The Nucleon Energy Correlators

Introduction. The femtoscale structure of the nucleon has been the central scientific importance of nuclear physics for decades. The next generation QCD facilities [1, 2, 3] will boost the revelation of the nucleon/nucleus partonic structure in great detail. Conventional approach to the nucleon/nucleus tomography is to probe its transverse momentum dependent (TMD) structure functions through either the semi-inclusive deep inelastic scattering (SIDIS) [4, 5, 6, 7, 8, 9] or the jet-based studies [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. However SIDIS calls for the knowledge of the TMD fragmentation functions, while jets involve the clustering procedure and require high machine energy, either probes seem to complicate the analysis in one way or another.

There are other substitutions to jets and identified hadrons, such as event-shape observables and the energy-energy correlator (EEC) [26, 27]. Recently, there has been ongoing efforts to reformulate jet substructure physics using the EEC and its higher point generalization [28]. This is largely inspired by the unprecedented detector resolution at the LHC, as well as insights from conformal collider physics [29, 30, 31, 32, 33, 34]. For recent application of energy correlators in jet substructure see for example [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49].

The EEC measures the correlation $\langle{\cal E}(n_{i}){\cal E}(n_{j})\rangle$ between the energy deposit in two detectors along directions $n_{i}$ and $n_{j}$ with angular separation $\theta_{ij}$, where ${\cal E}(n)=\lim_{r\to\infty}\int_{0}^{\infty}dtT_{0\vec{n}}(t,\vec{n}r)r^{2}$ is the asymptotic energy flow operator with $T_{\mu\nu}$ the energy-stress tensor [50, 51, 52, 53]. One particular interesting feature of the EEC is its collinear limit as $\theta_{ij}\to 0$, in which the EEC exhibits a universal scaling behavior (modulo running coupling effects), $\lim_{n_{j}\to n_{i}}{\cal E}(n_{i}){\cal E}(n_{j})\sim\theta^{\gamma(\alpha_{s})}$ as described by the light-ray OPE [29, 32]. The collinear limit is well encoded in the EEC jet function [34]

where $\chi_{n}$ is the gauge invariant $n$-collinear quark field in the soft collinear effective theory (SCET) [54, 55, 56, 57, 58] that serves the source to create collinear particles out of the vacuum $|\Omega\rangle$ and $Q$ represents the hard scale that initiates the process. $E_{i}$ and $E_{j}$ are the energies measured. The light like vectors $n=(1,\vec{0}_{\perp},1)$ and ${\bar{n}}=(1,\vec{0}_{\perp},-1)$. When $\theta Q\gg\Lambda_{\rm QCD}$, the EEC jet function can be predicted using the collinear splitting functions [34, 35]. When $\theta Q\sim\Lambda_{\rm QCD}$, a striking confining transition was observed in analyzing CMS Open Data [40].

The EEC has also been adapted to the TMD studies in DIS, where the back-to-back limit $\theta_{ij}\to\pi$ is probed instead [59, 60, 61]. It was shown that when the EEC is measured in DIS in this limit, the unpolarized TMD parton distribution arises [59, 61]. However, given that the EEC essentially measures the final state correlations and carries no information on the nucleon, the current EEC probe of the TMDs is indirect, in the sense that the EEC is used as a simple replacement of the jets or hadrons, while its power and features are not yet fully exploited.

In this manuscript, we propose a novel energy correlator named the nucleon EEC that probes the initial-final state correlation. The nucleon EEC encodes the information on the nucleon three dimensional microscopic structures, meanwhile inherits the fascinating features of the conventional final-state EEC. As one of the major results of this work, we will demonstrate the accessibility to the nucleon EEC via the measurement in DIS

where $N\geq 1$ is a positive power and $d\sigma$ is the differential cross section. $x_{B}$ is the Bjorken variable and $\theta_{i}$ is the polar angle of the calorimeter measured with respect to the beam. $p_{i}$ denotes the momentum flow into the detector and $P$ the momentum of the incoming nucleon while $Q^{2}$ the virtuality of the photon. We leave the detailed explanation to the rest of this manuscript.

The nucleon energy-energy-correlator. We first generalize the EEC jet function to introduce the (unpolarized) nucleon EEC, whose definition is

Here $z_{b}$ is the partonic momentum fraction with respect to the incoming hadron $P$ that enters the hard interaction at the hard scale $Q$. Here $\theta_{k}$ is the polar angle of the caloremiter $k$ with respect to the beam. The gluon nucleon EEC can be defined similarly using the gauge invariant gluonic field.

The nucleon EEC correlates the energy $E_{k}$ from the initial state radiation that flows into the calorimeter and the energy $z_{b}$ that participates the hard interaction, as illustrated in Fig. 1. The nucleon EEC has several fascinating features:

The $x_{B}$ weighted deep-inelastic scattering. To see how the nucleon EEC can be measured, we consider the DIS process $l+P\to l^{\prime}+X$ in the frame where the virtual photon $\gamma^{\ast}$ acquires no transverse momentum, e.g., the $\gamma^{\ast}$-$P$ center of mass frame or the Breit frame. We assume the nucleon is moving along the $+z$-direction. We measure the Bjorken $x_{B}=\frac{-q^{2}}{2P\cdot q}$ and the momentum flow $p_{i}^{\mu}$ deposit (including the ones from the beam remnants) in a calorimeter along the direction $i$. Here $q=l^{\prime}-l$ is the momentum carried by the virtual photon. In this manuscript, we will be particularly interested in the scenario where the detector is placed in the far-forward region and therefore the transverse momentum flow $p_{i,t}^{2}\sim\theta^{2}E_{i}^{2}$ is very small compared with $q^{2}=-Q^{2}$, as depicted in Fig 3.

The measurement probes the weighted cross section $\Sigma_{N}(Q^{2},\theta)$ in Eq. (2). We note that the polar angle $\theta_{i}$ is related to the transverse momentum as $\sin\theta_{i}=\frac{p_{i,t}}{E_{i}}$. For the time being, we assume the proton is unpolarized.

The weighted cross section in Eq. (2) can be calculated via

where $j^{\mu}$ is the conserved current. $\epsilon_{\lambda}$ is the virtual photon polarization vector with $\lambda=L,T$ for the longitudinal and transverse polarization, respectively. $f_{\lambda}(y)$ is the photon flux such that $f_{T}=1-y+y^{2}/2$ and $f_{L}=2(1-y)$ where $y=\frac{2p\cdot q}{2p\cdot l}$ is the inelastcity. We note that the property of the similar matrix element with $|P\rangle$ replaced by the vacuum state has been discussed in context of the conformal collider physics [29].

When $\theta\ll 1$ and thus $i$ is close to the beam, it is ready to show by using SCET [54, 55, 56, 57, 58] that $\Sigma_{N}$ takes the factorized form at LL

where we see the occurrence of the nucleon EEC $f^{i}_{EEC}(N,\theta)$ and therefore the proposed measurement does probe the nucleon EEC. The $z_{b}^{N-1}$ within $f^{i}_{EEC}(N,\theta)$ in Eq. (3) enters through the $x_{B}^{N-1}$ weight. We note that the coefficient of $f_{EEC}^{i}$ is nothing but the Mellin-moment $d\hat{\sigma}_{i}(N,\mu)$ of the partonic DIS cross section ¹¹1Strictly speaking, this is only true if we integrate $x_{B}$ down to $0$ in Eq. (2). However, if we choose sufficiently large $N$, to suppress the contribution from the small values of $x_{B}$, the integral will be well approximated by the Mellin-moment., satisfy $d\hat{\sigma}_{i}(N,\mu)/d\ln\mu^{2}=-\gamma^{N}_{ij}\hat{\sigma}_{j}(N,\mu)$. Here $i$ and $j$ can either be a quark or gluon.

When $\theta Q\gg\Lambda_{\rm QCD}$, the nucleon EEC is further factorized following Eq. (5). Thus the scale dependence of the coefficient $I_{ij}$ in Eq. (6) is an immediate consequence of the scale independence of the weighted cross section $d\Sigma_{N}/d\ln\mu=0$.

Now we estimate the requirement of the forward detector for this measurement. Suppose we want to probe the intrinsic transverse momentum of the nucleon, we will demand the detector to detect transverse momentum flow $p_{i,t}\sim\Lambda_{\rm QCD}$ and thus to cover polar angles down to $\theta\sim p_{i,t}/Q$. Hence for $Q\sim{\cal O}(5\>{\rm GeV})$, the estimated $\theta\sim{\cal O}(0.2\>\text{rad})$, which is well covered by the EIC far-forward particle detection plan [1, 2, 62, 63] and will be even better favored if the coverage proposals such as the Zero Degree Calorimeter [62, 64] down to and below $5\,{\rm mrad}$ would be realized. We emphasize that since we only count the energy deposit in the calorimeters, no jet clustering procedure is needed. Meanwhile, instead of using the calorimetry, the track-based measurements can be carried out [28, 38, 39] to offer better pointing and angular resolution.

Here we predict the normalized differential distribution $\langle\text{EEC}\rangle_{N}=1/\sigma\theta^{2}d\Sigma_{N}(\theta^{2})$ in the Breit frame. We define the rapidity $y=\ln\tan\theta/2$. For the prediction, we use Pythia82 [65] with the proton $P=275\>{\rm GeV}$ and the incoming lepton $l=10\>{\rm GeV}$.

In Fig. 4, we show the predictions for $\langle\text{EEC}\rangle\equiv\langle\text{EEC}\rangle_{2}$, i.e., with $N=2$. We vary the values of $Q$ with $Q>10\>{\rm GeV}$, $Q>30\>{\rm GeV}$, $Q>50\>{\rm GeV}$. We see from the upper panel of Fig. 4 that although the $Q$’s are different, the predicted $\langle\text{EEC}\rangle$’s display similar features, which implies that the normalized distributions reflect the property of the nucleon itself at different scales $\mu\sim Q$.

We note that Fig. 4 exhibits an interesting “phase transition” between the perturbative-phase for $\theta\gtrsim 0.2\>{\rm rad}$ and the “free-particle-phase” for $\theta\lesssim 0.005\>{\rm rad}$, connected by the non-perturbative transition region. In the perturbative region, the distribution is almost flat, largely independent of $Q$’s, which is a direct manifestation of Bjorken scaling in the space of polar angle (rapidity), as can be seen from Eq. (The Nucleon Energy Correlators).

The feature is more evident by looking at the slope $\gamma$ showed in the lower panel of Fig. 4, where in the perturbative region $\gamma\sim 0$, while in the deep non-perturbative region for $\theta\lesssim 0.005\>{\rm rad}$, $\gamma\sim 2$. It will be very interesting if we can confirm such phase transition at future experimental facilities.

We again notice that all the slope $\gamma$ distributions with different values of $Q$ shares similar behaviour which indicates it reveals the intrinsic property of the nucleon EEC at different scales $\mu$. The transition region moves to the right as we decrease $Q$, which is expected since the transition occurs when $\theta\sim{\cal O}(\Lambda_{\rm QCD}/Q)$. The $\gamma$ in the perturbative region can be predicted using the LL result in Eq. (The Nucleon Energy Correlators) and the factorization in Eq. (9), which is showed in the red line. All the $Q$ values are covered within the band, obtained by a dramatic variation of $\mu$, from $\mu=50\>{\rm GeV}$ to $\mu=300\,{\rm MeV}$. We find good agreement between the Pythia simulations and the analytic LL result in the perturbative phase. We emphasize that future observed deviation from the predicted slope could be used to extract the nature of the initial state source that induces the modification to the collinear splitting kernel, such as the hadronization and the hot/cold nucleus medium effects. The theory precision can be further improved and we leave it to future works.

In Fig. 5, we show the $\langle\text{EEC}\rangle_{N}$ for different $N$. The smaller values of the $N$ increase the sensitivity to the small-$x$ component of a nucleon while larger $N$ probes more into the large $x$ regime. Again the plot manifests the polar angle (rapidity) scaling behavior of the nucleon EEC, similar to the famous Bjorken scaling rule of the PDFs. We note that for larger $N$, therefore effectively larger $x$ closer to $1$, the transition region appears at larger angle, consistent with the expectation that $\theta\sim{\cal O}(\Lambda_{\rm QCD}/Q)$ increase as $Q$ decrease. Also since the normalization of nucleon EEC is propotional to moment of PDFs, we expect to see a decrease in the magnitude as $N$ increase, as a result of small-$x$ enhancement of PDFs.

Transversely polarized EEC. Once the incoming nucleon is polarized, we can also probe the spin asymmetry by measuring the azimuthal modulation. Here the beam and the out-going lepton span the $x$-$z$ plane.

For instance, for the transversely polarized nucleon beam, if we measure the $\sin(\phi-\phi_{S})$ distribution with $\phi-\phi_{S}$ the azimuthal angle between the detector and the nucleon spin, we are probing the spin dependent distribution $d\Sigma_{N}(\vec{n}_{t},S_{T})$ whose $\sin(\phi-\phi_{S})$ dependent part factorized similarly as the unpolarized case in Eq. (9), with the replacement of $f^{i}_{EEC}(\theta)$ by

for quark, where $\vec{n}_{t}=\sin\theta(\cos\phi,\sin\phi)$, $S_{T}$ is the nucleon spin and $M_{P}$ is the nucleon mass. The non-vanishing of $f_{T,EEC}$ is owing to the same mechanism that gives rise to the Sivers effect [66, 67]. The Sivers-like EEC $f_{T,EEC}$ induces the $\sin(\phi-\phi_{S})$ azimuthal asymmetry $A_{N}=\frac{d\Sigma_{N}(S_{T})-d\Sigma_{N}(-S_{T})}{\Sigma(S_{T})+\Sigma(-S_{T})}$. The predicion of $A_{N}$ relies on the non-perturbative input of the $f_{T,EEC}$ which requires further studies in the future. Since there is no Sudakov surprresion, we anticipate a better chance to observe the asymmetry at the EIC.

Conclusion. In this manuscript, we introduced the nucleon energy-energy-correlator (EEC) that measures the correlation of the energy flows from the initial nucleon. The nucleon EEC reflects the parton angular distribution in a nucleon. This new object is novel both theoretically and phenomenologically. Theoretically, we have demonstrated that the microscopic details of the nucleon such as the faithful vacuum collinear splitting behavior as well as the nucleon internal transverse momentum and spin degrees of freedom are imprinted in the energy correlation function, and meanwhile the nucleon EEC may offer additional possibilities to understand the nucleon structures using the light-ray OPE in QCD. Phenomenologically, we showed how the nucleon EEC can be probed at EIC with a far-forward detector. We have set the theoretical foundation for the observable and predict the measured distribution at EIC to exhibit power law scaling behaviours. A novel phase transition between the free-particle and the perturbative phases is observed. In the perturbative region, the polar angle (rapidity) version of the Bjorken scaling behavior is also predicted.

One advantage of the nucleon EEC is that its measurement involves no jet clustering procedure nor additional non-perturbative object other than the nucleon EEC itself. Besides, the factorization in Eq. (9) only involves a product instead of a convolution should make the extraction of the nucleon EEC a lot easier. Therefore it serves a clean complement to the conventional TMDs to the nucleon structures, good for either high energy or low energy machines.

Other than the scenarios considered in the manuscript, we expect the proposed nucleon EEC to have a wide application to future nucleon/nucleus studies. Extensions to other observables sensitive to the various TMD distributions will follow straightforwardly. By suitably choosing the weight $N$, the nucleon EEC can be made sensitive to the small-$x$ phenomenology. The nucleon EEC can also be used to study the cold nuclear effect in $eA$ collisions or to extract the hot medium effect with heavy ion data. All these effects will leave its foot print in the deviations from the $\langle\text{EEC}\rangle_{N}$ and its slope introduced in this work. As long as one charm is tagged in the detected forward event, the nucleon EEC can offer a direct look into the intrinsic charm content. Furthermore, the generalization of the EEC to multiple point correlations will allow for more delicate differentiation of the nucleon/nucleus microscopic details. We thus anticipate that the nucleon EEC introduced in this work will stimulate further theoretical developments along these directions.