Quantum Interference in Jet Substructure from Spinning Gluons

Collimated sprays of hadrons, called jets, are an emergent phenomenon of Quantum Chromodynamics (QCD) at collider experiments, whose detailed internal structure encodes valuable information about the interactions of high energy quarks and gluons, and their confinement into color-neutral hadrons. The flow of energy within jets is characterized by correlation functions of energy flow operators, with the three-point correlator, being the first correlator with non-trivial shape dependence, playing a special role in unravelling the dynamics of QCD. In this Letter we initiate a study of the three-point energy correlator to all orders in the strong coupling constant, in the limit where two of the detectors are squeezed together. We show that by rotating the two squeezed detectors with respect to the third by an angle $\phi$, a $\cos (2\phi)$ dependence arising from the quantum interference between intermediate virtual gluons with $+/-$ helicity is imprinted on the detector. This can be regarded as a double slit experiment performed with jet substructure, and it provides a direct probe of the ultimately quantum nature of the substructure of jets, and of transverse spin physics in QCD. To facilitate our all-orders analysis, we adopt the Operator Product Expansion (OPE) for light-ray operators in conformal field theory and develop it in QCD. Our application of the light-ray OPE in real world QCD establishes it as a powerful theoretical tool with broad applications for the study of jet substructure.

Collimated sprays of hadrons, called jets, are an emergent phenomenon of Quantum Chromodynamics (QCD) at collider experiments, whose detailed internal structure encodes valuable information about the interactions of high energy quarks and gluons, and their confinement into color-neutral hadrons. The flow of energy within jets is characterized by correlation functions of energy flow operators, with the three-point correlator, being the first correlator with non-trivial shape dependence, playing a special role in unravelling the dynamics of QCD. In this Letter we initiate a study of the three-point energy correlator to all orders in the strong coupling constant, in the limit where two of the detectors are squeezed together. We show that by rotating the two squeezed detectors with respect to the third by an angle φ, a cos(2φ) dependence arising from the quantum interference between intermediate virtual gluons with +/− helicity is imprinted on the detector. This can be regarded as a double slit experiment performed with jet substructure, and it provides a direct probe of the ultimately quantum nature of the substructure of jets, and of transverse spin physics in QCD. To facilitate our all-orders analysis, we adopt the Operator Product Expansion (OPE) for light-ray operators in conformal field theory and develop it in QCD. Our application of the light-ray OPE in real world QCD establishes it as a powerful theoretical tool with broad applications for the study of jet substructure.
Introduction.-Jet substructure, which was originally developed to exploit the flow of energy within jets of particles at the Large Hadron Collider (LHC) to enhance new physics searches [1], has since emerged as a primary technique for studying QCD. By focusing on the internal structure of jets, one obtains a clean probe of the dynamics of QCD on the light cone, which is measured with exquisite angular precision by the LHC detectors. This has led to a wealth of theoretical and experimental progress with the goal of fully exploiting this remarkable data set [2][3][4].
Much more non-trivial information about the dynamics of QCD is encoded in higher-point correlation functions. In this Letter we show that the squeezed limit of the three-point correlator [21], E(n 1 )E(n 2 )E(n 3 ) , where two detectors are brought together,n 2 →n 3 , is a direct probe of quantum interference between helicity λ = ± gluons in the jet, i.e. the transverse spin structure of QCD, and that this is imprinted on the detector as a cos(2φ) interference pattern. While the study of transverse spin in QCD has a long history (see e.g. [22][23][24][25][26][27][28][29][30][31][32][33][34][35]), the squeezed limit of the three-point correlator provides a qualitatively new observable for studying transverse spin. Unlike standard probes of transverse spin that rely on non-perturbative fragmentation processes, energy correlators are infrared and collinear safe, and for the high energy jets at the LHC, exhibit a regime where the transverse spin structure can be computed in perturbative QCD. In the extreme collinear limit they are sensitive to non-perturbative fragmentation, allowing for the study of the transition between perturbative and nonperturbative physics. Furthermore, they do not involve polarized beams or hadrons, or reference directions, and so are ideally suited for the LHC.
The description of the complicated all orders perturbative structure of multi-point correlators calls for the application of new theoretical tools. Much in analogy with how the local operator product expansion [36], , describes the behavior of local operators as they are brought together, in the limit of interest in jet substructure where the light-ray operators E(n) become collinear, there exists a recently developed light-ray OPE [9,12,17,37,38] E(n 1 )E(n 2 ) = θ γi O i (n 1 ), where the O i (n) are nonlocal light-ray operators [9,12,[39][40][41][42]. However, the use of the light-ray OPE has so far been restricted to conformal field theories (CFTs) [9,17,37,38].
In this Letter we show that the light-ray OPE can be arXiv:2011.02492v1 [hep-ph] 4 Nov 2020 i (n) with arbitrary collinear spin-J, but restricted transverse spin-j = 0, 2, and we explicitly compute the E(n 1 )E(n 2 ) and O [J] i (n 1 )E(n 2 ) OPEs. The all orders structure of spin interference effects in the threepoint correlator then arises naturally from the transverse spin structure of the light-ray OPE.
Interference in the Squeezed Limit.-The physics of the squeezed limit of the three-point correlator in a weakly coupled gauge theory can be described as a double slit experiment in spin space, see Fig. 1. The interference pattern in the usual double slit experiment is due to the interference in |A L (x) + A R (x)| 2 , where A L(R) (x) is the amplitude for going through the left (right) slit from the light source to position x on the detector. Similarly, in the squeezed limit of the three-point correlator, the interference terms in |A + (φ) + A − (φ)| 2 are the source of an interference pattern, where A +(−) is the splitting amplitude with a nearly on-shell virtual gluon with positive (negative) helicity. Therefore the slits in the standard double slit experiment are replaced by the intermediate +/− helicity gluons, and varying the distance x is replaced by varying the angle φ of the squeezed energy correlators.
We parametrize the squeezed limit symmetrically, using (θ S , θ L , φ) as shown in Fig. 1, to eliminate linear power corrections in θ S /θ L . The squeezed limit is characterized by θ S θ L , with φ arbitrary, and the expansion in this limit takes the form where the dots denote terms less singular in the squeezed limit. Expanding the full result for the three-point correlator in [21], we find for quark and gluon jets, Here we see cos(2φ) interference terms at leading twist, which at this order are identical for quark and gluon jets, since they arise only from an intermediate gluon, and have opposite signs for g → qq (in blue) and g → gg (in red). Positivity of the cross section guarantees that the cos(2φ) terms are smaller than the constant terms, analogous to the conformal collider bounds [9]. Due to the singular structure of the squeezed limit, the all orders resummation of these spin interference effects is required to describe the three-point correlator, as well as for limits of higher-point correlators.
Despite their importance for observables relevant to jet substructure, spin interference effects are not included in the standard parton shower simulations used to this point by experiments at the LHC, which, as illustrated through detailed studies in [43], can lead to large errors for multiemission jet substructure observables. Recently, significant progress was made with the implementation of algorithms for including spin interference effects in parton showers [44][45][46][47][48][49] in a new version of Herwig [49,50]. The goal of this Letter will be to understand the all orders I + One of the simplest observables from the theoretical perspective is the Energy-Energy Correlator (EEC), defined as [2,3] Here E i and E j are the energies of final-state partons i and j in the center-of-mass frame, and their angular separation is ij . d is the product of the squared matrix element and the phase-space measure. The EEC can also be defined in terms of correlation function of ANEC operators [4][5][6][7] where it is given by for some source operator O. This provides a connection between event shape observables and correlation functions of ANEC operators allowing the study of event shapes to profit from recent developments in the study of ANEC operators, and conversely, the EEC provide a concrete situation for studying the behavior of ANEC operators. There has recently been significant progress in the understanding of the EEC from a number of di↵erent directions. For generic angles, the EEC has been computed at next-toleading order (NLO) in QCD [8,9] for both an e + e source, and Higgs decaying to gluons, and up to NNLO in N = 4 SYM [7,10]. It has also been computed numerically in QCD at NNLO [11,12].
There has also been progress in understanding the singularities of the EEC, which occur as z ! 0 (the collinear limit) and z ! 1 (the back-to-back limit). In the back-to-back limit, the EEC exhibits Sudakov double logarithms, whose all orders logarithmic structure is described by a factorization formula [13,14]. In the z ! 0 limit, which will be studied in this paper, the EEC exhibits single collinear logarithms, originally studied at leading logarithmic order in [15][16][17][18][19]. Formulas describing the behavior of the EEC in the collinear limit were recently derived in [20] for a generic field theory, and in [21][22][23][24] for the particular case of a CFT. This limit is of theoretical interest for studying the OPE structure of non-local operators, and of phenomenological interest as a jet substructure observable.
The two-point correlator is particularly simple since it depends on a single variable, z. Indeed, in a conformal field theory (CFT), its behavior in the collinear limit is fixed to be a power law One of the simplest observables from the theoretical perspective is the Energy-Energy Correlator (EEC), defined as [2,3] Here E i and E j are the energies of final-state partons i and j in the center-of-mass frame, and their angular separation is ij . d is the product of the squared matrix element and the phase-space measure. The EEC can also be defined in terms of correlation function of ANEC operators [4][5][6][7] where it is given by for some source operator O. This provides a connection between event shape observables and correlation functions of ANEC operators allowing the study of event shapes to profit from recent developments in the study of ANEC operators, and conversely, the EEC provide a concrete situation for studying the behavior of ANEC operators. There has recently been significant progress in the understanding of the EEC from a number of di↵erent directions. For generic angles, the EEC has been computed at next-toleading order (NLO) in QCD [8,9] for both an e + e source, and Higgs decaying to gluons, and up to NNLO in N = 4 SYM [7,10]. It has also been computed numerically in QCD at NNLO [11,12].
There has also been progress in understanding the singularities of the EEC, which occur as z ! 0 (the collinear limit) and z ! 1 (the back-to-back limit). In the back-to-back limit, the EEC exhibits Sudakov double logarithms, whose all orders logarithmic structure is described by a factorization formula [13,14]. In the z ! 0 limit, which will be studied in this paper, the EEC exhibits single collinear logarithms, originally studied at leading logarithmic order in [15][16][17][18][19]. Formulas describing the behavior of the EEC in the collinear limit were recently derived in [20] for a generic field theory, and in [21][22][23][24] for the particular case of a CFT. This limit is of theoretical interest for studying the OPE structure of non-local operators, and of phenomenological interest as a jet substructure observable.
The two-point correlator is particularly simple since it depends on a single variable, z. Indeed, in a conformal field theory (CFT), its behavior in the collinear limit is fixed to be a power law -2 -I + One of the simplest observables from the theoretical perspective is the Energy-Energy Correlator (EEC), defined as [2,3] Here E i and E j are the energies of final-state partons i and j in the center-of-mass frame, and their angular separation is ij . d is the product of the squared matrix element and the phase-space measure. The EEC can also be defined in terms of correlation function of ANEC operators [4][5][6][7] where it is given by for some source operator O. This provides a connection between event shape observables and correlation functions of ANEC operators allowing the study of event shapes to profit from recent developments in the study of ANEC operators, and conversely, the EEC provide a concrete situation for studying the behavior of ANEC operators. There has recently been significant progress in the understanding of the EEC from a number of di↵erent directions. For generic angles, the EEC has been computed at next-toleading order (NLO) in QCD [8,9] for both an e + e source, and Higgs decaying to gluons, and up to NNLO in N = 4 SYM [7,10]. It has also been computed numerically in QCD at NNLO [11,12].
There has also been progress in understanding the singularities of the EEC, which occur as z ! 0 (the collinear limit) and z ! 1 (the back-to-back limit). In the back-to-back limit, the EEC exhibits Sudakov double logarithms, whose all orders logarithmic structure is described by a factorization formula [13,14]. In the z ! 0 limit, which will be studied in this paper, the EEC exhibits single collinear logarithms, originally studied at leading logarithmic order in [15][16][17][18][19]. Formulas describing the behavior of the EEC in the collinear limit were recently derived in [20] for a generic field theory, and in [21][22][23][24] for the particular case of a CFT. This limit is of theoretical interest for studying the OPE structure of non-local operators, and of phenomenological interest as a jet substructure observable.
The two-point correlator is particularly simple since it depends on a single variable, z. Indeed, in a conformal field theory (CFT), its behavior in the collinear limit is fixed to be a power law -2 -

I
One of the simplest observables from the theoretical perspective is the Energy-Energy Correlator (EEC), defined as [2,3] Here E i and E j are the energies of final-state partons i and j in the center-of-mass frame, and their angular separation is ij . d is the product of the squared matrix element and the phase-space measure. The EEC can also be defined in terms of correlation function of ANEC operators [4][5][6][7] where it is given by for some source operator O. This provides a connection between event shape observables and correlation functions of ANEC operators allowing the study of event shapes to profit from recent developments in the study of ANEC operators, and conversely, the EEC provide a concrete situation for studying the behavior of ANEC operators. There has recently been significant progress in the understanding of the EEC from a number of di↵erent directions. For generic angles, the EEC has been computed at next-toleading order (NLO) in QCD [8,9] for both an e + e source, and Higgs decaying to gluons, and up to NNLO in N = 4 SYM [7,10]. It has also been computed numerically in QCD at NNLO [11,12].
There has also been progress in understanding the singularities of the EEC, which occur as z ! 0 (the collinear limit) and z ! 1 (the back-to-back limit). In the back-to-back limit, the EEC exhibits Sudakov double logarithms, whose all orders logarithmic structure is described by a factorization formula [13,14]. In the z ! 0 limit, which will be studied in this paper, the EEC exhibits single collinear logarithms, originally studied at leading logarithmic order in [15][16][17][18][19]. Formulas describing the behavior of the EEC in the collinear limit were recently derived in [20] for a generic field theory, and in [21][22][23][24] for the particular case of a CFT. This limit is of theoretical interest for studying the OPE structure of non-local operators, and of phenomenological interest as a jet substructure observable.
The two-point correlator is particularly simple since it depends on a single variable, z. Indeed, in a conformal field theory (CFT), its behavior in the collinear limit is fixed to be a power law structure of these spin interference effects analytically, and show how they emerge in a simple manner from the transverse spin structure of the light-ray OPE. Transverse spin can also be studied using the twopoint correlator with a polarized source [38]. However, for a source coupling to quarks, polarization effects appear first at O(α 2 s ). While the unpolarized cross section is known at this order in QCD [51,52] and beyond in N = 4 super-Yang-Mills (SYM) [53,54], the polarized cross-section is not (or is related by superconformal symmetry to the scalar case [38,55,56]). Spin interference effects should appear at O(α s ) for the two-point correlator in QCD with the stress tensor as a source, which would be interesting to calculate.
The Light-Ray OPE in QCD.-To our knowledge, the analytic resummation of spin interference effects in the collinear limit has not been achieved using standard perturbative QCD techniques. This is in contrast to the more well studied Sudakov (or back-to-back) region (see e.g. [57,58]). To achieve this resummation, we will apply the light-ray OPE formalism [9,12,17,37,38]. In addition to this particular phenomenological application, our calculation provides a highly non-trivial application of the light-ray OPE in a non-conformal field theory, and illustrates its potential for jet substructure at the LHC.
The OPE of light-ray operators was originally presented [9] in the context of gauge theories using the string operators of [39]. It has recently been developed into a rigorous OPE in CFTs [12,17,37,38]. Although the loss of symmetries relaxes the rigid structure of the OPE in a CFT, the intuition for its existence remains. Furthermore, it is well known that QCD often inherits structure from the conformal limit [42].
For multiple light-ray operators with small transverse separations, one expects an OPE onto a sum of light-ray operators each formed from potentially multiple operators smeared on the light cone, as shown in the Penrose diagram in Fig. 2. Characterizing the operators involved by their twist, collinear spin-J and transverse spin-j, in a weakly coupled gauge theory, such as QCD in the perturbative regime, we have the simplification that the leading twist contributions involve two fields, and therefore can only have transverse spin-0 or 2. This greatly simplifies the OPE, allowing it to be iterated to any number of light-ray operators. The structure beyond leading twist is known for the E(n 1 )E(n 2 ) OPE in CFTs [38].
The leading twist operators in QCD are (see e.g. [59]) where the plus component of a vector A µ isn · A, and n = (1, −n). We have projected the transverse spin-2 operators onto helicities λ = ± using polarization vectors, µ . These transverse spin-2 operators are purely gluonic, and will reproduce the interference terms in Eq. 3. Smearing on the light cone (the light transform [12]) gives rise to light-ray operators, which we write as a vector g,+ , O Through an explicit matching calculation using Wightman functions, we derive the leading twist E(n 1 )E(n 2 ) OPE in the leading logarithmic approximation where θ is the angle between the OPEd pair, and J = (1, 1, 0, 0) is a projector onto unpolarized quark and gluon twist operators, and the matrix of OPE coefficients is γ gg (J) γ gg (J)e −2iφ /2 γ gg (J)e 2iφ /2 γg q (J)e 2iφ γg g (J)e 2iφ γgg(J) γgg ,± (J)e 4iφ γg q (J)e −2iφ γg g (J)e −2iφ γgg ,± (J)e −4iφ γgg(J) While the general structure of this OPE was anticipated in [9,38], the explicit form of the OPE in QCD is a new result. An interesting feature of Eq. 6 is that the OPE coefficients of light-ray operators are themselves anomalous dimensions. Indeed, γ ij (J), i, j = q, g are the standard twist-2 spin-J anomalous dimensions, related to the unpolarized splitting functions, while γ i,g (J), i = q, g are moments of the spin correlated splitting functions of [60]. This builds a direct bridge between the language of splitting functions in perturbative gauge theories and the light-ray OPE, which deserves further exploration.
On the other hand, γg ,i (J) have, to our knowledge, not appeared previously in the QCD literature. They do not appear in the E(n 1 )E(n 2 ) OPE due to the projector J , but will appear in the more general O [J] (n 1 )E(n 2 ) OPE considered below. Denoting the perturbative expansion as γ ij = (α s /(4π))γ where ψ (0) is the Digamma function, and β 0 = 11C A /3 − 4T F n f /3 is the first order QCD β-function. The other anomalous dimensions are standard (see e.g. [20]). Plugging the explicit values of the OPE coefficients in Eq. 6 for J = 3 into the E(n 1 )E(n 2 ) OPE of Eq. 5 reproduces the result for the two-point correlator in QCD for both quark and gluon sources [19], providing a nontrivial check of Eq. 5.
In a CFT, the collinear spin J = 3 appearing in the E(n 1 )E(n 2 ) OPE is fixed by symmetry [9,17]. From the structure of higher logarithmic terms in [19], we expect that in QCD light-ray operators with J = 3 + O(α s ) appear at higher orders due to the breaking of conformal symmetry and the non-trivial manifestation of Basso-Korchemsky reciprocity [61,62] in the EEC [19].
Squeezed Limits from Light-Ray OPEs.-To compute the squeezed limit of the three-point correlator, E(n 1 )E(n 2 )E(n 3 ) , we perform the successive OPE E(n 2 )E(n 3 ) → O [3] (n 2 ) followed by E(n 1 ) O [J] (n 2 ) → O [J+1] (n 1 ). The second OPE is computed analogously to the E(n 1 )E(n 2 ) OPE discussed above. Due to the fact that the maximal transverse spin at leading twist in QCD is 2, we find that the iterated OPE of light-ray operators closes onto the operators O [J] (n 1 ). To be able to apply our results to arbitrary iterated squeezed limits, we compute the OPE coefficients in the E(n 1 ) O [J] (n 2 ) OPE as analytic functions of the collinear spin-J. The Unlike the E(n 1 )E(n 2 ) OPE this involves the full matrix of OPE coefficients including the entries γg ,i (J).
In the squeezed limit, the hierarchy between the splitting angles, θ S θ L , gives rise to large logarithmic corrections in OPE coefficients that must be resummed to all orders to obtain a physical result. This resummation is performed by solving the renormalization group equations for the light-ray operators, wherê with 1 a 2×2 identity matrix. Combining Eqs. 5, 8 and 9, we derive a resummed result describing the leading twist singular behavior in the squeezed limit of the three-point correlator at leading-logarithmic order where the azimuthal angle φ in Eq. 2 is identified as φ S − φ L , and the overall rotation of the jet can be integrated out since we consider unpolarized sources. The dots denote higher twist and subleading logarithmic contributions. Plugging in the explicit values for the anomalous dimensions, Eq. 7, and expanding to leading order in α s , we find that Eq. 11 exactly reproduces the fixed order results in Eq. 3, providing a highly non-trivial test of our OPE formulas.
We can also consider the squeezed limit of the threepoint correlator in N = 4 SYM [21]. Here, the evolution matrix in Eq. 10, reduces to a scalar evolution with a universal anomalous dimension [63], and γgg = γ gg [9,64], so Eq. 11 agrees with the prediction for the scaling of the transverse spin-0 and spin-2 contributions from [9]. However, we find that the leading twist spin correlations vanish after summing over the multiplet, since [C φ (3)] i,g = 0, i = φ, q, g. This agrees with the perturbative prediction of [21]. This is one manifestation of the "classicality" of N = 4 SYM.
Numerical Results at the LHC.-Using Eq. 11, we make numerical predictions for unpolarized quark and gluon jets at the LHC. In Fig. 3 (a) and (b), we show the squeezed limit of the three-point correlator, weighted by θ 2 L θ 2 S , as a function of (φ, θ S ) for fixed θ L . The ripples in the distribution are clearly visible, and illustrate the direct imprint of quantum interference effects in the detector. They are modulated by the resummation in θ S , which has a qualitatively different structure for quark and gluon jets, as discussed in [19].
For the case of QCD with n f = 5 light flavors, the interference effects are at the few percent level due to a cancellation of g → qq and g → gg splittings. However, we believe that if measured using tracks, they are on the boundary of what can be achieved (see e.g. [65]). Furthermore, there are a number of ways to enhance the interference signals, including using charge information to identify the g → qq splitting, or b-tagging to perform the measurement. In Fig. 3 (c) and (d), we show predictions using an idealized b-tagging on the squeezed pair, which enhances the modulation to an O(1) effect. We leave a detailed phenomenological study of the optimal strategy to future work.
Conclusions.-In this Letter we have studied the threepoint energy correlator, E(n 1 )E(n 2 )E(n 3 ) , at the LHC. Our study is novel both phenomenologically, where we have proposed to use squeezed limits of energy correlators to probe quantum interference and transverse spin effects in jet substructure, as well as theoretically, where we have developed the light-ray OPE in QCD, and showed that it provides a transparent way of understanding the resummation of spin interference effects.
Our results for the E(n 1 )E(n 2 ) and E(n 1 )O [J] (n 2 ) OPEs in QCD allow the description of iterated squeezed limits of n-point correlators and opens the door for these observables to be used as precision probes of QCD at the LHC. The measurement of these multi-point event correlators in the deep non-perturbative regime is also fascinating as they are sensitive to polarization effects in the non-perturbative fragmentation process.
There are numerous avenues for further theoretical development of the light-ray OPE in QCD, including understanding the applicability of celestial blocks [17,38], the appearance of light-ray operators with non-integer spin at higher logarithmic orders, the structure of the 1 → 3 light-ray OPE, and the inclusion of electromagnetically charged correlators [66]. We believe that the application of the light-ray OPE to problems of phenomenological interest at the LHC can lead to fruitful collaboration between the CFT and jet substructure communities, leading to significant advances in our understanding of QCD.