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Advances in High Energy Physics
Volume 2019, Article ID 2526897, 11 pages
https://doi.org/10.1155/2019/2526897
Research Article

Nonperturbative Uncertainties on the Transverse Momentum Distribution of Electroweak Bosons and on the Determination of the Boson Mass at the LHC

1Dipartimento di Fisica, Università di Pavia, Via Bassi 6, I-27100 Pavia, Italy
2INFN, Sezione di Pavia, Via Bassi 6, I-27100 Pavia, Italy
3Physics Division, Argonne National Laboratory, 9700 S. Cass Avenue, Lemont, IL 60439, USA

Correspondence should be addressed to Andrea Signori; vog.lna@irongisa

Received 10 December 2018; Accepted 14 February 2019; Published 17 March 2019

Guest Editor: Daniel Pitonyak

Copyright © 2019 Giuseppe Bozzi and Andrea Signori. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Abstract

In this contribution we present an overview of recent results concerning the impact of a possible flavour dependence of the intrinsic quark transverse momentum on electroweak observables. In particular, we focus on the spectrum of electroweak gauge bosons produced in proton-proton collisions at the LHC and on the direct determination of the boson mass. We show that these effects are comparable in size to other nonperturbative effects commonly included in phenomenological analyses and should thus be included in precise theoretical predictions for present and future hadron colliders.

1. Introduction

Electroweak precision observables are interesting benchmarks to test the limits of the Standard Model and to discriminate between different scenarios for new physics. The mass of the boson, , is an example of such an observable.

The Standard Model prediction for the boson mass from the global fit of the electroweak parameters ( GeV) [1] has a very small uncertainty that represents a natural target for the precision of the experimental measurements of at hadron colliders.

Direct measurements of at hadronic colliders have been performed at the Tevatron collider with the D0 [2] and CDF [3] experiments and at the LHC collider with the ATLAS [4] experiment, with a total uncertainty of 23 MeV, 19 MeV, and 19 MeV, respectively. The current world average, based on these measurements and the ones performed at LEP, is  GeV [5]. Figure 1 presents an overview of these measurements compared to the electroweak global fit. The CPT theorem [6, 7] implies that the mass and lifetime of a particle and its antiparticle are the same. The ATLAS measurement of the and mass difference yields  MeV [4]. The experimental determinations are based on a template-fit procedure applied to differential distributions of the decay products: in particular, the transverse momentum of the final lepton, , the transverse momentum of the neutrino (only at the Tevatron), and the transverse mass of the lepton pair (where , with being the azimuthal angles of the lepton and the neutrino, respectively). The transverse momentum of the lepton pair, though not directly used in the template-fit procedure, is relevant for reweighing purposes (see, for instance, Sec. 6 of Ref. [4]).

Figure 1: Overview of the measurements of the boson mass. The indirect determination via the electroweak fit sets the precision for the measurements via direct determinations. Figure from Ref. [4].

At leading order the boson is produced with zero transverse momentum (), but perturbative and nonperturbative corrections give rise to nonvanishing values of . While perturbative and flavour-independent nonperturbative corrections have received much attention and reached a high level of accuracy (see, for instance, Ref. [8, 9] and references therein), a possible flavour dependence of the intrinsic transverse momentum () of the initial state partons has been less investigated.

In Figure 2, we examine the decomposition in flavour channels of the cross section for and production differential with respect to , . A nontrivial interplay among the different flavours and the gluon is observed. The role of the gluon becomes increasingly important at larger values of the transverse momentum. In the region of the peak, instead, the dominant channels involve combinations of , , , and (where and ). For this reason, we consider it important to study the impact of flavour-dependent effects on the production of electroweak bosons and on the determination of .

Figure 2: From top to bottom: the decomposition in flavour channels of the cross section for , , production differential with respect to the transverse momentum of the produced electroweak boson , . The rapidity and the collinear momentum fractions have been integrated over the kinematically allowed ranges. The cross section is calculated by means of CuTe [12] at LHC  TeV. The nonperturbative correction is implemented as a flavour-independent Gaussian smearing, governed by the parameter (see [12] and the Appendix). The channels add to one.

In this contribution we give an overview of selected studies related to flavour-dependent effects, focusing in particular on the results obtained in [10, 11], showing that they can be nonnegligible compared to other sources of theoretical uncertainty and should thus be included in precision physics programs at hadron colliders.

2. Formalism

In processes with a hard scale and a measured transverse momentum , for instance, the mass and the transverse momentum of an electroweak boson produced in hadronic collisions, we can distinguish three regions: a small region (), where large logarithms of have to be properly resummed; a large region (), where fixed-order perturbation theory provides reliable results; and an intermediate region, where a proper matching procedure between all-order resummed and fixed-order contributions is necessary. For a concise discussion (and for additional relevant references) on the development of the different frameworks available to resum the logs of and on their matching to fixed-order perturbative calculations, we refer the reader to [1825].

In the Transverse-Momentum-Dependent (TMD) factorisation framework [26], the unpolarized TMD Parton Distribution Function (TMD PDF) for a parton with flavour , carrying a fraction of longitudinal momentum at a certain scale , can be written in -space (where is the variable Fourier-conjugated to the partonic transverse momentum ) aswhere is the -dependent scale at which the collinear parton distribution functions are computed and is a hadronic mass scale. Equation (1) is a generic schematic implementation of the perturbative and nonperturbative components of a renormalized TMD PDF. Depending on the chosen perturbative accuracy, includes the UV-anomalous dimension of the TMD PDF and the Collins-Soper kernel. Also, in principle the TMD PDF depends on two kinds of renormalization scales, related to the renormalization of UV and light-cone divergences. Here we specify their initial and final values as and , respectively. Moreover, the perturbative scales can be chosen in position or momentum space [12, 14, 2730]. For the implementation of all these details, we refer the reader to the description of the public codes that we are going to discuss.

The coefficients in (1), also called Wilson coefficients for the TMD distribution, are calculable in perturbation theory and are presently known at order in the unpolarized case [21, 31, 32]. They are convoluted with the corresponding collinear parton distribution functions according to

The perturbative part of the evolution, the factor in (1), can be written asIt involves, in principle, the UV-anomalous dimension and the Collins-Soper kernel , which can be decomposed asThe and coefficients are known up to NNNLL (at least, their numerical value) and the integration of the Sudakov exponent in (4) can be done analytically up to NNNLL (for the complete expressions see, e.g., [3335]). The perturbative coefficients of the kernel are also known analytically up to NNNLL.

A well-known problem in the implementation of the QCD evolution of transverse-momentum-dependent distributions (TMDs) is the divergent behaviour at large caused by the QCD Landau pole. Two common prescriptions to deal with this divergence consist in replacing with a variable that saturates at a certain , as suggested by the CSS formalism [26, 36], or perform the integration on the complex plane in such a way that the Landau pole is never reached [37]. On the other hand, also the small region needs to be regularized, in order to eliminate unjustified contributions from the evolution of TMDs in the intermediate and large regions and to recover the expression for the cross section in collinear factorisation upon integration over . Several prescriptions exist [14, 20, 33, 38, 39] also in this case. In CuTe the cross section is calculated integrating on the real axis and the initial value for the renormalization scale is , where the scale screens the cross section from receiving long-distance contributions. The definition of is given analytically in (3.2) in [12] and its numerical value is  GeV. Also in DyRes the cross section is calculated integrating on the real axis. The initial scale is defined as [28] and a freezing prescription is given for in order to avoid the Landau pole; see in [28]. The value of the cutoff parameter in space is a function of the renormalization scale and the resummation scale : . In our case, . In DyqT, instead, the integration over is performed using the complex-b prescription [37]; thus no freezing parameter is needed and is just defined as [27].

Two intrinsically nonperturbative factors are introduced in (1) in order to account for the large behavior. The first one is named in the TMD/CSS literature [26]. It embodies the nonperturbative part of the evolution, which is flavour-independent. The second one, , accounts for a flavour-(in)dependent intrinsic transverse momentum of the parton with flavour . We note that, in principle, this contribution can be also -dependent (see, e.g., [15]), but in this treatment we choose to neglect this feature. The and are (vectors of) nonperturbative parameters that can be fit to data. The parameters are related to the quantity . For example, in case of a simple Gaussian functional form, , we have . For both the nonperturbative factors and , several implementations have been discussed; see, e.g., [14, 23] and references therein. In particular, a kinematic- and flavour-dependent Gaussian parametrisation has been proposed in [15, 29].

The studies that we discuss make use of three different computational tools: CuTe [12], DyqT [27], and DYRes [28]. CuTe is based on resummed expressions calculated using soft-collinear effective theory (SCET). It gives the transverse momentum spectrum of on-shell electroweak bosons up to NLO ()) accuracy in the Wilson coefficients and up to NNLL in the Sudakov exponent (CuTe is labelled NNLL in the SCET language but NNLL′ in standard pQCD language. The accuracy of NNLL′ is considered lower than that of the full NNLL, in which Wilson coefficients are computed at NNLO).

DyqT and DYRes are based on [27, 28] and perform soft gluon resummation in -space. The first computes the spectrum of an electroweak boson produced in hadronic collisions. The second also provides the full kinematics of the vector boson and of its decay products, allowing for the application of arbitrary cuts on the final-state kinematical variables and giving differential distributions in form of bin histograms. The accuracy of both codes is up to NNLL in the resummed part and up to NLO () at large .

A simple Gaussian parametrisation of the nonperturbative effects is present in these codes, as in most of the computational tools used to analyse the electroweak observables relevant for the determination of the boson mass. A single nonperturbative parameter, , usually encodes both the (flavour-independent) effect of and the distribution in the (potentially flavour-dependent) intrinsic transverse momentum (ResBos[40] is a counter-example, but it does not account for the flavour dependence of the intrinsic transverse momentum):

The values of the nonperturbative parameters used in fitting the boson mass are usually obtained through fits on production data [40], for which the relevant partonic channels are of the type , and then used to predict production, despite the process being sensitive to different partonic channels, . This procedure essentially neglects any possible flavour dependence of the intrinsic partonic transverse momentum.

In order to introduce the flavour dependence, one can simply decompose in the LHS of (5) into the sum , where the flavour indices span the range (the subscripts referring to the valence and sea components, respectively), additionally disentangling the nonperturbative contribution to the evolution and the intrinsic transverse momentum distribution. Thus, for each parton with flavour , the nonperturbative contributions and in (1) and (5) are included in the corresponding term in the flavour sum of the TMD factorisation formula. More details regarding the nonperturbative parameters in the codes under consideration have been collected in the Appendix.

3. Effects on the Spectrum of the

The impact of a flavour-dependent intrinsic on the spectrum of the electroweak bosons has been first studied in [10] and here we partly summarize the findings therein. Part of the analysis is devoted to the shifts induced in the position of the peak for the distribution in , and . Flavour-independent (f.i.) and flavour-dependent (f.d.) variations of the average intrinsic transverse momentum squared are considered, together with the uncertainties associated to other nonperturbative factors, such as the collinear PDFs, the renormalisation scale, and the value of the strong coupling constant. As justified in Section 1 and Figure 2, it is assumed that the intrinsic transverse-momentum depends on five flavours only: , where collectively refers to the strange, charm, and bottom quarks and to the gluon.

The numerical results are obtained by means of a modified (i.e., flavour-dependent) version of CuTe [12]. Namely, the nonperturbative parameter (see the Appendix), which corrects the whole cross section at large , is split into a sum of two flavour-dependent nonperturbative contributions, , such that . This decomposition reabsorbs the nonperturbative contribution to QCD radiation into . The flavour dependence of is compatible with the ratios fitted in [15]. The goal is to combine flavour dependent parameters in such a way to respect the values of fitted on the data, generating at the same time different values to be used in the calculation of the differential cross section for (we refer the reader to [10] for the precise values of used in the study).

The shifts (quantified in GeV) induced by different perturbative and nonperturbative contributions are summarized in Table 1. The renormalisation scale is varied between and , with , where is the cutoff introduced in the Cute to avoid the Landau pole [12]. The scale in the hard part has not been varied. Regarding the impact of the collinear PDFs, the result shown in the table is the smallest interval which contains or of peak positions, computed for every member of the NNPDF3.0 set [41]. The strong coupling is varied by from the central value of 0.118.

Table 1: Summary of the shifts in GeV induced on the peak position in spectra of , generated by different effects. “f.i.” stands for flavour-independent, whereas “f.d.” for flavour-dependent. “Max ” effect indicates the maximum shift induced on the peak position of the spectrum by flavour-dependent variations of that keep the peak of the spectrum unchanged. For the values of the flavour-dependent non-perturbative parameters we refer the reader to [10].

The shift induced in the peak position from flavour-dependent is smaller than that induced by scale variation, variation, and flavour-independent , but comparable in magnitude. It is also bigger than the uncertainty from the PDF set, which is the only other uncertainty where the shifts are not almost perfectly correlated between the three vector bosons. With flavour-dependent variations of , the peaks of the and distributions shift in different directions. Since the parameters are selected under the constraint that the -distribution is left unchanged (see Table 1), the channels for and move in different directions. The anticorrelation of the shifts between and is a peculiarity of the uncertainty generated by flavour-dependent variations of the intrinsic . This means that the uncertainty stemming from the nonperturbative hadron structure in the transverse plane can affect the determination of and in different ways. Indeed, this feature nicely emerges in the analysis summarized in Section 4.

The analysis in [10] thus shows that the uncertainty on the peak position for bosons arising from the flavour dependence of the intrinsic transverse momentum is not negligible with respect to the other sources of theoretical uncertainties and comparable in magnitude with the uncertainties due to the collinear PDFs.

We now analyse the ratios of the -differential cross section calculated with a flavour-independent set of nonperturbative parameters in over the same cross section calculated with flavour-dependent parameters. The results are presented in Figure 3 for , , . The calculation has been performed by means of a flavour-dependent modification of DyqT, where the nonperturbative contributions in (1) have been coded asThe values for , , are taken from [14] and the flavour-dependence in is inspired to the flavour ratios in [15]. The curves in Figure 3 correspond to 50 sets of flavour-dependent nonperturbative parameters built according to these criteria. The chosen perturbative accuracy is NLL [11, 13] and the collinear PDF set used is NNPDF3.1 [42].

Figure 3: In these figures the ratio is plotted for the three different electroweak bosons (, , , respectively), with a single set of flavour-independent (f.i.) nonperturbative parameters in the transverse part of the TMD PDFs and 50 different flavour-dependent (f.d.) sets of the same parameters. The analysis has been performed at NLL [11, 13]. The values of the nonperturbative parameters have been chosen from the results in [14, 15].

As predicted by the TMD formalism, the effect induced by the non-perturbative corrections is more evident at low . In particular, it is stronger for  GeV but sizable up to  GeV. The flavour dependence of the intrinsic transverse momentum can modify the shape of by at very low transverse momentum. This observable affects the cross section differential with respect to the kinematics of the final state particles, namely, the distributions in , , , and thus has an impact also on the determination of the boson mass.

4. Impact on the Determination of the Boson Mass

As previously mentioned, the measurements of at hadron colliders rely on a template-fit procedure performed on selected observables, i.e., the distributions in the transverse mass of the lepton pair and the lepton/neutrino transverse momentum. Both CDF and D0 experiments at Tevatron use data from all the three observables. In the ATLAS case, however, the transverse momentum of the (anti)neutrino is used for consistency checks only, since it is affected by larger uncertainties with respect to and .

In this section we consider selected results concerning the estimate of the uncertainties of nonperturbative origin on the determination of . In particular, we focus on shifts of the mass induced by possible configurations for the flavour dependence of the intrinsic transverse momentum, and we will compare them with the corresponding shifts generated by the uncertainties in the collinear PDFs.

In the template-fit procedure, several histograms are generated with a specific theoretical accuracy and description of detector effects, letting the fit parameter(s) (only , in this case) vary in a range: the histogram best describing the experimental data selects the measured value for . The details of the theoretical calculations used to compute the templates (the choice of the scales, of the collinear PDFs, of the perturbative order, the resummation of logarithmically enhanced contributions, the nonperturbative effects, etc.) affect the result of the fit and define the theoretical systematics.

This procedure can also be used to estimate the effect of each single theoretical uncertainty, by generating sets of pseudodata (with the same event generator used for the templates, but at a lower statistics) differing by the value of the parameter(s) controlling that uncertainty [43, 44]. Figure 4 contains a graphical illustration of the flowchart for the template-fit procedure, specified to the comparison of one set of pseudodata generated with flavour-dependent parameters with 30 templates generated with one set of flavour-independent parameters and 30 values of ( MeV with steps of 1 MeV). This method has been also used to estimate the shift in induced by the variation of the collinear PDF set in fitting the transverse mass [16, 45] and the lepton [17, 45] both at Tevatron and at the LHC in the central rapidity region of the produced electroweak boson ( for Tevatron and for the LHC). A similar study dedicated to LHCb and its forward acceptance has been performed in [46].

Figure 4: Flowchart for a template-fit procedure to estimate shifts in induced by the flavour dependence of the intrinsic quark transverse momentum.

In the transverse mass case, the total error (envelope) induced by three different PDF sets (CTEQ6.6 [47], MSTW2008 [48], and NNPDF2.1 [49]) is less than 10 MeV both at the Tevatron and at the LHC [16]. The results are shown in the left plot of Figure 5. The analysis has been performed at fixed-order NLO QCD ()), thus without all-order resummation, since the -shape is mildly sensitive to soft gluon emission from the initial state. The key factor in reducing the PDF uncertainty is the use of normalised differential distributions in the fitting procedure, in such a way to eliminate normalisation effects which are irrelevant for .

Figure 5: Shifts induced on by the choice of different PDF sets, obtained through a template-fit performed on the transverse mass (left) and the lepton (right) observables (left figure from [16], right figure from [17]).

A similar analysis applied to the lepton observable reveals a much larger error due to PDF variations (CT10 [50], MSTW2008CPdeut [48], MMHT2014 [51], NNPDF2.3 [52], and NNPDF3.0 [41]), as shown in the right plot of Figure 5. While the individual sets provide nonpessimistic estimates (), the distance between the best predictions of the various sets ranges between 8 and 15 MeV, and the total envelope ranges between 16 and 32 MeV (depending on the collider, the energy, and the final state) [16]. While soft gluon emission already provides a nonvanishing transverse momentum, additional contributions may come from the intrinsic transverse momentum of the colliding partons. The study of the impact of a possible flavour-dependent intrinsic on the determination of has been first performed in [11], using the same template-fit procedure described above and sketched in Figure 4, performed with modified versions of the DYqT [27] and DYRes [28] codes. In this case, the pseudodata are built with the Gaussian widths associated with the different flavours in (6).

In order to estimate the impact of the flavour dependence, it is necessary to first identify those sets of flavor-dependent parameters which perform equally well in describing the boson -spectrum, despite having potentially a very different flavor structure. This is motivated physically by the fact that the boson is produced from a pair whereas the bosons are produced from pairs, with . For this reason, the is less sensitive to the flavor structure with respect to the , and there might be flavor combinations that perform equally well in describing the -spectrum of the but produce very different results when applied to the case of . We identify as “-equivalent” those sets of flavor-dependent parameters in agreement with the transverse momentum distribution measured at hadron colliders [11]. To this extent(i)a single flavour-independent (i.e., using a version of Eq. (6) without -dependence) -spectrum for the boson is produced based on the parameters presented in Ref. [14];(ii)each bin of this flavour-independent spectrum is assigned an uncertainty equal to the one quoted by the CDF and ATLAS experiments, which includes statistical and systematic components, neglecting the correlations;(iii)several flavour-dependent sets for in Eq. (6) are generated randomly within a variation range consistent with the information obtained in previous TMD fits (in particular, taking into account the estimate for the flavour-independent contribution to the non-perturbative part of the evolution obtained in Ref. [14]);(iv)a flavour-dependent set is defined “-equivalent” if the associated spectrum for the has a with respect to one generated by the flavour-independent set.

We note that the boson data alone are not able to discriminate between flavor-independent and flavor-dependent sets of nonperturbative parameters. Data from flavor-sensitive processes are needed, in particular from SIDIS.

The flavour-dependent sets for CDF and ATLAS who pass this filter are treated as the pseudodata of the template-fit procedure, while the flavour-independent one is used for the generation of the templates at high statistics. The number of events corresponds to 135M for the pseudodata and 750M for the templates. Only 9 sets out of the 30 ones which are “-equivalent” both with respect to CDF and ATLAS uncertainties have been investigated. The values of the flavour-dependent parameters for each set are given in Table 2. A summary of the shifts obtained through this procedure is given in Table 3.

Table 2: Values of the parameter in (6) for the flavours . Units are GeV2.
Table 3: Shifts in (in MeV) induced by the corresponding sets of flavour-dependent intrinsic transverse momenta outlined in Table 2 (statistical uncertainty: 2.5 MeV).

The statistical uncertainty of the template-fit procedure has been estimated by considering statistically equivalent those templates for which . Overall, the quoted statistical uncertainty on the results in Table 3 is ±2.5 MeV.

Being the transverse mass mildly sensitive to the modeling of the transverse momentum, the corresponding shifts are compatible with zero considering the statistical uncertainty of the template-fit procedure. On the contrary, in the case the shifts can be incompatible with statistical fluctuations and are comparable to the ones induced by collinear PDFs, with an envelope of 15 MeV in the case of production and 11 MeV for production. We also notice a hint of a possible anti-correlation between the shifts in the and cases, as it was also noticed in Section 3.

Along this line, we also stress that ATLAS measured  MeV [4]. From Table 3, we can infer that part of the discrepancy between the mass of the and the can be artificially induced by not considering the flavour structure in transverse momentum. For example, the sets 1 and 2 in Table 2 feature (induced by ). This implies that for templates built with sets 1 and 2, instead of flavour-independent values, the difference between the two masses would be reduced. An opposite result would be obtained if building templates with flavour-dependent sets for which (e.g., sets 3 and 5, for the case).

5. Outlook and Future Developments

The selected results presented in this contribution point out that the impact of a possible flavour dependence of the intrinsic partonic transverse momentum should not be neglected, even in the kinematic region where nonperturbative effects are expected to be small [5355], such as for electroweak boson production at the LHC.

This kind of uncertainty directly affects the electroweak observables relevant for the measurement of : the transverse momentum distribution for the and the decay lepton and the transverse mass distribution of the lepton pair. The numerical results presented in Sections 3 and 4 indicate that flavour-dependent effects are comparable in size to other uncertainties of (non-)perturbative origin (for example, the choice of collinear PDF set). Thus, a flavour-blind analysis is not a sufficiently accurate option for a program of precision electroweak measurements at the LHC and at future colliders.

Moreover, in hadron colliders at a lower energy such as RHIC and a possible fixed-target experiment at the LHC, the non-perturbative effects can play an even more significant role (due to the larger -values probed) and affect the study of polarised TMDs [56] and the structure of the light sea quarks [57].

A detailed knowledge of TMD distributions is thus important, not only for nucleon tomography beyond the collinear picture [5866], but also to constrain fundamental parameters of the Standard Model, thus providing a direct connection between hadron physics and the high-energy phenomenology.

In light of these results, we call for improved investigations of the impact of nonperturbative effects linked to the hadron structure at hadron colliders and for the inclusion of these effects in the event generators employed in experimental and theoretical investigations of high-energy physics.

Appendix

Conventions for Nonperturbative Parameters

For convenience, we collect in this Appendix the naive translation of the nonperturbative parameters used in the numerical codes cited in the text. In the conventions of [15, 29], the nonperturbative parameters appear asIn CuTe [12] there is a single nonperturbative parameter entering the cross section:The same happens in DyqT [27] and DYRes [28], in terms of the nonperturbative parameter :We obtain the parameter employed in CuTe asand similarly for the DYqT parameter:The default value for discussed in [12] is 0.60 GeV2, whereas the conservative value for discussed in [27] is 0.8 GeV2 and in [28] is 1.2 GeV2.

Data Availability

The data used to support the findings presented in this study (authors: G. Bozzi, A. Signori) have been produced by means of the following public codes: (1) CuTe: https://cute.hepforge.org/ (2) DyqT: http://pcteserver.mi.infn.it/~ferrera/dyqt.html (3) DyRes: http://pcteserver.mi.infn.it/~ferrera/dyres.html. The data are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

We thank Alessandro Bacchetta and Marco Radici for the fruitful collaboration on this topic, Alessandro Vicini for many suggestions and discussions, Chao Shi for carefully reading the manuscript, and Piet Mulders and Mathias Ritzmann for contributing to the investigations summarized in this article. Andrea Signori acknowledges support from the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Contract no. DE-AC02-06CH11357. Giuseppe Bozzi acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement no. 647981, 3DSPIN).

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