<figure><img src="image_106.png" /><figcaption>**Fig. 91** The ratio of the $Z^0$-boson yield measured by ALICE [600] and ATLAS [598] in Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV and free-nucleon PDF (CT14) predictions. The ratios are compared to theoretical calculations including EPPS16 parameterisation of the nuclear modification effects</figcaption></figure>

particle momentum spectra, and integrated flow measurements simultaneously, which has been demonstrated in earlier chapters. However, around the time of the first LHC collisions, it was stated that modelling uncertainties in the initial state, which in turn provided an extraction of $\eta/s$ from RHIC data, led to uncertainties for $\eta/s$ of a factor of around 2.5 [959]. That factor arose from the ambiguity of whether the MC Glauber nucleon model [42], or the MC-KLN model [954,955], should be used to determine QGP transport properties based on comparisons to data. Therefore, assessing whether these modelling uncertainties still apply is critical for the determination of QGP transport properties. The situation has significantly improved in the past few years, after new developments arising from measurements on event-by-event flow fluctuations (and the construction of probability density function), namely the fluctuations of flow-vectors, the correlations between different-order flow coefficients via Symmetric Cumulants [310] that has been introduced in Sect. 2.2.5, and measurements sensitive to the non-linear hydrodynamic response of higher-order flow with the corresponding correlations between different-order symmetry planes. All of these observables have varying sensitivities to the initial-state anisotropy, which is a key feature in all of the initial-state models described, and which will be addressed in Sect. 4.3.

## 4.1 Electroweak-boson measurements

Electroweak bosons are predominantly produced via the Drell-Yan process, i.e. $q\bar{q}$ annihilation into a lepton pair, therefore they can be used as particularly clean probes of light-quark PDFs since leptons are not affected by final-state

effects. The resolution scale $Q^2$ is determined by the boson mass $Q^2 = M_{Z,W}^2$ while Bjorken-$x$ values of light quarks are directly related to the rapidity $y$ and the transverse mass $m_T$ of the dilepton pair as $x = \frac{m_T}{\sqrt{s}} \exp(\pm y)$. ALICE performed detailed studies of $Z^0$ and $W^\pm$ boson production in p-Pb and Pb-Pb collisions with the muon spectrometer at forward rapidity. These measurements are sensitive to nuclear effects at typical $x$ values either below $10^{-3}$, in the shadowing region, or above $10^{-1}$ where antishadowing and EMC effects play a role.

While ALICE measurements of $Z^0$ boson production in p-Pb collisions are described by models that include both free-nucleon and nuclear-modified PDFs [600,960], the Pb-Pb results at $\sqrt{s_{NN}} = 5.02$ TeV appear to be more sensitive to nuclear shadowing effects, showing a clear deviation from calculations based on free-nucleon PDFs [600]. $Z^0$ boson production is measured in the dimuon channel for muons with $p_T > 20$ GeV/c and $2.5 < y^{\mu\mu} < 4$ in a broad centrality range from 0 to 90%. The ratio of $Z^0$ boson yield and corresponding free-nucleon PDF (CT14) predictions is shown in Fig. 91 as a function of rapidity together with ATLAS results at midrapidity [598] The ratio is compared to theoretical calculations including the EPPS16 parameterisation of the nuclear modification effects. The ALICE data are described by nPDF calculations (CT14+EPPS16) but the free-nucleon PDF calculations (CT14) overestimate the data. A $3.4\sigma$ deviation is found between the rapidity-integrated yield and free-nucleon PDF calculations indicating that nuclear shadowing effects play an important role.

The $W^\pm$-boson production has been measured by ALICE via muonic decays requiring muon tracks with $p_T > 10$ GeV/c in the muon spectrometer acceptance [601]. Figure 92 shows the ratio of the cross section of muons from $W^+$ measured by ALICE [601] and CMS [961] in p-Pb collisions at $\sqrt{s_{NN}} = 8.16$ TeV to the free-nucleon PDF (CT14) predictions. While the ALICE measurement agrees with predictions based on EPPS16 nPDFs, it deviates by $2.7\sigma$ from the free-nucleon PDF calculation at positive rapidities, corresponding to Bjorken-$x$ below $10^{-3}$ in Pb nuclei. These measurements may provide significant constraints to future nuclear PDF global fits.

## 4.2 Photon-induced processes in heavy-ion collisions

Due to the short range of the strong force, collisions where the incoming nuclei pass by each other at a distance larger than the sum of their radii, called ultra-peripheral collisions (UPCs), are dominated by photon-induced processes. The intensity of the photon flux depends on the square of the electric charge of the incoming particle, so Pb nuclei are intense sources of photons at the LHC. Recent reviews of UPC physics can be found in Refs. [962–964].
<figure><img src="image_107.png" /><figcaption>**Fig. 92** The ratio of W⁺-boson cross section measured in p-Pb collisions at $\sqrt{s_{NN}}$ = 8.16 TeV to the free-nucleon PDF (CT14) predictions for ALICE [601] and CMS [961] data. The ratios are compared to theoretical calculations including nuclear PDF modification effects (CT14+EPPS16)</figcaption></figure>

One important process accessible in UPC is the diffractive photoproduction of vector mesons off hadronic targets, which is particularly sensitive to the gluon distribution in the target, because it proceeds through a colourless exchange and thus involves at least two gluons. ALICE has measured three variations of this process. The exclusive production of a vector meson off a proton target, and the production of the vector meson by the interaction with the full nucleus (coherent production), or with one of the nucleons in the nucleus (incoherent production). Measurements where a $J/\psi$ meson is produced allow for a pQCD treatment of the process at a scale $\approx M_{1/\psi}/2$, while those where a $\rho^0$ meson is produced are an excellent tool to study the approach to the black-disk limit of QCD at a semi-hard scale [949].

The exclusive photoproduction of $J/\psi$ off protons has been measured by ALICE in p-Pb UPC at $\sqrt{s_{NN}}$ = 5.02 TeV [965,966]. The cross section, shown in Fig. 93, covers a broad range of centre-of-mass energies in the $\gamma p$ frame, $W_{\gamma p}$, extending from 20 GeV to above 700 GeV, which correspond to three orders of magnitude in Bjorken-$x$ from $2 \times 10^{-2}$ to $2 \times 10^{-5}$, where $x = (M/W_{\gamma p})^2$ with M being the mass of the vector meson. The behaviour of the cross section in this large kinematic range can be described by a power law in $W_{\gamma p}$ with an exponent 0.70 ± 0.05. In the leading order pQCD, this behaviour implies that the gluon distribution in the proton keeps rising as a power law with decreasing $x$, without a clear signal of becoming saturated. Data can be described by models based on NLO BFKL equations [967] or the standard DGLAP pQCD without saturation effects, such as JMRT NLO shown in Fig. 93 [968]. Models including saturation effects, e.g. CGC [969] in the figure, predict a similar cross section to those predicted by models without saturation in the considered $W_{\gamma p}$ range. Nonetheless, a recent analysis claims that the data indicate the onset

<figure><img src="image_108.png" /><figcaption>**Fig. 93** Cross section of exclusive $J/\psi$ photoproduction off protons as a function of $W_{\gamma p}$ energy measured by ALICE in p-Pb UPC [965, 966], compared to HERA [971,972] and LHCb [973,974] results and theoretical models [967–969]</figcaption></figure>

of gluon saturation in the proton [970]. The ongoing analysis of Run 2 data from p-Pb UPC at $\sqrt{s_{NN}}$ = 8.16 TeV, and of future larger samples, will increase the experimental reach to the region above 1 TeV in $W_{\gamma p}$ with smaller uncertainties than in the current results. These measurements may provide a definitive answer about the presence of gluon saturation in the proton.

The photoproduction of $J/\psi$ in Pb-Pb UPC has been studied by ALICE at $\sqrt{s_{NN}}$ = 2.76 TeV [975,976] and $\sqrt{s_{NN}}$ = 5.02 TeV [977–979]. The results reported in Ref. [975] are the first measurement of photon-induced processes at the LHC. They demonstrated that this type of analysis could be carried out at the LHC with good precision and in a kinematic domain not accessible to other machines. ALICE has measured the coherent production of $J/\psi$ in UPC in a wide rapidity range as shown in the top plot of Fig. 94 [978]. The comparison of the rapidity dependence of data with that of the impulse approximation [948], computed neglecting all nuclear effects under the assumption that the scattering of photons on nuclei is given by the coherent superposition of the scattering on the individual nucleons, provides one of the cleanest, and clearest, signatures of gluon shadowing and its Bjorken-$x$ dependence. Data rise quickly towards midrapidity and then seem to saturate. This behaviour is not completely reproduced by models. The STARlight prediction [980], that includes only Glauber-like suppression, is above the data, particularly at midrapidity where the colour-dipole models including saturation, IPsat [981] and BGK-I [982] in Fig. 94, also overshoot the data by a large margin.

On the other hand, the leading-twist approximation [983], LTA in the figure, and the energy-dependent hot-spot model GG-HS [984], which also includes saturation, give the best overall description of the rapidity dependence. However,
<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /><figcaption>**Fig. 94** (Top) Cross section of coherent J/ψ production measured by ALICE in Pb–Pb UPC at $\sqrt{s_{NN}}$ = 5.02 TeV as a function of rapidity [978] and (Bottom) $|t|$-differential J/ψ photoproduction cross section extracted from the $p_T^2$ distribution of coherent J/ψ at midrapidity (right panel) [979]. The data are compared to various theoretical models</figcaption></figure>

these models underpredict the data at semi-forward rapidities in the range $2.5 < |y| < 3.5$, indicating that the nuclear shadowing might have a smaller effect at the Bjorken $x \sim 10^{-2}$ or $x \sim 5 \times 10^{-5}$ corresponding to this rapidity range. Figure 94 also shows a wide green band corresponding to the uncertainties of the model [983] based on the EPS09 LO parameterisation of nuclear PDFs [985], illustrating that the ALICE data have a great potential to improve the uncertainties of global nPDF fits.

In Pb–Pb UPCs there is an ambiguity regarding which of the nuclei is the source of the photon and which the target [986]. There are thus two contributions to the cross section, one from a high, the other from a low, energy photon. Only at midrapidity the two contributions are equal and the $\gamma$Pb cross section can be extracted directly from the UPC measurement. Comparing the $\gamma$Pb cross section with the

<figure><img src="image_111.png" /><figcaption>**Fig. 95** Coherent J/ψ photoproduction cross section measured by ALICE in hadronic Pb–Pb collisions as a function of $N_{part}$ [990]. Results are compared with theoretical calculations from Ref. [984] (GG-hs), Ref. [991] (S2 and S3 scenarios within the IIM and GBW dipole models) and from Ref. [992] (VDM)</figcaption></figure>

impulse approximation, a gluon shadowing factor can be defined [948] and determined from data to be around 0.6 for $x \approx 10^{-3}$. Future measurements of coherent J/ψ production, accompanied by secondary exchanges of photons and resulting in electromagnetic dissociation of nuclei, can serve as one of the most promising tools to disentangle low- and high-energy contributions at forward rapidities, since coherent J/ψ cross sections with and without nuclear dissociation are sensitive to different impact parameter ranges [987]. A proof-of-principle of this approach is provided by the recent measurement of Pb–Pb electromagnetic dissociation cross sections using the detection of forward neutrons in the ALICE Zero Degree Calorimeters [988].

The bottom plot in Fig. 94 shows the distribution of the momentum squared transferred at the target vertex, $|t|$, measured by ALICE at midrapidity in Pb–Pb UPC at $\sqrt{s_{NN}}$ = 5.02 TeV [979]. These data correspond to $x = 0.6 \times 10^{-4}$ and are obtained from the measured distribution of the square of the transverse momentum of the J/ψ, which, for colliders, is quite close to $|t|$. As the transferred $p_T$ is the Fourier conjugate of the impact parameter, this measurement provides information on the gluon distribution in the plane perpendicular to the motion of the nuclei. The shape of the measured $|t|$ distribution deviates from the simple form-factor-based dependence used in STARlight [980], but is well reproduced by models including gluon shadowing according to the leading-twist approximation [952], or gluon saturation effects from the impact-parameter-dependent Balitsky–Kovchegov equation [989].

ALICE has also discovered an excess of J/ψ at low-transverse momenta, $p_T < 0.3$ GeV/c, in peripheral and semi-central Pb–Pb collisions [722,990], which was inter-
preted as a sign of the coherent J/ψ photoproduction in nucleus-nucleus collisions that also include hadronic interactions [991–994]. Coherent J/ψ cross sections are shown in Fig. 95 as a function of $N_{\text{part}}$ corresponding to the range of centralities from 30 to 90% in comparison to various models based on different assumptions on how spectator and non-spectator nucleons participate in the coherent reaction. The IIM and GBW dipole model predictions [991] steadily increase with centrality in the scenario with unmodified photonuclear cross section (S2), while the use of an effective cross section where the overlap region between the two nuclei is assumed not to contribute to coherent photoproduction (S3) results in a reduction of the cross section toward more central collisions, providing a better description of the data. However, the reduction of the coherent cross section in central collisions has been also obtained in the VDM model [992] with unmodified photonuclear cross sections where the photon flux has been calculated differently compared to the S2 scenario. Though the mechanism of the coherent photoproduction in hadronic events still needs to be clarified, these measurements are considered as a promising tool to decouple low- and high-energy $\gamma$Pb cross sections in UPC measurements [994] and a potential probe of the quark-gluon plasma in more central events, see e.g. Ref. [995].

The knowledge of the structure of nuclei has been extended to even smaller momentum-exchange scales with the measurements of the coherent photoproduction of $\rho^0$ vector mesons at midrapidity in Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV [998] and $\sqrt{s_{NN}} = 5.02$ TeV [996], and Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV [997]. The coherent $\rho^0$ cross section in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [996] without forward-neutron selection is shown in Fig. 96, top, where it is compared with theoretical calculations. The GKZ model [983], based on a modified vector-dominance model, in which the hadronic fluctuations of the photon interact with the nucleons in the nucleus according to the Gribov–Glauber prescriptions, describes correctly the data. A slightly worse, but still quite satisfactory, description of data is obtained with the energy-dependent hot-spot model CCKT [984]. The GMMNS [999] and STARlight [1000] models underpredict the data. Furthermore, in [996] the cross section is presented in classes defined by the presence of forward neutrons produced by electromagnetic dissociation of the interacting nuclei due to secondary photon exchanges. The measurements in several neutron-emission classes were proposed as a tool to decouple high- and low-energy contributions to the UPC cross sections [987], since the presence of additional photon interactions effectively reduces the range of impact parameters and modifies the flux of photons participating in the coherent photoproduction process. ALICE measurements of the coherent $\rho^0$ production in neutron-emission classes are well described by models implemented in STARlight [1000–1002] and the NOON MC [1003], thus confirming the fac-

<figure><img src="image_112.png" /><figcaption>**Fig. 96** (Top) The coherent $\rho$ photoproduction cross section as a function of rapidity compared to theoretical calculations [996]. (Bottom) A-dependence of the coherent $\rho$ photoproduction cross section with a power-law fit shown by the band and general expectations for several extreme cases discussed in the text [997]</figcaption></figure>

torisation of the coherent photoproduction process and the additional photon exchanges assumed in these models. As the measurement at midrapidity can be converted from the UPC to the $\gamma$Pb cross section without ambiguity, the agreement between data and models suggests that the method proposed in [987] can be applied to disentangle the high- and low-energy photon contribution to the UPC cross section at other rapidities. The comparison of models with the measurement in Xe–Xe UPC produces a similar message. Furthermore, as the energy dependence of the $\gamma$Pb cross section is quite mild, the small difference in centre-of-mass energy in the Pb–Pb and Xe–Xe systems can be ignored and the measurements can be used to study the A-dependence of the cross section as shown in Fig. 96, bottom. The trend in data is quite different from that expected for coherent production without any other effect, represented in the figure by the model scaling as $A^{4/3}$. Indeed, the data are compatible with a power-law
<figure><img src="image_113.png" /><figcaption>**Fig. 97** Heavy-ion $dN_{ch}/d\eta$ data (at midrapidity) [174,176] scaled by the degrees of freedom from a variety of initial state models. The dotted lines are to guide the eye. The circular markers represent $dN_{ch}/d\eta$ divided by the number of participating nucleons in the MC Glauber nucleon model [42], diamonds use the number of constituent quarks from the MC Glauber constituent quark model [958], squares the entropy from $T_R\text{ENTo}$ [462], and stars the gluons from IP-Glasma [44,956]</figcaption></figure>

behaviour with exponent $\alpha = 0.963 \pm 0.019$, demonstrating clear shadowing, but still far from the black-disk limit that predicts $A^{2/3}$.

The measurements described in the preceding paragraphs can still be improved substantially with the data samples from the LHC Run 3 and 4, which are expected to be at least three orders of magnitude larger than those from Run 2 [139]. This huge increase in the available number of events will not only reduce the statistical uncertainties, but will permit a series of studies to reduce the systematic uncertainties. The expected data samples will also allow us to explore multidimensional investigations of the current observables as well as to study new signatures beyond the reach of our current data.

## 4.3 Multiplicity and flow measurements

In this section, we begin with a discussion of ALICE data from two different heavy-ion collision systems, namely Pb–Pb and Xe–Xe, that have yielded various measurements [174,176,279,809], which can be directly compared to the initial state models described earlier in this review. Figure 97 shows the measured $dN_{ch}/d\eta$ divided by the degrees of freedom in each initial state model. It has been demonstrated that entropy is approximately conserved through the

<figure><img src="image_114.png" /><figcaption>**Fig. 98** Comparisons of ALICE $v_2\{4\}/v_2\{2\}$ results from [279,809] to $\varepsilon_2\{4\}/\varepsilon_2\{2\}$ from the same initial state models shown in the previous figure: the MC Glauber nucleon model [42], the MC Glauber constituent quark model [958], $T_R\text{ENTo}$ [462], and IP-Glasma [44,956]</figcaption></figure>

evolution of the QGP using hydrodynamic simulations for $\eta/s$ values around $1/4\pi$ [1004]. The number of degrees of freedom in the initial state (or initial state entropy) should therefore be proportional to the number of particles produced in the final state. It is clear that this criterion is not met for the MC Glauber nucleon model since the scaled $dN_{ch}/d\eta$ increases for more central collisions, and based on the more central data, the MC Glauber constituent quark model also seems to fail. On the other hand, both the $T_R\text{ENTo}$ (which provides as an output the initial state entropy directly) and IP-Glasma models show a much better scaling, since the scaled $dN_{ch}/d\eta$ appears more constant as a function of centrality.

Figure 98 shows ALICE measurements of the ratio of elliptic flow coefficients estimated from 2- and 4-particle correlations $v_2\{4\}/v_2\{2\}$, compared to the same model predictions for the ratio of eccentricities $\varepsilon_2\{4\}/\varepsilon_2\{2\}$. The second order eccentricity $\varepsilon_2$ in the initial state models is determined based on the configuration of the overlap density. Both $v_2\{m\}$ and $\varepsilon_2\{m\}$ are sensitive to event-by-event fluctuations of $v_2$ and $\varepsilon_2$ respectively. Such fluctuations have a positive contribution to $v_2\{2\}$ and $\varepsilon_2\{2\}$, and have a negative contribution to $v_2\{4\}$ and $\varepsilon_2\{4\}$. Assuming $v_2 \propto \varepsilon_2$, the ratios $v_2\{4\}/v_2\{2\}$
and $\varepsilon_2\{4\}/\varepsilon_2\{2\}$ should therefore be equivalent. Hydrodynamic calculations have indeed shown these two quantities are compatible within 5% (denoted by the gray area). It is clear here that the $T_{\text{RENT}}$ and IP-Glasma models offer the most competitive descriptions, which is also the case for the multiplicity measurements shown in Fig. 97.

The success of both the $T_{\text{RENT}}$ and IP-Glasma models poses some key questions, but also provides reassurances when these models are used to extract QGP properties, as discussed in Sect. 2, since they appear to model the initial state quite well. As mentioned previously, each model has a fundamentally different approach to modelling the initial state, and the differing ways these models combine nuclear matter can be understood by considering the nuclear matter density projected in the $xy$ plane. In this case, $T_a(x, y)$ represents such a distribution for the projectile nucleus, while $T_b(x, y)$ for the target nucleus. The IP-Glasma model has its roots in a saturation QCD based approach [44], where the assumption of a high density of gluons in the initial state leads to the prediction that the overlap density should be characterised by $T_a T_b$. This is a feature of both weak and strong coupling theories in QCD [1005]. The $T_{\text{RENT}}$ model assumes that this combination should be $\sqrt{T_a T_b}$, and such an approach has been suggested by a 3D modelling of the initial state [1006]. It therefore seems clear that measurements sensitive to both initial state transverse and longitudinal effects might provide further distinguishing power.

Measurements of correlations between the average transverse momentum of all particles in a single event [$p_T$] and their anisotropic flow coefficients $v_n$, reflecting the correlations between energy density (therefore the size) and the shape of the initial conditions, offer another avenue to resolve this question. They are quantified by a modified Pearson correlation coefficient $\rho(v_n^2, [p_T])$. Because the physical mechanism driving $\rho(v_n^2, [p_T])$ arises from the initial state conditions [1007–1009], measurements of $\rho(v_n^2, [p_T])$ can provide valuable information in this regard. Figure 99 shows the centrality dependence of $\rho(v_2^2, [p_T])$ in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV measured by ALICE [1010]. It has a weak centrality dependence and is positive for the presented centrality range. This means that $v_2$ and $[p_T]$, hence the size and shape of the system in the initial stages, are positively correlated. Hydrodynamic model calculations from v-USPhydro [1011], Trajectory [834], JETSCAPE [1012] based on $T_{\text{RENT}}$ initial conditions, and IP-Glasma+MUSIC [1008], are shown for comparison. The IP-Glasma+MUSIC calculation quantitatively reproduces the measured $\rho(v_2^2, [p_T])$. For the calculations using $T_{\text{RENT}}$ initial conditions, but with different tuned input parameters, both v-USPhydro and Trajectory show a strong centrality dependence and underestimate the data by more than 50% for centrality above 30%. Furthermore, these models predict an opposite sign of $\rho(v_2^2, [p_T])$ with respect to

<figure><img src="image_115.png" /><figcaption>**Fig. 99** Centrality dependence of $\rho(v_2^2, [p_T])$ in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [1010]. The statistical (systematic) uncertainties are shown with vertical bars (filled boxes). The initial state estimations are represented by lines in the figures, while IP+Glasma+MUSIC+UrQMD [1008], v-USPhydro [1011], Trajectory [834], and JETSCAPE [1012] hydrodynamic model calculations are shown with hatched bands</figcaption></figure>

data for centralities above 40%. The discrepancies are more pronounced for the JETSCAPE predictions [1012], which become negative for centralities above 20%. Such discrepancies between IP-Glasma and $T_{\text{RENT}}$ cannot be attributed to the effect of the initial momentum anisotropy predicted by the CGC framework in IP-Glasma, as its impact is insignificant in the presented centrality ranges. Instead, the discrepancies are expected to arise from different geometric effects in the initial state. In particular, $\rho(v_2^2, [p_T])$ measurement favours smaller values of the width of the colliding nucleons [1013]. Such smaller values are also supported by a recent study based on the Trajectory framework [1014] that uses the ALICE measurement of the hadronic Pb–Pb cross section [1015]. These observations provide significant distinguishing power between different initial state models, which was not possible via the multiplicity or anisotropic flow fluctuations measurements described previously. Moreover, $\rho(v_2^2, [p_T])$ is found to be sensitive to the nuclear quadrupole deformation [1016] and has the potential to probe nuclear triaxial structure [1017]. Such measurements therefore open a unique window for the study of nuclear structure in heavy-ion collisions at the LHC. Full exploitation of the LHC as an imaging tool will advance our understanding of strongly-correlated nuclear systems via probes and techniques complementary to those utilized in low-energy applications. Such studies will ultimately yield unique insight into the behavior of quantum chromodynamics (QCD) across systems and energy scales [1018].

Finally, assuming that $v_2$ ($v_3$) has a linear response to the initial $\varepsilon_2$ ($\varepsilon_3$), it is expected that the normalised symmetric cumulant $NSC(3, 2)$ measured in the final state could reflect the initial correlations between $\varepsilon_2^2$ and $\varepsilon_3^2$. This has been validated in hydrodynamic calculations [307,313] where a
<figure><img src="image_116.png" /><figcaption>**Fig. 100** Normalised Symmetric Cumulants of the second and third orders from a variety of collisions as a function of the number of charged hadrons [280]. They are compared to predictions from the IP-Glasma+MUSIC model [48,953]</figcaption></figure>

good agreement between initial $NSC^\epsilon(3, 2)$ and final state $NSC(3, 2)$ is seen, independent of the type of the initial state models or the transport coefficients applied. Most of hydrodynamic model calculations [309,313,326,474] can only qualitatively or at best semi-quantitatively describe the ALICE $NSC(m, n)$, as discussed in detail in Sect. 3.3. If such a linear response of $v_2$ ($v_3$) to $\epsilon_2$ ($\epsilon_3$) holds also in small collision systems, then one can use the measured $NSC(3, 2)$ to probe the initial $\epsilon_2^2$ and $\epsilon_3^2$ correlation, and thus constrain the initial state for small collision systems, which is still poorly known so far. In Fig. 100, normalised $SC(3, 2)$ measurements are presented in both small and large systems [280], where the 3-subevent method is used to largely suppress the non-flow contamination [875,1019]. Negative $NSC(3, 2)_{3\text{-sub}}$ is observed in Xe-Xe and Pb-Pb collisions down to multiplicity of $N_{ch} \approx 100$, which suggests an anti-correlation between $v_2^2$ and $v_3^2$. There is a hint of a change to a positive sign of $NSC(3, 2)_{3\text{-sub}}$ in Pb-Pb collisions for multiplicity below 100. This tendency is also seen at even lower multiplicities in p-Pb and pp collisions. It is not observed in the measurement using larger $\eta$ acceptances in ATLAS [1020] and CMS [1021]. This could be due to different contributions of non-flow and longitudinal decorrelations when different kinematic regions in pseudorapidity are used. To further understand non-flow contamination, calculations from PYTHIA 8 model [856] were used, which show that the estimated non-flow effect cannot describe the ALICE data, nor the difference of ALICE with respect to the other measurements [1020, 1021]. To improve the understanding of the origin of the measured $NSC(3, 2)_{3\text{-sub}}$, several hydrodynamic calculations are presented for comparisons. It is found that the IP-Glasma+MUSIC+UrQMD [48,953] calculations for Xe-Xe and Pb-Pb collisions reproduce the negative correlation at large multiplicities. This negative sign persists

in simulations down to the lowest multiplicities, the change of the sign is not observed even in pp and p-Pb collisions. However, different results are seen in $T_{RENTO}+\nu\text{USPhydro}$ model calculations [1022], which predicted sign changes at low-multiplicity ($N_{part}$) for Pb-Pb, Xe-Xe, Ar-Ar and also O-O collisions, although there is no prediction for p-Pb and pp available. For pp collisions, the positive $NSC(3, 2)_{3\text{-sub}}$ qualitatively agrees with the iEBE-VISHNU calculations using HIJING initial conditions [876,887]. Last but not least, two calculations from the initial state correlation could also reproduce the sign changes in pp collision [1023, 1024]: one is based on the initial momentum correlations from colour domains [1023] and the other is based on spatial correlations between gluonic hot spots [1024].

The differences between different experiments and between different theoretical models are not yet fully understood. However, the differences in various model calculations show that the $NSC(3, 2)$ observable has a good sensitivity to initial conditions. Thus, the presented ALICE measurements should be very useful to further pin down the uncertainty in the initial state models and significantly improve the overall understanding of the initial conditions in large and small collision systems. This is essential to extract precise information on the properties of the QGP created in heavy-ion collisions at the LHC, as discussed in Sect. 2.2.5, and to reveal the origins of the collectivity observed in small collision systems (Sect. 3.3).

## 4.4 Conclusions

**Nuclear shadowing.** $Z^0$ and $W^\pm$ boson measurements by ALICE show clear evidence of nuclear shadowing effects at a resolution scale $Q^2 \sim M_{Z,W}^2$ increasing towards forward rapidity where smaller $x$ values are probed. First measurements of the coherent $J/\psi$ photoproduction in Pb-Pb UPCs provide strong constraints on gluon nPDFs, in particular, a gluon shadowing factor $R_g \approx 0.65$ at $x \approx 10^{-3}$ and $Q^2 \sim \frac{1}{4}m_{J/\psi}^2$ can be determined from the ALICE data. The first measurement of the $t$-dependence of the coherent $J/\psi$ photoproduction illustrates the potential of $t$-differential measurements for nuclear shadowing constraints in the impact parameter plane. Coherent $\rho^0$ photoproduction measurements in Pb-Pb and Xe-Xe UPCs show an approximately linear dependence of the $\gamma A$ cross section as a function of atomic mass number $A$ revealing an importance of Gribov-Glauber inelastic shadowing effects at LHC energies. ALICE measurements of coherent $\rho^0$ photoproduction in neutron emission classes confirmed that forward neutron emission can be used to tag different impact parameter ranges in UPCs.

**Gluon saturation.** Exclusive $J/\psi$ photoproduction cross section off protons, measured by ALICE in p-Pb UPC over 3
orders of magnitude in photon–proton centre-of-mass energy, keeps rising as a power law with energy showing no clear evidence of gluon saturation effects in the proton down to $x \sim 10^{-5}$. On the other hand, suppression of vector meson photoproduction observed by ALICE in Pb–Pb UPCs can be well described by both calculations with nuclear modified PDFs and by saturation-based models.

*Nuclear structure and overlap.* The charged particle multiplicity $dN_{ch}/d\eta$ is expected to scale with the number of degrees of freedom and/or the entropy in the initial state. Such scaling fails for the MC Glauber nucleon and constituent quark models, but holds for the $T_{\text{RENT}}$ and IP-Glasma models. In this case, the densities in the initial state are assumed to scale as $\sqrt{T_a T_b}$ and $T_a T_b$, respectively, where $T_i$ is the nuclear thickness function. The ratio of 2- and 4-particle flow coefficients $v_n$ gives direct access to information on the initial eccentricity and its fluctuations. As with the multiplicity measurements, the $T_{\text{RENT}}$ and IP-Glasma models offer the most competitive descriptions. On one hand, this cements their role in extracting the $\eta/s$ and $\zeta/s$ QGP transport properties. On the other hand, our measurements of $\rho(v_n^2)$, [$p_T$] provide further distinguishing power, and significantly favour a small nucleon width for the initial condition, which is consistent with the hadronic nucleus-nucleus cross section study. Such observations have broader implications regarding the structure of a nucleus at high energies in terms of its constituents, and how these constituents combine in a heavy-ion collision to form the QGP.

*Initial state geometry in small and large systems.* The study of correlations between different order flow vectors, especially with the normalised symmetric cumulant $NSC(3, 2)$ in large and small collision systems, provides new possibilities to investigate the role of initial state geometry and possible initial momentum correlations. These will eventually help to pin down the origin of the flow signal observed in small systems.

# 5 Nuclear physics at the LHC: (anti)nuclei formation and hadron–hadron interactions

Besides being at the forefront in the study of the QGP evolution and its properties, in recent years the ALICE experiment has demonstrated to be uniquely sensitive to specific aspects of low-energy nuclear and hadronic physics. The matter–antimatter symmetry that governs hadron production at LHC collision energies allows for the study of production yields, and of the elastic and the inelastic interaction cross sections with the detector materials of light nuclei and antinuclei at the same time [1025]. The production of (anti)deuteron, (anti)triton, (anti)$^3$He, (anti)$^4$He and

even (anti)hypertriton ($^{3}_{\Lambda}$H) nuclei have been studied in several collision systems [387,389,863,1026–1029], providing a solid test for microscopic models of (anti)nucleus production and their inelastic interactions. Such capabilities position ALICE as a discovery machine for exotic nuclei.

The residual strong interaction between hadron pairs, which is responsible for the stability of atomic nuclei, also acts among all hadrons produced in collisions at accelerators. To infer the properties of this residual strong interaction, the ALICE experiment has measured correlations in the momentum space among particle pairs with the femtoscopy technique [1030–1041]. This method has been applied to several hadron–hadron pairs, including kaon–nucleon pairs, all ground states of hyperons, and to baryon–antibaryons pairs. In particular, the strong interaction has been quantified for several hadron pairs for which it was not previously measured [1031, 1032, 1034–1040].

Information on the strong interaction among hadrons can also be inferred from the measurement of the lifetime and binding energy of hypernuclei. The measurement of the $^{3}_{\Lambda}$H lifetime achieved by ALICE [389, 1042, 1043], which has steadily acquired increasing precision in recent years and represents the most precise measurement ever carried out, provides a value compatible with the free $\Lambda$ lifetime. This has contributed to solving a long-standing puzzle, i.e. an observed hypertriton lifetime rather short compared to the free $\Lambda$ lifetime.

This chapter illustrates the main results achieved by ALICE in the field of nuclear physics with and without strangeness, showing how the different observables can be connected to each other. The highlighted connections lay the groundwork for future and more accurate studies which will enable ALICE to contribute with unprecedented precision to nuclear physics within the $SU(3)$ flavour sector.

## 5.1 Production of nuclei from small to large collision-system size

A detailed understanding of the formation of light (anti)nuclei and hypernuclei was achieved by investigating their production as a function of the system size. In small collision systems, such as pp or p–Pb, the nucleon emitting source radius is about 1–2 fm [1044]. The size of the source is thus smaller than (or similar to) the size of the (anti)nuclei that are formed from it, as for instance the RMS charge radius of the deuteron is 2.14 fm [1045]. In large collision systems like Pb–Pb the opposite is true as the radius of the homogeneity region of the emitting source (see Sect. 2.1.5) is about 4–9 fm [1046]. A special role can be attributed to the hypertriton, whose wave function is wider than the fireball produced in central Pb–Pb collisions ($^{3}_{\Lambda}$H RMS radius $\sqrt{\langle r_{d\Lambda}^2 \rangle} \approx 10.6$ fm [1047]). In addition to this, effects of baryon number conservation
on the particle production yields are best studied by varying the system size, because the yield of protons per event and unit of rapidity varies from about 0.12 in inelastic pp collisions [1048] to about 33 in central Pb–Pb collisions [273]. In practice, such studies are often performed as a function of the event multiplicity expressed in terms of the charged-particle pseudorapidity density (dN<sub>ch</sub>/dη) and the particle-source radius is found to be proportional to (dN<sub>ch</sub>/dη)<sup>1/3</sup> [246] (see Sect. 2.1.5).

### 5.1.1 Light-nuclei yield measurements

The measured yields of (anti)nuclei relative to those of protons as a function of (dN<sub>ch</sub>/dη) are summarised in the left panel of Fig. 101. Similarly to other light-flavoured hadrons (see Sect. 3.2), a smooth trend with (dN<sub>ch</sub>/dη) across different collision systems and independence on the centre-of-mass energy are observed. The nucleus-to-proton ratios exhibit an increasing trend with increasing multiplicity for low-(dN<sub>ch</sub>/dη) and they tend to saturate for (dN<sub>ch</sub>/dη) ≳ 100. The yield ratios at high-multiplicity are in good agreement with predictions from statistical hadronisation models (SHM) in the grand canonical limit (see Sect. 2.3.1). In these models, the production yield per unit of rapidity of a given particle species, dN/dy, is mostly determined by the chemical freeze-out temperature (T<sub>chem</sub>) [122] and particle mass m with an approximately-exponential dN/dy ∝ exp(-m/T<sub>chem</sub>) dependence. The decreasing trend towards smaller multiplicities is also explained in the model as a suppression due to baryon number conservation, i.e. a limitation of the available configurations that are compatible with a given canonical ensemble [1052]. This picture only holds if correlation volumes V<sub>c</sub> between one and three units of rapidity are allowed. However, these values are in contradiction with those obtained for proton-to-pion ratios [868].

Besides the SHM, the data in the left panel of Fig. 101 are also compared to predictions from the coalescence model, in which nuclei are formed by protons and neutrons that are close in phase space after the kinetic freeze-out [419]. In this approach, the invariant yield of a (anti)nucleus with mass number A can be written as

$$E_A \frac{d^3 N_A}{d p_A^3} = B_A \left( E_p \frac{d^3 N_p}{d p_p^3} \right)^A, \quad (27)$$

where p<sub>p</sub> is the momentum of the nucleon, p<sub>A</sub> = A p<sub>p</sub> is the nucleus momentum, and the proportionality factor B<sub>A</sub> is referred to as the coalescence parameter. The dependence of B<sub>A</sub> at a given value of p<sub>T</sub>/A on the event multiplicity is shown in the right panel of Fig. 101 for deuteron and <sup>3</sup>He. A strong decrease with increasing multiplicity is observed for both B<sub>2</sub> and B<sub>3</sub> that is explained in the coalescence model by an

inverse proportionality to the source volume: the coalescence probability of two or three nucleons is suppressed if they are largely separated in configuration space. This effect becomes less apparent in small collision systems where the size of the produced nucleus is larger than the source size. This effect becomes less apparent in small collision systems where the size of the produced nucleus is larger than the source volume. In general, the measured yields of light nuclei and their multiplicity dependence across different collision systems can be qualitatively described by the SHM for particle production at the chemical freeze-out and by the coalescence of nucleons at the kinetic freeze-out, even though some open questions still need to be addressed (see Sect. 3.2).

Due to its wide wave function, hypertriton yield measurements are fundamental to distinguish between the models, because of the relevant role of the size of the produced hypernucleus in the coalescence model predictions [1053, 1054]. Recent results on the <sup>3</sup>ΛH/Λ yield ratio in p–Pb collisions [1055] are shown in the left panel of Fig. 102 along with the same measurements in Pb–Pb collisions at √s<sub>NN</sub> = 2.76 TeV [389]. The measured ratio excludes with high significance the canonical versions of the SHM with V<sub>c</sub> ≥ 3 dV/dy. Nevertheless, it remains to be seen whether advanced versions of the SHM using the S-matrix approach to account for the interactions among hadrons [418] will be able to solve this discrepancy. The <sup>3</sup>ΛH/Λ ratio is well described by the two-body coalescence prediction (dark blue band), while the three-body formulation (light blue band) is disfavoured by the measurement.

The right panel of Fig. 101 shows the strong sensitivity of coalescence studies to the source volume, which will be discussed in more detail in Sect. 5.2. The coalescence process and the femtoscopic correlation between nucleons are closely related to each other [1056,1057] and simultaneous measurements of both quantities in future analyses will lead to precise understanding of the formation process of QCD bound states in high-energy particle collisions.

Recently, the measurement of d production fluctuation has been measured [1058]. Simple coalescence models fail to fit simultaneously the measurement of the cumulants ratios and the proton deuteron correlation coefficient. The state of-the-art SHM models can describe the data, but with a value of V<sub>c</sub> significantly smaller with respect to that describe p/π ratio and consistent with that obtained from deuteron yield measurement.

### 5.1.2 Flow of light nuclei in Pb–Pb collisions

The measurements of flow harmonics can provide additional insights into the production mechanism of antinuclei. The v<sub>2</sub> of (anti)deuteron has been measured in Pb–Pb collisions at √s<sub>NN</sub> = 2.76 [1027] and √s<sub>NN</sub> = 5.02 TeV [1059]. The measured v<sub>2</sub> as a function of p<sub>T</sub> for two centrality intervals
<figure><img src="image_117.png" /><figcaption>**Fig. 101** (Left) d/p ratio (Top) and $ ^{3}\text{He/p} $ ratio (Bottom) in pp, p-Pb, and Pb-Pb collisions [387,863, 1026, 1028, 1029, 1049–1051] as a function of the mean charged-particle pseudorapidity density. (Right) Coalescence parameters $B_2$ (Top) and $B_3$ (Bottom) as a function of the mean charged-particle pseudorapidity density for $p_T/A = 0.75 \text{ GeV/c}$ (0.73 $\text{GeV/c}$) calculated using the average of particles and antiparticles.</figcaption></figure>

<figure><img src="image_118.png" /></figure>

In all panels the expectations for the canonical statistical hadronisation model (SHM) evaluated with Thermal-FIST [398] and coalescence approaches [1052–1054] are shown. For the SHM, two different values of the correlation volume $V_c$ [1053] are displayed. The uncertainties of the coalescence calculations, which are due to the theoretical uncertainties on the emission source radius, are denoted as shaded bands

in Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV is shown in the right panel of Fig. 102. The $v_2$ of (anti)deuteron has been evaluated by means of the scalar-product method [262, 1060], which is a two-particle correlation technique. In the measured $p_T$ interval, $v_2$ increases with increasing $p_T$ and going from central to more peripheral Pb-Pb collisions, as expected from the relativistic hydrodynamic description of the collective expansion of a hot and dense medium [1004]. The measurement shows that these effects, related to initial geometrical anisotropy and also observed for most hadron species at LHC energies [290, 293], are also visible for (anti)deuterons.

The elliptic flow of (anti)deuterons is compared with the blast-wave model [276, 1061, 1062], which is a simplified version of a full relativistic hydrodynamical calculation (see also Sect. 2.2.2). In the lower right panel, the ratios between the measured (anti)deuteron $v_2$ and the one calculated with the blast-wave parameterisation for the two considered centrality intervals are shown as markers. The blast-wave model underestimates the measured (anti)deuteron elliptic flow in semi-peripheral collisions for $p_T > 1.4 \text{ GeV/c}$, while it describes the data for central events. The (anti)deuteron $v_2$ is also compared to a coalescence model based on mass-

number scaling and isospin symmetry, for which the proton and neutron $v_2$ are identical, and the $v_2$ of deuteron can thus be calculated from the measured $v_2$ of protons. The width of the band represents the combination of the statistical and systematic uncertainties on the measured proton $v_2$. The coalescence model overestimates the (anti)deuteron $v_2$ by about 20–30% in central collisions and is closer to the data for semi-peripheral collisions, as illustrated in the lower panel.

The two simplified models bracket an interval in which the measured deuteron $v_2$ is located and can be used to describe the data in different multiplicity regimes, indicating that currently none of these two models is able to describe the (anti)deuteron anisotropic flow in the low- and high-multiplicity intervals at the same time.

## 5.2 Determination of a universal source for small colliding systems

The size of the particle emitting source created in different collision systems plays an important role in the study of (anti)nucleus formation [1056, 1057]. It can be measured by employing the femtoscopy techniques (see Sect. 2.1.5).
<figure><img src="image_119.png" /><figcaption>**Fig. 102** (Left) $^3\Lambda$H/$\Lambda$ measurements in p–Pb (in red) and Pb–Pb collisions (in blue) as a function of $(dN_{ch}/d\eta)$ [389, 1055]. The vertical lines and boxes are the statistical and systematic uncertainties, respectively. The expectations for the SHM and coalescence models are shown as lines and bands, respectively. (Right) (anti)deuteron $v_2$ as a function of $p_T$ for Pb–Pb at $\sqrt{s_{NN}} = 5.02$ TeV in two centrality intervals. Results for 10–20% and 40–50% centrality intervals shown as green circles</figcaption></figure>

and magenta squares, respectively. Data are compared with the expectations from simple coalescence model and predictions obtained from a blast-wave fit to the $v_2(p_T)$ of pions, kaons, and protons [402], shown as shaded bands and dashed lines, respectively. In the lower panel, the ratios between the coalescence model predictions are shown as bands, while the ratio to the blast-wave predictions are shown as markers

This source size plays also a fundamental role in the study of the residual strong interaction between pairs of hadrons. The correlation function employed in both kinds of studies reads

$$C(k^*) = \int S(r^*) |\Psi(\vec{k}^*, \vec{r}^*)|^2 d^3r^*, \quad (28)$$

where $r^*$ is the relative distance between the two particles and $k^*$ is half of their relative momentum, both measured in the pair rest frame (*). The source function $S(r^*)$ is, in general, the generic expression for a three dimensional Gaussian source. In case of a symmetric source, as considered here, the relevant one-dimensional probability density function of $r^*$ is obtained with a trivial angular phase space factor. The source size $r_0$ obtained from the fit to the correlation function of p–p and p–$\Lambda$ pairs for different $m_T$ intervals is shown on the left panel of panel of Fig. 103. Theoretical approaches, such as chiral effective field theory ($\chi$EFT) [1063–1065] or lattice QCD computations [1066, 1067], provide information on the wave function making it possible to compare their predictions directly to the measured $C(k^*)$, given a known source function. Small collision systems, pp in particular, are beneficial for study the source function knowing the interaction among particles. It is known that strongly decaying resonances may lead to significant exponential tails of the source distribution, modifying the measured source radii. A new procedure was developed [1044] to quantify these modifications. Within this model, the core radius $r_{core}$, which represents the universal

Gaussian core source common to all the produced particles, is a free fit parameter. The abundances of the resonances which modify the source size are taken from the statistical hadronisation model [1068] and their branching ratios and lifetimes from the PDG [440]. The kinematic distributions related to the emission are based on the EPOS transport model [1069]. For both proton and $\Lambda$ baryons, approximately 2/3 of the total production yield is associated with intermediate resonances with a mean lifetime of 1.7 fm/c and 4.7 fm/c, respectively. The fit to the data employing the new source model leads to an identical $m_T \in (1.1–2.2)$ GeV/c$^2$ scaling of $r_{core} \in (0.85–1.3)$ fm for p–p and p–$\Lambda$ pairs (right panel of Fig. 103). This result supports the assumption of a common source parameterisation with $r_{core}(m_T)$ for any hadron pair produced in pp collisions.

## 5.3 Hyperon–nucleon and kaon–nucleon interactions

### 5.3.1 Hypertriton lifetime

The measurement of the properties of hypernuclei gives access to the study of interactions among hyperons and nucleons. The hypertriton $^3\Lambda$H, for example, is a weakly bound state of a proton, a neutron and a $\Lambda$ hyperon. The binding energy, or more specifically the separation energy of the $\Lambda$ to the pn core, was recently measured by ALICE [1043]. The measurement is consistent with the previous world-average $B_\Lambda = (130 \pm 50)$ keV [1070, 1070–1073]. This small sep-
<figure><img src="image_120.png" /><figcaption>**Fig. 103** (Left) Source radius $r_0$ as a function of $\langle m_T \rangle$ for the assumption of a purely Gaussian source. (Right) Source radius $r_{core}$ as a function of $\langle m_T \rangle$ for the assumption of a Gaussian source with added resonances. The blue crosses result from fitting the p–p correlation function with the strong Argonne $v_{18}$ [1063] potential. The green squared crosses</figcaption></figure>

(red diagonal crosses) result from fitting the p–Λ correlation functions with the strong χEFT LO [1064] (NLO [1065]) potential. Statistical and systematic uncertainties are indicated by vertical bars and boxes, respectively [1044]

aration energy leads to the expectation that the lifetime of the hypertriton is very close to that of the free Λ hyperon, $\tau_Λ = (263.2 \pm 2.0)$ ps [440]. Similar conclusions can be expected considering the $^3_\Lambda$H wave function. In a simple quantum-mechanical model, the RMS radius of this hypernucleus is about $\sqrt{\langle r_{d\Lambda}^2 \rangle} = 10.6$ fm, if a deuteron-Λ bound state is assumed [1074, 1075]. Similar values are extracted from more sophisticated theoretical models for the RMS radius of the hypertriton [1047, 1076]. Since this means that the Λ is, with a very high probability, several femtometres away from the other nucleons, the lifetime can be expected to be close to the one of the Λ.

A compilation of the available measurements of the lifetime of hypertriton is displayed in Fig. 104. The data points, shown as markers, were obtained using several experimental techniques and collisions systems. Specifically, emulsions, bubble, and tracking chambers were used to detect $^3_\Lambda$H produced using meson beams on different targets [1077–1082] and general purpose detectors were used to detect $^3_\Lambda$H produced in heavy-ion collisions at different energies [389, 1042, 1043, 1083–1086]. It should be noted that the data from heavy-ion collisions are characterised by statistical and systematic uncertainties of less than 15%, smaller by a factor three than bubble chambers experiment results. The early measurements by STAR [389, 1085] and ALICE [1083] provided values typically lower than the free Λ lifetime, which are in contrast with the small value of the Λ separation energy. This raised the so-called “hypertriton lifetime puzzle”. The most recent ALICE measurement (shown as a red marker in Fig. 104), with unprecedented precision, is consistent with the Λ lifetime and conclusively settles the puzzle.

Besides the experimental results, the theoretical predictions for the $^3_\Lambda$H lifetime are reported in Fig. 104 for comparison with the data. The calculation performed by Rayet and Dalitz [1090] takes into account the phase space factors and the Pauli principle, including also corrections to account for final state pion scatterings and the non-mesonic weak decay channel. More recently, Congleton [1087] published a prediction for the $^3_\Lambda$H lifetime, which was obtained using updated values for nucleon-nucleon and hyperon-nucleon potentials and is expected to be close to one of the free Λ hyperon. The prediction by Kamada et al. [403] was obtained with a rigorous determination of the hypernucleus wave function and of the three nucleons scattering states, thus finding a value of 256 ps, which is the closest to the free Λ lifetime value. Recently, Garcilazo and Gal performed a calculation [1088] using the wave function generated by solving three-body Faddeev equations. This calculation is the first one which includes also a final-state interaction effect of the pion. This leads to a reduction of the expected hypertriton lifetime down to 81% of the free Λ value due to additional attraction from the final-state interaction between the hypertriton decay products. A more recent approach [1089] incorporates a hypertriton wave function that is based on chiral effective field theory, distorted waves for the pions and p-wave interactions, while in Ref. [1088] only s-waves were included. In addition, in Ref. [1089] show for the first time that the off-shell $\Sigma \to N + \pi$ weak decay contribution to the hypertriton decay rate reduces its lifetime by about 10%. Finally, the most recent calculation by Hildenbrand and Hammer [1091] show a value of the hypertriton lifetime very close (about 0.9 $\tau_\Lambda$) to the one of the free Λ. In this theory, a pion-less effective field theory approach with Λd degrees of freedom and the commonly accepted separation energy
**Fig. 104** Collection of the $^3\Lambda$H lifetime measurements obtained with different experimental techniques. The lowest point corresponds to the latest ALICE measurement [1043]. The horizontal lines and boxes are the statistical and systematic uncertainties respectively. The orange band represents the average of the lifetime values and the lines at the edge correspond to $1\sigma$ uncertainty. The dashed-dotted lines represent five theoretical predictions [403, 1087–1089]

<figure><img src="image_121.png" /></figure>

of $B_\Lambda = (130 \pm 50)$ keV is used. The dependence of the lifetime on the separation energy was studied: increasing $B_\Lambda$ up to 2 MeV, led to a reduction of $^3\Lambda$H lifetime down to about $0.9 \tau_\Lambda$.

### 5.3.2 Nucleon–hyperon and hyperon–hyperon interactions: $p-\Lambda, \Lambda-\Lambda$ and $p-\Sigma^0$

A direct measurement of the two-body nucleon–hyperon (N–Y) interaction was carried out in recent years. The strangeness $|S| = 1$ sector, p–Λ in particular, is already partially constrained by the measurement of scattering processes [1092–1094], yet the precision of these data is rather limited. Theoretical methods allow for the extrapolation of the interaction from the vacuum to large baryonic densities [1095]. Thus, it is of utmost importance to determine the two-body N–Y and three body N–N–Y interactions to study different interacting systems (see Sect. 7.1). The recent measurements of the p–Λ [1038] and p–$\Sigma^0$ [1036] interaction in high-multiplicity pp collisions at $\sqrt{s} = 13$ TeV by ALICE provided new and valuable input to the study of both channels and their strong coupling $N\Sigma \leftrightarrow N\Lambda$.

The measurement for the p–Λ system, shown in Fig. 105, represents the experimental result with highest statistical precision at low relative momentum $k^*$, about a factor of 25 better with respect to the available scattering-experiment results. Moreover, this measurement provided the first direct observation of the kinematic cusp at $k^* \approx 290$ MeV/c, corresponding to the $N\Sigma$ threshold. The sensitivity of the data at $k^* < 100$ MeV/c is sufficient to probe the correlation signal due to feed-down of $p-\Xi^- \to p-\Lambda$ and $p-\Sigma^0 \to p-\Lambda$. The former is modelled using the results from lattice calculations

<figure><img src="image_122.png" /></figure>

**Fig. 105** p–Λ correlation function measured in pp collisions at $\sqrt{s} = 13$ TeV [1038] compared to theoretical predictions obtained using the NLO19 version of $\chi$EFT. On the right fit is performed assuming a residual $p-\Sigma^0$ interaction as predicted by $\chi$EFT, while on the right a negligible interaction is assumed. The middle panels show a zoomed version along the y-axis, to further investigate the kinematic cusp, around 290 MeV/c corresponding to the $N\Sigma \leftrightarrow N\Lambda$ coupling. The bottom panels show the discrepancy of model to the data

by the HAL QCD collaboration, which are in agreement with ALICE data (see Sect. 5.3.3). Due to the coupling $N\Sigma \leftrightarrow N\Lambda$, the $p-\Sigma^0 \to p-\Lambda$ has to be treated within the same theoretical framework as p–Λ. Thus, the prediction of $\chi$EFT has been employed. The fit to the data using the $\chi$EFT predictions is shown in the left panel of Fig. 105 where the bottom panel represents the deviation between the theory and the data in each $k^*$ interval. The kinematic cusp is well
described. However, there is a systematic deviation related to the slope for $k^* < 100$ MeV/c. The right panel of Fig. 105 shows the fit result obtained by modifying the p-$\Sigma^0$ contribution to a flat function, mimicking a very weak interaction. A similar effect can be achieved by slightly reducing the strength of the p-$\Lambda$ attraction. The main conclusion is that the current data call for more precise theoretical calculations.

The N-$\Sigma$ interaction was also directly probed employing correlation studies. The pioneering measurement by ALICE in high-multiplicity pp collisions [1036] analysed the decay channel $\Sigma^0 \to \Lambda\gamma$ and the subsequent decay of the $\Lambda$ to $p\pi^-$. The soft photon from the decay is detected via conversions to an electron-positron pair in the ALICE detector material. The resulting purity of the $\Sigma^0$ sample is 34.6%, thus rather low, and it is owed to the dominant contribution from uncorrelated primordial $\Lambda\gamma$ combinations. The data were compared to state-of-the-art theoretical descriptions of the interaction, such as fss2 [1096], two versions of the soft-core Nijmegen models (ESC16 [1097], NSC97f [1098]), and results of $\chi$EFT at Next-to-Leading Order (NLO) [1099]. The current precision of the experimental data does not yet allow for a discrimination among the different models. Nevertheless, these studies demonstrate the feasibility of the measurement and the prospects of the analysis of the larger data samples to be collected during the ongoing LHC Run 3 [902].

The $\Lambda$-$\Lambda$ interaction was investigated by directly measuring the correlation among the hyperons and by measuring the $\phi$-N correlation. The direct measurement of the $\Lambda$-$\Lambda$ correlation in pp collisions at $\sqrt{s} = 13$ TeV and p-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV evidenced a shallow attractive strong interaction potential but could not exclude the presence of a $\Lambda$-$\Lambda$ bound state [1033]. The most precise upper limit for the binding energy of this state was extracted and found to be equal to $B_{\Lambda\Lambda} = [3.2_{-2.4}^{+1.6}(\text{stat})_{-1.0}^{+1.8}(\text{syst})]$ MeV. An alternative method to study the hyperon-hyperon interaction is to measure the coupling between the $\phi$ meson and either nucleons or hyperons [1100]. The first measurement of the p-$\phi$ correlation in pp collisions at $\sqrt{s} = 13$ TeV was also carried out and the Lednický-Lyuboshits approach fit [229] was used to extract the scattering parameters [1040]. The small scattering parameters indicate a rather shallow p-$\phi$ interaction potential, which is consistent with the observed shallow $\Lambda$-$\Lambda$ interaction potential.

### 5.3.3 Multistrange hyperon-nucleon interactions

The strangeness $|S| = 2$ and $|S| = 3$ sectors have been also recently studied using phenomenological approaches [1097, 1101] and lattice QCD calculations [1066, 1067]. Experimentally little is known about the interaction of $\Xi$ and $\Omega$ baryons with nucleons. The practical difficulties of creating a beam of such particles prevent the construction of a reliable data base of scattering data, hence the experimen-

<figure><img src="image_123.png" /><figcaption>**Fig. 106** Measured p-$\Xi^-$ (Top) and p-$\Omega^-$ (Bottom) momentum correlation functions in pp collisions at $\sqrt{s} = 13$ TeV [1037]. The experimental data are shown as black symbols. The black vertical bars and the grey boxes represent the statistical and systematic uncertainties, and the square brackets show the $k^*$ bin width. The measurements are compared with theoretical predictions, shown as coloured bands, that assume either Coulomb or Coulomb+strong HAL QCD interactions. The width of the curves including HAL QCD predictions represents the uncertainty associated to the calculation and the grey shaded band represents the uncertainties associated with the determination of the source radius</figcaption></figure>

tal information relies on data on $\Xi$ hypernuclei [1102, 1103] and on a pioneer study of p-$\Omega$ two-particle correlation by STAR collaboration using Au-Au interactions [1104]. Data on $\Xi$ hypernuclei indicate the possibility that $\Xi$ baryons can be bound in a nucleus; data from p-$\Omega$ tend to favour the existence of a p-$\Omega$ bound state.

The ALICE collaboration has measured with high precision the p-$\Xi$ correlation function in p-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [1032] and the p-$\Xi$ and p-$\Omega$ correlation functions in pp collisions at $\sqrt{s} = 13$ TeV using high-multiplicity events [1037], demonstrating the attractive character of both interactions. The correlation function obtained by ALICE [1037] for p-$\Xi^-$ $\oplus$ p-$\Xi^+$ and p-$\Omega^-$ $\oplus$ p-$\Omega^+$ using high-multiplicity pp collisions at $\sqrt{s} = 13$ TeV are shown in Fig. 106. The data are corrected for experimental artefacts
and feed-down contributions, displaying the genuine correlation function of the pairs of interest and it can be directly compared to the theoretical predictions for a given source.

The data shown in Fig. 106 are compared with the correlation function corresponding to the Coulomb-only expectation (green curves) and with predictions that consider also the strong interaction calculated on the lattice by the HAL QCD collaboration [1066,1067]. The lattice QCD calculations predict an attractive interaction for both the p-Ξ and the p-Ω systems. The results obtained in pp confirms what was observed in the analysis of p-Pb collisions [1032]. In the p-Ω system, the interaction is attractive at all distances due to the absence of a repulsive core [1066]. The attractive core can accommodate the presence of a p-Ω bound state with a binding energy of 1.54 MeV [1067]. The presence of the bound state has a direct effect on the expected correlation function causing a depletion that, for the given radius, reduces the correlation function to a value lower than unity in the $k^*$ range between 100 MeV/c and 200 MeV/c, as highlighted in the inset in the right panel of Fig. 106. The blue and orange coloured bands corresponding to the HAL QCD prediction for the p-Ω system in Fig. 106 reflect the current uncertainty of the calculations due to the presence of coupled-channels in the p-Ω system.

In the case of the p-Ω the lattice QCD predictions lie below the experimental data, and the depletion characteristic of the presence of a bound state is not supported by the ALICE data. In order to obtain firm conclusions on the possible existence of the p-Ω bound state, and, if existent, quantify experimentally its binding energy, a differential analysis of the p-Ω correlations in systems with slightly different source sizes is necessary. A recent measurement of the Λ-Ξ⁻ correlation function in high-multiplicity pp collisions at $\sqrt{s} = 13$ TeV supports a very small coupling strength for this channel [1105] and hence the inelastic contribution to the p-Ω state should be rather small. More precise measurement of both the Λ-Ξ⁻ and Σ⁰-Ξ⁻ correlations will further reduce the uncertainties of the calculations. Such measurements will be performed at the LHC by ALICE in pp, p-Pb, and peripheral Pb-Pb collisions.

### 5.3.4 K-p interaction: evidence of coupled-channel contributions

Coupled-channel processes are widely present in hadron-hadron interactions whenever pairs of particles, relatively close in mass, share the same quantum numbers: baryonic charge, electric charge and strangeness. The coupling translates into on/off-shell processes from one system to the other. The multi-channel dynamics deeply modifies the hadron-hadron interaction and is at the origin of several phenomena, such as bound states and resonances, which crucially depend on the coupling between the inelastic channels. A key example

can be found in the Λ(1405) resonance, a molecular state arising from the coupling of antikaon-nucleon ($\overline{K}$-N) to Σ⁻-π [1106,1107].

The advantage of the femtoscopic measurement with respect to scattering experiments for the study of coupled channels is that the final state is fixed by the measured particle pair. Hence, the corresponding correlation function represents an inclusive quantity sensitive to all the available initial inelastic channels produced in the collision [1108,1109].

Estimates for the weights to be assigned to the different initial channels can be obtained using combined information on yields from statistical hadronisation models [391,398,1110] and on the kinematics of the produced pairs from transport models [1069]. A coupled-channel contribution can modify the $C(k^*)$ in two different ways, depending on whether the minimum energy at which they can be produced occurs below or above the reduced mass of the pair. Inelastic channels opening below threshold do not introduce any shape modification on the $C(k^*)$, acting as an effective attraction and increasing the signal of the correlation function. On the other hand, channels appearing above threshold lead to a modification of the $C(k^*)$ in the vicinity of the opening, which is typically translated into a cusp-like structure, whose height is driven by the coupling strength. These two main differences are illustrated in Fig. 107 for the K⁻-p system.

The K-p system contains couplings to several inelastic channels below threshold such as π-Λ, π-Σ and, due to the breaking of isospin symmetry, to charge-conjugated $\overline{K}^0$-n at roughly 4 MeV above threshold. In the left panel of Fig. 107, a schematic representation of the collision is shown. The correlation of K-p pairs composing the final-state is measured and its decomposition into contributions of different channels contributions is shown for two different source sizes in middle and right panel, respectively. The largest contributions on the $C(k^*)$ from coupled-channels occur for a small emitting source with Gaussian radius $r_G = 1$ fm as shown in middle panel. The $C(k^*)$ signal increases as the inelastic contributions are added and the cusp structure, visible when the $K^0$-n channel is explicitly added, indicates the opening of this channel above threshold. For both source radii, this structure already appears when the mass difference between K⁻ and $\overline{K}^0$ is considered, and it is present also in the elastic K⁻-p → K⁻-p contribution (dotted line). The effect of coupled-channels is suppressed when the source size is increased up to $r_G = 5$ fm, as in central heavy-ion collisions (right panel in Fig. 107).

This theoretical scenario and the extreme sensitivity of the correlation function to coupled-channel dynamics have been confirmed by the measurements of the K⁻-p correlation function, measured in different colliding systems, shown in Fig. 108. From left to right, the size of the emitting source increases as the correlation function is measured respectively
<figure><img src="image_124.png" /><figcaption>**Fig. 107** (Left) Sketch of the system configuration in femtoscopic measurements, where only the final $K^{-}p$ channel is measured. (Middle and right) Theoretical correlation function for $K^{-}p$, from the pure elastic term (dotted line) to the full $C(k^*)$ (solid line) with all coupled-</figcaption></figure>

channels ($\bar{K}^{0}-n, \pi-\Sigma, \pi-\Lambda$) included. The results are shown for two different radii, typically achieved in pp collisions (1 fm) and in heavy-ion collisions (5 fm)

<figure><img src="image_125.png" /><figcaption>**Fig. 108** Measured correlation function versus the relative momentum $k^*$ for $K^{-}p \oplus K^{+}\bar{p}$ pairs in pp collisions at $\sqrt{s} = 13$ TeV (left), in 40–60% centrality interval in p–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV (middle) and in 60–70% centrality Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV</figcaption></figure>

(right) [1112]. In all the panels, data are compared with the $\chi$EFT-based potential [1109] with fixed (blue bands) and free (red bands) coupling weights

in pp collisions [1034] and Pb–Pb collisions [1039] and the shape of the measured correlation function varies significantly. The cusp signature at $k^* = 58$ MeV/c is not present in large systems. As the source size increases, the enhancement at low momenta due to coupled-channel contributions follows the same trend, becoming less pronounced. In large collision systems such as Pb–Pb, the asymptotic part of the wave function is probed, hence the effects of coupled-channels on $C(k^*)$ are noticeably suppressed and a partial direct access to the elastic term can be obtained. This leads to a good agreement between the scattering length $f_0$ obtained fitting the Pb–Pb data and the values extracted from kaonic atom measurements, in which coupled-channel effects are typically not considered [1111]. The small colliding systems on the other hand provide the opportunity to test the coupling strength of different initial channels to the final channel $K^{-}p$ with unprecedented precision [1112].

Coupled-channel studies have been recently extended to the charm sector by studying the p–D$^{-}$ system in high-multiplicity pp collisions [1058]. This study suggests the possibility of the formation of a N–D bound state, but the current statistical significance of the data is not enough to draw a clear conclusion. Precise measurements of correlation functions between charm and light-flavour or strange hadrons will become accessible with the Run 3 data samples.

## 5.4 Baryon–antibaryon interactions in Pb–Pb collisions: possible existence of bound states

The study of the strong interaction in the baryon–antibaryon sector is very challenging, because obtaining beams or targets of antimatter is very difficult. Hence, very little is known about the baryon–antibaryon interaction. From
<figure><img src="image_126.png" /><figcaption>**Fig. 109** Example of correlation functions (points) of $p-\bar{p}$, $p-\bar{\Lambda} \oplus \bar{p}-\Lambda$, and $\Lambda-\bar{\Lambda}$ pair for the 10–20% centrality class [1031]. Dashed lines show a part of the global fit, performed simultaneously to correlation functions of all three pair types in 6 centrality classes</figcaption></figure>

proton–antiproton scattering it is known that the formation of protonium (or antiprotonic hydrogen) occurs [1115,1116]. Its 1S and 2P states are of particular interest as there is evidence of a contribution from the strong force. Nevertheless, the nature of protonium, whether it can be considered a nuclear bound state or a result of the Coulomb interaction, remains an open question. Much less is known for baryon–antibaryon pairs with non-zero strangeness, because only few experimental measurements from scattering experiments exist.

Two-particle momentum correlations have been also employed to study the strong interaction in the baryon–antibaryon sector in ultrarelativistic pp and Pb–Pb collisions at the LHC, where approximately the same amount of baryons and antibaryons are produced [1120].

Correlation functions for $p-\bar{p}$, $p-\bar{\Lambda} \oplus \bar{p}-\Lambda$ and $\Lambda-\bar{\Lambda}$ were measured in Pb–Pb collisions at energies of $\sqrt{s_{NN}} = 2.76$ TeV and $\sqrt{s_{NN}} = 5.02$ TeV [1031] and in pp collisions at $\sqrt{s} = 13$ TeV [1041]. The analysis in Pb–Pb was performed in several centrality intervals; an example for the 10–20% centrality interval is reported in Fig. 109. A simultaneous fit of all the correlation functions was performed to extract the scattering parameters of the strong interaction. The spin-averaged scattering parameters, i.e. $\Re f_0$ the real and $\Im f_0$ imaginary parts of the spin-averaged scattering length, and $d_0$ for the real part of the spin-averaged effective range

<figure><img src="image_127.png" /><figcaption>**Fig. 110** Comparison of extracted spin-averaged scattering parameters $\Im f_0$ and $\Re f_0$ (Top) and $d_0$ and $\Re f_0$ (Bottom) for $p-\bar{\Lambda} \oplus \bar{p}-\Lambda$, $\Lambda-\bar{\Lambda}$ pairs and effective parameters accounting for the contribution of heavier $B-\bar{B}$ pairs [1031]. Results are compared with theoretical models [1099,1113–1118] and with results by the STAR collaboration [1119]</figcaption></figure>

of the interaction for $p-\bar{\Lambda} \oplus \bar{p}-\Lambda$ and $\Lambda-\bar{\Lambda}$ was obtained and the scattering parameters for baryon–antibaryon pairs, which were not measured directly, were estimated. All of the extracted parameters from the femtoscopic fit are summarised in Fig. 110 and compared with theoretical models and with other measurements.

The real and imaginary parts of the scattering length, and the effective interaction range extracted from the Pb–Pb fits have similar values for all baryon–antibaryon pairs at low-$k^*$. The significant magnitude of $\Im f_0$ shows that inelastic processes (annihilation) can occur for baryon–antibaryon systems. This finding was verified by the analysis in pp collisions, where an even larger contribution to the inelastic part of the interaction was found for the $p-\bar{p}$ and $p-\bar{\Lambda} \oplus \bar{p}-\Lambda$ pairs, while the same scattering parameters found in Pb–Pb collisions provided a good description of the $\Lambda-\bar{\Lambda}$ correlation [1041]. In general, the $\Im f_0$ is very different for the three baryon–antibaryon pairs when looking at both colliding systems and it gets smaller for baryon–antibaryon pairs of higher total pair mass. The elastic part of the interaction exhibits also the same mass hierarchy and the negative values of $\Re f_0$ show either that the interaction between baryons and antibaryons is repulsive, or that baryon–antibaryon bound states can be
formed. These findings can not exclude the presence of a bound state in the baryon-antibaryon channels but the presence of sizeable inelastic components makes it very difficult to disentangle the two effects.

## 5.5 Conclusions

**Light nuclei.** The production mechanism of light (anti)nuclei was investigated in small and large collision systems and the measured yields allowed one to test different production models. A tension remains between the coalescence and the statistical hadronisation approaches since a coherent description of all observables in all colliding systems is yet to be achieved. An ingredient recognised as essential for future studies is the accurate measurement of the nucleon source radii in the different colliding systems, which could be achieved with femtoscopy techniques. The measurement of the flow parameter $v_2$ for light (anti)nuclei in Pb-Pb collisions provided a complementary tool to study the production mechanisms. So far, the simplified modelling of such an observable via coalescence or blast-wave approaches describe the data qualitatively.

**Particle emitting source in small collision systems.** The precision of the system size measurement for small colliding systems has been significantly improved by means of a new model that accounts for the contributions of strong decays. A universal source for all hadrons produced in small colliding systems was characterised and its properties hint to the presence of collective effects in these collisions.

**Hypertriton lifetime and binding energy.** The most accurate measurement of the hypertriton lifetime was obtained by ALICE and is consistent with the free $\Lambda$ lifetime, thus solving in a definite way a long-standing puzzle about this observable. The precise determination of the binding energy of the hypertriton confirms the weakly-bound nature of $^{3}_{\Lambda}H$ and is in agreement with the binding energy value describing the p-$\Lambda$ momentum correlations measured with the femtoscopy technique.

**Proton-Hyperon interactions.** Correlations in momentum space measured in pp and p-Pb collisions provided the first high precision data on the strong p-$\Lambda$ interaction (25 times more precise than previous scattering data) and the first assessment of strong p-$\Sigma^0$ and p-$\phi$ interaction. These precise measurements challenge the current versions of effective field theoretical calculations. The $\Lambda$-$\Lambda$ and p-$\phi$ correlations allowed for assess a rather shallow hyperon-hyperon interaction potential. The so far unknown p-$\Xi^{-}$, $\Lambda$-$\Xi^{-}$ and p-$\Omega^{-}$ residual strong interactions have been measured for the first time via the analysis of correlations in the momentum space in pp and p-Pb collisions. These measurements confirmed

the attractive interaction predicted by lattice QCD calculations and allowed to search for the existence of di-baryon bound states. So far the precise data measured by ALICE provide no evidence that supports the formation of either a p-$\Xi^{-}$ or a p-$\Omega^{-}$ bound state.

**Kaon-proton interactions.** The K$^{-}$p interaction was also revisited with unprecedented precision and the combined measurements of correlations in small and large colliding systems allowed one to extract scattering parameters with a precision matching that of measurements with kaonic atoms. A direct measurement of the coupled channels that characterise the K$^{-}$p interaction as a function of the system size was also carried out. The measurements in different collisions systems are the only ones available to calibrate the parameters within effective field theories.

**Baryon-antibaryon correlations.** Baryon-antibaryon correlations were investigated in pp and Pb-Pb collisions and the real and imaginary part of the scattering parameters for p-$\bar{\Lambda}$ and $\Lambda$-$\bar{\Lambda}$ were be extracted for the first time. No evidence of bound states has been observed.

# 6 QCD studies with high-$Q^2$ processes in pp collisions

Perturbative QCD (pQCD) is one of the cornerstones of high-energy physics. A thorough understanding of the phase space in which pQCD is under precise control provides a baseline for the discovery of new physics, including searches for physics Beyond the Standard Model (BSM) [639, 1121–1123], and emergent behaviors of the quark-gluon plasma, as detailed in Sect. 2.

Theoretical descriptions of high-$Q^2$ processes in QCD employ various perturbative approximations to compute cross sections. Fixed-order expansions in $\alpha_S$ include all terms up to a given power in $\alpha_S$, with the precision of many recent calculations achieving Next-to-Next-to-Leading-Order (NNLO) [1124, 1125]. However, for certain observables, terms appearing at all orders in $\alpha_S$ contribute significantly, due to large logarithms that arise from characteristic scales in the observable's phase space. These scale-dependent terms can be computed analytically by all-order resummations [640, 1126] or by matching fixed-order calculations with parton showers [497, 856, 1127–1131]. Other controlled approximations, such as the leading colour approximation [1132, 1133] or the power suppression of other characteristic scales [639, 1123], are similarly employed. In all cases, scale variations are used to estimate the contributions of higher-order terms. General-purpose Monte Carlo (MC) event generators are built with these first-principles calculations in mind, and attempt to pro-
vide approximate descriptions of broad classes of observables.

The factorisation approach in QCD enables the connection of perturbative calculations to measurements, by relating perturbatively calculable cross sections to non-perturbative initial and final states [55,1134]. Observables from high-$Q^2$ processes therefore intrinsically involve long-distance QCD interactions parameterized by parton distribution functions (PDFs), fragmentation functions (FFs), and other non-perturbative objects assumed to be universal across collision systems and observables [1135–1137]. The relevant non-perturbative objects involved depend on the observable considered; for example, parton-to-hadron fragmentation is typically described using FFs, whereas parton-to-jet fragmentation is often computed purely perturbatively. Measurements of high-$Q^2$ processes in pp collisions test the importance of higher-order terms in state-of-the-art pQCD calculations, and constrain these non-perturbative objects [1122].

ALICE has made significant contributions to studying high-$Q^2$ processes in pp collisions. While the experimental design of ALICE limits its data-taking rate and high-$p_T$ capabilities, and thereby its sensitivity to the rarest processes produced at the LHC, ALICE records collisions in a low-pileup environment ideal for precision QCD studies. The ALICE detector design is complementary to the other LHC detectors; its high precision tracking and particle identification capabilities, with a focus on low- to moderate-transverse momentum $p_T \lesssim 100$ GeV/c, are unique at the LHC (see Sect. 1.4). ALICE also puts special emphasis on measurements in pp collisions of high-$Q^2$ observables that are of importance for the study of the QGP, which provides a complementary focus to the observables studied by experiments that concentrate more on BSM physics. In this chapter, we discuss several such measurements of high-$Q^2$ processes in pp collisions which contribute to the fundamental understanding of QCD. We begin with jet measurements that test the perturbative accuracy of state-of-the-art pQCD calculations in Sect. 6.1, before highlighting identified particle measurements which inform the understanding of the fragmentation process and its universality in Sect. 6.2, and finally discuss an innovative jet substructure measurement used to reveal the QCD dead-cone in Sect. 6.3.

## 6.1 Testing perturbative accuracy with jets

Jets are collimated sprays of hadrons arising from the fragmentation of energetic quarks and gluons generated in high-$Q^2$ interactions. Jets are fundamental objects in QCD, and are produced copiously at collider energies [164,563,567,1138–1151]. Inclusive and heavy-flavour jet production cross sections, as well as jet substructure observables, can be calculated analytically in pQCD. ALICE measurements test the perturbative accuracy of these calculations and pro-

vide insight into the regime of applicability of perturbative approximations. ALICE reconstructs full jets using its tracking system combined with information from the EMCal (MC corrections are applied for missing neutral hadron energy, due to the absence of a hadronic calorimeter), as well as charged-particle jets using the tracking system alone. The reconstruction of full jets allows a more direct comparison to theoretical calculations, and are frequently used for measurements of inclusive jet production, whereas charged-particle jets allow better angular resolution, and are often used by ALICE for jet substructure measurements. The ALICE approach to jet measurements is discussed further in Sect. 2.4.2.

### 6.1.1 Inclusive jet production

The inclusive jet production cross section in pp collisions is one of the most fundamental quantities in QCD; it was recently calculated to NNLO [1124,1125]. Inclusive jet measurements in elementary collisions are used in global fits to determine PDFs and the strong coupling constant $\alpha_S$ [1145, 1148, 1152–1154]. Historically, jet measurements and calculations with large- and moderate-$R$ (resolution parameter), $R \gtrsim 0.4$, have been preferred for high precision studies because they minimise hadronisation effects. Calculation of the inclusive jet production cross section for small-$R$ jets is challenging, due to large logarithms of the jet resolution parameter $R$ that can contribute at all orders in $\alpha_S$. However, this issue was addressed recently by calculations of the inclusive jet cross section with all-order resummation of logarithms of the jet resolution parameter [1155–1160]. Small-$R$ jets are of current interest both because of their importance in measuring jet quenching effects in the high-background environment of heavy-ion collisions (Sect. 2.4.2), and because theoretical techniques for all-order resummation are best tested with small-$R$ jets due to their large resummation effects. Furthermore, the jet cross section as a function of $R$ can disentangle the contributions from pQCD, the underlying event, and hadronisation [1156].

In order to explore these theoretical developments, ALICE has measured the inclusive jet cross section for a wide range in $R$ in pp collisions at $\sqrt{s} = 2.76$ TeV [164] ($0.2 < R < 0.4$) and $5.02$ TeV [563] ($0.1 < R < 0.6$), with the latter shown in the left panel of Fig. 111. The right panel of Fig. 111 shows comparisons of these measurements to four pQCD calculations: NLO and NNLO fixed-order calculations [1124], NNLO fixed-order calculations taken from [1124] matched to a leading-logarithmic (LL) resummation of $R$ (NNLO+LL) [1155, 1156], and NLO fixed-order calculations with resummation of logarithms of $R$ and threshold logarithms to next-to-leading-logarithmic (NLL) accuracy (NLO+NLL) [1157, 1159, 1160]. Non-perturbative corrections due to hadronisation and the underlying event are
applied using MC event generators.<fnref n="13" /> The non-perturbative corrections are large for small-$R$ and low-$p_T$, reaching approximately a factor 2 for $R = 0.1$ for $p_T < 50$ GeV/c [563, 1156]; the comparisons at low-$p_T$ should be interpreted with this in mind.

For $R = 0.4$ and 0.6, all four theoretical calculations describe the data well, suggesting that NNLO and resummation effects are small in these cases. In contrast, for $R = 0.1$ and 0.2, the level of description of the data varies significantly depending on the type of calculation. The NLO fixed-order calculation has poorest description, while the NNLO calculation is closer to the data, demonstrating the crucial importance of NNLO contributions. However, at high-$p_T$ for $R = 0.1$, the NNLO calculations overestimate the data, suggesting that resummation effects are important when $R$ becomes sufficiently small. Furthermore, the theoretical uncertainties in the NLO and NNLO calculations do not overlap, corroborating the importance of resummation effects. This is shown directly by the NLO+NLL and NNLO+LL calculations, which describe the data consistently within uncertainties. Note that while the NLO and NLO+NLL calculations are dramatically different, the NNLO and NNLO+LL calculations differ only slightly, suggesting that further theoretical work is needed to clarify the magnitude of LL vs. NLL contributions and differences in the calculational frameworks employed. Together, these observations show clearly the importance of both NNLO contributions and resummation contributions for small-$R$ jets, both to ensure a good description of data and to ensure theoretical consistency between calculational orders.

The $R$-dependence of the inclusive jet cross section is determined most precisely by measuring the ratio of experimental cross sections with different $R$, taking account of correlated systematic uncertainties which cancel in the ratio [164,563]. Initial considerations suggested that theoretical uncertainties may also cancel significantly in such ratios [1161]. Comparison of recent ALICE cross section ratio measurements [563] with the calculations described above (not shown here) indicate that for the ratio $R = 0.2/R = 0.4$ all calculations describe the data within uncertainties. For the ratio $R = 0.2/R = 0.6$, the NLO+NLL and NNLO calculations describe the data, while the NNLO+LL calculation exhibits a tension. Note, however, that scale variations can reach into the non-perturbative regime at low-$p_T$ and prevent theoretical uncertainty cancellation, which is treated differently in each calculation. The experimental

data are now precise enough that theoretical uncertainty is the limiting factor in the comparison [1162,1163].

### 6.1.2 Heavy-flavour jet production

The production of jets containing charm or beauty quarks can be similarly tested against pQCD calculations [1164–1166]. Heavy-flavour jets in pp collisions are relevant both to understand the flavour-dependence of jet quenching in heavy-ion collisions (Sect. 2.4.2) and as a Standard Model background to the decay of massive particles, such as $H \to b\bar{b}$ [1167].

Figure 112, left panel, shows the ALICE measurement of the $p_T$-differential production cross section of charged-particle jets containing a $D^0$ meson among its constituents in pp collisions at $\sqrt{s} = 7$ TeV, in the range $5 \le p_{T, \text{ch jet}} < 30$ GeV/c, at midrapidity [1168]. The $D^0$-tagged jet over inclusive jet fraction was also measured (not shown here), and found to increase with $p_{T, \text{ch jet}}$ from $\approx 4.2\%$ at 5 GeV/c to $\approx 8.0\%$ at 30 GeV/c. The $p_{T, \text{ch jet}}$ dependence of the $D^0$-tagged jet production cross section and of its ratio to inclusive-jet cross section are consistent with NLO pQCD calculations provided by POWHEG [1130,1131] matched with PYTHIA 6 [76,1169] for the generation of the parton showering, hadronisation and particle decays. The absolute magnitude of the description is less satisfactory, with discrepancies up to a factor 2, though data and model uncertainties are rather large. The production cross section of $D^0$-tagged jets was also evaluated as a function of the jet-momentum fraction carried by the $D^0$ meson in the jet-axis direction ($z_{\parallel}^{\text{ch}}$) (not shown here). The predictions provide good agreement with the data for the $z_{\parallel}^{\text{ch}}$ dependence of the cross section, apart from a small tension for $15 \le p_{T, \text{ch jet}} < 30$ GeV/c, where data tend to favour a softer charm-quark fragmentation (see Sect. 6.2.2 for further discussion of charm fragmentation). Measurements of $D^0$-tagged jets were also carried out by ALICE in pp collisions at $\sqrt{s} = 5.02$ and 13 TeV [1170].

Additionally, ALICE has reconstructed charged jets at midrapidity produced by beauty-quark fragmentation in pp collisions at $\sqrt{s} = 5.02$ TeV [1171], as shown in right panel of Fig. 112. The jet is tagged using two techniques, each of which exploits the relatively long lifetime of beauty hadrons: the reconstruction of displaced secondary vertices from jet constituent tracks, and the search for jet constituent tracks with a large impact parameter. Both techniques exploit typical features of the decay of long-lived beauty hadrons. The $p_T$-differential production cross section of beauty jets has been measured in a jet transverse-momentum interval $10 \le p_{T, \text{ch jet}} < 100$ GeV/c. These are the lowest $p_{T, \text{ch jet}}$ beauty-jet measurements at the LHC. These measurements also demonstrate agreement with ATLAS and CMS in their region of overlapping kinematics [1172,1173]. The fraction of charged jets originating from beauty quarks has also been

<footnote n="13">¹³ In the NNLO+LL calculation, the non-perturbative corrections are constructed from an envelope of several MC generators (PYTHIA, SHERPA, HERWIG), and are included in the uncertainty bands. For all other calculations, PYTHIA8 Monash 2013 alone is used, and no uncertainty is included; the difference in central values between these approaches is < 6% at all data points.</footnote>
<figure><img src="image_128.png" /><figcaption>**Fig. 111** (Left) Inclusive full jet cross sections in pp collisions at $\sqrt{s} = 5.02$ TeV for $R = 0.1–0.6$, measured by ALICE [563]. (Right) Ratio of various pQCD calculations to data [1124, 1155–1157, 1159, 1160]. The systematic uncertainties in the ratio, shown as boxes, are the quadratic</figcaption></figure>

sum of the systematic uncertainties in data and calculations. No systematic uncertainty for non-perturbative corrections are included, except for the case of NNLO+LL

measured in the same collision system, and found to increase from ≈ 2% at 10 GeV/c to ≈ 3.5% at 100 GeV/c. Such measurements were compared with NLO pQCD calculations provided by POWHEG, using the dijet process implementation, with subsequent parton showering provided by PYTHIA 8. The resulting predictions are able to correctly reproduce the absolute value and the $p_T$ dependence of both the beauty-jet production cross section and the beauty-jet to inclusive-jet fraction. These measurements can be compared in the future to analytical calculations of beauty-tagged jet production [1164–1166].

Overall, the fair description of the measurements indicates that perturbative QCD is capable, albeit within large theoretical uncertainty at the current level of perturbative accuracy, of describing the production of jets obtained from heavy-quark fragmentation.

### 6.1.3 Jet substructure

Jet substructure, defined by observables constructed from the distribution of constituents within a jet, provides the ability to access specific regions of QCD radiation phase space [639]. Jet reclustering techniques enable jet radiations to be organised into a phase space diagram known as the Lund plane [642]; traditional jet substructure observables tar-

get specific regions of this phase space, and thereby emphasise certain features of jets. The N-subjettiness [669], for example, characterises the number of prong-like structures in the jet, while the jet mass [569] characterises the virtuality of the jet-initiating parton. Jet substructure is therefore used as a tool to study rare event topologies in pp collisions, such as boosted objects that decay into jets [1174, 1175]. A detailed understanding of jet substructure is therefore needed in order to distinguish rare signals from the prevailing QCD background. Studies of jet substructure observables across a wide range of phase space enable differential tests of our understanding of pQCD. Jet substructure can also be used to remove pileup and study non-perturbative effects including hadronisation [1176, 1177].

The ALICE tracking system is ideal for jet substructure measurements, for which fine angular resolution is paramount. While track-based jet observables are collinear-unsafe [1136, 1178, 1179], they can be measured with greater precision than calorimeter-based jet observables, and recent measurements have demonstrated that for many substructure observables track-based distributions are compatible with the corresponding collinear-safe distributions [1180]. With a focus on low-$p_T$ jets, the ALICE jet substructure program in pp collisions tests the applicability of pQCD approximations as $\alpha_s$ increases and the theory becomes increasingly non-
<figure><img src="image_129.png" /><figcaption>Fig. 112 (Left) $p_T$-differential cross section of charged-particle jets tagged with $D^0$ mesons [1168] in pp collisions at $\sqrt{s} = 7$ TeV, compared to the POWHEG heavy-quark and di-jet implementations matched to PYTHIA 6 parton shower. (Right) $p_T$-differential cross</figcaption></figure>

<figure><img src="image_130.png" /></figure>

section of b-tagged charged-particle jets in pp collisions at $\sqrt{s} = 5.02$ TeV [1171], compared to NLO pQCD prediction by the POWHEG di-jet implementation matched to PYTHIA8 parton shower

perturbative. ALICE has measured a variety of jet substructure observables, including longitudinal and transverse fragmentation distributions, the groomed jet momentum fraction and radius, and N-subjettiness [571,572,662,1181–1184]. These results have been compared to MC generators, and provide future input for constraints on FFs and the tuning of MC event generators.

One of the most important classes of jet substructure observables are the infrared- and collinear- (IRC-) safe jet angularities [663], defined as

$$ \lambda_{\alpha} \equiv \sum_{i} z_{i} \theta_{i}^{\alpha}, \qquad (29) $$

where the sum runs over the jet constituents, and $\alpha > 0$ is a continuous free parameter. The first factor, $z_i \equiv p_{T,i}/p_T^{jet}$, describes the momentum fraction carried by the constituent, and the base of the second factor, $\theta_i \equiv \Delta R_i/R$, denotes the rapidity (y) – azimuth ($\varphi$) separation of the constituent from the jet axis, where $\Delta R_i \equiv \sqrt{\Delta y_i^2 + \Delta \varphi_i^2}$. By varying $\alpha$, one can systematically vary the contribution of collinear radiation within the shower, providing stringent tests of pQCD calculations. The IRC-safe angularities include the case $\lambda_1$, commonly referred to as the radial moment or “girth” [571,670,1185,1186], and the case $\lambda_2$, commonly referred to as the “thrust”, which is closely related to the jet mass [569,630,670,1180,1185–1189].

ALICE has performed a systematic measurement of the IRC-safe jet angularities in pp collisions [1190]. Figure 113 shows measurements of $\lambda_{1.5}$, $\lambda_2$, $\lambda_3$ for $R = 0.2$ jets. As $\alpha$

increases, the distributions skew towards small $\lambda_\alpha$, since $\theta_i$ is smaller than unity. The measured angularities are compared to analytical calculations with all-order resummations of large logarithms to next-to-leading-logarithmic (NLL') accuracy [552,1191,1192]. The angularity distributions are expected to be dominated by pQCD only at sufficiently large values of $\lambda_\alpha$, with the non-perturbative regime estimated by $\lambda_\alpha^{NP} \le \Lambda/(p_{T, ch jet} R)$, for an energy scale $\Lambda$ at which $\alpha_S$ becomes non-perturbative. A dashed vertical line is included as a rough estimate of the boundary between perturbative- and non-perturbative-dominated regions, with $\Lambda = 1$ GeV/c. Overall agreement is found between the theoretical calculations and the measurements in the perturbative regime, while discrepancies are found in the non-perturbative regime. For $\alpha = 1.5$, the majority of the distributions can be described perturbatively – especially for $R = 0.4$ (see Ref. [1190]) – while as $\alpha$ increases to $\alpha = 3$, the majority of the distributions become strongly non-perturbative. It was also demonstrated that by applying jet grooming algorithms, the perturbative region can be extended and good agreement of perturbative calculations can be achieved to even smaller values of $\lambda$ [1190,1193].

In addition to testing the limits of the applicability of pQCD, these considerations provide critical guidance for jet substructure measurements in heavy-ion collisions. If jet substructure distributions such as those in Fig. 113 are normalised to unity over the full range of the observable, as is typically done in heavy-ion collisions, then disagreement with theory predictions in the non-perturbative region
<figure><img src="image_131.png" /><figcaption>**Fig. 113** Measurements of ungroomed jet angularities $\lambda_{\alpha}$ in pp collisions at $\sqrt{s} = 5.02$ TeV for $R = 0.2$ charged-particle jets [1190] compared to analytical NLO+NLL predictions [1191,1192]. The theoretical calculations are corrected with a folding procedure to account for hadronisation and underlying event effects as well as charged particle</figcaption></figure>

jets, using PYTHIA 8 Monash 2013 and HERWIG 7. The distributions are normalised such that the integral of the perturbative region defined by $\lambda_{\alpha} > \Lambda/(p_{T, ch\,jet} R)$ is unity. Note that the transition from perturbative to non-perturbative is smooth, and this vertical line is merely a visual guide

– which is expected – will necessarily induce apparent disagreement with theory predictions in the perturbative region. This demands rigorous and careful choice of the observables used to quantify jet quenching effects and of the range of values of the observable that can be described by pQCD, such as the dependence of the applicability of pQCD on $\alpha$, $R$, and $p_{T, ch\,jet}$ in the case of jet angularities. These measurements bring guidance from first-principles pQCD and illustrate that additional scrutiny is needed when interpreting measurements of jet substructure in heavy-ion collisions. The importance of capturing both perturbative and non-perturbative processes is evident, as well as the need to discriminate between the two regimes in order to achieve a well controlled theoretical understanding of observed jet quenching modifications.

## 6.2 Studying fragmentation with identified particles

Measurements of hadron cross sections involve not only perturbatively calculable production cross sections, as in the case of jets, but also non-perturbative parton-to-hadron FFs. The study of identified single particle production in high-$Q^2$ processes provides a ground for simultaneously testing both of these elements, as well as QCD factorisation and the universality (across collision systems and observables) and evolution of both FFs and PDFs. In particular, measurements

at the LHC extend to unprecedentedly large values of centre-of-mass energy and hadron $p_T$, and are useful for studying the gluon-to-hadron FFs and their universality [1194,1195].

### 6.2.1 Light flavour hadrons

ALICE has measured identified hadron production at high-transverse momentum with different techniques and detectors:

- Charged pions, charged kaons, and protons with the TOF and HMPID detectors (up to $p_T$ of about 5 GeV/c) and using the relativistic rise of the specific energy loss in the TPC (up to 20 GeV/c) [402,458,511,1196]

- Charged kaons with kink topologies (up to about 7 GeV/c) [402,511,1196]

- Strange particles ($K_S^0$, $\Lambda$ and also $\Xi$ and $\Omega$) with topological reconstruction of their weak decays into charged particles (up to about 10–20 GeV/c) [1196,1197]

- Hadronic resonances such as $K^{*0}(892)$ and $\phi(1020)$ based on invariant mass analyses and relying on particle identification for daughter tracks (up to about 20 GeV/c) [505,1198]

- $\pi^0$ and $\eta$ with the electromagnetic calorimeters (up to about 200 GeV/c and 30 GeV/c, respectively) [163,1199–1201].
Measurements in these $p_T$ ranges are optimal to study the gluon-to-hadron fragmentation, given that the $p_T$ is high enough for the production to be already dominated by pQCD processes, but low enough that the fraction of hadrons coming from gluon fragmentation remains dominant. In fact, model calculations [1202,1203] indicate that the fraction of pions originating from gluon fragmentation is above 75% for $p_T < 30$ GeV/c at LHC energies – while the same happens only for $p_T < 5–7$ GeV/c at top RHIC energy [1204] – and remains dominant (> 50%) up to $p_T = 100$ GeV/c. In the following the measurements of the $\pi^0$ and $\eta$ mesons are discussed along with their implications in this context.

ALICE has measured the invariant differential cross sections of inclusive neutral pion and eta meson production at centre-of-mass energies $\sqrt{s} = 0.9, 2.76, 7$, and 8 TeV, covering a maximum transverse momentum range of $0.3 < p_T < 200$ GeV/c [163,1199–1201]. Figure 114 shows the ratio of the measured cross sections to the fit performed with the two-component model (TCM) proposed in Ref. [1205], which describes the spectra of the $\pi^0$ and $\eta$ meson over the full $p_T$ range at all energies. The invariant differential production cross sections are compared with NLO pQCD calculations [1206,1207] using MSTW08 (PDF) [1208] with DSS14 (FF) [1206] for $\pi^0$ and CTEQ6M5 (PDF) [1209] with AESSS (FF) [1207] for the $\eta$ meson. The same scale value $\mu$ ($0.5p_T < \mu < 2p_T$) is chosen for the factorisation, renormalisation, and fragmentation scales used in the NLO pQCD calculations. The variations of the $\mu$ values reflect the uncertainty of the NLO pQCD calculation.

For the $\pi^0$, the combination of the NLO PDF, production cross section, and FF describes the RHIC data [1210] and the lowest energy LHC data at $\sqrt{s} = 0.9$ TeV [163] rather well, whereas at $\sqrt{s} = 2.76$ TeV pQCD overpredicts the data by 30% at moderate $p_T$ and describes the data at higher $p_T$ [1200]. For all $\mu$ values, these calculations overestimate the measured cross sections for both $\pi^0$ and $\eta$ mesons for all centre-of-mass energies above $\sqrt{s} = 2.76$ TeV, although better description of data is achieved for $\mu = 2p_T$, for which calculations overestimate the data by 10–40%, depending on $p_T$. This suggests that higher order corrections are necessary to describe the data, as already argued when considering inclusive full jet cross sections for small value of cone radius. It has to be noted that FF uncertainties in the NLO pQCD calculations have been considerably reduced after including the published $\pi^0$ measurement at $\sqrt{s} = 7$ TeV [163] in the calculation of DSS14. Including precise new data for $\eta$ meson production measured at $\sqrt{s} = 2.76, 7$, and 8 TeV [163,1200] will also help to considerably reduce fragmentation function uncertainties in this case.

In addition, the neutral meson measurements are compared to PYTHIA 8 [856] Monash 2013 tune [1211]. PYTHIA reproduces the $\pi^0$ spectrum well at the two lower centre-of-mass energies, while it overpredicts the data by

$10–20\%$ for $\sqrt{s} = 7$ and 8 TeV. Concerning the $\eta$ meson, the Monash 2013 tune reproduces the measured spectra for $p_T > 1.5$ GeV/c within uncertainties at all energies. At lower transverse momenta it deviates significantly in magnitude and shape from data. Consequently, at higher transverse momenta, the $\eta/\pi^0$ ratio is underestimated by about 10–15% in this PYTHIA 8 tune. The tuning parameters of the soft QCD part of PYTHIA apparently fail to describe the measured $\eta$ meson spectrum below $p_T < 1.5$ GeV/c, whereas further tension is observed up to $p_T \approx 3.5$ GeV/c [1201]. This suggests that these data will provide significant constraints on fragmentation functions and for modeling of hadronisation effects in MC event generators.

Using the electromagnetic calorimeter, ALICE has also measured the inclusive production cross section of isolated photons at midrapidity in pp collisions at $\sqrt{s} = 7$ TeV for the transverse momentum interval $0.3 < p_T < 60$ GeV/c [1212]. This cross section is dominated by prompt photons from hard-scattering processes (the isolation cut suppresses photons from parton fragmentation) and it is well described by NLO pQCD calculations. The data sample collected during LHC Run 2 at $\sqrt{s} = 13$ TeV will enable an extension of the measurement up to a $p_T$ of about 200 GeV/c.

### 6.2.2 Open heavy flavour hadrons

The study of the production of hadrons containing heavy quarks, i.e. charm and beauty, in pp collisions at LHC energies is a sensitive test of QCD calculations based on the factorisation approach. Factorisation is implemented in terms of the squared momentum transfer $Q^2$ (collinear factorisation) [55] or of the partonic transverse momentum $k_T$ [1213]. At LHC energies, calculations based on collinear factorisation are available in the general-mass variable-flavour-number scheme, GM-VFNS [1214–1217], and in the fixed-order plus next-to-leading logarithms approach, FONLL [1218,1219], both of them having NLO accuracy with all-order resummation of next-to-leading logarithms. Recently, fixed-order calculations with NNLO accuracy became available for charm-quark [1220] and beauty-quark [1221] production cross sections. Within the $k_T$-factorisation framework, heavy-flavour production cross section calculations exist only at LO in $\alpha_s$ [1213,1222,1223]. The FONLL, GM-VFNS, and $k_T$-factorisation frameworks utilise different fragmentation functions, all based on dedicated fits to $e^+e^-$ data [1224–1226].

Precise measurements of the $p_T$-differential cross sections of charm and beauty hadrons provide a fundamental test for pQCD-based calculations, to validate the factorisation assumption and, indirectly, the ingredients used in the calculations. This is important also to constrain the charm- and beauty-quark $p_T$-differential spectra, which are used as input
<figure><img src="image_132.png" /><figcaption>**Fig. 114** Ratio of the neutral pion (Left) and eta meson (Right) invariant cross sections to their respective TCM parametrisations [163, 1199–1201]. Additionally, the ratios of NLO pQCD calculations and PYTHIA 8 Monash 2013 predictions to the TCM data fit are shown</figcaption></figure>

at the respective centre-of-mass energies. The NLO pQCD blue band in the left panel for $\pi^0$ is obtained considering the factorisation scale values $0.5 p_T < \mu < 2 p_T$

ingredients in models describing the heavy-quark interaction in the QGP. These models often rely on FONLL calculations.

In Fig. 115 the $p_T$-differential cross section of prompt and non-prompt $D^0$ mesons, the latter coming from the decays of beauty hadrons, in $|y| < 0.5$ in pp collisions at $\sqrt{s} = 5.02$ TeV is reported and compared to theoretical predictions. For prompt $D^0$ mesons [483], the data points are systematically higher than the FONLL central values by a factor 1.5–2 but remain compatible within the theoretical uncertainty, while the measured non-prompt $D^0$-meson cross section [484] is closer to the central value of the prediction. This can be better appreciated from the ratio of the data to the theoretical predictions, shown in the right panel of Fig. 115. For the GM-VFNS calculation, the central values tend to underestimate the data at low- and intermediate-$p_T$ and to overestimate them at high-$p_T$. The $k_T$-factorisation predictions describe the data at low- and intermediate-$p_T$ within uncertainties, but overshoot them for $p_T > 7$ GeV/c. The data-to-theory ratios indicate that the shape of the $p_T$-differential cross section is better reproduced by the FONLL calculation, which shows a flatter ratio than GM-VFNS and $k_T$-factorisation for $p_T > 2$ GeV/c. It is worth noting that in the case of charm production, the uncertainties on theoretical

predictions, which are dominated by the choice of the factorisation and renormalisation scales, are significantly larger than the uncertainties on the measured data points.

At forward rapidity ($2.5 < y < 4$), similar observations are derived from the ($p_T, y$)-double-differential comparison of FONLL predictions with the cross section of muons from semi-leptonic decays of charm and beauty hadrons measured in pp collisions at $\sqrt{s} = 5.02$ TeV [1227]. The $p_T$- and $y$-differential cross sections as well as the $p_T$-differential cross section ratios between different centre-of-mass energies and different rapidity intervals are described, within experimental and theoretical uncertainties, by FONLL. The data, significantly more precise than FONLL calculations, set important constraints for pQCD calculations in a kinematic region important for probing PDFs at low Bjorken-$x$ values, down to about $10^{-5}$.

In the mentioned theoretical calculations, the FFs are typically parametrised from measurements performed in $e^+e^-$ or ep collisions [485,486,1228], under the assumption that the hadronisation of charm quarks into charm hadrons is a universal process across different colliding systems. This assumption may be broken by higher-twist effects or other effects related to the heavy-quark kinematics and the pp underly-
<figure><img src="image_133.png" /><figcaption>**Fig. 115** (Left) $p_T$-differential cross sections of prompt and non-prompt D⁰ mesons in pp collisions at $\sqrt{s} = 5.02$ TeV [483,484] compared with various pQCD-based theoretical calculations. (Right) Ratios</figcaption></figure>

of $p_T$-differential production cross sections of prompt and non-prompt D⁰ measurements to theoretical predictions

ing event [1229–1231]. Recent measurements of charmed-baryon production at midrapidity in pp collisions at various collision energies evidence higher baryon-to-meson cross section ratios for $\Lambda_c^+/D^0$ [481,482,526], $\Sigma_c^{0,++}/D^0$ [487], and $\Xi_c^{+,0}/D^0$ [488,489,527] at low-$p_T$ with respect to $e^+e^-$ and ep collisions. It is worth noting that those by ALICE are the first and, to date, unique measurements of $\Xi_c^{+,0}$ and $\Sigma_c^{0,++}$ cross sections in hadronic collisions. The $\Omega_c^0/D^0$ ratio was recently measured as well [1232], although without an absolute normalisation because of the lack of absolute measurements of the branching ratios of $\Omega_c^0$ decays.

The results at $\sqrt{s} = 13$ TeV are reported in the left panel of Fig. 116. The predictions from the default tune of PYTHIA 8.243 (Monash2013), essentially driven by $e^+e^-$ measurements, severely underestimate the ratios. Hadronisation in PYTHIA is based on the Lund string fragmentation model [495]. In the default PYTHIA tune strings are formed by colour-connecting partons in the Leading-Colour (LC) approximation, which allows to reproduce adequately $e^+e^-$ data. However, in pp collisions at LHC energies several partons are created via multiple-parton interactions and colour reconnections beyond LC topologies become important [1240]. Among these, 3-leg or “Y-shaped junction” topologies have larger probabilities to fragment into baryons, increasing baryon production. Indeed, a tune of PYTHIA 8 that implements colour reconnection beyond the leading-colour approximation (CR-BLC Mode2) [1240] better describes the $\Lambda_c^+/D^0$ and $\Sigma_c^{0,++}/D^0$ ratios, though it

underestimates the $\Xi_c^{+,0}/D^0$ data. These measurements suggest that charm hadronisation involves mechanisms in pp collisions at LHC energies that do not play a role in $e^+e^-$ collisions at LEP energies. Overall, the hadronisation process is not fully understood. Further investigations are needed to fully elucidate charm-hadron production across different colliding systems. In this respect, a detailed discussion of the $\Lambda_c^+/D^0$ measurements in pp and Pb-Pb collisions is reported in Sect. 2.3.2.

A consequence of the significant difference between the charmed baryon-to-meson ratios measured in pp and $e^+e^-$ and ep collisions is that charm-quark fragmentation fractions, $f(c \rightarrow H_c)$, i.e. the probabilities of a c quark to hadronise as a given charmed hadron species $H_c$, estimated from $e^+e^-$ and ep data, cannot be used to calculate the total charm-quark production cross section from the measurement of D-meson production cross section alone. Therefore, the $c\bar{c}$ production cross section per unit of rapidity at midrapidity ($d\sigma^{c\bar{c}}/dy|_{|y|<0.5}$) was calculated by summing the $p_T$-integrated cross sections of all ground-state charmed hadrons ($D^0$, $D^+$, $D_s^+$, $\Lambda_c^+$, and $\Xi_c^0$) measured in pp collisions at $\sqrt{s} = 5.02$ TeV. The contribution of $J/\psi$ and $\Omega_c^0$ is considered negligible with respect to the other hadron species, as specified in Ref. [726]. A 7% uncertainty was assigned to account for the possibility that $\Omega_c^0$ production may be significantly larger than expectations from $e^+e^-$ collisions, as a recent ALICE measurement and the Catania coalescence model suggest [491,1232].
<figure><img src="image_134.png" /><figcaption>**Fig. 116** (Left) Charmed baryon-to-meson $\Lambda_c^+/D^0$, $\Sigma_c^{0,++}/D^0$, and $\Xi_c^{0,-}/D^0$ ratios in pp collisions at $\sqrt{s} = 13$ TeV, in comparison with PYTHIA 8. (Right) Charm and beauty production cross sections per unit of rapidity at midrapidity as a function of the collision</figcaption></figure>

energy [484,538,726,1233–1239]. STAR and PHENIX measurements in pp collisions at $\sqrt{s} = 200$ GeV are slightly displaced along the horizontal direction for visibility. Comparison to the FONLL model calculations [1218,1219] and NNLO calculations [1220,1221] are reported

<table><caption>**Table 4** Charm fragmentation fractions into charmed hadrons, $f(c \to H_c)$, measured in pp collisions at $\sqrt{s} = 5.02$ TeV [726], in comparison with the average of the LEP measurements [485]. For the former, the statistical and systematic uncertainties are reported separately; for the latter, the first quoted uncertainty is the combination of statistical and systematic errors, while the second uncertainty originates from the limited knowledge of the relevant branching fractions</caption><thead><tr><th>H<sub>c</sub></th><th>ALICE f(c → H<sub>c</sub>)%</th><th>Average LEP f(c → H<sub>c</sub>)%</th></tr></thead><tbody><tr><td>D<sup>0</sup></td><td>39.1 ± 1.7(stat)<sup>+2.5</sup><sub>-3.7</sub>(syst)</td><td>54.2 ± 2.4(unc.) ± 0.7(BR)</td></tr><tr><td>D<sup>+</sup></td><td>17.3 ± 1.8(stat)<sup>+1.7</sup><sub>-2.1</sub>(syst)</td><td>22.5 ± 1.0(unc.) ± 0.5(BR)</td></tr><tr><td>D<sup>+</sup><sub>s</sub></td><td>7.3 ± 1.0(stat)<sup>+1.9</sup><sub>-1.1</sub>(syst)</td><td>9.2 ± 0.8(unc.) ± 0.5(BR)</td></tr><tr><td>Λ<sup>+</sup><sub>c</sub></td><td>20.4 ± 1.3(stat)<sup>+1.6</sup><sub>-2.2</sub>(syst)</td><td>5.7 ± 0.6(unc.) ± 0.3(BR)</td></tr><tr><td>Ξ<sup>0</sup><sub>c</sub></td><td>8.0 ± 1.2(stat)<sup>+2.5</sup><sub>-2.4</sub>(syst)</td><td>—</td></tr></tbody></table>

The charm fragmentation fractions $f(c \to H_c)$ are then obtained by dividing the $p_T$-integrated cross section of each measured hadron species by the total charm cross section [726]. They are listed in Table 4. The fragmentation fractions measured in pp collisions at the LHC are different from those measured in $e^+e^-$ and ep collisions [485], providing evidence that the assumption of universality (colliding-system independence) of parton-to-hadron fragmentation functions is not valid for charm production.

In the right panel of Fig. 116, ALICE measurements of the $c\bar{c}$ [726] and $b\bar{b}$ [484,538,1233,1234] production cross sections in pp collisions are shown as a function of the collision energy and compared to FONLL and NNLO [1220,1221]

predictions, as well as to results in pp and $p\bar{p}$ at lower collision energies [1235–1239].

For the $c\bar{c}$ measurements at $\sqrt{s} = 2.76$ and 7 TeV, the D-meson cross sections and the FF determined at $\sqrt{s} = 5.02$ TeV are used to calculate the total cross section, as explained in [726]. The measurements are higher than the upper edge of the FONLL and NNLO calculation, though compatible within ~ $1\sigma$ of the experimental uncertainty. The STAR [1235] and PHENIX [1236] results estimated in pp collisions at $\sqrt{s} = 200$ GeV, assuming fragmentation fractions measured at $e^+e^-$ collisions, are also shown.

For the $b\bar{b}$ cross section, the ALICE results at different energies are obtained in different manners: at $\sqrt{s} = 2.76$ TeV the results are extrapolated from the production cross section of electrons from semi-leptonic decays of beauty hadrons, at $\sqrt{s} = 5.02$ TeV the results are estimated from the averaged results of non-prompt D-meson and non-prompt $J/\psi$ measurements, and at $\sqrt{s} = 7$ and 13 TeV they are obtained via non-prompt $J/\psi$ measurements. The ALICE measurements are shown along with other existing measurements at lower centre-of-mass energies, by PHENIX [1236], CDF [1238] and UA1 [1239] Collaborations. The experimental results are found to be compatible with FONLL and NNLO calculations. Though, especially for the charm case, the data uncertainties are significantly smaller than the theory ones, it is worth noting that FONLL as well as NNLO calculations can reproduces within uncertainties heavy-quark cross sections varying by 3 orders of magnitude over a factor 65 variation in collision energy.
These measurements not only provide further constraints to pQCD calculations but are also fundamental as a reference for the investigation of the charm- and beauty-quark interaction with the medium formed in heavy-ion collisions.

### 6.2.3 Quarkonium

Quarkonia are bound states of either a $c\bar{c}$ (charmonia) or a $b\bar{b}$ (bottomonia) quark pair (indicated generically with $Q\bar{Q}$ in the following). As for open heavy flavour production, the initial hard parton-parton scattering process (mostly gluon-gluon scattering at LHC energy) that produces the $Q\bar{Q}$ pair can be described within pQCD, while the subsequent formation and evolution of the bound state are non-perturbative processes that involve long distances and soft momentum scales. Due to the large rest mass of both Q and $\bar{Q}$ valence quarks, their relative motion in bound states is non-relativistic thus making the formation and evolution of the quarkonium a unique test bench to study the properties of the fundamental strong force. Different theoretical approaches have been developed to describe the quarkonium production, for comprehensive reviews see e.g. Ref. [68,680]. Among these, in the colour evaporation model [920,1241] it is assumed that all $Q\bar{Q}$ pairs with an invariant mass above the threshold for heavy-quark pair production and below twice the open heavy flavor (D or B meson) threshold production evolve into quarkonium states. In the Colour Singlet Model (CSM) [1242] the quantum state of the $Q\bar{Q}$ pair does not have any transition from its production to its hadronisation: only colour-singlet heavy-quark pairs are thus assumed to be produced. Instead, in the so called “Non-Relativistic QCD” (NRQCD) approach [20] also colour-octet heavy-quark pairs can be initially formed and evolve towards colour-singlet bound states. In this approach, so called “Long Distance Matrix Elements” (LDMEs), which are postulated to be universal, i.e. independent of the energy and production process, express the probability for a $Q\bar{Q}$ pair to evolve into a given quarkonium state, and are determined from existing measurements.

Experimentally, prior to the LHC start, charmonium production in $p\bar{p}$ and $pp$ collisions was studied at the Tevatron [1238, 1243–1245] and RHIC [1246–1249] colliders. The available approaches were not able to describe in a consistent way the cross section, the $p_T$ distribution and the polarisation measurements. The LHC experiments have provided a wealth of results for ground and excited charmonium states in a wide range of phase space, and also for bottomonium states. ALICE contributed with competitive and complementary measurements, thanks to its moderate magnetic field, low material budget of the barrel detectors, and the event selection strategy based on either minimum-bias trigger or on a selection of the muon candidates in the forward muon arm down to transverse momenta as low as 0.5 GeV/c. In particular, ALICE measured production cross sections of

$J/\psi$ down to $p_T = 0$ at midrapidity in the di-electron channel and of $J/\psi, \psi(2S)$ and $\Upsilon(1S, 2S, 3S)$ states at forward rapidity in the dimuon channel [162, 1234, 1250–1259]. Additionally, thanks to the very low-$p_T$ threshold on single muon in the triggers, ALICE also measured the polarisation of the $J/\psi$ meson reconstructed in the dimuon channel at forward rapidity and down to low-$p_T$ [1260, 1261].

Despite significant recent progress on the theory side, see e.g. [1262–1267], none of the approaches can still provide a satisfactory description of the different observables. The case of the NRQCD approach is illustrative: while on the one hand a global fit to $J/\psi$ production cross sections measured in hadroproduction, photoproduction, $e^+e^-$ collisions and $\gamma\gamma$ scattering constrained the LO colour octet LDMEs and demonstrated their relevance, the resulting predictions for the $J/\psi$ polarisation in $pp$ at the LHC energy are in striking disagreement with measurements. Figure 117 shows the ALICE results on the $J/\psi$ polarisation in $pp$ collisions at $\sqrt{s} = 8$ TeV as compared to predictions from the NRQCD model and CSM [1261]. The averages of the $p_T$-integrated inclusive $J/\psi$ polarisation parameters $(\lambda_\theta, \lambda_\phi, \lambda_{\theta\phi})$ in the allowed 2-D regions (white areas) are displayed as $1\sigma$ ($2\sigma$) full (dashed) contour ellipses in Collins-Soper (in red) and helicity (in green) frames. On the other hand, within the same NRQCD approach, fits to both $J/\psi$ yield and polarisation measurements in hadro-production work reasonable well, but predictions based on these fits have drastically failed in describing either $J/\psi$ results from other than hadronic collisions or the LHCb results on $\eta_c$ production [1268–1271].

It is evident that, also in the quarkonium sector, we do not have yet a thorough understanding of the hadronisation mechanism of heavy flavour quarks produced in different colliding systems at high energies.

## 6.3 Direct experimental observation of the QCD dead cone

The angular distribution of radiation emitted by a massive moving particle is predicted to depend on its mass-to-energy ratio. In particular, the radiation is to be suppressed for angles $\theta < m/E$. This “dead cone” effect known from classical theories conforming to special relativity and gauge theories in general has been worked out in QCD for massive quarks in Ref. [1272,1273]. As an indirect evidence of the effect, a difference in the charged particle multiplicity between events containing jets initiated by beauty quarks and those by light quarks was first reported at the LEP [1274], but a direct observation was still missing. More recently, new methods traditionally applied to jet substructure measurements have been proposed in order to expose the dead cone of heavy quarks [1275,1276]. Characterising the dead-cone effect experimentally is not only critical to the understanding of parton showers in vacuum, but also plays a central role in the studies of the physical properties of the QGP. The assess-
<figure><img src="image_135.png" /><figcaption>**Fig. 117** Inclusive J/ψ polarisation parameters measured for $3 < p_T < 15$ GeV/c and $2.5 < y < 4$ in pp collisions at $\sqrt{s} = 8$ TeV. Full (dashed) contour lines show the $1\sigma$ (2σ) confidence intervals in the Collins–Soper (CS, red contours) and helicity (HX, green contours))</figcaption></figure>

frames [1261]. Filled contours correspond to predictions of either the CSM (full filled contours) or the NRQCD model (shaded filled contours) in the corresponding frames (CS in red and HX in green)

ment of mass and flavour dependence of jet quenching effects is an area of active exploration in ultrarelativistic heavy-ion collisions (see also Sect. 2.4).

The ALICE Collaboration has experimentally implemented the method proposed in Ref. [1276], using pp collisions recorded at $\sqrt{s} = 13$ TeV [1277]. Jets were reconstructed from charged particles with the anti-$k_T$ algorithm and a resolution parameter of $R = 0.4$. Charm quark initiated jets were tagged via the presence of a $D^0$ (or $\bar{D}^0$) meson among their constituents, reconstructed via its hadronic decay channel into $K^+\pi^-$ (K$^-\pi^+$). The jets were then re-clustered using the Cambridge–Aachen [1278] algorithm, giving access to the angular ordered splitting tree. A primary Lund Plane [642] was populated by following the branch containing the $D^0$ meson at each reclustering step and registering the kinematics of the emission. The branch containing the $D^0$ meson at each splitting step coincided with the hardest branch to percent level accuracy. This allowed for comparison to a reference measurement in a (predominantly gluon-initiated) inclusive jet sample, where the reclustering procedure follows the hardest branch. To mitigate the bias imposed by the presence of a $D^0$ meson, a minimum $p_T$ selection on the leading particle of the leading branch was imposed for each accepted splitting. To suppress hadronisation effects a cut on the $k_T$ of the reconstructed splittings [1276] was applied. The dead-cone effect was then studied through projections of the Lund plane onto the splitting angle axis $\theta$.

The ratios $R(\theta)$ of these angular distributions in intervals of the radiating prong energy $E_{\text{radiator}}$ are shown in Fig. 118 as a function of $\ln(1/\theta)$. A suppression of emissions at small angles is observed for charm-tagged jets compared to inclusive jets, with the magnitude of suppression increasing with decreasing radiator energy, as predicted by the dead-cone effect. Furthermore, the measurement was compared to MC simulations (PYTHIA 8 and SHERPA) for both the dead-

cone and no dead-cone limits. These comparisons corroborate the observation that the suppression seen in data is compatible with the dead-cone effect directly related to the mass of the quarks. This measurement in the charm sector paves the way for future measurements both of the beauty quark dead cone, which will allow the mass dependence of the effect to be explored, as well as in relativistic heavy-ion collisions where a new dimension to studies of the mass dependence of energy loss can be added [1279–1281].

## 6.4 Conclusions

In this chapter we have highlighted ALICE measurements of high-$Q^2$ processes in pp collisions that contribute to our fundamental understanding of QCD:

*Precision pQCD tests.* ALICE jet measurements provide stringent tests of state-of-the-art pQCD calculations, demonstrating that NNLO and resummation contributions are important for inclusive jet production, and investigating the boundary at which pQCD applies to jet substructure observables.

*Identified particle production.* ALICE measurements of identified particles including neutral mesons, charm and beauty hadrons, and quarkonia test descriptions of hadronisation in both the light and heavy flavour sector. Precise measurements of charm hadron production, including the measurements of some of the charmed baryons for the first time in hadronic collisions, allow us to compute the charm cross section at midrapidity and to measure the fragmentation fractions in pp collisions, finding a large increase in charm baryon fractions and demonstrating the non-universality of charm fragmentation compared to $e^+e^-$ and ep collisions.
<figure><img src="image_136.png" /><figcaption>**Fig. 118** The ratio of the angular distributions of emissions from D⁰-tagged to inclusive charged-particle jets in pp collisions at $\sqrt{s} = 13$ TeV [1277], shown in three radiator energy intervals of $5 \le E_{\text{radiator}} < 10$ GeV/c, $10 \le E_{\text{radiator}} < 20$ GeV/c and $20 \le E_{\text{radiator}} < 35$ GeV/c. A selection on the splitting scale of $k_T > \Lambda_{QCD}$ is applied to suppress hadronisation effects. The shaded regions correspond to the angles within which emissions are suppressed by the dead-cone effect, assuming a charm-quark mass of $1.275$ GeV/c².</figcaption></figure>

**Dead-cone effect.** Using reclustering techniques on charm-tagged jets, ALICE reported the first direct experimental observation of the QCD dead-cone effect in pp collisions.

# 7 ALICE contributions beyond QCD physics and synergies of heavy-ion physics with other fields

The ALICE Collaboration has contributed not only to the field of QCD physics but also to some other areas of physics. The corresponding measurements are discussed in the following and show also synergies of heavy-ion physics with other fields. These range from connections of ALICE results to the physics of neutron stars, cosmic particles and exotic searches, to the test of fundamental symmetries, theoretical concepts as the AdS/CFT correspondence, the chiral magnetic effect, ultracold gases, Bose-Einstein condensates, and new developments in machine learning.

## 7.1 Neutron stars and the nuclear equation of state

The interaction of hyperons (Y) with nucleons (N) is one of the key ingredients needed to understand the composition of the most dense stars in our Universe: neutron stars (NS) [1282, 1283]. Neutron stars are one possible final outcome of supernova explosions and are characterised by large masses ($M \approx 1.2–2.2$ solar masses $M_\odot$) and small radii ($R \approx 9–13$ km) [1284–1286].

In the standard scenario, the gravitational pressure is typically counter-balanced by the Fermi pressure of neutrons in the core, which, along with electrons, are the only remnants from the mother-star collapse. A large interest in this topic has been triggered also from the recent measurements of gravitational wave signals from NS mergers [1287, 1288], which opened a new gate to the properties of matter inside NS.

The high-density environment ($\rho \approx 4 \rho_0$, with $\rho_0$ being the nuclear density [1289–1294]) supposed to occur in the interior of NS leads to an increase in the Fermi energy of the nucleons, translating into the appearance of new degrees of freedom such as hyperons. This energetically-favoured production of strange hadrons induces a softening of the Equation of State (EoS).<fnref n="14" /> The mass as a function of the radius is defined by the EoS through the solution of the Tolman-Oppenheimer-Volkoff equations, hence the mass-radius relation strongly depends on the constituents of the NS and on their interactions.

The inclusion of hyperons leads to NS configurations unable to reach the current highest mass limit from experimental observations of close to $3 M_\odot$ [1287, 1295, 1296].

<footnote n="14">¹⁴ A stiff (or hard) EoS is one where the pressure increases strongly for a given increase in density. Such a material would be harder to compress and offers more support against gravity. A soft equation of state leads to a smaller increase of pressure for a change in density and the material is more easy to compress.</footnote>
For this reason, the presence of hyperons inside the inner cores of NS is still under debate, and this “hyperon puzzle” is far from being solved [1294,1297]. A possible way out is represented by two-body and three-body repulsive YN and YNN interactions. In both cases, a sufficiently strong YN or YNN repulsive interaction can push the appearance of hyperons to larger densities, limiting the possible presence of these particle species inside NS, stiffening the EoS and leading to larger star masses.

The hyperon-nucleon two-body and three-body interactions in vacuum are used as the input for calculations at finite density, relevant for determining the baryon content inside NS, hence high-precision data are required in order to provide solid constraints on the available theoretical models. Currently, ALICE femtoscopic measurements in small systems (pp, p-Pb) for baryon-baryon pairs involving hyperons deliver the most precise data on the residual strong interaction between nucleons and strange hadrons (see Sects. 5.3.2 and 5.3.3).

The femtoscopic measurements of the $p-\Lambda$ strong interaction represents one of the major achievements in this sector. The $\Lambda$ baryons are typically the first hyperon species that are produced inside NS, due to their low mass. Their appearance is also theoretically favoured by the overall attractive potential, $U_\Lambda = -30$ MeV, that a $\Lambda$ feels at the saturation density [1298]. The results obtained by the ALICE Collaboration on this system support recent $\chi$EFT calculations in which an even more attractive interaction of the $\Lambda$ with the surrounding nucleons, due to the $\Lambda-N \leftrightarrow \Sigma-N$ dynamics, is predicted. In this case the early appearance of $\Lambda$ hyperons in neutron matter will lead to a too soft EoS and ultimately to a too low mass limit of NS configurations. Such a scenario, in order to be consistent with the astrophysical constraints on NS masses, requires the introduction of repulsive forces that might be present in other Y-N systems and the inclusion of three-body interactions.

Repulsive hyperon-nucleon-nucleon interactions, such as $\Lambda$-N-N, have already been included in several approaches to obtain a stiffer EoS [1299,1300]. At the moment, however, calculations of three-body forces rely on the experimental measurements of hypernuclei ($^{4}_{\Lambda}$H, $^{4}_{\Lambda}$He) binding energies, in which the determination of the genuine $\Lambda$-N-N interaction is not straightforward and can be affected by many-body effects. For this reason, the question of the role of three-body terms in the strangeness $|S| = 1$ sector inside NS remains open. ALICE has started to pioneer the study of the p-p-p and $\Lambda$-p-p interaction with three-body femtoscopic measurements [1301] and this sector will be further investigated with the much larger data samples expected during the LHC Runs 3 and 4.

Neutron star properties can also be studied in heavy-ion collisions at lower energies, e.g. HADES at GSI ($\sqrt{s_{NN}}$ ≈ 2.4 GeV) [1302] and STAR at RHIC in the Beam Energy

Scan [1303,1304]. These energies correspond to high-baryon densities and one directly tests certain regions of the EoS there. In particular, measurements of the direct flow observable $v_1$ can be connected to either momentum-dependent potential models or chiral effective field theory calculations involving protons, $\Lambda$ or even hypernuclei [1305,1306].

A major improvement in the understanding of the role played by heavier strange hadrons within NS has been achieved by the validation of lattice QCD predictions for the $N-\Xi$ interaction. The ALICE measurements on $p-\Xi^-$ pairs [1032] (see also Sect. 5.3.3) confirmed a strong attractive interaction between these two hadrons and provided a direct confirmation of lattice potentials [1066]. Using this two-body interaction to extrapolate to a neutron-rich dense system, a repulsive interaction potential of about +6 MeV is obtained [1310]. Currently, models for EoS including $\Xi$ hyperons assume large variations in the values of the single particle potential (-40, +40 MeV) [1307] and hence the validated lattice predictions impose a much more stringent constraint. In Fig. 119, the chemical potentials $\mu_i$ of different particles, obtained from mean-field calculations [1290,1307-1309], produced in the inner part of NS are shown as a function of the energy density. The isospin coupling strengths of $\Sigma$ and $\Xi$ hyperons to the nucleons have been rescaled in order to reproduce the current values constrained from scattering data and hypernuclei, and confirmed by ALICE femtoscopic measurements. The isovector couplings to the $\Xi$ have been adjusted to reproduce the predicted results in pure neutron matter (PNM) obtained from HAL QCD calculations at finite density [1310], stemming from the predictions in vacuum validated by ALICE femtoscopic measurements on $p-\Xi^-$ [1032,1037]. In the figure the corresponding values for the symmetric nuclear matter case are reported. The production of neutral hyperons such as $\Lambda$ in the NS core becomes energetically favoured when the corresponding chemical potential is equal to the chemical potential of neutrons $\mu_n$. In the left panel of Fig. 119, this is occurring at an energy density $\epsilon \approx 2.5 \epsilon_0$ where $\epsilon_0$ is the energy density of nuclear matter. For charged hyperons, as the $\Sigma^-$ and $\Xi^-$, the contribution of the electrons present in the medium has to be taken into account, so that the condition to be fulfilled to start the conversion reads $\mu_i = \mu_n + \mu_e$. The slight repulsion acquired by a $\Xi^-$ in pure neutron matter directly translates into the larger energy densities, and hence larger nuclear densities, for the appearance of this hyperon species. In the right panel of Fig. 119, the resulting mass-radius relation obtained by assuming the predicted HAL QCD $\Xi$ interaction in medium is shown. The production of cascade hyperons occurring at higher densities leads to a maximum NS mass of 2.13 $M_\odot$, which is compatible with the recent measurements, indicated by the coloured bands, of NS close to and above two solar masses [1284-1286].
<figure><img src="image_137.png" /><figcaption>**Fig. 119** (Left) Chemical potential $\mu_i$ of hyperons produced in the inner core of a NS as a function of the energy density, in units of energy density $\varepsilon_0$ at the nuclear saturation point. The single-particle potentials depths in symmetric nuclear matter (SNM) for $\Lambda$, $\Sigma$ and $\Xi$ hyperons are displayed. The vertical dotted lines indicate the central energy densities reached for a standard NS of $1.4 M_\odot$ and for the maximum mass, $2.13 M_\odot$, reached within this specific EoS. The mean-field calculations [1290, 1307–1309] have been tuned in order to reproduce the lattice predicted value of $U_\Xi$ in pure neutron matter (PNM)</figcaption></figure>

obtained in [1310], using the in-vacuum results validated by ALICE data in [1032]. The EoS obtained with these constraints provides a stable NS with a maximum mass of $M_{\text{max}} = 2.13 M_\odot$, as seen on the mass-radius plot on the right and is compatible with recent astrophysical measurements of heavy NS, indicated by the orange [1284], green [1285] and blue [1286] bands. See also Ref. [1311]

The results obtained in recent years from femtoscopic measurements in small colliding systems have proven that femtoscopy can play a central role in understanding the dynamics between hyperons and nucleons in vacuum. A comparison between hadronic models and these data is necessary in order to constrain calculations at finite density and to pin down the behaviour of hyperons in a dense matter environment. The great possibility to investigate, within the femtoscopy technique, different Y–N interactions and to extend the measurements to three-body forces, can finally provide quantitative input to the long-standing hyperon puzzle.

## 7.2 Cosmogenic dark matter

Thanks to the study of light nucleus and antinucleus production in high-energy collisions and to the measurement of low-energy antinuclei inelastic cross section (see Sect. 5), ALICE indirectly contributes to the search of dark matter. These measurements providing valuable input for the tuning of the cosmic-ray transport codes used in Monte Carlo simulations to interpret the data collected with experiments for dark matter searches. In addition, the predicted flux of antinuclei from dark-matter annihilation depends on the production mechanism and antinuclei transport properties within the interstellar medium, where ALICE data also is useful as input. The comparison of coalescence models with the ALICE production data on antinucleus production shed light on the results achieved in the indirect search of dark matter, since most of the utilised models are based on a coalescence process. Measurements of high-energy antimatter in the cos-

mic ray spectrum provide a powerful probe of new physics, including the annihilation or decay of dark-matter particles in the halo of the Milky Way [1312–1316]. An excess of 10–20 GeV cosmic-ray antiprotons [1317–1320] has been identified in data from AMS-02 [1321] (and PAMELA [1322]) with characteristics that are consistent with the annihilation of 50–90 GeV dark matter particles. This excess can, in a particular model, also be explained without annihilation of dark matter particles [1323]. In addition to $\gamma$ rays and antiprotons, dark matter annihilations can produce potentially detectable fluxes of heavier antinuclei, including antideuterons and antihelium [1324]. As kinematic considerations strongly suppress the production of heavy antinuclei in astrophysical processes, the detection of such particles could constitute a smoking-gun for dark matter annihilation. Intriguingly, the AMS-02 Collaboration has reported preliminary evidence of a few candidate antihelium events. Such a rate would be very surprising, as it would significantly exceed predictions from standard astrophysical processes or from annihilating dark matter [1325,1326]. A possible additional contribution to a $^3\overline{\text{He}}$ signal observed in near-Earth experiments is the decay of heavy-flavoured hadrons, e.g. $\overline{\Lambda}_b \rightarrow {}^3\overline{\text{He}} + X$ [1327].

At the LHC, the ALICE Collaboration has studied $^3\text{He}$ and $^4\text{He}$ production in pp, p–Pb and Pb–Pb collisions at centre-of-mass energies per nucleon pair from 0.9 to 13 TeV [387,388,863,1026,1028,1029,1328] and the measured yields were interpreted by means of coalescence and statistical hadronisation models [1053, 1054, 1329] (see also Sect. 5.1). The ALICE measurements, combined with different coalescence models, have been employed to estimate the
antideuteron and antihelium flux from cosmic-ray interactions at the AMS-02 and GAPS experiments [1330–1333]. In addition, annihilation cross sections for antinuclei are barely known and for antideuterons the inelastic cross sections have been measured for several materials only for two momentum values, $p = 13.3 \text{ GeV/c}$ [1334] and $p = 25 \text{ GeV/c}$ [1335]. For antihelium, the inelastic cross section had never been measured. ALICE has contributed in this field by measuring for the first time the antideuteron inelastic cross section in p–Pb data at 5.02 TeV [1025]. The measurement used the ALICE detector material as an effective target, with mean charge number $\langle Z \rangle = 8.5$ and mass number $\langle A \rangle = 17.4$ for the antideuteron momentum range $0.3 < p < 0.9 \text{ GeV/c}$, and $\langle Z \rangle = 14.8$ and $\langle A \rangle = 31.8$ for $0.9 < p < 4.0 \text{ GeV/c}$. Recently, these absorption measurements were extended to $^{3}\overline{\text{He}}$ with pp and Pb–Pb data and using two different techniques [1336]: one was the same as antideuterons and the another one used one of the ALICE subdetectors as “target” ($\langle A \rangle = 34.7$ in $1.17 < p < 10 \text{ GeV/c}$).

The measured inelastic cross sections and their uncertainties can now be used in propagation models of antinuclei within the interstellar medium for dark-matter searches. Figure 120 shows an example of such a calculation using the GALPROP software package [1338]<fnref n="15" /> and showing the comparison between the implemented GEANT4 [1339] absorption cross section [1340] and the measurement of ALICE [1336]. The GEANT4 estimation is consistent with the ALICE measurements, but the latter provide for the first time an direct experimental support and experimental uncertainties for this estimation. The obtained transparencies show that $^{3}\overline{\text{He}}$ nuclei can propagate for long distances in our galaxy and can thus be used to study cosmic-ray interactions or possible dark-matter decays. Future studies with larger data samples during the next LHC runs will represent a unique opportunity to extend these measurements and for the first time measure the inelastic cross sections of heavier antinuclei like $^{3}\overline{\Lambda}\text{He}$ and $^{4}\overline{\text{He}}$.

## 7.3 Cosmic rays

ALICE contributes to cosmic-ray measurements in two ways. First, ALICE provides data that can be used to constrain hadronic interaction models (Monte Carlo) that are critical for interpreting measurements on particles reaching the ground, the height of shower maximum, etc., in terms of incident cosmic-ray energy and nuclear composition. Second, ALICE makes direct measurements of cosmic-ray muon multiplicities and charges.

### 7.3.1 Data for cosmic-ray models

At energies above about 1 PeV (10¹⁵ eV), cosmic rays cannot be studied directly; instead, the resulting particle showers are measured in the atmosphere and at ground level. Relating these observables to the incident cosmic-ray energy and nuclear composition requires a model for the resulting hadronic shower. The shower energy is estimated based on the density of particles reaching the ground (seen by scintillation and Cherenkov detectors), and at high energies via a quasi-calorimetric method, by measuring the amount of visible light and/or radio waves emitted by the shower. The composition is usually inferred from the height at which the shower contains the maximum number of particles, $X_{\text{max}}$ and the number of muons of different energies seen in the shower [1341–1343]. The muon density, in particular, is sensitive to the hadronic physics as the air shower develops.

The development of the shower as it travels through the atmosphere depends on the frequency of interactions (i.e. the cross section) and the number of particles produced at each interaction. ALICE measurements of multiplicities of several hadron species in pp and p–A collisions are important benchmarks for the air showers models. The baryon-to-meson ratio and the strangeness content also play a role, since baryons have a larger interaction cross section than pions, whereas kaons interact less, and therefore penetrate more deeply. The EPOS code, for example, has been tuned against ALICE data [1069].

Most muons and neutrinos in air showers come from the decay of pions and kaons. Kaons have a shorter lifetime, so are more likely than pions to decay (producing a muon and a neutrino); this also affects the zenith angle distribution [1344]. Therefore, strangeness enhancement in light systems (as from hadron-oxygen collisions) is critical for air showers. ALICE provides important data on strangeness production (see Sects. 2.3 and 3.2), allowing for tuning the corresponding models.

In a slightly different field, conventional neutrinos $\nu$ (from pion and kaon) and neutrinos produced directly in the primary p–A cosmic collision are an important source of background in astrophysical neutrino studies. Although conventional neutrinos are fairly well studied [1345], prompt neutrinos have not been seen [1346, 1347]. Because of their harder energy spectrum, prompt $\nu$ are easier to confuse with astrophysical neutrinos, and are often included as a nuisance parameter in studies of diffuse astrophysical $\nu$. Measurements of charm production utilising pp collisions (see Sect. 6.2.2) are important in pinning down prompt neutrino calculations [1348]. Tighter constraints on prompt neutrinos would lead to better estimates of the astrophysical neutrino flux and energy spectrum [1349]. These can also improve models of neutrino production in astrophysical sources which include significant charm production [1350].

<footnote n="15">¹⁵ GALPROP version 56 was used, available at https://galprop.stanford.edu.</footnote>
<figure><img src="image_138.png" /><figcaption>**Fig. 120** Expected $^{3}\overline{\text{He}}$ cosmic-ray fluxes near Earth before (Left) and after (Right) solar modulation [1336]. The latter are obtained using Force Field with modulation potential $f = 400$ MV. Upper panels show the fluxes of antihelium nuclei for signal (dark-matter decay, in red) and background (ordinary cosmic rays, in blue) for various cases</figcaption></figure>

<figure><img src="image_139.png" /></figure>

of inelastic cross section used in the calculations. Bottom panels show the transparency of our galaxy to the propagation of $^{3}\overline{\text{He}}$. Shaded areas on the top-right panel show expected sensitivity of the GAPS [1337] and AMS-02 [1330] experiments

High-$p_T$ (here $p_T \ge 2$ GeV/c) particle production has also been considered as measure of cosmic-ray composition. Proton interactions reach higher $\sqrt{s_{NN}}$ than iron nuclei of the same total (not per nucleon) energy, leading to more high-$p_T$ muons. The advantage of this method is that the hadronic interactions can be described in perturbative QCD, rather than the more model-dependent approaches required at lower $p_T$. The IceCube Collaboration has studied high-$p_T$ muons (those with large lateral separation) and observed a power-law behaviour at large lateral separations [1351]. This is as expected from pQCD, but the zenith angle distribution was not in particularly good agreement with any of the models. More precise data on charm production might improve the understanding, and lead to composition measurements. This refers to both, published measurements that have yet to be taken into account in model tuning, and also data with higher precision to be recorded in Run 3 of the LHC.

### 7.3.2 ALICE cosmic muon measurements

Located 52 m underground with 28 m of rock overburden, ALICE has also studied the atmospheric muons coming from Extensive Air Showers (EAS), i.e. the shower of particles created by the collisions of cosmic rays with the nuclei of the atmosphere. The use of collider detectors to study atmospheric muons was pioneered by the LEP experiments ALEPH [1352], DELPHI [1353] and L3 [1354]. These experiments concluded that the high-muon-multiplicity events

occur with a frequency at almost one order of magnitude above expected. In this context, ALICE began a cosmic-ray physics program taking 30.8 days of live time of cosmic data in the period 2010–2013, collecting more than 20 million events and more than 7000 multimuon events. The study of the muon multiplicity distribution (MMD) and the measurement of the rate of high muon multiplicity (HMM) events, defined as the events with more than 100 muons ($N_{\mu} > 100$) detected in the Time Projection Chamber (TPC), was described in [1355]. In this work, the experimental results on the MMD and the HMM events were compared with simulations using the CORSIKA$^{16}$ code [1356] that utilizes the latest version of QGSJET [1357] to model hadronic interactions. The data have been compared with pure proton composition, representing the lightest cosmic ray composition that hits the atmosphere and produces the high-multiplicity muon events, and pure iron representing the heaviest composition. As an example, Fig. 121 shows a comparison of the measured MMD with the simulations in the intermediate range of muon multiplicity. The limited sample size does not allow for a quantitative study of the composition, but the measurement suggests a mixed ion primary cosmic-ray composition with an average mass increasing with energy in accordance with most of the previous experiments. The observed rate of HMM events is consistent with the predictions using a pure

<footnote>$^{16}$ CORSIKA is a package for detailed simulations of extensive air showers initiated by high-energy cosmic-ray particles, that allows different models for the hadronic interaction to be used with it.</footnote>
<figure><img src="image_140.png" /><figcaption>**Fig. 121** Number of cosmic-muon events per interval of multiplicity 4 in 30.8 days of data-taking [1355], compared with a model calculation</figcaption></figure>

iron primary composition up to MMD values of about 70 and energies $E > 10^{16}$ eV. For the first time the rate of HMM events is well reproduced using conventional hadronic interaction models and a plausible primary flux extrapolation. This measurement places significant constraints on alternative, more exotic, production mechanisms.

## 7.4 Atomic physics through lepton-pair production

Lepton-pair production, $\gamma A \rightarrow \ell^{+}\ell^{-}A$ ($AA \rightarrow \ell^{+}\ell^{-}AA$ in ultra-peripheral collisions (UPCs) [962]) studies atomic physics at very high energies, testing QED in very strong fields. Because they involve high-Z nuclei at large Lorentz boosts, UPCs are important event classes that enable to study of pair production in ultra-intense electromagnetic fields. In these high fields, higher-order diagrams should make a substantial contribution. This was studied intensively in a series of calculations using coupled-channel equations and then all-orders QED calculations. These calculations found very different cross sections for $AA \rightarrow \ell^{+}\ell^{-}AA$ before converging on a generally accepted result [1358]. However, theorists have not generally considered higher order corrections to the cross section in a restricted acceptance such as that for ALICE. Data from STAR [1359,1360] and ATLAS [1361,1362] are in general agreement with the expectations for lowest-order QED, except for the pair $p_T$ spectrum, which is broader than is expected in the naive calculation. This is likely due to the limited UPC impact parameter ($b$) range, i.e. requiring $b > 2R_A$, with $R_A$ being the nucleus radius; the STAR trigger imposes a more sophisticated restriction) for UPCs; when $b$ is limited, $\langle p_T \rangle$ should rise. However, the calculations are not in complete agreement [1363–1365] and this is still under active study. Effects such as incoherent photon emission ($A \rightarrow \gamma A^*$) and Sudakov resummation [1366] can also increase $\langle p_T \rangle$.

So far, ALICE has mostly studied the production as a background to photoproduction of the $J/\psi \rightarrow \mu^{+}\mu^{-}$ [977]

and $J/\psi \rightarrow e^{+}e^{-}$ [976]. In both cases, the pair mass regions away from the $J/\psi$ mass provide a fairly pure sample of lepton pairs. The $\mu^{+}\mu^{-}$ mass spectrum was found to be consistent with the expectations from lowest order QED [975], as implemented in the STARlight event generator [1000]. For $e^{+}e^{-}$ production at midrapidity (with lepton $|\eta| < 0.9$) at $\sqrt{s_{NN}} = 2.76$ TeV, quantitative results were presented: $154 \pm 11(\text{stat.})^{+17}_{-11}(\text{syst.})$ $\mu b$ for $2.2 < M_{ee} < 2.6$ GeV/c, and $91 \pm 10(\text{stat.})^{+11}_{-8}(\text{syst.})$ $\mu b$ for $3.7 < M_{ee} < 10$ GeV/c$^2$. These are in good agreement with the STARlight lowest-order QED predictions of 128 $\mu b$ and 77 $\mu b$ respectively. Figure 122 shows that the $M_{ee}$ distribution is in good agreement with the predictions.

Looking ahead, ALICE can study leptons with much lower $p_T$ than ATLAS, and at much smaller $M_{\ell\ell}/\sqrt{s_{NN}}$ than STAR; the latter condition increases the typical impact parameter, limiting the effect of the UPC condition $b > 2R_A$. The small $M_{\ell\ell}/\sqrt{s_{NN}}$ also increases the cross section, and so may allow for observation of new phenomena, such as multiple pair production (when one pair of interacting ions produces two pairs), or interference effects in the angular distributions [1367]. A more detailed study of the pair $p_T$ spectrum with different conditions on neutrons in the ZDCs would also be of interest. Finally, interesting connections can be made to hadronic interactions (i.e. $b < 2R_A$); the STAR collaboration has observed apparent $\gamma A \rightarrow \ell^{+}\ell^{-}A$ processes in peripheral collisions; they are visible as a large excess of pairs with $p_T < 100$ MeV/c [1368]. These pairs may provide some information about the overall reaction plane.

## 7.5 Searches for new particle states and new physics

ALICE has contributed to the understanding of the antimatter production mechanism in heavy-ion collisions and has also provided insight into the production and lifetime measurement of the hypertriton (i.e. the lightest known hypernucleus) (see Sects. 5.1 and 5.3.1). Besides the nuclei containing one hyperon, more exotic forms of deeply bound states with charm have been proposed as states of matter, either consisting of baryons or quarks. In the last years, this field has attracted much attention with the unexpected observation at electron-positron colliders of the new X, Y and Z states with masses around $4 \text{ GeV/c}^2$ (see for instance the reviews on non-$q\bar{q}$ mesons and spectroscopy of mesons containing two heavy quark in Ref. [440]). These heavy particles show very unusual properties, whose theoretical interpretation is entirely open. Above the $D-\bar{D}$ threshold the situation is quite different. A number of new states have been recently discovered by BaBar, Belle and CLEO. One of the most established among the XYZ states is the narrow X(3872) (later denominated $\chi_{c1}(3872)$ [440]) state of less than $2.3 \text{ MeV/c}^2$ width. It has been discovered by Belle and the main hadronic decay
<figure><img src="image_141.png" /><figcaption>**Fig. 122** Invariant-mass distribution for two-photon production of $e^{+}e^{-}$ pairs with the leptons within $|\eta| < 0.9$, at $\sqrt{s_{NN}} = 2.76$ TeV [976], compared with the lowest-order QED prediction</figcaption></figure>

<figure><img src="image_142.png" /></figure>

modes are $\pi^{+}\pi^{+}J/\psi$ and $D^{0}\bar{D}^{0}\pi^{0}$ [1369, 1370]. The small width and the decay properties disfavor a $c\bar{c}$ structure and the very close vicinity of the $D^{0}\bar{D}^{0*}$ threshold favors a molecular interpretation with these constituents. Measurements on the production of the $\chi_{c1}(3872)$ at the LHC have been performed by LHCb [1371–1376], ATLAS [1361, 1377] and CMS [1378, 1379]. The debate on this topic is still open and ALICE will be able to contribute in the next LHC runs in such a search thanks to measurement of the deuteron production in pp collisions with very high statistics. The comparison with the deuteron (as a loosely bound two-baryon system) transverse momentum spectrum is seen as a possible check between the two possibilities of the $\chi_{c1}(3872)$ being either a compact tetraquark [1380–1382] or a loosely-bound D-meson (molecular) state [1383, 1384].

Another field of interest connected to exotic states is the search of the H-dibaryon, a six-quark state (uuddss) predicted by Jaffe in 1977 [1385] and later. The discovery of the H-dibaryon or the $\Lambda n$ bound state would be a breakthrough in hadron spectroscopy as it would imply the existence of a six-quark state and provide crucial information on the $\Lambda$-nucleon and $\Lambda-\Lambda$ interaction. ALICE has investigated the existence of a possible $\Lambda n$ bound state and the H-dibaryon by reconstructing the invariant mass for the $\Lambda p\pi^{-}$ decay channel of the H-dibaryon, under the assumption of a weakly bound state [1386]. Since no evidence of a signal for the H-dibaryon was found in the invariant mass distribution, an upper limit (99% CL) of the production yield was obtained, assuming a $BR = 64\%$ and the free $\Lambda$ lifetime. The limit $\approx 3.0 \times 10^{-4}$ is a factor 20 below values predicted by the statistical-thermal models (see Sects. 2.3 and 5). The $\Lambda n$ bound state [1387] was investigated in the decay $d\pi^{+}$, but no signal was found in the invariant mass distribution. Assuming a $BR = 54\%$ and the free $\Lambda$ lifetime, this led to an upper limit (99% CL) of the production yield $\approx 2.0 \times 10^{-3}$ that also in this case is a factor 20 below values predicted by thermal models [1386]. The possible existence of the H-dibaryon was

also investigated through $\Lambda-\Lambda$ correlations [1033] and this significantly shrank the phase space of the predicted bound state further, but left some space for a resonance structure. A compact six-quark state (uuddss), called S (Sexaquark), was proposed [1388] as a possible long-lived or even stable version. It would only be possible to detect it from interaction with (detector) material [1388]. Searches for this elusive dark-matter candidate have been started at the LHC.

Dark matter is a hypothetical form of matter that is responsible for accounting for approximately 85% of the matter in the Universe. Since dark matter cannot be incorporated into the Standard Model, new interactions between dark matter particles and the ordinary Standard Model (SM) particles have to be introduced via unknown dark-sector forces. The dark photon is an extra U(1) gauge boson and acts as a messenger of a dark sector with the residual interaction to the SM particles. A number of fixed-target and collider experiments have searched for a dark photon, where the dark photon can decay back into charged SM particles with a coupling strength regulated by a 'mixing parameter'. ALICE can contribute to dark photon searches by examining the electron-positron invariant mass in very large samples of $\pi^{0}$ Dalitz decays ($\pi^{0} \to \gamma A'$. $A' \to e^{+}e^{-}$) for $20 \le M_{ee} \le 100$ MeV/c$^{2}$. The Run 2 data analysis is ongoing and the data samples to be collected with the upgraded detector in Runs 3-4 are expected to reduce the limit on the mixing parameter down to $10^{-4}$ below 100 MeV/c$^{2}$ [139].

## 7.6 Tests of fundamental symmetries

After the discovery of the P, CP and T violation, CPT invariance is the only discrete exact symmetry of the Standard Model that still holds, and it is thought to be one of the most fundamental symmetries of nature. The conservation of CPT invariance is theoretically guaranteed by the CPT theorem, which is based on three assumptions: unitarity, locality and Lorentz invariance, formulated in the framework of a
quantum field theory. The experimental search of CPT vio-
lating processes thus probes our fundamental understanding
of nature. Most of the experiments exploit one of the most
important consequences of CPT invariance: i.e., that implies
that the fundamental properties of antiparticles are identi-
cal to those of their matter-conjugates, e.g. the inertial mass,
the lifetime, the absolute value of the electric charge and the
magnetic moment. The same applies to systems of particles
and to their dynamics. Experimental tests of CPT invariance
have been carried out for elementary fermions and bosons,
and for QED and QCD systems (a recent compilation can be
found in Ref. [440]). Despite the different levels of precision
reached, which span over several orders of magnitude, each
of these distinct tests provides important information. Indeed,
effects in different systems might not be directly compara-
ble. The copious production of light antimatter nuclei at LHC
energies and the excellent ALICE tracking and PID perfor-
mance enabled a precise and unique measurement of the mass
difference between the deuteron, the ³He nucleus and their
antimatter partners, with relative accuracies of 1.5 × 10⁻⁴
and 1.3 × 10⁻³, respectively [1328]. The first represents the
most stringent experimental constraint for CPT invariance
in the nuclei sector, probing any matter/antimatter asymme-
try in nucleon-nucleon interactions. In the upcoming LHC
Runs, the higher luminosity and the increased ALICE read-
out rate will allow the sensitivity to be improved and the
measurement to be extended to ³H and ⁴He nuclei.

7.7 AdS/CFT correspondence and heavy-ion collisions

The formation, evolution, and dynamics of a heavy-ion colli-
sion are complex (see Sect. 2.2). Starting from the initial-state
parton distributions in the projectile nuclei, partonic interac-
tions over a wide range in momentum scale drive the sys-
tem rapidly towards equilibration and formation of the QGP,
which then expands and cools. Similarly, jet interactions with
the QGP occur over a wide range of momentum scales (see
also Sect. 1.2.4). The theoretical description of these dynami-
cal processes is extremely challenging, both because of their
non-equilibrium nature, and because the coupling α_S runs
with momentum scale. Soft, strongly-coupled interactions
play a crucial role in both QGP equilibration and jet-medium
interactions, but such processes are not amenable to a pertur-
bative treatment. Therefore, other theoretical tools are needed
to provide insight into these aspects of the QGP.

A key tool in this program is the AdS/CFT correspon-
dence, which provides a connection between certain confor-
mal super-symmetric gauge theories and classical gravity in
curved spacetime with one additional dimension [1389]. This
remarkable correspondence transforms very difficult out-of-
equilibrium calculations in finite-temperature quantum field
theory to tractable calculations in the holographic classical
gravitational space. While the gauge theories for which this

correspondence is known to apply have different symme-
tries, degrees of freedom, and coupling than QCD, they are
nevertheless “QCD-like”, and can provide insight into uni-
versal features of strongly-coupled non-Abelian gauge the-
ories [1390]. Indeed, the landmark observation that the spe-
cific shear viscosity, η/s, has a lower bound at strong cou-
pling (see Sect. 1.2.4), was first made using the AdS/CFT
correspondence [107].

The AdS/CFT correspondence has been used to study the approach to equilibrium, or “hydrodynamisation”, that characterizes the earliest phase of a nuclear collision. Such calculations show that, for strongly-coupled gauge theories, the far-from-equilibrium medium can be described by hydrodynamics after only a short evolution time of order 1/T, where T is the temperature [1390–1393]. These calculations are in good agreement with phenomenological studies comparing flow data to relativistic hydrodynamic calculations (see also Sect. 1.2.4).

The AdS/CFT correspondence has likewise been applied
to the process of a far-out-of-equilibrium colour charge (a
jet) propagating through the QGP, under the assumption of
strong coupling; i.e. that the coupling constant is large at
all momentum scales relevant to the jet-medium interac-
tion [1390,1394]. Such calculations provide detailed pictures
of the jet energy loss process, including the generation of a
wake [1395] and the response of the medium ([1396] and
references therein). However, the strong-coupling assump-
tion is in practice not a good approximation for modelling
jet-medium interactions in data. A hybrid model, combining
strong coupling using holography and weakly-coupled QCD
using PYTHIA, has therefore been developed which incor-
porates both weak and strong coupling approaches to calcu-
late a broad array of experimental observables [1396,1397].
Comparison of Hybrid Model calculations with ALICE jet
quenching data is discussed in Sect. 2.4.

7.8 Ultracold gases and Bose–Einstein condensates

Heavy-ion collisions are one of only a few known strongly-
interacting, weakly-coupled systems that can be studied
experimentally, and it is of particular interest to investigate
and compare with other such model systems in order to fully
understand strong interactions in their extreme. Remark-
ably, aside from the quark–gluon plasma, the most “per-
fect” liquids found in nature are ultracold Bose and Fermi
gases, despite having a temperature twenty orders of mag-
nitude lower than the QGP! Like the QGP, these degener-
ate atomic gases are also characterised by high occupation
numbers, a shear viscosity to entropy density ratio near the
proposed quantum lower bound (η/s ≥ ℏ/(4πk<sub>B</sub>)), and
can be described by fluid dynamic equations over a wide
range of length scales. Similar theoretical approaches, such
as within the kinetic theory and holographic duality frame-
works, have been used to describe the transport parameters and thermodynamic parameters of both the QGP and ultracold quantum gases (for more details, see the reviews in Refs. [32, 1398, 1399] and references therein); however, there have been fewer attempts to compare experimental measurements of the two systems.

Ultracold atomic gases are an important experimental testing ground for strongly-interacting systems, since they can be manipulated in optical traps, and observed at all stages of the system evolution. Such fine control over the initial conditions and evolution of the system is not possible in ultrarelativistic nucleus-nucleus collisions. However, roughly analogous tools exist: for example, event shape engineering techniques [1400] and centrality selection [1401] can give some control over the initial shape and size of the energy-density distribution, and various experimental probes demonstrate sensitivity to different stages of the collision evolution (for example, the azimuthal anisotropy coefficients $v_n$ are self-attenuating probes which give higher sensitivity to the early stages of a heavy-ion collision, while femtoscopic source size measurements are uniquely sensitive to the final state). While qualitatively similar flow behaviour has been observed in ultracold gases [1402] as in heavy-ion collisions, in recent years great progress has been made in developing new flow observables which give unique information on the hydrodynamic properties of the QGP, such as the shear ($\eta/s$) and bulk ($\zeta/s$) viscosities [307, 809, 1403]. It would be particularly interesting to consider whether some of these measurements of particle correlations could be translated into measurements in ultracold gas systems. Furthermore, in ultracold dilute Fermi gases, the strength of the interaction can be tuned using Feshbach resonances to span from the weakly attractive BCS regime to the strongly attractive Bose-Einstein condensate (BEC) regime (see for instance Ref. [1404]). In the latter, the second-order phase transition occurs near the degeneracy temperature and can be studied in detail. Meanwhile, in heavy-ion collisions, measurements of net-particle fluctuations [407, 1405, 1406] are being used to probe the crossover transition between the QGP and hadron gas phases, which coincides with the chemical freeze-out temperature. Future advancements in understanding the phase structure of strongly-interacting systems may come from considering similar or analogous observables in heavy-ion collisions and ultracold gases.

## 7.9 Chiral magnetic effect and new materials

The observation of the chiral magnetic effect (CME) [772, 773, 1407] in heavy-ion collisions would be the first direct proof of the topological fluctuations in QCD (ALICE searches are discussed in Sect. 2.6). This notion motivated the prediction of the effect in condensed matter systems in Dirac and Weyl semimetals, and eventually led to the discovery

of the CME by the BNL-Stony Brook-Princeton-Berkeley group [802] in a Dirac semimetal ZrTe$_5$. The existence of the CME was later also confirmed in dozens of other materials. The effect also has important potential applications in sensor development and quantum computing ("chiral qubit"). CME is of great interest for both QCD and condensed matter physics in particular as it allows to investigate the role of quantum anomalies.

The process of baryon asymmetry generation in the early Universe in theory is presumably also very similar to the process of chirality generation in heavy-ion collisions; both are described by the topological transitions between the vacuum sectors of the non-Abelian gauge theory.

Another emerging area of cross-disciplinary interest for both the heavy-ion and condensed-matter communities is the role of vorticity in the dynamical evolution of the systems. In heavy-ion collisions, where large vorticity can be generated in off-central collisions, the vortices are expected to play a very important role in the dynamical evolution of the system.

## 7.10 Machine learning

Machine learning (ML) represents a powerful tool for uncovering features in large and complex datasets. In the past decade, the number of machine learning applications in high-energy physics has grown tremendously. ML techniques are currently adopted for event classification, event reconstruction and selection, particle identification, detector calibration and monitoring [1408, 1409].

As described in the previous sections, a large fraction of heavy-ion measurements rely on complex strategies designed to select rare probes like heavy-flavour hadrons down to very low transverse momenta in presence of a large combinatorial background. The ALICE Collaboration has recently published several analyses that rely on ML techniques. As an example, the first measurement of the $\Lambda_c^+$ production in central (0-10%) Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [437] is discussed in the following. The $\Lambda_c^+$ selection uses a Boosted Decision Tree (BDT) classifier trained on a sample of signal $\Lambda_c^+$ candidates simulated with PYTHIA and background candidates extracted from data in the side-band regions. In the BDT training, both topological and particle-identification selection variables are considered, and the optimised set of BDT (hyper)parameters is identified by using a Bayesian optimisation procedure. The impact of these techniques is highlighted in the left panel of Fig. 123, where the invariant-mass distribution of $\Lambda_c^+$ candidates measured in central Pb-Pb collisions obtained with the BDT analysis is compared to the results obtained with a selection based on rectangular cuts (in the upper panel the actual counts and in the lower panel the residuals after polynomial-background-fit subtraction). The BDT analysis achieves a substantial increase of the signal selection efficiency, up to about four times larger,
and of the statistical significance, by 50% at low-transverse momentum. Similar techniques were adopted in the study of other heavy-flavour hadrons or of hypernuclei in heavy-ion collisions, resulting in similar increases in the statistical significance of the signals (see also Sects. 6.2.2 and 5.1). The use of ML classifiers has also been applied to the study of prompt and non-prompt production of D mesons in pp and Pb–Pb collisions [484,498,541]. In these measurements, a BDT classifier was trained to separate three classes of D-meson candidates: prompt D mesons from c-quark hadronisation, non-prompt D mesons from B-meson decays, and combinatorial background. A fit of the total D-meson yield as a function of the BDT score with templates for the prompt and non-prompt components provides an accurate estimation of the fraction of D mesons from B decays.

ALICE has also made significant strides in using machine learning techniques for jet physics. One such project is the use of machine learning to correct jets for the large fluctuat- ing background in heavy-ion collisions from the underlying event [1410]. In this method, the machine learning algorithm is used to build a data-driven mapping from jet properties, including properties of the constituents of the jet, to the back- ground corrected jet p$_{T}$. The estimator is trained on PYTHIA jets embedded into a realistic Pb–Pb background. A shallow neural network is used with 3 layers and [100, 100, 50] nodes. The performance of this method is evaluated using the δp$_{T}$ distributions, where a narrowing of the distribution corre- sponds to a more precise determination of the jet signal. As shown in Fig. 123 on the right, the machine learning based method has a much reduced width in δp$_{T}$ as compared to the standard area-based method [1411], corresponding to a reduction in the residual fluctuations remaining after back- ground subtraction.

The interest in the use of machine learning techniques in heavy-ion collisions and in soft-QCD proton–proton physics is constantly growing. Machine learning techniques are likely to replace standard optimisation routines in the upcoming years for the large majority of the analyses and provide new tools to design new experimental observables. The ALICE Collaboration is currently exploring the impact of ML tech- niques on a wide range of applications, which go from par- ticle identification of single hadrons and decays to heavy- flavour jet tagging or Monte Carlo reweighting. Deep neural network techniques are also being applied to critical aspects of the calibration and maintenance of the upgraded ALICE detector, from the correction of the electric field distortion fluctuations during data-taking in the ALICE Time Projec- tion Chamber to the data control and quality monitoring of the event reconstruction.

## 8 Summary

Since the start of LHC collisions in 2009, the ALICE detec- tor has carried out a very successful data taking programme. The experiment is dedicated to exploring QCD at the LHC in the context of the largest collision energies available in the laboratory. Our main focus is the study of many-body QCD interactions at the largest temperatures possible via the formation of the QGP from heavy-ion collisions. We have also extensively explored hadron–hadron strong interactions and several aspects of QCD in pp collisions, using the unique capabilities that the LHC and our detector have provided us. In terms of heavy-ion physics, starting from the BEVALAC to the LHC, this has allowed for some of the most precise explo- rations of effects discovered previously, and has unraveled its own set of discoveries. Here we summarise the main findings by addressing the questions posed at the end of Sect. 1:

1. What are the thermodynamic and global properties of the QGP at the LHC?

Our measurements have shown that heavy-ion collisions at the LHC create conditions that very much exceed those needed to form the QGP. The initial QGP temper- ature implied by our results in conjunction with theoret- ical modelling is up to 5 times higher than the predicted QCD deconfinement temperature $T_{pc} = 155–158$ MeV, and of the order of a trillion Kelvin, ~ $10^5$ times hot- ter than the centre of the Sun. An estimate can be made of the largest QGP energy densities from central Pb– Pb collisions, which yields ~ 12 GeV/fm³ at the early time of 1 fm/c. This is about seventy times higher than the energy density of atomic nuclei and about five times higher than the core of the most massive neutron stars. The lifetime of the system in central Pb–Pb collisions is found to be about 10–13 fm/c, which is about 40% larger than at RHIC. As will be addressed in later questions, such a system reaches equilibrium within this lifetime of the order of $10^{-23}$ s, and no other system in nature has been observed to do so in such a short time. The volume at freeze-out is about 7000 fm³ and about twice higher than at top RHIC energy. We therefore conclude that the matter created in heavy-ion collisions at the LHC has the largest temperature and energy density ever observed, and the longer lifetime and larger volume compared to lower energy heavy-ion collisions. These observations provide a critical backdrop for conclusions that will be made in this summary.

2. What are the hydrodynamic and transport properties of the QGP?

We have demonstrated that the QGP formed at LHC energies undergoes the most rapid expansion ever observed for a many-body system in the laboratory. The radial flow velocities derived from $p_T$ spectra approach
<figure><img src="image_143.png" /><figcaption>**Fig. 123** (Left) Invariant mass of $\Lambda_c^+$ candidates using standard (cut-based) and BDT selection techniques in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV for an example $p_T$ interval [437]. The upper panel shows the actual counts and the lower panel the residuals after polynomial back-</figcaption></figure>

ground subtraction for the two distributions. (Right) $\delta p_T$ distribution for charged jets with $R = 0.4$ in Pb–Pb collisions at 5.02 TeV. Comparison of ML-based and area-based correction

about 70% of the speed of light. Elliptic flow in non-central collisions has been observed for all measured hadron species, including the light nuclei and $^3$He, with the only exception of the $\Upsilon(1S)$ bottomonium state. Our measurements of charmed-hadron $v_2$ and spectra have revealed, for the first time, that low-momentum heavy quarks participate in the collective motion of the QGP, despite being produced out of equilibrium in the initial stages of the collision. Based on a set of ALICE measurements performed over the last ten years, a suite of new observables has unveiled an extremely rich pattern for the dynamical evolution of the QGP as a whole. These include higher-order anisotropies ($v_{n \ge 3}$), correlations between different order anisotropies, and a translation of the angular momentum of the QGP to the polarisation of its outgoing hadrons. Hydrodynamic calculations, using the QGP equation of state and which assume that the collision system behaves as a liquid about 1 fm/c after the initial collision, describe a wide variety of these observables. This description, which encompasses many more observables compared to hydrodynamic descriptions prior to the LHC, is achieved with the crucial inclusion of small but finite viscous effects, and has shown that hydrodynamics has emerged as a successful effective theory of many-body non-perturbative QCD interactions at high temperature. QGP viscosities explored in these hydrodynamic calculations imply that the system is strongly-coupled at the scale of the QGP temperatures probed at the LHC, with the extracted shear viscosity over entropy density ($\eta/s$) values in the range

$1/4\pi < \eta/s < 0.3$, which are at least five times smaller than superfluid helium, and establishes that the QGP is the most perfect liquid ever observed. In the heavy-flavour sector, comparisons of our open-charm hadron measurements with transport models have enabled an estimation of the charm spatial-diffusion coefficient $D_s$ in the range $1.5 < 2\pi D_s(T)T < 4.5$ at $T_{pc}$. These values provide evidence that charm quarks couple strongly with the QGP at low momenta, and our measurements in conjunction with the transport model description, indicate how equilibration in the QGP can occur on a microscopic level during the extremely small time scales associated with QGP formation and expansion.

### 3. How does the QGP affect the formation of hadrons?

Our measurements of hadron yields over all momenta have provided an extremely extensive mapping of hadrochemistry in heavy-ion collisions at LHC energies. Models which assume thermalisation at the time of the transition from QGP to a hadron gas (Statistical Hadronisation Models) describe the integrated yields of various light-flavour hadron species over many orders of magnitude. Besides the (u,d,s) sector, ALICE has discovered that, for heavy-ion collisions at the LHC, charmed hadron and quarkonium relative yields follow thermal expectations i.e. they can be understood in the context of a common temperature. The light-flavour chemical freeze-out temperature of about 156 MeV is within the predicted range of the deconfinement temperature, suggesting chemical equilibrium cannot be maintained easily after the QGP hadronises. The measurements of iden-
tified hadron spectra at intermediate-transverse momen-
tum, where non-equilibrium processes emerge, indicate
that an assumption of quark coalescence from the QGP
captures the main features of the data for both the light-
and heavy-flavour sectors. This is consistent with what
was observed prior to the LHC for light flavours, with
ALICE measurements revealing that such patterns are
also seen in the heavy-flavour sector. At high-transverse
momentum, where non-equilibrium processes domi-
nate, we have shown that hadron production remains
understood in terms of in-vacuum parton fragmenta-
tion. Finally, our investigations into the hadronic phase
via measurements of resonance yields and femtoscopic
radii have demonstrated complex dynamics at work.
They have indicated that this phase is prolonged, and
the decoupling of particles from the expanding hadron
gas is likely to be a continuous process, rather than a
sudden kinetic freeze-out of all particle species at the
same temperature. These observations have important
implications on how hadrons form and interact at high
temperatures, such as those in the early Universe.

4. *How does the QGP affect the propagation of energetic partons?*

Energetic, highly-virtual partons generated in hard scat-
terings in nuclear collisions produce a parton shower
that interacts with the surrounding QGP as it expands
and cools. The in-medium modification of the jet shower
("jet quenching") results in several distinct observable
effects, enabling a broad multi-messenger exploration
of QGP structure and dynamics using jets. ALICE has
measured significant yield suppression for a wide range
of hadrons and reconstructed jets in both inclusive and
coincidence channels, showing that in-medium energy
loss occurs at the partonic level and quantifying its mag-
nitude in terms of energy shift. ALICE measurements
of heavy-flavour yield suppression provide insight into
details of this process: the energy loss is reduced for
beauty with respect to charm quarks (e.g. by the dead-
cone effect) and both radiative and collisional processes
are necessary for the models to describe the data. ALICE
jet substructure measurements indicate preferential sup-
pression of wide-angle radiation in the jet shower, which
is sensitive to colour coherence of the QGP and the
space-time structure of jets. Finally, ALICE measure-
ments of di-jet acoplanarity constrain the rate of jet scat-
tering off quasi-particles in the medium, thereby prob-
ing the micro-structure of the QGP. A comprehensive
description of this rich set of jet measurements in a well-
constrained physics picture of the QGP is still work-
in-progress. Inclusive jet-suppression measurements are
described by a broad range of QGP models incorporat-
ing different underlying physics that include both weak
and strong-coupling descriptions of the QGP. Overall,

ALICE has observed the energy loss of energetic par-
tons in the presence of the QGP and the modification of
their shower.

5. *How does deconfinement in the QGP affect the QCD force?*

Significant modifications of the quarkonium binding in the QGP have been observed, with our results at the LHC showing new and striking features. For the J/ψ charmonium state, the QGP-induced suppression was shown to be counterbalanced by a strong regeneration effect, taking place during and/or at the fireball chemical freeze-out, that dominates charmonium production at low transverse momenta. This mechanism constitutes a proof of deconfinement, as it implies that coloured partons can move freely over distances much larger than the hadronic scale. At the same time, a significant elliptic flow was observed for the J/ψ: an unequivocal signal that charm quarks participate in the medium expansion. The excited ψ(2S) state, which has a binding energy lower by one order of magnitude with respect to the J/ψ, was found to be more suppressed than the J/ψ, by a factor about two. Like for the J/ψ, a hint of reduced suppression towards zero transverse momentum supports the presence of the regeneration effect. In the bottomonium sector, a suppression of Υ(1S) and Υ(2S) with a clear ordering related to binding energy was observed. While the observed suppression of the strongly bound ground state (~1 GeV binding energy) can be attributed to some extent to cold nuclear matter effects and feed-down from higher mass states, QGP effects are evident for the 2S state.

6. *Can the QGP lead to discovery of novel QCD effects?*

Our charged-hadron dependent directed flow measure-
ments indicate that, in heavy-ion collisions, extremely
large electromagnetic fields are created, strong enough
to influence electrically charged degrees of freedom in
the QGP. In addition, our Λ polarisation measurements
provide an upper limit (95% CL) for the magnetic field
at freeze-out of 14 × 10¹² T at the top LHC heavy-ion
collision energy. Multiple searches were carried out to
investigate whether parity is violated in the strong inter-
action, a process which is facilitated by the magnetic
fields produced in heavy-ion collisions. The experimen-
tal signal is known as the Chiral Magnetic Effect, and has
been observed in the QED sector for Type II supercon-
ductors at sub-critical temperatures. Our measurements
provided upper limits of 15–33% for the contribution of
the CME, consistent with the results at RHIC energies,
and constitute state-of-the-art limits.

7. *What are the minimal conditions of QGP formation?*

Several of our measurements in high-multiplicity pp
and p-Pb collisions exhibit features similar to those
observed in heavy-ion collisions, where these are associ-