ton LArTPC with the same drift distance and full-scale engineering parts as a DUNE Far Detector Horizontal Drift module. The detector has two drift volumes and has dimensions of 7.2 m wide, 6.1 m high, and 7 m long. It measures the tracking and calorimetry of charged particles by detecting the ionization electrons that drift toward three layers of wire planes. The H4-VLE beamline, a tertiary beam from the CERN Super Proton Synchrotron, is referred to as simply the “beam” in many places in this paper. ProtoDUNE-SP collected data from the beam, using many beamline momentum settings, over two months from September 2018 to November 2018.


The data from ProtoDUNE-SP can be used to inform predictions from event generators that simulate hadron-nucleus interactions, like the neutrino event generator GENIE <cit.> and the transport and interaction simulation program Geant4 <cit.>, and thus improve the modeling of kaon interactions on argon nuclei. The cross section has never been measured as a function of energy on argon. Therefore, the purpose of this analysis is to provide the first measurement of the total inelastic cross section of kaons on argon at these high energies. Neither GENIE nor Geant4 has recommended uncertainties for kaon-argon interactions, providing a unique opportunity for ProtoDUNE-SP to inform inputs on associated modeling uncertainties. 


In this work, the kaon-argon total inelastic cross section is reported as a function of kaon energy within the limits of the detection threshold, described in Section <ref>. Figure <ref> shows the total inelastic and the elastic cross section predicted by the Geant4 Bertini cascade model <cit.>. Charged kaons produced by the beam with kinetic energies of approximately 4.5 to 7 GeV are capable of reaching the liquid argon of ProtoDUNE-SP. Using the Geant4 prediction from Figure <ref>, the simulated total inelastic cross section at the relevant energies should be approximately 450 millibarns (mbarns).  




 Section <ref> discusses ProtoDUNE-SP more broadly, and Section <ref> outlines the simulation and reconstruction of ProtoDUNE-SP data. Section <ref> explains the thin slice method used in this measurement. This method divides the detector into thin targets, referred to as thin slices, using the wires of the LArTPC to demarcate the slices. An incident slice is counted if a particle reaches a particular wire. Within that slice, it may also interact on the argon, which means the slice contains both an incident and an interacting slice. After an interacting slice is detected, the counting for the event stops as the outgoing particles have unknown identities and energies. The cross section is measured using the counts of the incident and interacting slices as a function of kinetic energy.


Section <ref> describes the selection of candidate kaon interaction events, and Section <ref> shows energy-related measurements using selected kaons. Section <ref> reports the kaon-argon cross section with comparisons to models. Section <ref> discusses the evaluations of the statistical and systematic uncertainties.





§ PROTODUNE-SP AND THE H4-VLE BEAMLINE


ProtoDUNE-SP is a 770-ton liquid argon detector with two TPCs, each with a drift distance of 3.6 m <cit.>. The detector contains six readout wire planes called Anode Plane Assemblies (APAs). These APAs contain three readout wire planes, the U, V, and X wire planes, and are 6.2 m high, 2.3 m wide, and 0.1 m thick <cit.>. The U and V wires are the first two planes and detect drifting electrons via the currents induced on the wires as the charges drift past them, creating bipolar signals. The X wires, known as collection wires, have unipolar signals where the drifting electrons collect on the wires and stop drifting in the TPC <cit.>. 


The U, V, and X wires are oriented 35.7^∘, -35.7^∘, and 0^∘ relative to the vertical direction, respectively. The pitch between wires is 0.467 cm for induction wires and 0.479 cm for collection wires. Each APA has 960 X wires, 800 U wires, and 800 V wires. Three APAs lie on one wall of the cryostat, and the other three APAs lie on the opposite wall of the cryostat. These APAs are 7.2 m away from each other, and the Cathode Plane Assembly (CPA) sits midway between the two separate walls of APAs. The CPA provides a high voltage of 180 kV, leading to a nominal electric field strength of 500 V/cm across the 3.6 m separating each APA from the CPA, which allows the ionization electrons to drift to the APAs. The H4-VLE beam pipe connects to the upstream face of LArTPC via a low-density beam plug that allows the beam to enter without scattering off the material in the cryostat <cit.>. 

The beam only enters one TPC of the detector. The beam side of the detector contains APAs separated by electron diverters intended to improve the charge-collection efficiency for electrons drifting near the gap between neighboring APAs. Unfortunately, these electron diverters exhibited high-voltage shorts and were left electrically grounded during operations, distorting the track images and causing some loss of collected charge. 

As a surface-based detector, ProtoDUNE-SP is exposed to an intense flux of cosmic-ray muons, which create electron-ion pairs in the detector. The argon ions drift slower than the ionization electrons, leading to an excess of ions around the surface of the detector. The excess of ions creates a space charge effect that alters the local electric field, leading to distorted calorimetry and tracking <cit.>. 

A calibration of the space charge effect is completed by measuring the tracking distortions on the surfaces of the detector, where the effect is maximal, with cosmic-ray muon data <cit.>. The distortions measured are then used to correct for local electric field fluctuations by using a linearly interpolated three-dimensional map. An “inverted” map using these data measurements is used to recreate the space charge effect in simulation. The original three-dimensional map is utilized to calibrate this effect in simulation.


From September 2018 to early November 2018, the H4-VLE  beamline settings were adjusted to emit positively-charged particles at 0.3, 0.5, 1, 2, 3, 6, and 7 GeV/c beamline momentum settings. The beamline trigger operates at a rate of 25 Hz, which qualitatively translates to beam particles being observed one-at-a-time within ProtoDUNE-SP. The beam consists of positively charged protons, positrons, kaons, pions, and muons. The beam particle species is identified using a time-of-flight system and Cherenkov detectors. The beam particle momentum is measured from the bend of the particle's trajectory through a well-known magnetic field using data from tracking fibers <cit.>. At 6 GeV/c and 7 GeV/c beam momentum settings, particle identification comes from only the Cherenkov detectors, explicitly requiring a signal in the high-pressure Cherenkov detector but no signal in the low-pressure Cherenkov detector <cit.>. 



§ SIMULATION AND RECONSTRUCTION



A simulation of the beam, including its transport to and through the LArTPC, is implemented using Geant4 <cit.>, with the entire CERN H4-VLE facility simulated from the primary beam to the tertiary beam that reaches ProtoDUNE-SP <cit.>. The selection of kaon inelastic scattering events starts with the beamline instrumentation discussed in Section <ref>. An event is defined as any time the beamline instrumentation has a signal recorded by the high-pressure Cherenkov detector and the absence of a signal recorded by the low-pressure Cherenkov detector <cit.>. 

The rest of the selection steps rely on information from the reconstruction of tracks and showers in the TPC to select relevant events. Additionally, the beamline instrumentation also has tracking fibers to reconstruct a beam track that can be extrapolated to the TPC <cit.>. 


ProtoDUNE-SP uses the Pandora multi-algorithm reconstruction package to identify the beam particle, reconstructing particle hierarchies using pattern recognition <cit.>. It then employs a boosted decision tree to select beam particle candidates that enter through the beam pipe and beam plug into the liquid argon detector. A full description of the software used in ProtoDUNE-SP is given in References <cit.>. 

Figure <ref> shows the observed and simulated distributions of the reconstructed track lengths for events with a beam kaon, as reported by only the beamline instrumentation, for the 6 GeV/c samples. The corresponding distributions for the reconstructed track lengths and all other event selection distributions for the 7 GeV/c samples showed similar agreement and are included in Appendix <ref>. The spikes in Figure <ref> at around 230 cm and 460 cm correspond to the electron diverters that inadvertently break tracks between APAs in the detector, as discussed in the previous section, with the last spike at around 700 cm corresponding to the end of the active volume. Most TPC tracks are secondary particles without any TPC-related selection steps. An excess of short reconstructed track lengths is observed in the data, likely driven by background secondary particles.



The interaction point, or track endpoint, is determined using clustering and vertex-finding algorithms that are almost identical to those from the MicroBooNE reconstruction and are described in detail in Reference <cit.>. Initial clusters are created in a way to ensure that the clusters only contain signals from one particle but are unlikely to contain all TPC signals from that particle. Numerous algorithms are applied to topologically merge clusters with the goal of one cluster containing all of the activity from one particle. In addition, algorithms are applied to split clusters if kinks are found or where the topology suggests that there may be contributions from multiple particles. These clusters are classified as either tracks or showers based on their topologies. Candidate 3D interaction points are produced by comparing pairs of clusters from two 2D views and reconstructing their start and end points as candidate interaction points. A boosted decision tree is then used to select the candidate vertex most likely to be the point where the beam particle interacts. Using Pandora vertex-finding algorithms, interaction points are observed. The signal process is an inelastic interaction of the incident kaon. An inelastic interaction is defined as any process where either:


  *  the angle between the beam kaon and leading outgoing particle is greater than 11 degrees
    
  *  two or more particles emerge from the interaction point.


The kinetic energy threshold for observing a final-state proton or charged kaon in the detector is 40 MeV, and for a charged pion, it is 20 MeV. We apply these restrictions to our signal definition.





§ METHODOLOGY



The cross section measurements presented in this paper use the thin slice method pioneered by the LArIAT experiment <cit.>. The approach treats the detector as a series of thin argon targets (slices). The number of surviving particles (N_surv) is:


    N_surv(d)=N_incexp(-d/l)=N_incexp(-σ  d n )

where N_inc is the number of incident particles, d is the distance traveled, and l=(nσ)^-1 is the interaction length of a kaon in argon, where n is the number density and σ is the cross section <cit.>. 

 A natural way to develop slices in a LArTPC with a wire readout is to use the individual wires to demarcate thin target slices from one another. Therefore, a slice is a three-dimensional box of argon between wires. For each particle, the incident energy at each thin slice is estimated. The total number of particles at each incident energy (N_inc) is counted, as are the total number of interactions (N_int). Regardless of whether or not there was an interaction, if an energy deposit from the kaon is registered in a thin target slice, then the incident energy of the kaon is counted (N_inc). The cross section, using Equation <ref>, is:



    σ(E_kin)=M_Ar/N_A r ρln[N_inc(E_kin)/N_inc(E_kin)-N_int(E_kin)]

where E_kin is the kinetic energy of the particle, N_A is the Avogadro constant, M_Ar is the atomic mass of argon, ρ is the density of liquid argon, and r is the three-dimensional distance the particle travels from one wire to the next. This value is 0.498 cm, given the wire spacing between the collection plane wires of 0.479 cm and that the beam travels at a 16-degree angle in the detector <cit.>. 


The kinetic energy at a given slice (E_kin,j) is defined as: 


    E_kin,j=E_kin,beam-∑^j-1_i=0Δ E_i

where E_kin,beam is the initial beam particle kinetic energy and Δ E_i is the energy lost in slice i. The total Δ E is summed from all slices up to slice j. 


Background subtractions, unsmearing, and efficiency corrections are required to convert the measured interaction and incident spectra into a cross section. These corrections are applied via RooUnfold with unfolding done using a Bayesian-like unfolding algorithm implemented based on Richardson-Lucy deconvolution <cit.>. The background subtractions, unsmearing, and efficiency corrections are done by RooUnfold. These corrections are applied on the incident and interacting spectra separately, an approach similar to that previously used by LArIAT <cit.>. These unfolded distributions of the incident and interacting slices are then used in Equation <ref> to measure the cross section as a function of kinetic energy.




§ EVENT SELECTION



There are three event selection steps to select candidate kaons and an additional step to select a candidate kaon with an inelastic interaction. They include the following selection steps for events where the beamline trigger reports a kaon candidate:


  *  The event must have a reconstructed TPC track.
    
  *  The endpoint of the TPC track must enter the fiducial volume by being at least 30 cm downstream of the start of the active volume of the detector. This selection step is motivated by significant inefficiencies and impurities in correctly identifying and reconstructing the beam particle with a TPC track in the first 30 cm of the detector. 
    
    
  *  The TPC track must be matched to the trajectory of the beam track from the beamline instrumentation. A match requires that their positions and angles agree within three times the resolutions of these measurements at the start of the fiducial volume.




Because the electron diverters tend to break tracks, as discussed in Section <ref>, only the interaction and incident slices contained before the point of 220 cm across the detector length are considered in the cross-section measurement. This is the final step. At each collection wire, the kaon energy is estimated per Equation <ref>, and the kaon either undergoes an interaction or does not. Thus, for each incident particle, we observe many “slices” and record the interaction as a function of energy. The interaction point, or vertex, identification occurs through Pandora as described in Section <ref>. Event displays of some selected kaon inelastic interaction candidates are shown in Figure <ref>. In these events, the beam enters the TPC at time tick 4750, where a time tick represents the 500 ns sampling intervals of the analog-to-digital converters for the wires, and then travels over 50 cm before interacting with the argon. The beam particles, highlighted by the black ovals, travel in approximately straight lines from the left to the right before scattering, creating complicated final states with many showers. The top two event displays show little shower activity, indicating they may be 1K^++X candidate events. The third event display shows a complex interaction with many showers and tracks in the final state. 



Figure <ref> shows the distributions of reconstructed track lengths for selected TPC tracks that will form the incident and interacting slice spectra from the 6 GeV/c beamline setting. Secondary kaons, which are byproducts of true beam kaons interacting off the argon and travel with some unknown kinetic energies, are the most significant background for the event selection. As these secondary kaons will have similar characteristics to beam kaons, they are an irreducible background. The breakdown of the data and simulation samples through each selection step are shown in Table <ref>.








The selection efficiency and purity are evaluated as a function of kinetic energy from simulation. An inefficiency in measuring a slice of kinetic energy occurs when no TPC track corresponds to the beam particle in the slice. A background slice occurs when there is a TPC track in a slice that the true kaon does not reach. The definition for a background slice is used regardless of whether the TPC track is from a true kaon or not, which allows unfolding to fully recover the truth distribution after unfolding. Results are shown in Figure <ref>. The purity is close to 95% for interacting slices and 85% for incident slices. The lower purity is because a single background particle entering the detector contributes to many non-interacting slices, but only the first interaction is recorded. The efficiency varies between 35-40% as a function of energy. The inefficiencies are dominated by events with a true kaon in the fiducial volume, but the event did not have a TPC track identified as the beam particle.





§ ENERGY MEASUREMENTS AND BINNING



As referenced in Equation <ref>, the initial kinetic energy is determined using measurements from the beamline instrumentation. Figure <ref> displays the beamline kinetic energy measurements of the selected beam kaons. The systematic uncertainty error bars are found by shifting the data distribution by the 1.2% kinetic energy modeling uncertainty of the beamline simulation, which will be discussed in greater detail in Section <ref>.





The TPC calorimetry is calibrated by applying corrections to the electric field variations, corrections for the spatial variations, and an overall charge scale using through-going and stopping cosmic-ray muons <cit.>. The energy resolution was evaluated to find the binning resolution, and done by measuring the difference between the true and reconstructed kinetic energies at the interaction points in the simulation. The resolution is measured to be 124 MeV, as seen in Figure <ref>.






The binning of the analysis was developed to address the kinetic energy resolution and ensure equal statistics in each bin for the reconstructed interacting slice distributions in the data sample for both beam momentum settings. The minimum bin size is then 260 MeV. The reconstructed slice distributions, highlighting both the binning and slice distributions as a function of energy, are shown for incident slices in Figure <ref> and for interacting slices in Figure <ref>. These distributions include calorimetric-related uncertainties discussed in Section <ref>.





§ RESULTS


 The kinetic energy distributions for all kaons, and for interacting kaons, are separately unfolded using the method of D'Agostini with four iterations <cit.>. The smearing matrices are shown in Figure <ref> and Figure <ref>. Studies were done to test unfolding by altering the regularization, not correcting for bin-to-bin smearing, and changing the background subtraction and efficiency corrections. All had an impact of less than a percent on the average cross section compared to the nominal unfolding process described above. The response matrix is obtained using only 66% of the simulated data, which was done to use the remaining 33% as statistically-independent fake data samples for investigating systematic uncertainties.





The reconstructed slice spectra, shown in Figure <ref> and Figure <ref>, are unfolded and then used to measure the cross section with Equation <ref>. Figure <ref> displays the result for the data of the 6 GeV/c sample with comparisons to predicted cross sections from Geant4, GENIE v3.2.0 hA2018, and GENIE v3.2.0 hN2018 <cit.>. GENIE calculates the total cross section using data and partial wave analysis <cit.>. It simulates interactions with either an empirical model (hA2018) or a fully simulated cascade (hN2018) <cit.>. Geant4 applies alterations of the base model cross section using data sets included in the Particle Data Group summary cross-section measurements <cit.>. The reduced chi-squared statistics between these models over four bins are 11.26 for Geant4 and 13.33 for GENIE v3.2.0 hA2018. The 6 GeV/c data sample flux-averaged cross section is measured at 380±26 mbarns. Table <ref> shows the final results with the breakdown of the uncertainties applied.





Figure <ref> presents the cross section measured with data at the 7 GeV/c beam setting. The reduced chi-square statistic measured divided by the number of bins is 2.64/2 bins for Geant4 and 4.05/2 bins for GENIE v3.2 hA2018. Table <ref> displays the final result with uncertainties broken down by category. The flux-averaged cross section is 379±35 mbarns for the 7 GeV/c sample. Encouragingly, the bin whose energy range overlaps with the 6 GeV/c sample has a similar measured cross section which is within uncertainties.




§ TREATMENT OF UNCERTAINTIES



Uncertainties are propagated by randomly sampling statistical and systematic parameters 1000 times within their a priori uncertainties <cit.>.
The impact of the statistical uncertainty on the individual kinetic energy bins of the incident and interaction spectra is assessed by randomly Poisson-fluctuating the number of entries in each bin independently according to the measured counts. It is done this way as the statistical uncertainty of the cross section is not directly proportional to the statistical uncertainty of interaction points given that the interaction and incident distributions are inside a logarithm, as seen in Equation <ref>. However, there are enough statistics in the incident and interacting distributions to assume both are Poisson-distributed and uncorrelated; therefore, doing many independent fluctuations of each bin can acquire the statistical uncertainty on the cross section by remeasuring the cross section with each Poisson-fluctuated sample. The resulting uncertainty on the measured cross section is approximately 2.7%, according to Table <ref> and Table <ref>.

The finite statistics of the simulation sample primarily impacts the analysis via the background subtraction, unsmearing, and efficiency corrections. This effect is propagated by varying the number of counts in each kinetic energy bin for the backgrounds, inefficiencies, and within the response matrix. Values from sampling a Poisson distribution are used to regenerate the response matrices and affiliated corrections, with each bin treated as independent from the others. As statistics in the incident slices are significantly larger than those of the interacting slices, by a factor of around 200 according to Figure <ref>, this uncertainty is only applied to the response matrix and affiliated corrections for interacting slices. 


The systematic uncertainties considered in this analysis are related to the simulation of the beamline and beamline instrumentation, the TPC response, the hadron transport model, and instances in which modeling and reconstruction of the TPC data are ill-posed. The results from unfolding with the new response matrices are used to address the total impact of the systematic effects on the analysis.



 The systematic term associated with the beamline momentum scale arises from uncertainties in terms of the position of the fiber monitors and the magnetic field strength in the beamline instrumentation. This value was calculated as 1.2% <cit.> of the beam particle energy. Therefore, the energy of the beam kaon is fluctuated according to a Gaussian distribution with width of 1.2%, resulting in an approximately 2% uncertainty on the measured cross section.


Furthermore, the particle itself can “scrape” against material upon entering the liquid argon, losing energy in the process. These beam “scrapers” should appear in selections if their position is greater than 1.5 times the radius of the beam away from the beam center (r_beam). There are 3.15 times more selected events in data that exceed this 1.5r_beam metric than in simulation. Therefore, the systematic uncertainty treatment alters the frequency of the beam “scrapers” with a central value weight of 3.15 and a standard deviation of 2.15 to address the difference between data and simulation. The weight upscales simulation events whereby the beamline instrumentation system momentum and truth information momentum at the TPC differ by over 200 MeV, the estimated minimal energy lost for a beam “scraper.”



A 3% calorimetric uncertainty from the TPC is assumed in the energy determination. This value is taken from evaluations of the calorimetry calibration uncertainty <cit.>. That study observed the spread in charge calibration results from subsamples of cosmic-ray muons separated by their trajectories in the detector, based on a similar analysis from MicroBooNE <cit.>. Although the ProtoDUNE-SP study measured 2% deviations in calibration values, a 3% uncertainty is applied to address time-dependent fluctuations in the spread of calibration results. 



The space charge effect, as discussed in Section <ref>, may alter the reconstructed positions of particles. The fiducial volume is defined to reduce the impact of the space charge effect on the track reconstruction efficiency. Mismodeling of the space charge distribution in the TPC can be effectively treated as shifts in the boundaries of the fiducial volume. The impact of the uncertainty of the space charge modeling is estimated using the spread in the mean distortion at the surfaces of the detector over time. The spread, and subsequently the uncertainty, was measured to be 8%. 


The systematic uncertainty in this analysis arising from mismodeling of charged kaon scattering is assessed by using the Geant4Reweight package <cit.> to reweight events based on the total signal cross section, which intends to probe how the underlying simulated cross section impacts the background subtraction and efficiency corrections. The total inelastic cross section was varied by 20%. Additionally, due to differences in vertexing observed based on the final state, the kaon multiplicity in the final state of the interaction is reweighted by 20%, and that weighting is done in a manner to hold the total cross section constant. The impact of all these modeling uncertainties is 2-6% on the cross section per bin.




The impact of mismodeling the effect of the electron diverter is determined by changing the frequency of these events occurring in the incident response matrices. The following uncertainty reweights the number of tracks broken between 220 and 234 cm inside the detector. It only applies to tracks that do not end within the fiducial volume, which means this weight only impacts the incident slice spectrum. The uncertainty on this effect is set to 100% as the rate at which the electron diverter breaks reconstructed tracks is not well simulated and overpredicts the number of broken tracks, as seen in Figure <ref>.




While Pandora employs various algorithms to find the endpoint of the beam particle, they may miss the vertex, as Figure <ref> shows that the endpoint of the reconstructed track may not exactly be the true endpoint. These events may still have kaon inelastic scatters in the fiducial volume. However, these events may underestimate or overestimate the number of incident slices of the beam particle, biasing the results. These tracks are called either broken or extended tracks. An additional uncertainty term was introduced to address the miscounting of incident slices from inaccurate vertex positions, changing the “flux” in a thin slice measurement. The uncertainty alters the frequency of reconstructed tracks with displacements larger than 5 cm away from the true endpoint along the axis of the detector length. A 100% uncertainty on their frequencies in simulation is assumed for both broken and extended tracks, as data-driven constraints cannot be provided on vertexing. 





The fiducial volume is chosen to minimize the impact of the tracking inefficiency. However, the TPC track reconstruction may still have discrepancies in performance between data and simulation not covered by the space charge effect systematic uncertainty. Therefore, an uncertainty of 6% is applied to events without a TPC track, which is a conservative value from the measurements of the efficiency for selecting a beam particle in Reference <cit.>.


Table <ref> shows the ±1σ shifts for data from the 6 GeV/c beamline setting. Table <ref> reveals the same shifts for data from the 7 GeV/c beamline setting. The dominant uncertainties are the extended track uncertainty and the uncertainty on the Geant4 model used in the simulation. The former can be improved with in-depth vertexing studies on how the reconstruction delineates vertices and how much energy is required to create a vertex or stitch the parent and secondary. The latter can be improved with the reduction of the background of secondary kaons reconstructed as the beam particle as the uncertainty alters the frequency of all kaons in simulation, even those in events where the background is selected by the TPC track reconstruction. There is currently no known way to reduce the TPC track selection choosing a secondary kaons, as the dE/dx would be nearly identical to that of beam kaons.



§ CONCLUSIONS


This paper describes a measurement of the total inelastic cross section of positively-charged kaons on argon with the ProtoDUNE-SP detector using the thin slice method <cit.>. This measurement was done with data taken at the H4-VLE at the CERN Neutrino Platform. A simple event selection achieved a purity of approximately 85-90% between kinetic energies of 4.5 and  7.0 GeV (Table <ref>). The results, at around 380 mbarns, have a precision of approximately 14%, according to Table <ref> and Table <ref>. The total uncertainty almost entirely comes from systematic uncertainties addressing the modeling of the detector and uncertainties regarding the kaon cross-section model used in Geant4. The measurements translate to Geant4 overestimating the cross section by 16% and GENIE overestimating the cross section by 19%. For the 6 GeV/c data sample, the reduced chi-square statistic measured is 11.26/4 bins with Geant4 and 13.33/4 bins with GENIE hA2018, suggesting tension with both models.


Future studies can utilize the results for tuning kaon reaction scattering models incorporated in various hadron interaction event generators. The upcoming ProtoDUNE Horizontal Drift detector will also exist in the same detector hall with a wire-based readout and can measure similar cross sections with nearly identical methods. It may also run with the beam polarity reversed, allowing for a cross-section analysis of negatively-charged kaons. 










§ ACKNOWLEDGEMENTS






The ProtoDUNE-SP detector was constructed and operated on the CERN Neutrino Platform.
We gratefully acknowledge the support of the CERN management, and the
CERN EP, BE, TE, EN and IT Departments for NP04/Proto­DUNE-SP.



This document was prepared by the DUNE collaboration using the
resources of the Fermi National Accelerator Laboratory 
(Fermilab), a U.S. Department of Energy, Office of Science, 
HEP User Facility. Fermilab is managed by Fermi Research Alliance, 
LLC (FRA), acting under Contract No. DE-AC02-07CH11359.



This work was supported by
CNPq,
FAPERJ,
FAPEG and 
FAPESP,                         Brazil;
CFI, 
IPP and 
NSERC,                          Canada;
CERN;
MŠMT,                       Czech Republic;
ERDF, 
H2020-EU and 
MSCA,                           European Union;
CNRS/IN2P3 and
CEA,                            France;
INFN,                           Italy;
FCT,                            Portugal;
NRF,                            South Korea;
CAM, 
Fundación “La Caixa”,
Junta de Andalucía-FEDER,
MICINN, and
Xunta de Galicia,               Spain;
SERI and 
SNSF,                           Switzerland;
TÜBİTAK,                    Turkey;
The Royal Society and 
UKRI/STFC,                      United Kingdom;
DOE and 
NSF,                            United States of America.





§ DISTRIBUTIONS OF THE 7 GEV/C BEAM EVENT SELECTIONS


This appendix contains the distributions for the 7 GeV/c samples for the event selection in simulation and data. The distribution of tracks without the event selection are shown in Figure <ref>. The distribution for all selected kaons and selected kaons with interactions in the fiducial volume are shown in Figure <ref>. The initial beamline kinetic energy distributions for all selected beam kaons at this beamline setting, as measured by the beamline instrumentation, is shown in Figure <ref>.

All distributions show agreements with similar distributions in the 6 GeV/c sample, as seen in Figure <ref>. Furthermore, similar purities and slightly lower efficiencies in incident and interacting slice distributions can be observed in Figure <ref>.

