Quenching of jets tagged with $W$ bosons in high-energy nuclear collisions

In our calculations, jet productions in association with a $W$ boson in p+p collisions are simulated within a MC event generator SHERPA 2.24 Gleisberg:2008ta , which can perform NLO ME calculations matched to the PS with several merging schemes. AMEGIC++ Krauss:2001iv and COMIX Gleisberg:2008fv are SHERPA’s original matrix-element generators which provide tree-level matrix-elements and create the phase-space integration as well. MC programs OpenLoops Cascioli:2011va is interfaced with SHERPA to provide the virtual matrix-elements. MEPS@NLO merging method Hoeche:2009rj ; Hoche:2010kg ; Hoeche:2012yf is used to yield improved matrix elements for multiple jets production at NLO matched to the resummed parton showers Gleisberg:2007md ; Schumann:2007mg . LHAPDF is interfaced with SHERPA and the parton distribution function (PDF)set ‘CT14 NLO’ Hou:2016sho is used to provide the PDF for partons that participate in the hard interaction in p+p collisions.

Though $W$ and $Z$ bosons have a lot in common, $W$ bosons have some unique properties as compared to $Z$ bosons. For instance, $W$ bosons carry electric charge and would change the flavor of the jet parton. As a result, the flavors of jets associated with W’s are different from a $Z$ boson. We have classified the flavor of the leading parton triggered by a $Z$ boson and $W$ bosons in Fig. 1. As can be seen, the quark fractions increase significantly with the transverse momentum of the gauge boson. At high gauge boson energy, the jets are dominated by quark jets. Compared to jet production associated with a $Z$ boson, the fraction of the leading parton flavor tagged with $W$ bosons is quite different. For instance, the fraction of $u$ quark associated with $W$ bosons is almost the same as the fraction of $d$ quark associated with a $Z$ boson and the fraction of $s$ quark associated with a $Z$ boson is almost the same as the fraction of $c$ quark associated with a $W$ boson as a result of isospin symmetry. The different parton flavor fractions would lead to different fractions of hadrons as well as different jet properties. The comparison between $W$+jet and $Z$+jet would provide new opportunities to explore the jet tomography of the QGP. The difference in the flavor fraction is a result of different production mechanisms of $W$ and $Z$ boson in hard scattering. In this regard, W’s associated with jets or hadron can be used to constrain non-perturbative hadronization models in both p+p and Pb+Pb collisions as well as flavor dependence of jet quenching, which is beyond the scope of this paper and will be discussed in the future.

In order to compare with experimental measurements, we select the $W$ bosons and associated jets according to the kinematic cuts adopted by ATLAS experiment Aad:2014qxa . The electrons are constrained in the phase space $p_{T}>25$ GeV/$c$ and $|\eta|<$ 2.47 and are rejected in the transition region ($1.37<|\eta|<1.52$). Muons are required to have $p_{T}>25$ GeV/$c$ and $|\eta|<2.4$. Additionally, jets are reconstructed using the anti-kt algorithm Cacciari:2011ma ; Cacciari:2008gp with a radius parameter $R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$ = 0.4 using all the final state partons. Jets are required to have $p_{T}>30$ GeV/$c$ and $|y|<4.4$ and are removed if a jet is within $\Delta R$ = 0.5 of an electron or muon. Furthermore, since the $W$ boson eventually decays into an electron and a neutrino, events are required to have significant missing transverse momentum and large transverse mass to compensate the missing information of the neutrino which can not be detected directly by experiment. The missing transverse momentum is defined as the negative vector sum of the transverse momentum of leptons, photons, and jets as well as the soft deposits in the calorimeter, and is required to have $E^{miss}_{T}=-|\vec{p}_{T}^{\ l}+\vec{p}_{T}^{\gamma}+\sum\vec{p}_{T}^{jets}+\vec{p}_{T}^{\ soft}|>25$ GeV/$c$ Aad:2014qxa ; Aad:2019sfe . Transverse mass is defined as $m_{T}=\sqrt{2p_{T}^{l}p_{T}^{\nu}(1-\cos(\phi^{l}-\phi^{\nu}))}$ and required to have $m_{T}>40$ GeV/$c$.

The differential cross section of jet production associated with a $W$ boson as a function of jet transverse momentum calculated by SHERPA is compared with the experimental data Aad:2014qxa in Fig. 2a. The distribution in di-jet invariant mass $m_{jj}=\sqrt{(E^{L}+E^{SubL})^{2}-(\vec{p}^{L}+\vec{p}^{SubL})^{2}}$ between the two leading jets in $N_{jets}\geq$ 2 events is also calculated and compared with experimental data in Fig. 2b. The jet distributions in association with a $W$ boson production from SHERPA show excellent agreement with the experimental data and can be used as references and inputs for the energy loss models to study jet-medium interactions in heavy-ion collisions. The jet spectrum monotonically decreases as a function of jet transverse momentum. However, the distribution as a function of the dijet invariant mass increases when $m_{jj}<M_{W}$ and decreases steeply when $m_{jj}>M_{W}$, which is similar with the $W$ boson mass distribution and quite different from inclusive dijet mass distribution which is a monotonic function of $m_{jj}$.

For a quantitative investigation of the jet properties associated with a vector gauge boson in heavy-ion collisions, cold nuclear matter (CNM) effects should also be taken into consideration. In our calculations, we use the EPPS16 Eskola:2016oht nuclear parton distribution functions (nPDF’s) in the LHAPDF library to investigate the cold nuclear matter effect due to nuclear modification of the parton distribution functions. Cold nuclear matter effects are negligible in the distribution of gamma/$Z$+jets in the kinematic ranges we are interested in. However, the cross section of jet production associated with a $W$ boson is rather sensitive to the isospin dependence of nPDF due to the production of the charged $W$ gauge bosons. The nuclear modification factors for the jet $p_{T}$ distribution due to CNM are calculated and shown in Fig. 3. As one can see, $W^{-}$ is enhanced by 20$\%$ while $W^{+}$ is suppressed by 20$\%$ due to the isospin dependence of nPDF in Pb nuclei. However, the CNM effect beyond the isospin dependence of nPDF is negligible as seen in the modification factor for the sum of $W^{+}$ and $W^{-}$ triggered jets. Similar conclusions are reached in Ru:2014yma . Therefore, the isospin dependence of nPDF must be taken into account for the study of nuclear modification of jet production associated with $W^{+}$ or $W^{-}$ bosons. However, CNM effect becomes negligible when the final results are averaged over $W^{+}$ and $W^{-}$.

In our study, jet propagation, parton energy loss and the medium response in hot/dense QCD medium due to jet-medium interactions are simulated within the Linear Boltzmann transport (LBT) model Li:2010ts ; He:2015pra ; Cao:2016gvr , which is based on the Boltzmann equation,

for parton propagation in the QGP medium, where $f_{i}$’s are phase-space distributions of medium parton. $S_{2}(s,t,u)$ is Lorentz-invariant regulation condition to regulate all soft and collinear divergency. Elastic scatterings are simulated with the corresponding matrix elements $|M_{ab\rightarrow cd}|$ which include the complete set of leading order $2\rightarrow 2$ elastic scattering processes.

The induced gluon radiation from inelastic scattering is numerically incorporated into LBT according to the High-Twist formalism Guo:2000nz ; Zhang:2003yn ; Zhang:2003wk :

where, $x$ and $k_{\perp}$ are the energy fraction and transverse momentum of the radiated gluon, respectively, $P(x)$ is the splitting function, $\hat{q}$ is the jet transport coefficient which is calculated from the elastic scattering, and $\tau_{f}=2Ex(1-x)/(k_{\perp}^{2}+x^{2}M^{2})$ is the formation time of the radiated gluon. The medium information is provided by 3+1D CLVisc hydrodynamics Pang:2012he ; Pang:2014ipa with the initial condition provided by the AMPT Lin:2004en Monte Carlo model. LBT has been successful in describing experimental data on the suppression of large $p_{T}$ hadrons He:2015pra ; Cao:2016gvr , inclusive jets He:2018xjv , $\gamma$-hadron/jets Chen:2017zte ; Chen:2020tbl ; Luo:2018pto correlations and $Z$+jet production Zhang:2018urd .

We first investigate the distribution in dijet invariant mass $m_{jj}$ between the two leading jets in $W$-jets events with $N_{jets}\geq 2$ and medium modification of the $m_{jj}$ distribution, which is defined as

In Fig. 4, we present the normalized dijet invariant mass distribution for events passing the $W$-jets selection in p+p and the scaled distributions in Pb+Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV as well as the nuclear modification factor. Since the dijet invariant mass is proportional to the virtuality  Majumder:2014gda ; Connors:2017ptx of the initial hard scattering, the suppression of the modified invariant mass distribution in Pb+Pb relative to p+p collisions is mainly due to the effects of jet quenching. We note that the $m_{jj}$ distribution is significantly suppressed due to jet quenching and the modification factor tends to decrease with increasing $m_{jj}$ as shown in Fig. 4. The suppression of this dijet invariant mass distribution is due to the reduction of the dijet events that pass all the selection cuts in Pb+Pb collisions due to jet quenching. The $m_{jj}$ dependance of the suppression factor also indicates that the effective invariant mass of the dijets that pass the selection cuts is suppressed due to the broadening of each individual jet and their relative momentum.

We also calculate another nuclear modification factor for the double differential cross section of W+jet produciton:

which is defined as the ratio of the double differential tagged jet spectra in central Pb+Pb collisions to that in p+p collisions. The double differential tagged jet spectra in both 0-30% central Pb+Pb and p+p collisions are shown in Fig. 5a and the modification factors are shown in the bottom plot of Fig. 5b in four $p_{T}^{W}$ intervals.

In LO calculations, the jet is produced in the opposite direction of the recoil $W$ boson with the same momentum in the transverse plane. The tagged jet spectra will fall off rapidly above the cutoff value of $p_{T}^{W}$. With high-order corrections from NLO perturbative matrix element calculations of hard emissions as well as resummation of soft and collinear radiations, the tagged jet spectra are smeared but have a maximum value at around the $p_{T}^{W}$ interval. The jet energy loss in Pb+Pb collisions will lead to a shift of the tagged jet spectra to a smaller value of $p_{T}$. This results in the suppression at low $p_{T}^{jet}$ and the enhancement at high $p_{T}^{jet}$ of the nuclear modification factor $I_{AA}$. Consequently, the nuclear modification factor is quite sensitive to the transverse momentum cut for the $W$ boson and reach its minimum value in $p_{T}^{jet}\simeq p_{T}^{W}$ region. This is similar to the jet spectra tagged by direct photon or $Z$ boson.

Since the $W$ boson eventually decays into an electron and a neutrino, the existence of the neutrino with missing energy would make the reconstruction of the $W$ boson relatively more difficult than that of $Z^{0}$ boson, particularly in Pb+Pb collisions with enhanced production of low $p_{T}$ particles Aad:2019sfe . When correlation of $W$+jets in heavy-ion collisions is concerned, the situation may be further complicated due to the attenuation of jet energies in the QGP.

To facilitate the experimental study of $W$+jets in Pb+Pb, we define

which represents the vector sum of the lepton and jets in a $W$+ jets event, and propose to measure the nuclear modification of $\vec{p}_{T}^{Miss}$ distribution as given by:

The missing transverse momentum $\vec{p}_{T}^{Miss}$ excludes neutrino, and only includes lepton and jets. Therefore it should be much easier to be measured. In p+p collisions, it is equal to the transverse momentum of the neutrino because of momentum-energy conservation. In Pb+Pb collisions, it represents the vector sum of the transverse momentum that is outside of the jet cone and the neutrino. This missing energy in Pb+Pb collisions reflects directly the amount and the direction of energy that jets loses in the $W$+jets event in Pb+Pb collisions. The distributions of events passing the $W$ + jets kinematic selection cut as a function of $\vec{p}_{T}^{Miss}$ in Pb+Pb to that in p+p collisions at $\sqrt{s}=5.02$ TeV is plotted in Fig. 6a while their ratio is illustrated in Fig. 6b. One observe that both distributions peak around $p_{T}^{W}$, and jet quenching effect in Pb+Pb may shift the peak to a smaller value of $p_{T}^{miss}$. This shift is caused by transverse energy transfer outside the jet cone due to elastic and inelastic scattering with the medium and the direction of the energy carried by radiated partons outside the jet cone in the opposite direction of the neutrino or $W$ boson. As a consequence, the modification factor $R_{AA}(p_{T}^{Miss})$ increases as a function of $\vec{p}_{T}^{Miss}$ in the region $\vec{p}_{T}^{Miss}<$ 50 GeV/$c$ and decreases with increasing $\vec{p}_{T}^{Miss}$ in the region $\vec{p}_{T}^{Miss}>$ 50 GeV/$c$, and is greater than one in the region 30 $<\vec{p}_{T}^{Miss}<$ 60 GeV/$c$.

To quantify the relative energy loss of $W$+jet events due to jet-medium interaction, we start with the nuclear modification for the summed scalar $p_{T}$ of all reconstructed jets that pass the kinematic cut in an event $S_{T}$, which should be sensitive to the total transverse momentum broadening of $W$+jets events.

The distributions in Pb+Pb and p+p collisions and the nuclear modification factor,

as a function of $S_{T}=\sum p_{T}^{jets}$ at $\sqrt{s_{NN}}$=5.02 TeV are shown in Fig. 7. We note that $R_{AA}(S_{T})$ is smaller than one if no cut is adopted on the transverse momentum of the $W$ boson. However, the distributions of $S_{T}$ is enhanced in the region $S_{T}<$ 60 GeV/$c$, and suppressed in the region $S_{T}>$ 60 GeV/$c$ if we adopt a kinematic cut $p_{T}^{W}>$ 60 GeV/$c$. $R_{AA}(S_{T})$ has similar behaviors as $I_{AA}$ for tagged jet spectra because of the steeply falling cross section when $S_{T}>p_{T}^{W}=$ 60 GeV/$c$. Compared to inclusive jet transverse momentum, the suppression of $R_{AA}$ is a result of the reduction of jet yields as well as the reduction of the jet energy in the QGP. However, $S_{T}$ is the scalar summed of all the final states jets, the difference of $S_{T}$ between p+p and Pb+Pb collisions is the total transverse momentum loss or broadening due to jet-medium interactions.

In this section we turn to correlations between the jet and the recoil $W^{\pm}$ boson. First, the imbalance in the transverse momentum between a $W$ boson and the associated jet $x_{jW}=p_{T}^{jet}/p_{T}^{W}$ is calculated in four $p_{T}^{W}$ bins and is shown in Fig. 8. We only consider the most back-to-back $W$+jet pairs which are required to have azimuthal angle difference $\Delta\phi_{jW}>7\pi/8$. Even in p+p collisions, the jet energy does not exactly balance the $W$ boson energy because of next-to-leading order effects and some of the quark’s energy may extend outside of the jet cone. Compared to p+p collisions, there is a significant displacement of the peak position of the momentum imbalance $x_{jW}$ towards a smaller value in Pb+Pb collisions. The shift of the transverse momentum asymmetry is a direct consequence of the energy loss of the jet associated with the $W$ boson with energy above the threshold. The transverse momentum of the $W$ boson is unattenuated in the QGP, while the jet loses energy to the outside of the jet cone due to elastic and inelastic interactions with the hot medium constitutions. This leads to a smaller value of $x_{jW}$ in Pb+Pb compared to that in p+p collisions.

To quantify the relative shift in the asymmetry distribution between p+p and central Pb+Pb collisions, the mean value of the momentum imbalance $\langle x_{jW}\rangle$ in different transverse momentum interval of the recoil $W$ boson is calculated as shown in Fig. 9a. We see that the mean value decreases as a function of the transverse momentum of the $W$ boson. When $p_{T}^{W}$ is in the interval 40-50 GeV/$c$, the mean value is about 0.98, the energy of the $W$ trigger is almost equal to the momentum of the associated jet in the transverse plane. When $p_{T}^{W}>80$ GeV/$c$, the mean value is about 0.85, the jet energy is noticeably smaller than the energy of the recoil $W$ boson as a result of additional soft or hard emissions from high order corrections. We can also calculate the mean value for a $W$ plus only one jet process (denoted as “$W$+1jet”) and a $W$ in association with more than one jets (denoted as “$W$+ ($\geq 2$)jets”) as shown by the dotted lines and dash dotted lines. We see that the mean value in $W$+($\geq 2$)jets processes is about 0.5 for a high energy $W$ boson, indicating that the jet energy in the multi-jet event is only half of the energy of the $W$ boson. The mean value in $W$+($\geq 2$)jets processes is greater than 1 when $p_{T}^{W}<$ 60 GeV/$c$. There is a very small probability that a $W$ boson is associated with more than one jets in the back-to-back region with energy greater than $p_{T}^{W}$. In those processes, a $W$ boson may be radiated from one of the jets as a fragmentation $W$ boson.

It is not a surprise that the mean value of the momentum imbalance in Pb+Pb collisions is much smaller than that in p+p collisions due to jet-medium interactions. The reduction of this mean value in Pb+Pb collisions from that in p+p collisions $\Delta\langle x_{jW}\rangle$ and fraction of the reduction of the mean value $\Delta\langle x_{jW}\rangle/\langle x_{jW}\rangle_{pp}$ are tabulated in Table. 1. We see the reduction increases as a function of the transverse momentum of the $W$ boson. It indicates that jets tagged by higher energy $W$ bosons lose a larger fraction of their energy.

The amount of jet energy loss in Pb+Pb collisions is also shown in the last line in Table 1. We see that the average jet energy loss increases with the energy of the recoil $W$ boson. With the increased energy of the trigger $W$ boson, the initial transverse momentum of the recoil jet is also larger and it has a higher probability to interact with the medium and loses larger fraction of its energy to the outside of the jet cone. However, our results of average jet energy loss in $W+$jets processes is smaller than the Bayesian extraction from $\gamma$+jet He:2018gks . The underlying reason of the difference come from two aspects. First, the quark fraction in $W+$jets processes is larger than that in $\gamma$+jet. In addition, jet cone size $R$ used in our calculation is 0.4 while $R=0.3$ is used in He:2018gks .

Another direct consequence of jet quenching is the reduction of the absolute jet yields above the kinematic threshold in Pb+Pb collisions, which can be investigated through calculating the average number of jet partners per $W$ boson $R_{jW}$. The dependence of $R_{jW}$ on the transverse momentum of the $W$ boson $p_{T}^{W}$ is shown in Fig. 9b. As can be seen, the average number of tagged jets per $W$ boson that pass the selection cut is overall suppressed in Pb+Pb due to jet quenching compared to that in p+p collisions. We also calculated the fraction of jet that fall below the kinematic selection threshold and shown in Table. 2. We see that, high energy $W$ bosons lose smaller fraction of jets.

In addition to the transverse momentum correlations, we also calculate the azimuthal angle correlation of jets and the recoil $W$ boson in four $p_{T}^{W}$ intervals in both p+p and Pb+Pb collisions as shown in Fig. 10. Compared to p+p collisions, the correlation is moderately suppressed at small azimuthal angle (relative to the $W$ boson) in Pb+Pb collisions. The distribution is normalized to the number of $W$ bosons that pass the selection cut rather than the number of $W$+jet pair, and one boson may can not find any associated jets or can have more than one associated jets, so the integration of the azimuthal angle correlation may be less or greater than one. The suppression of the small angle correlation of $W$+jets is mainly due to the suppression of the secondary or multiple jets by jet quenching similar to the $\gamma$/$Z$+jets correlation Luo:2018pto ; Zhang:2018urd .

To illustrate the mechanism of this suppression, the contributions from $W$ plus only one jet and $W$ in association with more than one jet to the transverse momentum asymmetry and azimuthal angle correlation are also calculated as shown by dotted line and dash-dotted line both in p+p and Pb+Pb collisions respectively in Fig. 11. We see that $W$+1jet processes dominate the large angle region where the jet is opposite to the direction of the $W$ boson in the transverse plane. However, in the small angle region, it is the $W$ plus more than one jets processes that dominate the correlation. Compared to $W$ plus only one jet, the azimuthal angle correlation of $W$ associated with more than one jets is much broadened. This is because, $W$ plus only one jet processes mainly come from leading-order matrix element and the $W$ boson is balanced by only one jet with azimuthal angle around $180^{0}$ relative to the recoil $W$ boson. On the other hand, $W$ production associated with more than one jet mainly originates from NLO matrix elements which contain hard emissions at large angles. These multiple jets with relatively lower energy can easily lose energy due to jet quenching and shift their final energy below the kinematic cut. This leads to the suppression of $W$+jets correlations at small azimuthal angle.

We observe a moderately broadened W-jet correlation in these events in Pb+Pb relative to p+p collisions, which is different with the results of $Z$+jets Zhang:2018urd where no modification is observed in $Z$ plus only one jet process between p+p and Pb+Pb collisions. The underlying reason of the difference is that the kinematic threshold used in those calculations is different, especially the cut on the jet rapidity. Fig. 12 shows the azimuthal angle correlation $\Delta\phi_{jW}$ of $W$ plus only one jets (up) and $W$ associated with more one jets (bottom) with jet rapidity $|y^{jet}|<1.6$ and $|y^{jet}|<4.4$ and the comparison between p+p and Pb+Pb collisions. We see $W$+1jets is much broader with jet rapidity $|y^{jet}|<1.6$ compared to $|y^{jet}|<4.4$, while contributions from $W$+2jets is much enhanced with jet rapidity $|y^{jet}|<4.4$ compared to $|y^{jet}|<1.6$. This is because larger rapidity cut would include more jets, as a result of which, the $W$ event in association with only one jet with constraint $|y^{jet}|<1.6$ would become a event in which $W$ is associated with more than one jet with condition $|y^{jet}|<4.4$. With constraint $|y^{jet}|<1.6$, no significant difference of the $\Delta\phi_{jW}$ in $W$+1jet is observed between p+p and Pb+Pb collisions as in $Z$+jets Zhang:2018urd . However, a moderately broadened W-jet correlation is seen with constraint $|y^{jet}|<4.4$ in Pb+Pb compared to p+p collisions. This is because the energy of the jets with large rapidity in $W$ plus multi-jets events is relative small. When some of those jets lose energy and get easily lost in Pb+Pb collisions. Some of these $W$ plus multi-jets events in p+p collisions would become $W$+1jet process in Pb+Pb collisions due to jet quenching. This leads to the enhancement of $W$+1jet azimuthal correlation in the small angle region with kinematic constraint $|y^{jet}|<4.4$ in Pb+Pb relative to p+p collisions.