Tuning PYTHIA 8.1: the Monash 2013 tune
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Abstract
We present an updated set of parameters for the PYTHIA 8 event generator. We reevaluate the constraints imposed by LEP and SLD on hadronization, in particular with regard to heavyquark fragmentation and strangeness production. For hadron collisions, we combine the updated fragmentation parameters with the new NNPDF2.3 LO PDF set. We use minimumbias, Drell–Yan, and underlyingevent data from the LHC to constrain the initialstateradiation and multipartoninteraction parameters, combined with data from SPS and the Tevatron to constrain the energy scaling. Several distributions show significant improvements with respect to the current defaults, for both \(ee\) and \(pp\) collisions, though we emphasize that interesting discrepancies remain in particular for strange particles and baryons. The updated parameters are available as an option starting from PYTHIA 8.185, by setting Tune:ee = 7 and Tune:pp = 14.
1 Introduction
A truly impressive amount of results on QCD has been produced by the first run of the LHC. Most of these are already available publicly, e.g. via the data preservation site HEPDATA [1]. A large fraction has also been encoded in the analysis preservation tool RIVET^{1} [2]. Especially in the area of soft QCD, many of the experimental results have spurred further modelling efforts in the theory community (nice summaries of some of the current challenges can be found in [3, 4]), while there is also significant activity dedicated to improving (“tuning”) the parameters of the existing models to better describe some or all of the available new data (see, e.g., the recent review in [5]).
The PYTHIA event generator [6, 7] has been extensively compared to LHC data, and several tuning efforts have already incorporated data from Run 1 [5, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, in particular for the newest version of the model, PYTHIA 8 [7], it has been some time since the constraints imposed by \(ee\) colliders were revised (in 2009), and then only via an undocumented tuning effort (using the PROFESSOR tool [17]). One of the main aims of this paper is therefore first to take a critical look at the constraints arising from LEP, SLD, and other \(e^+e^\) experiments, reoptimize the finalstate radiation and hadronization parameters, and document our findings. We do this manually, rather than in an automated setup, in order to better explain the reasoning behind each parameter adjustment. This writeup is thus also intended to function as an aid to others wishing to explore the PYTHIA 8 parameter space.
We then consider the corresponding case for hadron colliders, and use the opportunity to try out a new PDF set, an LO fit produced by the NNPDF collaboration [18, 19, 20] which has recently been introduced in PYTHIA 8 (NLO and NNLO sets are also available, for people that want to check the impact of using LO vs (N)NLO PDFs in hardscattering events). In a spirit similar to that of the socalled “Perugia tunes” of PYTHIA 6 [8, 21], we choose the same value of \(\alpha _s(M_Z)=0.1365\) for both initial and finalstate radiation. (Though we do regard this choice as somewhat arbitrary, it may facilitate matching applications [21].) Again, we adjust parameters manually and attempt to give brief explanations for each modification. We also choose the \(\alpha _s(M_Z)\) value for hardscattering matrix elements to be the same as that in the PDFs, here \(\alpha _s(M_Z)=0.13\). (The difference between the value used for radiation and that used for hardscattering MEs may be interpreted as an artifact of translations between the CMW and \(\overline{{\mathrm {MS}}}\) schemes, see Sect. 3.3.)
Below, in Sect. 1.1, we begin by giving a brief general explanation of the plots and \(\chi ^2\) values that are used throughout the paper. Next, in Sect. 2, we describe the physics, parameters, and constraints governing fragmentation in hadronic \(Z\) decays (finalstate radiation and string fragmentation). We turn to hadron colliders in Sect. 3 (PDFs, initialstate radiation, and multiparton interactions). We then focus on the energy scaling between different \(ee\) and \(pp\) (\(p\bar{p}\)) collider energies in Sect. 4, including in particular the recently published highstatistics data from the Tevatron energy scan from 300 to 1960 GeV [22, 23]. We round off with conclusions and a summary of recommendations for future efforts in Sect. 5.
A complete listing of the Monash 2013 tune parameters is given in Appendix A. Appendix B contains a few sets of additional plots, complementing those presented in the main body of the paper.
1.1 Plot legends and \({\mathbf {\chi ^2}}\) values
In several places, we have chosen to use data sets/constraints that differ from the standard ones available e.g. through RIVET (as documented below). Since our tuning setup is furthermore manual, rather than automated, we have in fact not relied on RIVET in this work (though we have made extensive use of HEPDATA [1]). Instead, we use the VINCIAROOT plotting tool [24], which we have here upgraded to include a simple \(\chi ^2\) calculation, the result of which is shown on each plot.
2 Finalstate radiation and hadronization
Theoretically, a set of formally subleading terms can be resummed by using 2loop running of \(\alpha _s\) in the socalled MC (a.k.a. CMW) scheme [28]. However, in a leadingorder code like PYTHIA, this produces too little hard radiation in practice, due to missing NLO “K” factors for hard emissions (see, e.g., the study of NLO corrections in [29]). Empirically, we find that a better overall description is achieved with oneloop running, which, for a fixed value of \(\Lambda _{{\mathrm {QCD}}}\), can effectively mimic the effect of missing \(K\) factors via its relatively slower pace of running, leading to values of \(\alpha _s(M_Z)\) in the range \(0.135{}0.140\), consistent with other LO extractions of the same quantity. (See [29] for an equivalent extraction at NLO.)
For this study, we did not find any significant advantage in reinterpreting this value in the CMW scheme^{3} and hence merely settled on an effective \(\alpha _s(M_Z) = 0.1365\) (to be compared with the current default value of \(0.1383\)).
The resulting distribution of the Thrust eventshape variable was shown in Fig. 1, comparing the Monash 2013 tune to the current default tune and to an alternative contemporary tune by Fischer [30]. To avoid clutter, the other eventshape variables (\(C\), \(D\), \(B_W\), and \(B_T\)) are collected in Appendix B.1. There are no significant changes to any of the lightflavour tagged event shapes in our tune as compared to the current default one.
2.1 Lightflavour fragmentation
Given a set of postshower partons, resolved at a scale of \(Q_{\mathrm {had}}\sim \) 1 GeV, the nonperturbative stage of the fragmentation modeling now takes over, to convert the partonic state into a set of onshell hadrons. In the leadingcolour approximation, each perturbative dipole is dual to a nonperturbative string piece [31]. Quarks thus become string endpoints, while gluons become transverse kinks, connecting two string pieces [32]. The Lund string fragmentation model [33] describes the fragmentation of such string systems into onshell hadrons.
This value is obtained essentially from the first two bins of the Thrust distribution, Fig. 1, and from the bins near zero of the other event shapes, see Appendix B.1. Note that the \(\sigma _{\perp }\) value is interpreted as the width of a Gaussian distribution in the total \(p_{\perp }\) (measured transversely to the local string direction, which may differ from the global event axis), such that each of the \(p_x\) and \(p_y\) components have a slightly smaller average value, \(\sigma _{x,y}^2 = \frac{1}{2}\sigma _{\perp }^2 = (0.237\,{{\mathrm {GeV}}})^2\). Also note that each nonleading hadron will receive two \(p_{\perp }\) kicks, one from each of the breaks surrounding it, hence \(\left\langle p_{\perp {{\mathrm {had}}}}^2\right\rangle = 2 \sigma _{\perp }^2 = (0.474\,{{\mathrm {GeV}}})^2\).
The \(a\) and \(b\) parameters govern the shape of the fragmentation function, and must be constrained by fits to data. Eq. (3) expresses the most general form of the fragmentation function, for which the \(a\) parameters of the original stringendpoint quark, \(a_i\), and that of the (anti)quark produced in the string break, \(a_j\), can in principle be different, while the \(b\) parameter is universal. Within the Lund model, the \(a\) value is normally also taken to be universal, the same for all quarks, with the only freedom being that a larger \(a\) parameter can be assigned to diquarks [36], from which baryons are formed, and hence meson and baryon spectra can be decoupled somewhat. (See StringZ:aExtraDiquark below.)
Roughly speaking, large \(a\) parameters suppress the hard region \(z\rightarrow 1\), while a large \(b\) parameter suppresses the soft region \(z\rightarrow 0\). By adjusting them independently, both the average hardness and the width of the resulting fragmentation spectra can be modified. For example, increasing both \(a\) and \(b\) yields a narrower distribution, while changing them in opposite directions moves the average.
Both of the earlier tunes exhibit a somewhat too broad multiplicity distribution in comparison with the L3 data. The relatively large Lund \(a\) and \(b\) values used for the Monash tune, combined with its large \(\sigma _{\perp }\) value, produce a narrower \(n_{{\mathrm {Ch}}}\) spectrum, with in particular a smaller tail towards large multiplicities. All the tunes produce a sensible momentum spectrum. The dip around \(\left \ln (x)\right \sim 5.5\) corresponds to the extreme softpion tail, with momenta at or below \(\Lambda _{{\mathrm {QCD}}}\). We did not find it possible to remove it by retuning, since a smaller \(b\) parameter would generate significantly too high particle multiplicities and a smaller \(\sigma _{\perp }\) would lead to conflict with the eventshape distributions.
Further information to elucidate the structure of the momentum distribution is provided by the plot in the righthand pane of Fig. 4, which uses the same \(\left \ln (x)\right \) axis as the righthand plot in Fig. 3 and shows the relative particle composition in the Monash tune for each histogram bin. (The category “Other” contains electrons and muons from weak decays.) An interesting observation is that the relatively harder spectrum of Kaons implies that, for the highestmomentum bins, the charged tracks are made up of an almost exactly equal mixture of Kaons and pions, despite Kaons on average only making up about 10 % of the charged multiplicity.
2.2 Identified particles
Continuing on the topic of identified particles, we note that the extraction of the \(a\) and \(b\) parameters from the inclusive chargedparticle distributions is made slightly more complicated by the fact that not all observed particles are “primary” (originating directly from string breaks); many lowermass particles are “secondaries”, produced by prompt decays of more massive states (e.g., \(\rho \rightarrow \pi \pi \)), whose relative rates and decay kinematics therefore influence the spectra. In the \(e^+e^\) measurements we include here, particles with \(c\tau < 100~\hbox {mm}\) were treated as unstable, hence leading to secondaries. (For completeness, we note that the equivalent standard cut at the LHC is normally \(10~\hbox {mm}\).)
The particle composition in PYTHIA 8 was already tuned to a set of reference values provided by the PDG [39], and the default parameters do reasonably well, certainly for the most copiously produced sources of secondaries. Nonetheless, we have here reoptimized the flavourselection parameters of the stringfragmentation model using a slightly different set of reference data, combining the PDG tables with information provided directly by the LEP experiments via HEPDATA [1]. Based on the level of agreement or disagreement between different measurements of the same particles, we have made our own judgement as to the level of uncertainty for a few of the particles, as follows. (Unless otherwise stated, we use the value from the PDG. Particles and antiparticles are implicitly summed over, and secondaries from particles with \(c\tau < 100~{\mathrm {mm}} \) are included.)

The various LEP and SLD measurements of the \(\phi \) meson rate on HEPDATA are barely compatible. E.g., OPAL [40] reports \(\left\langle n_{\phi }\right\rangle = 0.091 \pm 0.002 \pm 0.003\) while ALEPH [38] quotes \(\left\langle n_{\phi }\right\rangle = 0.122 \pm 0.004 \pm 0.008\), a difference of 30 % with uncertainties supposedly less than 10 %. DELPHI [41] and SLD [42] fall in between. The PDG value is \(\left\langle n_{\phi }\right\rangle = 0.0963 \pm 0.003\), i.e., with a combined uncertainty of just 3 %. We choose to inflate the systematic uncertainties and arrive at \(\left\langle n_{\phi }\right\rangle = 0.101 \pm 0.007\).

For \(\Lambda \) production, we use the most precise of the LEP measurements, by OPAL^{7} [43], \(\left\langle n_\Lambda \right\rangle = 0.374\pm 0.002\pm 0.010\), about 5 % lower than the corresponding PDG value.

For \(\Sigma ^{\pm }\) baryons, we use a combination of the two most recent LEP measurements, by L3 [44] for \(\Sigma ^+ + \overline{\Sigma }^\) and by DELPHI [45] for \(\Sigma ^ + \overline{\Sigma }^+\), for an estimated \(\left\langle n_{\Sigma ^{\pm }}\right\rangle = 0.195 \pm 0.018\), which is roughly 10 % higher than the PDG value.

For \(\Sigma ^0\) baryons, we use the most recent measurement, by L3 [44], \(\left\langle n_{\Sigma ^0}\right\rangle = 0.095 \pm 0.015 \pm 0.013\); this is about 20 % larger than the PDG value. The L3 paper comments on their relatively high value by noting that L3 had the best coverage for lowmomentum baryons, hence smaller modeldependent correction factors.

For \(\Delta ^{++}\) baryons, there are only two measurements in HEPDATA [46, 47], which are mutually discrepant by about \(2\sigma \). The DELPHI measurement is nominally the most precise, but OPAL gives a much more serious discussion of systematic uncertainties. We choose to increase the estimated extrapolation errors of the DELPHI measurement by 50 % and obtain a weighted average^{8} of \(\left\langle n_{\Delta ^{++}}\right\rangle = 0.09 \pm 0.017\), 5 % larger than the PDG value, with a 20 % larger uncertainty.

For \(\Sigma ^*\), the three measurements on HEPDATA [38, 43, 48] are likewise discrepant by \(2\sigma 3\sigma \). We inflate the systematic uncertainties and arrive at \(\left\langle n_{\Sigma ^{*\pm }}\right\rangle = 0.050 \pm 0.006\), which is again 5 % higher than the PDG value, with twice as much uncertainty.

The measurements for \(\Xi ^{\pm }\) are in good agreement [38, 43, 48], with a weighted average of \(\left\langle n_{\Xi ^{\pm }}\right\rangle = 0.0266 \pm 0.0012\), slightly larger than the PDG value.

For \(\Xi ^{*0}\), however, the DELPHI measurement [48] gives a far lower number than the OPAL [43] and ALEPH [38] ones, and the weighted average differs by more than 10 % from the PDG value, despite the latter claiming an uncertainty smaller than 10 %. Our weighted average is \(\left\langle n_{\Xi ^{*0}}\right\rangle = 0.0059\pm 0.0012\).

Finally, for the \(\Omega \) baryon, the DELPHI [49] and OPAL [43] measurements are in agreement, and we use the PDG value, \(\left\langle n_{\Omega }\right\rangle = 0.0016 \pm 0.0003\).
Hadronic \(Z\) decays at \(\sqrt{s}=91.2~\hbox {GeV} \). Measured rates of lightflavour mesons and baryons, expressed as percentages of the average chargedparticle multiplicity, as used in this work. Multiply the numbers by 20.7/100 to translate the percentages to corresponding production rates. Source labels indicate: A (ALEPH), D (DELPHI), L (L3), O (OPAL), S (SLD), P (PDG)
Mesons \(\left\langle n\right\rangle /\left\langle n_{{{\mathrm {Ch}}}}\right\rangle \)  Our reference value (in %)  Our source  Baryons \(\left\langle n\right\rangle /\left\langle n_{{{\mathrm {Ch}}}}\right\rangle \)  Our reference value (in %)  Our source 

\(\pi ^{+}+\pi {}\)  \(82.2 \pm 0.9\)  P  \(p+\bar{p}\)  \(5.07 \pm 0.16\)  P 
\(\pi ^0\)  \(45.5\pm 1.5\)  P  \(\Lambda + \bar{\Lambda }\)  \(1.81 \pm 0.051\)  O 
\(K^++K^\)  \(10.8 \pm 0.3\)  P  \(\Sigma ^{+} + \Sigma ^{} + \bar{\Sigma }^{+} + \bar{\Sigma }^{}\)  \(0.942 \pm 0.087\)  DL 
\(\eta \)  \(5.06 \pm 0.38\)  P  \(\Sigma ^0 + \bar{\Sigma }^0\)  \(0.459 \pm 0.096\)  L 
\(\eta '\)  \(0.73 \pm 0.09\)  P  \(\Delta ^{+\,\!+} + \bar{\Delta }^{\,\!}\)  \(0.434 \pm 0.082\)  DO 
\(\rho ^{+} +\rho ^{}\)  \(11.6\pm 2.1\)  P  \(\Sigma ^{*\,\!+} + \Sigma ^{*\,\!} + \bar{\Sigma }^{*\,\!+} + \bar{\Sigma }^{*\,\!}\)  \(0.242 \pm 0.029\)  ADO 
\(\rho ^0\)  \(5.95 \pm 0.47\)  P  \(\Xi ^{+} + \bar{\Xi }^{}\)  \(0.125 \pm 0.0050\)  ADO 
\(K^{*\,\!+}+K^{*\,\!}\)  \(3.45 \pm 0.28\)  P  \(\Xi ^{*0} + \bar{\Xi }^{*0}\)  \(0.0285 \pm 0.0058\)  ADO 
\(\omega \)  \(4.90 \pm 0.31\)  P  \(\Omega ^ + \bar{\Omega }^+\)  \(0.0077 \pm 0.0015\)  P 
\(\phi \)  \(0.49 \pm 0.035\)  ADOS 
It is interesting, however, to note that all of these spectra indicate, or are at least consistent with, a modelling excess of soft identifiedparticle production below \(\ln (x)\sim 4.5\), corresponding to absolute momentum scales around \(500~\hbox {MeV} \), while we recall that the inclusive \(\ln (x)\) spectrum above showed an underproduction around \(\ln (x) \sim 5.5\). Within the constraints of the current theory model, we have not managed to find a way to mitigate these features while remaining consistent with the rest of the data. Nonetheless, it should be mentioned that these observations could have relevance also in the context of understanding identifiedparticle spectra at LHC, a possibility which to our knowledge has so far been ignored.
2.3 Heavyquark fragmentation

For \(D\) mesons, the average \(D^{\pm }\) rate given in sec. 46 of the PDG (0.175) is equal to the inclusive branching fraction for \(Z\rightarrow D^{\pm } X\) given in the \(Z\) boson summary table in the same Review (after normalizing the latter to the hadronic \(Z\) fraction of \(69.91~\%\) [39]). However, the former ought to be substantially larger given that some \(Z\rightarrow c{\bar{c}}\) events will contain two \(D^{\pm }\) mesons (counting once in the \(Z\rightarrow D^{\pm } X\) branching fraction but twice in the average \(D^{\pm }\) multiplicity). We therefore here use a measurement by ALEPH [54] to fix the \(D^{\pm }\) and \(D^0\) rates, resulting in a reference value for the average \(D^{\pm }\) multiplicity almost twice as large as that given by sec. 46 in the PDG.

For \(\Lambda _c^+\), the average multiplicity given in sec. 46 of the PDG is twice as large as that indicated by the branching fraction \({{\mathrm {BR}}}(Z\rightarrow \Lambda _c^+ X)\) in the \(Z\) boson summary table in the same Review. We here use the branching from the \(Z\) boson summary table as our constraint on the \(\Lambda _c^+\) rate, normalized to the total branching fraction \({{\mathrm {BR}}}(Z\rightarrow {{\mathrm {hadrons}}})\).

We also include the average rate of \(g\rightarrow c{\bar{c}}\) splittings, obtained by combining an ALEPH [55] and an OPAL measurement [56], but with an additional 10 % systematic uncertainty added to both measurements to account for possibly larger mismodeling effects in the correction factors [57, 58].

For \(B\) particles, we use the quite precise inclusive \(Z\rightarrow B^+X\) branching fraction from the \(Z\) boson summary in the PDG.

We also use the sum of \(B^{\pm }\) and \(B^0(\bar{B}^0)\) in sec. 46 of the PDG.^{10}

The \(B_s^0\) multiplicity given in sec. 46 of the PDG (\(0.057\pm 0.013\)) is more than twice the inclusive \({{\mathrm {BR}}}(Z\rightarrow B_s^0X)/{{\mathrm {BR}}}(Z\rightarrow {{\mathrm {hadrons}}})\) branching fraction (\(0.0227\pm 0.0019\)) quoted in the \(Z\) boson summary table. We find these two numbers difficult to reconcile and choose to use the inclusive \({{\mathrm {BR}}}(Z\rightarrow B_s^0X)/{{\mathrm {BR}}}(Z\rightarrow {{\mathrm {hadrons}}})\) branching fraction as our main constraint.

We also include the inclusive branching fractions for \(B\)baryons (summed over baryons and antibaryons), the rate of \(g\rightarrow b{\bar{b}}\) splittings obtained by combining ALEPH [59], DELPHI [60], and SLD [61] measurements (including an additional 10 % systematic to account for larger possible mismodeling effects in the correction factors [57, 58]) and the rate of \(Z\rightarrow bb{\bar{b}}{\bar{b}}\) from the PDG \(Z\) boson summary table [39].
Hadronic \(Z\) decays at \(\sqrt{s}=M_Z\). Measured rates and inclusive branching fractions of particles containing \(c\) and \(b\) quarks, as used in this work. Note The branching fractions are normalized to \(Z\rightarrow {{\mathrm {hadrons}}}\), and hence should be interpreted as, e.g., \({{\mathrm {BR}}}(Z\rightarrow B^+ X)/{{\mathrm {BR}}}(Z\rightarrow {{\mathrm {hadrons}}})\). Note 2 The sum over \(B^*\) states includes both particles and antiparticles. Note 3 The \(\Upsilon \) rate is multiplied by a factor 10. Source labels indicate: A (ALEPH), D (DELPHI), O (OPAL), P (PDG, section 46), S (SLD), Z (PDG Z Boson Summary Table)
Charm \(\left\langle n\right\rangle \hbox { or }{{\mathrm {BR}}}\)  Our reference value  Our source  Beauty \(\left\langle n\right\rangle \hbox { or }{{\mathrm {BR}}}\)  Our reference value  Our source 

\(D^{+} + D^\)  \(0.251 \pm 0.047\)  A  \({{\mathrm {BR}}}(Z\rightarrow B^{+}X)\)  \(0.087 \pm 0.002\)  Z 
\(D^0 + \bar{D}^0\)  \(0.518 \pm 0.063\)  A  \(B^{+} + B^0 + \bar{B}^0 + B^\)  \(0.330 \pm 0.052\)  P 
\(D^{*+} + D^{*}\)  \(0.194 \pm 0.0057\)  P  \(B^{*}_{u}+B^{*}_d+B^*_s\)  \(0.288 \pm 0.026\)  P 
\(D_s^+ + D_s^\)  \(0.131 \pm 0.021\)  P  \({{\mathrm {BR}}}(Z\rightarrow B_s^0X)\)  \(0.0227 \pm 0.0019\)  Z 
\({{\mathrm {BR}}}(Z\rightarrow \Lambda _c^+X)\)  \(0.0220 \pm 0.0047\)  Z  \({{\mathrm {BR}}}(Z\rightarrow B_{{\mathrm {baryon}}}X)\)  \(0.0197 \pm 0.0032\)  Z 
\({{\mathrm {BR}}}(Z\rightarrow X+c{\bar{c}})\)  \(0.0306 \pm 0.0047\)  AO  \({{\mathrm {BR}}}(Z\rightarrow X+b{\bar{b}})\)  \(0.00288 \pm 0.00061\)  ADS 
\(J/\psi \)  \(0.0052 \pm 0.0004\)  P  \({{\mathrm {BR}}}(Z\rightarrow bb{\bar{b}}{\bar{b}}X)\)  \(0.00051 \pm 0.00019\)  Z 
\(\chi _{c1}\)  \(0.0041 \pm 0.0011\)  P  \(\Upsilon ~(\times 10)\)  \(0.0014 \pm 0.0007\)  P 
\(\psi '\)  \(0.0023 \pm 0.0004\)  P 
Our parameters are slightly smaller than the current default values, leading to slightly smaller \(D^*\) and \(B^*\) rates, as can be seen from the plots in Fig. 7. Note also that the increased overall amount of strangeness in the fragmentation leads to slightly higher \(D_s\) and \(B_s\) fractions, in better agreement with the data. Uncertainties are, however, large, and some exotic onium states, like \(\chi _{c1}\), \(\psi '\), and \(\Upsilon \) are not well described by the default modeling. (It is encouraging that at least the multiplicity of \(J/\psi \) mesons is well described, though a substantial fraction of this likely owes to the feeddown from \(B\) decays, and hence does not depend directly on the stringfragmentation model itself.)
We also note that it would be desirable to reduce the rate of \(g\rightarrow b{\bar{b}}\) and \(Z\rightarrow bb{\bar{b}}{\bar{b}}\) events, while the \(g\rightarrow c{\bar{c}}\) one appears consistent with the LEP constraints. We suspect that this issue may be tied to the fixed choice of using \(p_{\perp }\) as the renormalization scale for both gluon emissions and for \(g\rightarrow q{\bar{q}}\) splittings in the current version of PYTHIA. A more natural choice for \(g\rightarrow q{\bar{q}}\) could be \(\mu _R\propto m_{q{\bar{q}}}\), as used e.g. in the VINCIA shower model [29].
We now turn to the dynamics of heavyquark fragmentation, focusing mainly on the \(b\) quark.
We emphasize that this is the only fragmentation function that is selfconsistent within the stringfragmentation model [33, 62]. Although a few alternative forms of the fragmentation functions for massive quarks are available in the code, we therefore here work only with the Bowler type. As for the massless function, the proportionality sign in Eq. (6) indicates that the function is normalized to unity.
which produces softer \(B\) spectra and simultaneously agrees better with the theoretically preferred value (\(r_b=1\)).
Comparisons to L3 event shapes in \(b\)tagged events are collected in Appendix B.1 (the left column of plots contains lightflavour tagged event shapes, the right column \(b\)tagged ones). In particular, the Monash tune gives a significant improvement in the soft region of the jetbroadening parameters in \(b\)tagged events, while no significant changes are observed for the other event shapes. These small improvements are presumably a direct consequence of the softening of the \(b\) fragmentation function; it is now less likely to find an isolated ultrahard \(B\) hadron.
We note that the low\(x\) part of the \(D^*\) spectrum originates from \(g\rightarrow c{\bar{c}}\) shower splittings, while the high\(x\) tail represents prompt \(D^*\) production from leading charm in \(Z\rightarrow c{\bar{c}}\) (see [55] for a nice figure illustrating this). The intermediate range contains a large component of feeddown from \(b\rightarrow c\) decays, hence this distribution is also indirectly sensitive to the \(b\)quark sector. The previous default tune had a harder spectrum for both \(b\) and \(c\)fragmentation, leading to an overestimate of the high\(x\) part of the \(D^*\) distribution. The undershooting at low \(x_{D^*}\) values, which remains unchanged in the Monash tune, most likely indicates an underproduction of \(g\rightarrow c{\bar{c}}\) branchings in the shower. We note that such an underproduction may also be reflected in the LHC data on \(D^*\) production, see e.g. [65]. We return to this issue in the discussion of identified particles at LHC, Sect. 3.5.
For completeness, the righthand pane of Fig. 11 shows the \(D^*\) spectra from the two other generalpurpose MC models, HERWIG [66] and SHERPA [67]. The HERWIG spectrum (dashed lines) is similar to the default PYTHIA one, with a deficit in the \(g\rightarrow c{\bar{c}}\) dominated region at low \(x_E\) and a significant overshooting in the hard leadingcharm region, \(x_E\rightarrow 1\). Interestingly, the \(D^*\) spectrum in SHERPA (dotted lines) exhibits an excess at small \(x_E\) values, suggesting relatively larger contributions from \(b\) decays and from \(g\rightarrow c{\bar{c}}\) splittings.
3 Hadron collisions
We discuss PDFs in Sect. 3.1, the choice of strong coupling (and total cross sections) in Sect. 3.2, initialstate radiation (and primordial \(k_T\)) in Sect. 3.3, minimumbias and underlying event in Sect. 3.4, and finally identifiedparticle spectra in Sect. 3.5. Energy scaling is discussed separately, in Sect. 4.
3.1 Parton distributions
In recent years there has been some discussion about possible modifications of the vanilla LO PDFs that could lead to improved predictions from LO event generators. Some possibilities for these improvements that have been explored include the use of the LO value of \(\alpha _s\) but with twoloop running, or relaxing the momentum sum rules constraint from the LO fits. These and other related ideas underlie recent attempts to produce modified LO PDFs such as MRST2007lomod PDFs [69] and the CT09MC1/MC2 [70] PDFs. The claim was that such improved LO (also called LO*) PDFs lead to a better agreement between data and theory in the LO fit and that their predictions for some important collider observables are closer to the results using the full NLO calculation. We note, however, that in the context of earlier multipartoninteractionmodel tuning studies undertaken by us [8] and by ATLAS [13], the large gluon component in LO* PDFs has been problematic (driving very high inclusivejet and MPI rates).
In the context of the NNPDF fits, which we shall use for the Monash 2013 tune, the above modifications were also studied. In particular, in the study of the NNPDF2.1LO fits in Ref. [18], it was found that, from the point of view of the agreement between data and theory, the standard LO PDFs provided as good a description as the other possible variations, including a different value of \(\alpha _s(M_Z)\), using the one or twoloop running or relaxing the momentum sum rule. The different results found by previous studies could be related to the limited flexibility in the input gluon PDFs in the CTEQ/MSRT LO fits: indeed, with a flexible enough parametrization such as that used in the NNPDF fits, the differences between these theory choices can always be absorbed into the initial condition.
Note that the NNPDF2.3 LO sets are provided for two values of the strong coupling, \(\alpha _s(M_Z)=0.119\) and \(0.130\); we use the latter here. The sets have also been extended in order to have a wider validity range, in particular they are valid down to \(x=10^{9}\) and \(Q=1\) GeV\(^2\), precisely with the motivation of using them in LO event generators.
3.2 The strong coupling and total cross sections
3.3 Initialstate radiation and primordial kT
We follow the approach of the Perugia tunes of PYTHIA 6 [6, 8] and use the same \(\alpha _s(M_z)\) value for initialstate radiation as that obtained for finalstate radiation. That is, we use oneloop running with \(\alpha _s(M_Z) = 0.1365\) for both FSR and ISR. This choice is made essentially to facilitate matching applications, see e.g. [21]. Nonetheless, we emphasize that we do not regard this choice as mandatory, for the following reasons.
Firstly, since each collinear direction is associated with its own singular (set of) diagram(s), one can consistently associate at least the collinear radiation components with separate welldefined \(\alpha _s\) values without violating gauge invariance. Secondly, while the LO splitting functions for ISR and FSR are identical, they differ at higher orders (beyond the shower accuracy), and there are important differences between the collinear (DGLAP) evolution performed in PDF fits and the (coherent, momentumconserving) evolution performed by parton showers; these differences could well be desired to be reflected in slightly different effective scale choices for ISR with respect to FSR, one possibility then being to absorb this in a redefinition of the effective value of \(\alpha _s(M_Z)\). Thirdly, and perhaps most importantly, while we agree that maintaining separate \(\alpha _s\) values (equivalent to making slightly different effective scale choices) for ISR and FSR is ambiguous for wideangle radiation, we emphasize that merely using the same \(\alpha _s(M_Z)\) value for the two algorithms does not remove this fundamental ambiguity. This is because, in the context of a shower algorithm, the value of the renormalization scale depends upon which parton is branching, and that assignment is fundamentally ambiguous outside the collinear limit. For instance, an emitted gluon with a certain momentum will have a different \(p_{\perp }\) with respect to the beam (ISR), than it will with respect to a finalstate parton (FSR), and hence the argument of \(\alpha _s\), typically taken to be proportional to some measure of \(p_{\perp }\), will be different, depending on who the emitter was. This effect is present in all partonbased shower algorithms and is not cured by arbitrarily setting \(\alpha _s(M_Z)\) to be the same for ISR and FSR. Using the same \(\alpha _s(M_Z)\) for both ISR and FSR (as we do here) should therefore not be perceived of as being more rigorous than not doing so; it is a choice we make purely for convenience. (The situation is slightly better in antennabased showers [79, 80, 81], where there is no distinction between radiator and recoiler in the soft limit, hence the renormalizationscale choice is unique, at leading colour.)
In the ATLAS spectra, the feature around \(p_{\perp }^{\mu \mu } \sim 35~\hbox {GeV} \) is repeated by all MCs in the comparisons shown on the MCPLOTS web site [25], hence we regard it as an artifact of the data. We note however that there is a tendency for PYTHIA to overshoot the data between \(p_{\perp }\) values of roughly 20 to \(100~\hbox {GeV} \), at both CM energies. This is an interesting region intermediate between low\(p_{\perp }\) bremsstrahlung and high\(p_{\perp }\) \(Z\)+jet processes, which will be particularly relevant to reconsider in the context of matrixelement corrections at the \(\mathcal{O}(\alpha _s^2)\) level and beyond [87].
3.4 Minimum bias and underlying event
The Monash 2013 tune has been constructed to give a reasonable description of both softinclusive (“minimumbias”) physics as well as underlyingevent (UE) type observables. The difference between the two is sensitive to the shape of the hadronhadron overlap profile in impactparameter space (the UE probes the most “central” collisions while minbias (MB) is more inclusive) and to the modeling of colour reconnections (CR). Most previous tunes, including the current default Tune 4C [9], have used a Gaussian assumption [76] for the transverse matter distribution, but this appears to give a slightly too low UE level (for a given average MB level).
For the Monash tune, we have chosen a slightly more peaked transverse matter profile [27], thus generating a relatively larger UE for the same average MB quantities. We note, however, that there are still several indications that the dynamics are not well understood, in particular when it comes to very low multiplicities (overlapping with diffraction), very high multiplicities (e.g., the socalled CMS “ridge” effect [88]), and to identifiedparticle spectra (e.g., possible modifications by rescattering [89], string boosts from colour reconnections [90], or other collective effects).
The relative dominance of the gluon PDF is illustrated by the bottom righthand pane of Fig. 17, showing the gluon fraction (relative to all MPI initiators) as a function of \(\log _{10}(x)\). Below \(x\sim 0.1\), the NNPPDF sampling is 80 % gluondominated, and the gluon fraction is higher than in CTEQ6L1 for both very small \(x<10^{5}\) as well as for very large \(x>0.2\).
A further consistency check is provided by the \(\bar{u}/u\) ratio, shown in the bottom lefthand pane of Fig. 17. This is consistent with unity (as expected for sea quarks) in the entire small\(x\) region \(x<10^{2}\). The valence bump appears to be slightly more pronounced in the NNPDF tune (relative to the sea), since the \(\bar{u}/u\) ratio drops off more quickly above \(10^{2}\). This trend persists until the very highest bin, at \(x\sim 1\), where the experimental uncertainties are extremely large. The CTEQ6L1 parametrization there forces the \(\bar{u}\) PDF to zero, while the NNPDF parametrization allows for a small amount of \(\bar{u}\) to remain even at the largest \(x\) values, though we note that they are still outnumbered by \(u\) quarks at a level of hundredtoone.
In the bottom two panes of Fig. 19, we focus on the forward region (with physical event selections). In particular, we see that the NNPDF set [20] generates a broader rapidity spectrum, so that while the activity in the central region (top pane) is reduced slightly, the activity in the very forward region actually increases, and comes into agreement with the TOTEM measurement [94], covering the range \(5.3<\eta <6.4\). The bottom righthand pane shows the forward energy flow measured by CMS [93], in the intermediate region \(3.23<\eta <4.65\). The dependence on \(\eta \) is a bit steeper in the Monash tune than in the previous one, and more similar to that seen in the data.
3.5 Identified particles at LHC
In the \(K^0_S\) rapidity distribution, shown in the lefthand pane of Fig. 23, we observe that tune 4C exhibits a mild underproduction, of about 10 %. Though it might be tempting to speculate whether this could indicate some small reduction of strangeness suppression in \(pp\) collisions, however, we already noted in Sect. 2.1 that the strangeness production in \(ee\) collisions also needed to be increased by about 10 %. After this adjustment, we see that the overall \(K^0_S\) yield in the Monash 2013 tune is fully consistent with the CMS measurement. Nonetheless, we note that the momentum distribution is still not satisfactorily described, as shown in the righthand pane of Fig. 23. Our current best guess is therefore that the overall rate of strange quarks is consistent, at least in the average minbias collision (dedicated comparisons in highmultiplicity samples would still be interesting), but that the phasespace distribution of strange hadrons needs more work. Similarly to the case in \(ee\) collisions, cf. Fig. 6, the model predicts too many very soft kaons, though we do not currently know whether there is a dynamic link between the \(ee\) and \(pp\) observations.
For strange baryons, we note that the increase in the \(\Lambda ^0\) fraction in \(ee\) collisions (cf. Fig. 5) does not result in an equivalent improvement of the \(\Lambda ^0\) rate in \(pp\) collisions, shown in Fig. 24. The Monash 2013 tune still produces only about 2/3 of the observed \(\Lambda ^0\) rate (and just over half of the observed \(\Xi ^\) rate, cf. Appendix B.2). We therefore believe it to be likely that an additional source of net baryon production is needed (at least within the limited context of the current PYTHIA modelling), in order to describe the LHC data. The momentum spectrum is likewise quite discrepant, exhibiting an excess at very low momenta (stronger than that for kaons), a dip between 1–4 GeV, and then an excess of very hard \(\Lambda ^0\) production. The latter hard tail is somewhat milder in the Monash 2013 tune than previously, and it may be consistent with the trend also seen in the \(\Lambda ^0\) spectrum at LEP, cf. Fig. 6. We conclude that baryon production still requires further modelling and tuning efforts.
4 Energy scaling
From the plots in Fig. 25, it is clear that there are no significant differences between the energy scaling of the three \(ee\) tunes considered here (mainly reflecting that they have been tuned to same reference point, at 91.2 GeV, and that their scaling is dictated by the same underlying physics model), and that their energy dependence closely matches that observed in data. However, the increased amount of nonperturbative strangeness production in the Monash tune leads to a better agreement with the overall normalization of the \(K^{\pm }\) and \(\Lambda \) rates at all energies.
5 Conclusions and exhortation
We have presented a reanalysis of the constraints on fragmentation in \(ee\) collisions, and applied the results to update the finalstate fragmentation parameters in PYTHIA 8. We combine these parameters with a tune to hadroncollider data, using a new NNPDF 2.3 LO QCD+QED PDF set, which has been encoded so it is available as an internal PDF set in PYTHIA 8, independently of LHAPDF [112].
In this PDF set as well as in our tune, the value of the strong coupling for hardscattering matrix elements is fixed to be \(\alpha _s(M_Z)=0.13\), consistent with other LO determinations of it. For initial and finalstate radiation, our tune uses the effective value \(\alpha _s(M_Z)=0.1365\). The difference is consistent with an effective translation between the \(\overline{{\mathrm {MS}}}\) and CMW schemes. We note that alternative (LO, NLO, and NNLO) NNPDF 2.3 QCD+QED sets with \(\alpha _s(M_Z)=0.119\) are also available in the code, for people who want to check the impact of using a different \(\alpha _s(M_Z)\) value and/or higherorder PDF sets on hardscattering events. For the purpose of such studies, we point out that it is possible, in PYTHIA 8, to preserve most of the features of the shower and underlyingevent tuning by changing only the PDF for the hardscattering matrix elements, leaving the PDF choice for the shower evolution and MPI framework unaltered (see the PYTHIA 8 HTML manual’s PDF section, under PDF:useHard).
Moreover, despite the comprehensive view of collider data we have attempted to take in this study, there still remains several issues that were not addressed, including: initialfinal interference and coherence effects [113, 114] (probably more a modelling issue than a tuning one); reliable estimates of theoretical uncertainties [8, 17, 24, 29, 97, 115]; diffraction^{14} [74, 116, 117] and other coloursinglet phenomena such as onium production; longdistance (e.g., forwardbackward, forwardcentral, and “ridge”type) correlations [88, 118, 119, 120, 121, 122, 123]; \(B\)hadron decays [124]; and tuning in the presence of matrixelement matching, at LO and NLO (see [21, 29, 115, 125, 126] for recent phenomenological studies). Especially in the latter context of matrixelement matching, we expect that in many cases PYTHIA 8 will be used together with codes such as ALPGEN [127], MADGRAPH [128], aMCatNLO [129], or POWHEG [130], either using the matching algorithms of those programs themselves, or via any of PYTHIA ’s several internal (LHEFbased [131]) implementations of matching schemes (POWHEG [86], CKKWL [132, 133, 134], MLM [135, 136], UMEPS [137], NL3 [138], UNLOPS [139]). The impact of such corrections on MC tuning depends on the details of the matching scheme (especially its treatment of unitarity), and there is in general a nonnegligible possibility of “mistuning” when combining a standalone tune with ME corrections. A simple example illustrating this is the effective value of \(\alpha _s(M_Z)\), which for a leadingorder tune is typically of order \(0.13\), while a consistent NLO correction scheme should be compatible with values closer to \(0.12\) [29]. There is also the question of the running order of \(\alpha _s\). The propagation of such changes from the level of hard matrix elements through the shower and hadronization tuning process are still not fully explored, and hence we advise users to perform simple crosschecks, such as checking the distributions presented in this paper, before and after applying matrixelement corrections. Parameters that appear on both sides of the matching, such as \(\alpha _s\), should also be checked for consistency [21].
We noted several issues concerning the \(ee\) data used to constrain the fragmentation modelling, that it would be good to resolve. In particular, we find some tensions between the identifiedparticle rates extracted from (1) HEDPATA, (2) Sec. 46 of the PDG, and (3) the \(Z\) boson summary table in the PDG, as discussed in more detail in Sect. 2, and concerning which we made some (subjective) decisions to arrive at a set of hopefully selfconsistent constraints for this work. We also note that the overall precision of the fragmentation constraints could likely be significantly improved by an FCCee type machine, such as TeraZ, a possibility we hope to see more fully explored in the context of future \(ee\) QCD phenomenology studies.
 1.
the net strangeness fraction (has been increased by 10 %, reflected not only in improved kaon and hyperon yields, but also in the \(D_s\) and \(B_s\) fractions),
 2.
the ultrahard fragmentation tail (has been softened, especially for leading baryons and for \(D\) and \(B\) hadrons),
 3.
the \(p_{TZ}\) spectrum (softened at low \(p_{TZ}\)),
 4.
the minimumbias charged multiplicity in the forward region (has increased by 10 %),
 5.
the underlying event at 7 TeV (is very slightly higher than before).
Some questions that remain open include the following. We see a roughly 20 % excess of very soft kaons in both \(ee\) and \(pp\) environments, cf. Figs. 6 and 23, despite the overall kaon yields being well described, and the overall baryon yields at LHC appear to be underestimated by at least \(30~\%\) despite good agreement at LEP. The momentum spectra of heavier strange particles are also poorly reproduced, in particular at LHC. It is interesting and exciting that some of the LHC spectra appear to be better described by allowing collective flow in a fraction of events (cf. the EPOS model [140]), though we believe the jury is still out on whether this accurately reflects the underlying physics. For instance, it has been argued that colour reconnections can mimick flow effects [90], and they may also be able to modify the yield of baryons if the creation/destruction of string junctions is allowed [141]. We look forward to future discussions on these issues.

Not only central tunes: experiments and other userend colleagues need more than central descriptions of data; there is an increasing need for serious uncertainty estimates. In the context of tune variations, it is important to keep in mind that the modelling uncertainties are often intrinsically nonuniversal. Therefore, the constraints obtained by considering data uncertainties only (e.g., in the spirit of PROFESSOR’s eigentunes [17]) can at most constitute a lower bound on the theoretical uncertainty (similarly to the case for PDFs). A serious uncertainty estimate includes some systematic modelling variation, irrespectively of, and in addition to, what data allows (e.g., in the spirit of the Perugia set of tunes for PYTHIA 6 [8]). We therefore hope the future will see more elaborate combinations of data and theorydriven approaches to systematic tune uncertainties;

Not only global tunes: the power of MC models lies in their ability to simultaneously describe a large variety of data, hence we do not mean to imply that one should give up on universality and tune to increasingly specific corners of phase space, disregarding (or deemphasizing, with lower weights) all others. However, as proposed in [97], one can obtain useful explicit tests of the universality of the underlying physics model by performing independent tunes on separate “physics windows”, say in the forward vs. central regions, for different eventselection criteria, at different collider energies, or even for different collider types. In this connection, just making one global “bestfit” tune may obscure tensions between the descriptions of different complementary data sets. By performing independent tunes to each data set separately, and checking the degree of universality of the resulting parameters, one obtains a powerful cross check on the underlying physics model. If all sets produce the same or similar parameters, then universality is OK, hence a global tune makes very good sense, and the remaining uncertainties can presumably be reliably estimated from data alone. If, instead, some data sets result in significantly different tune parameters, one has a powerful indication that the universality of the underlying modeling is breaking down, which can lead to several productive actions: (1) it can be taken into account in the context of uncertainty variations, (2) the nature of the data sets for which nonuniversal tune parameters are obtained can implicitly indicate the nature of the problem, leading to more robust conclusions about the underlying model than merely whether a tune can/cannot fit the data, and (3) the observations can be communicated to the model authors in a more unambiguous way, hopefully resulting in a speedier cycle of model improvements.
Footnotes
 1.
In particular, RIVET ensures that any (current or future) Monte Carlo eventgenerator codes can be compared consistently to the data, with exactly the same cuts, definitions, etc., as the original analysis.
 2.
We note that a similar convention is used on the MCPLOTS validation web site [25].
 3.
One slight disadvantage is that the CMW scheme produces somewhat larger \(\Lambda _{{\mathrm {QCD}}}\) values. Since the current formulation of the shower algorithm does not include a nonperturbative regularization of \(\alpha _s\), a higher \(\Lambda _{{\mathrm {QCD}}}\) value necessitates a larger IR cutoff in the shower, which can leave an undesirable gap between the transverse kicks generated by shower emissions and those generated by nonperturbative string splittings.
 4.
The IR shower cutoff must still remain somewhat above the Landau pole of \(\alpha _s\); a lower cutoff scale would activate a hardcoded protection mechanism implemented in the PYTHIA shower, forcing it to be higher than \(\Lambda _{{\mathrm {QCD}}}\).
 5.
Explicitly, \(\sigma _{\perp }\) expresses the \(p_{\perp }\) broadening transverse to the string direction, but implicitly its size also enters in the longitudinal fragmentation function, via the \(m_{\perp }^2\) term in Eq. (3), causing higher\(p_{\perp }\) hadrons to have relatively harder longitudinal spectra as well.
 6.
One might worry whether the effect could be due solely to the \(Z\rightarrow b{\bar{b}}\) events which are only present in the ALEPH measurement, and if so, whether this could indicate a significant mismodeling of the momentum distribution in \(b\) events. However, as we show below in the section on \(b\) fragmentation, the chargedparticle momentum distribution in \(b\)tagged events shows no excess in that region (in fact, it shows an undershooting).
 7.
We note that HEPDATA incorrectly gives the systematic error as \(0.002\) while the value in the OPAL paper is \(0.010\) [43]. This has been communicated to the HEPDATA maintainers.
 8.
Even with the inflated error, the uncertainty on the DELPHI measurement is still less than a third that of the OPAL one. DELPHI therefore still dominates the average.
 9.
For reference, the current default value of ProbStoUD is 0.19 while ours is 0.217. The increased value also improves the agreement with the \(D_s\) and \(B_s\) rates, see Sect. 2.3. The default values of mesonUDvector and mesonSvector are 0.62 and 0.725 respectively, while ours are 0.5 and 0.55.
 10.
Note that we have a factor 2 relative to the PDG, since it appears the PDG quotes the average, rather than the sum. Note also that all the average \(B\) meson multiplicities in sec. 46 of the PDG are accompanied by a note, “(d)”, stating that the SM \(B(Z\rightarrow b{\bar{b}}) = 0.217\) was used for the normalization. For completeness, the reader should be aware that this is the fraction normalized to hadronic \(Z\) decays; the branching fraction relative to all \(Z\) decays, is 0.151 [39].
 11.
The difference between this \(\alpha _s\) value and that used for ISR/FSR will be discussed in Sect. 3.3.
 12.
\(t\)channel gluon exchange gives an amplitude squared proportional to \(1/t^2\), which for small \(p_T\) goes to \(1/p_T^4\).
 13.
We note, however, that the value obtained for the 8TeV elastic cross section in PYTHIA is 20 mb, whereas the value measured by TOTEM is \(27.1\pm 1.4\) mb [72]. While this discrepancy does not influence the normalization or modelling of inelastic events and hence is a nonissue in that context, an update of the total crosssection expressions in PYTHIA may be timely in the near future, e.g. using the updated Donnachie–Landshoff analysis in [75]. We also note that the decomposition of the inelastic cross section into individual nondiffractive and diffractive components, which follows Schuler–Sjöstrand [74], may also be due for an update.
 14.
In particular, the constraints on fragmentation mainly come from SLD and LEP, where the nonperturbative parameters are clearly defined at the shower cutoff scale, \(Q_{{\mathrm {had}}}\), whereas diffraction is dominated by soft physics, for which the definition of the effective hadronization scale is less clear. The amount of MPI in hard diffractive events also requires tuning.
Notes
Acknowledgments
We thank S. Mrenna, M. Ritzmann, and T. Sjöstrand for comments on the manuscript and L. de Nooij for pointing us to the ALICE \(K^*\) and \(\phi \) measurements [142] and to the ATLAS \(\phi \) measurements in [143]. The work of J. R. is supported by an STFC Rutherford Fellowship ST/K005227/1. This work was supported in part by the Research Executive Agency (REA) of the European Commission under the Grant Agreements PITNGA2012315877 (MCnet).
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