Event shapes in N=4 super-Yang-Mills theory

We study event shapes in N=4 SYM describing the angular distribution of energy and R-charge in the final states created by the simplest half-BPS scalar operator. Applying the approach developed in the companion paper arXiv:1309.0769, we compute these observables using the correlation functions of certain components of the N=4 stress-tensor supermultiplet: the half-BPS operator itself, the R-symmetry current and the stress tensor. We present master formulas for the all-order event shapes as convolutions of the Mellin amplitude defining the correlation function of the half-BPS operators, with a coupling-independent kernel determined by the choice of the observable. We find remarkably simple relations between various event shapes following from N=4 superconformal symmetry. We perform thorough checks at leading order in the weak coupling expansion and show perfect agreement with the conventional calculations based on amplitude techniques. We extend our results to strong coupling using the correlation function of half-BPS operators obtained from the AdS/CFT correspondence.


Introduction
Recently, significant progress has been made in understanding properties of correlation functions and scattering amplitudes in maximally supersymmetric Yang-Mills theory (N = 4 SYM). There is growing evidence that the theory possesses a hidden integrability symmetry which is powerful enough to determine both quantities for an arbitrary value of the coupling constant, at least in the planar limit. Correlation functions and scattering amplitudes have different properties and carry complementary information about the dynamics of N = 4 SYM. Unlike the correlation functions, the on-shell scattering amplitudes are not well defined in four dimensions due to infrared (IR) singularities and, hence, they require regularization. This introduces a dependence on unphysical parameters (such as the dimensional regularization scale playing the role of the IR regulator) which break (super)conformal symmetry. At the same time, the correlation functions of protected (half-BPS) operators are well-defined functions of the coordinates of the operators in four-dimensional N = 4 SYM. As a consequence, they do not require regularization and enjoy the full unbroken N = 4 superconformal symmetry.
The main goal of this paper is to study a different class of gauge invariant quantities in N = 4 SYM which admit two equivalent representations: They are given by integrated correlation functions and, at the same time, they can be expressed as (infinite) sums over absolute squares of scattering amplitudes. These quantities are closely related to various observables which have been thoroughly studied in the context of QCD for the final states produced in e + e − annihilation [1,2,3]. In the latter case, the electron and positron annihilate to produce a virtual photon, which in turn creates an energetic quark-antiquark pair from the vacuum. The outgoing particles move away from each other and emit a lot of radiation before fragmenting into hadrons (see Fig. 1). The distribution of particles in the final state of e + e − annihilation can be characterized by a set of observables, the so-called event shapes (see, e.g., [2]). One of them, the energy-energy correlation [4], plays a distinct role in our analysis.
e + e − γ * (q) Figure 1: Final states in e + e − annihilation in QCD. The electron and positron annihilate to produce a virtual photon γ * (q) that decays into an arbitrary number of quarks and gluons which go through a hadronization process (shaded rectangle) to become hadrons (double lines). The dot denotes the electromagnetic QCD current.
Needless to say, QCD is quite different from N = 4 SYM. Due to the presence of a mass gap in the hadron spectrum, QCD scattering amplitudes are free from IR singularities but their calculation is still impossible due to our inability to control the confining (hadronization) regime in the theory. A remarkable property of the event shapes is that, for asymptotically large values of the center-of-mass energy q 2 , the hadronization corrections become negligible (for a review see, e.g., [5]). As a consequence, the event shapes can be approximated at high energy by a perturbative QCD expansion. It is in this context that N = 4 SYM arises as a simpler model of gauge dynamics in four space-time dimensions. It shares many features with perturbative QCD, on the one hand, and can be studied analytically using its symmetries, on the other. In particular, the AdS/CFT correspondence opens up a possibility to explore the previously unreachable regime of strong coupling.
To generalize the QCD process shown in Fig. 1 to N = 4 SYM, we have to find an appropriate analogue of the QCD electromagnetic current. For this purpose we can choose any local protected operator in N = 4 SYM, e.g., the half-BPS operator O 20 ′ (x) built from two scalar fields. At weak coupling one can think about the state created by this operator as follows. The operator O 20 ′ (x) produces out of the vacuum a pair of scalars that propagate into the final state and radiate on-shell particles of N = 4 SYM -scalars, gluinos and gluons. The fact that these particles are massless leads to a degeneracy of the final states. For instance, a single-particle state is undistinguishable from the state containing an additional gluon with vanishing momentum and from the state containing a pair of particles with aligned momenta and the same total charge. According to the Kinoshita-Lee-Nauenberg mechanism [6,7], the degeneracy of on-shell states leads to (soft and collinear) divergences in the perturbative expansion of the corresponding scattering amplitudes.
This phenomenon is quite general when massless particles are present in the spectrum. An important issue in the early days of QCD was whether one could define quantities that are free from infrared divergences at all orders of perturbation theory. The answer was found with the introduction of the so-called inclusive infrared safe observables. The latter are given by a sum over an infinite number of scattering amplitudes involving an arbitrary number of degenerate states [1,4]. Each amplitude has infrared divergences but they cancel in the sum, so that infrared safe observables are well defined in four dimensions order-by-order in the coupling. The question arises whether there is another way to compute the same observables that bypasses the introduction of any regularization and, therefore, makes all symmetries of the theory manifest at each step of the calculation.
As a simple example, consider the total probability of the transition O 20 ′ → everything. This is an infrared safe quantity, but it is given by (an infinite) sum over all final states, with each individual term being infrared divergent. The optical theorem allows us to express the same quantity as the imaginary part of the two-point (time-ordered) correlation function of the half-BPS operators O 20 ′ (x) (see Eq. (2.8) below). This two-point function is well defined in four dimensions and its form is uniquely fixed by N = 4 superconformal symmetry. In this way, we obtain a definite prediction for the total transition amplitude O 20 ′ → everything that agrees with the result of an explicit calculation [8] based on amplitudes.
In this paper, we deal with a special class of event shape distributions related to the flow of various quantum numbers (energy, charge) in the final state. A typical event contributing to such an observable is shown in Fig. 4 below. There, the particles propagate into the final state where the detectors measure the flow of their quantum numbers per solid angle in the directions indicated by the unit vectors n, n ′ , . . . . As was shown in Refs. [9,10,11,12] in the context of QCD, the optical theorem can be generalized to such differential distributions. For instance, the energy flow distributions can be expressed in terms of the correlation functions O 20 ′ (x)E( n)E( n ′ ) . . . O 20 ′ (0) containing additional energy flow operators E( n), E( n ′ ), . . . (one for each detector). Quantities of this type have been studied in the framework of conformal field theories in [13], particularly in connection with the AdS/CFT correspondence. An unusual feature of these correlation functions is that the operators are not time-ordered. In other words, we are dealing with correlation functions of the Wightman type defined on a space-time with Lorentzian signature. Notice, however, that significant advances have been made in the calculation of their Euclidean counterparts. The natural question arises whether we can make use of the latter to compute the weighted cross sections. The answer was presented in the companion paper [14], where we explained in detail how to obtain the charge flow correlators in a generic CFT, starting from the Euclidean correlation functions and making a nontrivial analytic continuation. We briefly review it below to make the exposition self-contained. In this paper, we apply the approach of [14] to the particular case of N = 4 SYM.
The paper is organized as follows. In Section 2, we consider the process O 20 ′ → everything in N = 4 SYM and introduce a set of infrared safe observables, determined by weighted cross sections, describing the flow of various quantum numbers into the final state in this process. We also work out a representation for these observables in terms of Wightman correlation functions involving insertions of flow operators. In Section 3, we consider weighted cross sections with one or two flow operators and evaluate them to lowest order in the coupling using the conventional amplitude techniques. In Sections 4 and 5, we elaborate on the main result of this work and explain how the same observables can be obtained from the known results for Euclidean correlation functions of half-BPS and other operators, such as the R−current and the energy-momentum tensor, in the same N = 4 supermultiplet. In particular, we derive a master formula which yields an all-loop result for the weighted cross sections as a convolution of the Mellin amplitude defined by the Euclidean correlation function with a coupling-independent 'detector kernel' corresponding to the choice of the flow operators. In Section 6, we demonstrate the efficiency of the formalism making use of the same examples as covered in Section 3, first at weak and then at strong coupling. Section 7 contains concluding remarks. Several technical details are deferred to appendices.

Correlations in N = 4 SYM
In the context of N = 4 SYM, we can introduce an analog of the quark electromagnetic current, the lowest-dimension half-BPS Hermitian operator O IJ 20 ′ (x) built from the six real scalars Φ I (x) (with SO(6) vector indices I, J = 1, . . . , 6), Here Φ I ≡ Φ Ia T a and the generators T a of the gauge group SU(N c ) are normalized as tr[T a T b ] = 1 2 δ ab (with a, b = 1, . . . , N 2 c − 1). The operator (2.1) possesses a protected scaling dimension, ∆ = 2, very much like the QCD electromagnetic current. Moreover, it is the lowest-weight state of the N = 4 stress-tensor supermultiplet and is related by supersymmetry to the R−symmetry current, which can be viewed as a 'cousin' of the electromagnetic current.
The operator (2.1) belongs to the real irrep 20 ′ of the R−symmetry group SO(6) ∼ SU (4). To keep track of the isotopic structure, it is convenient to consider the projected operator where Y I is an auxiliary six-dimensional (complex) null vector, Y 2 = 6 I=1 Y I Y I = 0, defining the orientation of the operator in the isotopic space. 1 Next, we can ask the question about the properties of the final states created by the operator (2.2) from the vacuum. To lowest order in the coupling, the final state consists of a pair of scalars. For arbitrary coupling, the state d 4 x e iqx O(x, Y )|0 can be decomposed into an infinite tower of asymptotic on-shell states, |ss , |ssg , |sλλ , . . . involving an arbitrary number of scalars (s), gluinos (λ) and gauge fields (g). Each on-shell state carries the same quantum numbers -the total momentum q µ , zero (color) SU(N c ) charge and R−charges of the irrep 20 ′ . We can then define the amplitude for creation of a particular final state |X out of the vacuum, where k X is the total momentum of the state |X . Defined in this fashion, the amplitude M O→X has the meaning of a form-factor, For a given on-shell state |X , it suffers from IR divergences that require a regularization procedure. In addition, this quantity depends on the number of colors N c and on the coupling constant g YM . For our purposes it proves convenient to introduce the 't Hooft coupling g 2 = g 2 YM N c and the analog of the fine structure constant, a = g 2 YM N c /(4π 2 ), familiar from QCD.

Total transition probability
In close analogy with the QCD process e + e − → everything, we can examine the transition O 20 ′ → everything. The total probability of this process is given by the sum over all final states where the summation runs over the quantum numbers of the produced particles including their helicity, color, SU(4) indices, etc. To lowest order in the coupling, it describes the production of a pair of scalars as shown in Fig. 2 6) 1 We can always reveal the index structure of the SO(6) tensor O IJ 20 ′ (x) by differentiating the final expressions involving O(x, Y ) with respect to the Y 's, bearing in mind the restriction Y 2 = 0. 2 We use the Minkowski signature (+, −, −, −) and the shorthand notation δ + (k 2 ) = δ(k 2 )θ(k 0 ).
where the ellipsis stand for omitted higher order corrections and the integration in the first term goes over the phase space of the two massless particles carrying the total momentum q µ . The prefactor accompanying the integral is the SO(6) invariant contraction of the auxiliary internal variables (Y Y ) = I Y I Y I .
Using the completeness condition on the asymptotic states, X |X X| = 1, we can rewrite (2.5) as . Notice that the operators in (2.7) are not time-ordered.
The optical theorem allows us to rewrite (2.7) as the imaginary part of the time-ordered correlation function The correlation function on the right-hand side is well defined in four dimensions and the same is true for σ tot (q). This is not the case however for each term on the right-hand side of (2.5) since M O 20 ′ →X suffers from IR divergences. In agreement with the Lee-Nauenberg-Kinoshita theorem [6,7], the infrared finiteness of σ tot (q) is restored in the infinite sum over the final states |X in (2.5).
Another advantage of the representation (2.8) is that it allows us to compute σ tot (q) exactly, to all orders in the coupling. Indeed, in N = 4 SYM the two-point correlation function of the half-BPS operators O(x, Y ) is protected from loop corrections 3 and is given by the Born level expression where D F (x) = 1/(4π 2 (−x 2 + i0)) is the Feynman propagator of the scalar field. Its substitution into (2.8) yields the leading tree-level term in (2.6). Performing the integration in (2.6) we arrive at The fact that σ tot (q) does not depend on the coupling constant implies that the perturbative corrections cancel order by order in N = 4 SYM. To two loop accuracy, this property has been verified in Ref. [19] by an explicit calculation. The product of the two step functions on the righthand side of (2.10) ensures that the cross section is different from zero for the total momentum q 0 > 0 and q 2 > 0. In what follows we tacitly assume that this condition is satisfied and we do not display the step functions in any formulas that follow.

Weighted cross section
The quantity (2.5) is completely inclusive with respect to the final states. We can define a less inclusive quantity by assigning a weight factor w(X) to the contribution of each state |X where the additional factor of 1/σ tot is inserted to obtain the normalization condition σ W (q) = 1 for w(X) = 1. Appropriately choosing the weight factors w(X) and evaluating the corresponding weighted cross section σ W (q), we can get a more detailed description of the flow of various quantum numbers of particles (energy, charge, etc) in the final state |X . For a generic final state |X , the scattering amplitude M O 20 ′ →X contains soft and collinear divergences as we reviewed in the introduction. They arise from the integration over the loop momenta of virtual particles and appear as poles in ǫ in dimensional regularization with D = 4−2ǫ. Taken by itself, each term in the sum in the first relation in (2.11) vanishes as |M O 20 ′ →X | 2 ∼ e −f (g 2 )/ǫ 2 → 0 for ǫ → 0 (with a positive-definite function f (g 2 ) related to the cusp anomalous dimension), due to the exponentiation of infrared singularities (see, e.g., Ref. [20]). However, these are not the only divergences that we encounter in the calculation of the cross section. Namely, additional poles in 1/ǫ come from the integration over the phase space of soft and collinear massless particles in the final state |X . For the cross section to be IR finite, the two effects, i.e., virtual and real singularities should cancel each other, thus producing a finite net result. In the case of the total transition probability σ tot , the cancellation of IR divergences follows from the Kinoshita-Lee-Nauenberg theorem. For weighted cross sections, the condition of infrared finiteness imposes a severe restriction on the weights w(X) [21]. Namely, the weight should be insensitive to adding one particle to the final state |X , with the momentum either soft, or collinear to the momenta of the parent particles in the state |X .

Energy flow
One of the well-known examples of a weight factor, which was introduced in the context of e + e − − annihilation and which is very useful for our purposes, corresponds to the energy flow. For a given final on-shell state |X = |k 1 , . . . , k ℓ , consisting of ℓ massless particles, k 2 i = 0, with the total momentum i k µ i = q µ , it is defined in the rest frame q µ = (q 0 , 0) as where k µ i = (k 0 i , k i ) and Ω k i = k i /| k i | is the solid angle in the direction of k i . The corresponding weighted cross section has a simple physical meaning -it measures the distribution of energy in the final state that flows in the direction of the vector n. Most importantly, the weight (2.12) can be identified with the eigenvalue of the energy flow operator, (2.13) As we show below, this relation allows us to simplify (2.11) along the same lines as (2.7) and to express the cross section σ E in terms of the correlation function 0|O(x, Y )E( n)O(0, Y )|0 with an insertion of the energy flow operator. The explicit expression for the operator E( n) is given in terms of the energy-momentum tensor of N = 4 SYM [9,10,11,12] (see also [13]) (2.14) where the unit vector n = (n 1 , n 2 , n 3 ) (with n 2 = 1) indicates the spatial direction of the energy flow. To get a better understanding of the action of the operator E( n) on the asymptotic states, we replace the energy-momentum tensor T µν (x) in (2.14) by its expression in terms of free fields and obtain the following representation for E( n) in terms of creation and annihilation operators: where the sum goes over all on-shell states (scalars, helicity (±1/2) gluinos and helicity (±1) gluons) carrying an SU(N c ) index b = 1, . . . , N 2 c − 1. To simplify the formulas, in what follows we do not display the SU(N c ) indices of the creation/annihilation operators. Making use of the (anti)commutation relations between a † i (k) and a i (k) (see, e.g., Eq. (A.3)), it is straightforward to verify the relations (2.13) and (2.12), as well as the commutativity condition The latter equality states that the energy flows in two different directions n and n ′ are independent from each other and can be measured separately. Making use of (2.16), we can define a weight which measures the energy flows in various directions n 1 , . . . , n ℓ simultaneously: (2.17) Substituting this relation into (2.11) we can apply the completeness relation X |X X| = 1 and obtain the following representation of the corresponding weighted cross section 4 which has the meaning of an energy flow correlation. Notice that the product of operators on the right-hand side of (2.18) is not time-ordered and, therefore, their correlation function is of the Wightman type.

Charge flow
In close analogy with (2.12) we can define a weight that measures the flow of the R−charges through the detector. We recall that in N = 4 SYM only the scalars and gluinos are charged with respect to the R−symmetry group SU (4). The flow of the R−charge is defined by the operator involving the time component of the R−current (J µ ) B A (x). An important difference as compared with (2.14) is that the operator transforms under SU(4). We shall come back to this point in a moment.
Replacing the R−current in (2.19) by its expression in terms of the free fields, we obtain (2.20) where (trace) denotes terms proportional to δ B A that are needed to ensure the tracelessness condition δ A B Q B A ( n) = 0. Here a † A,1/2 and a B † −1/2 are the creation operators of gluinos with helicity ±1/2, respectively, and a † AC (k) are the creation operators of the scalars in SU(4) notation (see Appendix B). Using (2.20) we can work out the action of the operator Q B A ( n) on the asymptotic states. For instance, for a single-particle gluino state |k E ≡ a † E,1/2 (k)|0 we get from where we conclude that the operator Q B A ( n) does not change the momentum of the particle but it rotates its SU(4) index. The same is true for the scalar states whereas the gluon state has zero R−charge and, therefore, is not affected by Q B A ( n). Relation (2.21) seems to contradict our definition of the flow operator (2.13), according to which the on-shell states should diagonalize the operator Q B A ( n). To restore the diagonal action of the operator Q B A ( n) on the asymptotic states we introduce an auxiliary traceless matrix Q A B and consider the following linear combination of the operators (2.20) To preserve the reality condition on the eigenvalues of the flow operator (2.22), the matrix Q A B should be Hermitian. This allows us to decompose it over its eigenvectors, Its real eigenvalues satisfy the tracelessness condition 4 α=1 Q α = 0. 5 Using the eigenvectors of Q A B , we can define the projected on-shell states |k α =ū A α |k A . Such states allow us to rewrite the action of the SU(4) flow operator (2.22) on, e.g., a single-particle gluino state |k A , in a form analogous to the diagonal action of the energy flow operator in (2.13). Indeed, the charge flow operator (2.22) acts diagonally on the projected states |k α , Q( n; Q)|k α = Q α δ (2) (Ω k − Ω n ) |k α .
The contribution of the gluino state to (2.11) can be written in two equivalent forms, A |k A k| A = α |k Aū A α u α B k| B = α |k α k| α . Then, the action of the charge flow operator Q( n; Q) takes a diagonal form in the new basis: According to (2.25), the charge detector, described by the flow operator Q( n; Q), decomposes the on-shell state of each particle, propagating in the direction of the vector n, over the four basis vectors u α A in the SU(4) space and assigns a charge Q α to each component. Along the same lines as before, we can define the charge flow along various directions n 1 , . . . , n ℓ and express the corresponding weighted cross section as Here each detector is specified by the unit vector n i and by the Hermitian matrix (Q i ) B i A i , where i = 1, . . . ℓ. Unlike the case of the energy correlations in (2.18), this weighted cross section has a non-trivial dependence on the isotopic variables Q and Y (see Eq. (3.9) below).

Scalar flow
The definition of the energy and charge flow, Eqs. (2.14) and (2.19), respectively, involves two of the conserved currents of the N = 4 SYM theory, the energy-momentum tensor T µν (x) and the R−current (J µ ) B A . As was already mentioned, they belong to the same N = 4 stress-tensor supermultiplet whose lowest-weight state is the half-BPS scalar operator O IJ 20 ′ (x), Eq. (2.1). This suggests to introduce, in addition to the energy and charge flow, a 'scalar flow' operator These three flow operators have scaling dimensions and T µν is that it is a Lorentz scalar and is not a conserved current. Since the operators O IJ , (J µ ) B A and T µν belong to the same supermultiplet, we anticipate that the correlations of the corresponding flow operators O, Q and E should be related to each other by supersymmetry. In Section 6, we provide a lot of evidence for such relations, but the precise mechanism will be worked out in our future work.
The expression for the scalar flow operator (2.28) in terms of the free scalar fields looks as 7 (2.31) Since O IJ ( n) is symmetric and traceless, the projection matrix S IJ has to have the same properties. In addition, the reality condition on the detector measurement leads to the reality condition S IJ = S * IJ . This allows us to decompose S IJ over its real eigenvectors, with real eigenvalues S i . The condition for S IJ to be traceless leads to 6 i=1 S i = 0. 8 Then, the scalar flow operator (2.30) takes the following form and it is diagonalized by the projected on-shell scalar states |k i = φ i I |k I : The interpretation of (2.34) is similar to that of (2.25) for the charge flow. The scalar flow operator decomposes the on-shell scalar state moving in the direction of the vector n over the basis of eigenvectors (or SO(6) harmonics) φ i I and assigns to each component an eigenvalue S i divided by (twice) the energy of the particle.
We emphasize the appearance of the inverse energy factor in the last relation in (2.34). It leads to some unusual properties of the scalar flow operator (2.33) as compared to its energy and charge counterparts. To show this, we examine the commutator [O( n; S), O( n ′ ; S ′ )]. Since each operator receives contributions from particles propagating along two different directions n and n ′ , we may expect that Hence, the commutator should vanish since the same particle cannot go through the two detectors simultaneously. This is correct unless the momentum of the particle vanishes. Indeed, for n = n ′ examining their contributions (see Appendix A), we find that, precisely due to the factor of 1/k 0 in (2.33), the commutator is different from zero, . The non-vanishing commutator (2.36) leads to a divergence in certain weighted cross sections, as explained later in the paper.
We observe that the expression on the right-hand side of (2.36) involves the commutator of the matrices defining the two scalar detectors. Therefore, we can restore the commutativity of the operators O( n; S) and O( n ′ ; S ′ ) by requiring [S, S ′ ] = 0 .
(2. 38) In physical terms, this condition prohibits the cross-talk between the two detectors mediated by the exchange of particles with zero momentum. Notice that while this is a necessary condition on the detector matrices, which eliminates potentially divergent contributions due to particle exchanges with zero energy, it is not sufficient when we try to match the weighted cross sections with the integrated correlation functions. The latter require further constraints on the projection matrices S, see Appendix D for details. Having defined the scalar flow operator (2.33), we can introduce the corresponding multipledetector weighted cross section It depends on a set of vectors n i determining the spatial orientation of the detectors, as well as on the projection matrices S i , Eq. (2.32). By construction, the scalar flow operators O( n i , S i ) should commute with each other. In addition to (2.18), (2.27) and (2.39), we can also define mixed correlations involving all three flow operators, O( n i , S i ), Q( n i , Q i ) and E( n i ). As was already mentioned, they are effectively related to each other by supersymmetry. The scalar correlations (2.39) play a special role in our analysis below. Firstly, they can be expressed in terms of the correlation function involving ℓ + 2 copies of the same half-BPS operator (2.1). Secondly, in the special case ℓ = 2 the correlation function of the half-BPS operators O(x) uniquely determines, by means of N = 4 supersymmetry transformations, all four-point correlation functions of the other operators from the stress-energy multiplet that generate charge and energy flow correlations. This is not the case for ℓ > 2 since the solution to the corresponding N = 4 superconformal Ward identities is not unique anymore due to the appearance of nontrivial N = 4 superconformal invariants depending on 2 + ℓ ≥ 5 points.
We would like to emphasize that, in virtue of the definition of the flow operators (2.14), (2.19) and (2.28), the weighted cross sections (2.18), (2.27) and (2.39) are related to the integrated Wightman correlation functions defined at spatial infinity. This makes the issue of infrared finiteness of the flow correlations extremely nontrivial. It is believed that the energy flow correlations are IR finite both at weak and strong coupling whereas for the scalar and R−charge flow observables the situation remains unclear. As a counterexample, we can recall that similar problem also arose in QCD, where the flavor observables in jet physics (closely related to charge flow correlations) while perfectly well-defined at leading order of the perturbative expansion, cease to stay finite once higher order corrections are accounted for [24]. The problem requires further studies and will be addressed elsewhere.

Weighted cross sections from amplitudes
In this section we employ the conventional approach based on the scattering amplitudes to evaluate the weighted cross sections introduced in the previous section to lowest order in the coupling in N = 4 SYM.
We start by computing the matrix elements (2.4) involving the operator defined in (2.2). At tree level, the final state consists of a pair of scalars denoted by |s I (k 1 )s J (k 2 ) : To first order in the coupling, the final state also contains three-particle states |X = |s, s, g and |X = |s, λ, λ . The corresponding transition amplitudes are where the notation was introduced for the Mandelstam invariants s ij = (k i + k j ) 2 with k 2 i = 0. The total transition probability (2.5) is given to order O(g 2 ) by where F virt (q 2 ) describes the one-loop (virtual) correction to the transition amplitude (3.1) and the notation was introduced for the Lorentz invariant integration measure on the phase space of ℓ massless particles with the total momentum q µ , The symmetry of this integration measure under the exchange of any pair of particles allows us to rewrite (3.3) as where we symmetrized the one-loop integrand with respect to the particle momenta and used the identity q 2 = s 12 + s 23 + s 13 . As was already mentioned, the total transition amplitude σ tot (q) is protected from perturbative corrections and, therefore, the terms proportional to g 2 on the right-hand side of (3.5) should vanish. This allows us to fix the virtual correction F virt (q 2 ). 9 Taking into account (3.1) we verify that the resulting expression for σ tot (q) coincides with (2.10).

Single detector
Let us now examine the weighted cross sections involving a single detector. We start with the energy flow. According to the definition (2.11), the corresponding cross section can be obtained from the first relation in (3.3) by inserting the energy weight factor (2.12) into the phase space integrals. The weight factor w E( n) (k 1 , k 2 ) for the transition M O 20 ′ →ss depends on the energy of the two scalars (in the rest frame of the source), whereas for M O 20 ′ →ssg and M O 20 ′ →sλλ it is given by w E( n) (k 1 , k 2 , k 3 ) that receives additive contributions from all produced particles, as seen from Eq. (2.12). In this way, we obtain (3.6) Replacing the matrix elements on the right-hand side by their explicit expressions (3.1) and (3.2), we symmetrize the integrands with respect to the particle momenta and find, after a simple calculation, Notice that this expression does not depend on the coupling constant. In Sect. 4 we show that E( n) is indeed protected from loop corrections. 10 We verify that the integral of (3.7) over n correctly reproduces the total energy, d 2 Ω n E( n) = q 0 , in agreement with the results in [13].
For the charge flow the analysis goes along the same lines, with the only difference that the non-trivial weight factor w Q is different from zero only for scalars and gluinos carrying nonzero R−charges. As explained in Sect. 2.2.2, the charge flow operator introduces the projection (or 'polarization') matrix Q B A for the SU(4) states of these particles. More precisely, at tree level, the contribution of the transition M O 20 ′ →ss to the total transition probability involves the R−symmetry factor (in SU(4) notation, see Here the sum over the SU(4) indices corresponds to the summation over the quantum numbers of the particles propagating in the final state. The contribution to the weighted cross section reads where the last expression is the corresponding weight factor. 11 Inserting this result in the phase space integral dPS 2 |M O 20 ′ →ss | 2 and taking into account (3.1) and (B.5), we obtain where Q is an isotopic factor keeping track of the R−charges, To first order in the coupling constant, the correction to (3.9) has the same form as (3.6) with the only difference that the corresponding charge weight factor is given by an expression analogous to (3.8). Calculating the O(g 2 ) correction to (3.9) we find that it vanishes (see footnote 10). Finally, for the scalar flow, the weight factor w O is different from zero only for scalar particles. According to (2.34), for a scalar with momentum k i in the final state, the insertion of the weight factor modifies the SO(6) tensor structure as follows: where S IJ is the projection matrix of the scalar detector. Repeating the calculation at Born level we obtain where S is an isotopic factor keeping track of the tensor structure, The calculation of the O(g 2 ) correction to this expression shows that it vanishes in the same manner as for E( n) and Q( n) (see footnote 10). We recall that the expressions for the one-point correlations (3.7), (3.9) and (3.12) were obtained in the rest frame of the source, q µ = (q 0 , 0). We present the same expressions in Lorentz covariant form in Sect. 4.3.

Double correlations
In the previous subsection we have shown that the single-detector weighted cross sections (3.7), (3.9) and (3.12) do not depend on the spatial orientation of the detector n. For weighted cross sections involving two detectors oriented along the vectors n and n ′ , the rotation symmetry implies that they can only depend on the relative angle 0 ≤ θ ≤ π between the vectors, ( n · n ′ ) = cos θ. We call such observables 'double correlations'. For θ = 0 the orientations of the two detectors coincide so that the same particle can go through them sequentially. For θ = π the detectors capture particles moving back-to-back in the rest frame of the source.
The calculation of the double correlations at one loop goes along the same lines as in the previous subsection. For the double energy correlation E( n)E( n ′ ) , the only difference as compared with (3.6) is that the weight factor w E ( n) gets replaced by the product w E ( n)w E ( n ′ ), which fixes the spatial orientation of the momenta of the two particles in the final state.
At Born level, the final state consists of two scalar particles moving collinearly in the rest frame q µ = (q 0 , 0), each carrying energy q 0 /2. As a consequence, their contribution to E( n)E( n ′ ) is localized at θ = 0 and θ = π, (3.14) Here the first delta-function describes the situation of aligned detectors capturing particles moving into the same direction, while the second delta-function corresponds to two particles moving back-to-back. For 0 < θ < π, the double-energy correlation receives a non-zero contribution starting from one loop. It comes from the three-particle transitions Using the explicit expressions for the matrix elements (3.2) we find (see Appendix C for details) where the logarithmic correction arises from the integration over the energy fraction of one of the particles, τ 1 = 2k 0 1 /q 0 . For θ → 0, the expression in the right-hand side of (3.16) scales as O(θ −2 ), whereas for w = π − θ → 0 it has the well-known Sudakov behavior O(w −2 ln(w −2 )). Both asymptotics are modified at higher loops in a controllable way [25].
It is convenient to rewrite (3.16) by introducing the scaling variable where 0 < θ < π is the angle between the detector vectors n and n ′ . Then, the double-energy correlation takes the following form at one loop with z varying in the range 0 < z < 1 and a = g 2 /(4π 2 ), as defined earlier. It is instructive to compare (3.18) with the analogous expression in QCD. In that case, the final state is created by an electromagnetic current and it consists of quarks and gluons. To lowest order in the coupling, the energy-energy correlation looks as [4] being the quadratic Casimir in the fundamental representation of the SU(N c )).
We observe that the N = 4 SYM result (3.18) can be obtained from the QCD expression (3.19) by discarding the rational term inside the square brackets in (3.19) and by retaining only the leading singularity for z → 1 in the ln(1 − z) term. In other words, the two expressions have the same leading asymptotic Sudakov behavior as z → 1. According to (3.17), this limit corresponds to the two detectors capturing particles moving back-to-back in the rest frame of the source (see Fig. 4).
For the double-scalar correlation O( n)O( n ′ ) the weights corresponding to the particles in the final state of M O 20 ′ →X also depend on the detector matrices S and S ′ . For 0 < θ < π, to one-loop order, it reduces to the product of isotopic factors S S ′ from the single detector correlation (3.12): where the additional factor of 2 comes from the symmetry of the weight factor under exchanging the detectors, n ↔ n ′ . Performing the integration over the phase space of the three particles in the final state of M O 20 ′ →ssg , we obtain (see Appendix C) with 0 < z < 1.
For the double-charge correlation Q( n)Q( n ′ ) , with 0 < θ < π, we have to take into account all possible correlations between the charges of scalars and gluinos in the final state where the last line describes the correlations between two gluinos. Note that tr[yQȳQ ′ ] = tr[yQ ′ȳ Q] due to the antisymmetry of the matrices y,ȳ. The integration over the final state phase space yields where Q is given by (3.10) and the notation was introduced for the (non-factorizable) correlation between the matrices of the two detectors In the above calculation we have tacitly assumed that the detectors measure two different particles. In general, we also have to examine the possibility for the same particle to go sequentially through the two detectors. As was already explained, for n = n ′ (or equivalently θ = 0) the momentum of the particle should be necessarily zero in this case and, as a result, its contribution to the charge correlations vanishes. The same is true for the scalar correlations provided that the projection matrices of the detectors satisfy the 'no-cross-talk' condition (2.38).
In a similar manner, we can also define mixed correlations involving various flow operators: Notice that the dependence of these expressions on the total energy q 0 is uniquely fixed by the scaling dimension of the flow operators. The non-trivial dynamical information resides in the z−dependence. Quite remarkably, the obtained one-loop expressions are all proportional to the same function ln(1 − z)/(1 − z). Its appearance is not accidental, of course, since it ensures the universal Sudakov behavior of the correlations for z → 1. The approach described in this section can be extended to higher loops. Namely, to any order in the coupling constant, following (2.11) we can express the correlations as a sum over all possible final states, evaluate the corresponding transition amplitudes (2.4) and, then, perform the integration over the phase space. However, such an approach becomes very cumbersome and inefficient beyond one loop for the following reasons. The number of production channels O 20 ′ → X grows rapidly at higher loops and, therefore, we have to deal with an increasing number of terms in the sum over the final states. Secondly, with many particles in the final state the integration over their phase space becomes very complicated and cannot be done analytically. Finally, each individual transition amplitude |M O 20 ′ →X | 2 suffers from infrared divergences and requires regularization. Infrared divergences cancel however in the sum over all final states between contribution involving different number of particles. 12 In the next section we describe another approach to computing the various double correlations in N = 4 SYM. It makes efficient use of the superconformal symmetry of the theory. It also allows us to go to higher loops and even to strong coupling, via the AdS/CFT correspondence.

Weighted cross sections from correlation functions
In this section we shall exploit the relation between physical observables and correlation functions involving two half-BPS operators as the source and sink, and various flow operators, Eqs. (2.18), (2.27) and (2.39), as the detectors. Unlike the more familiar Euclidean correlation functions, widely discussed in the N = 4 SYM literature, the operators on the right-hand side of (2.18), (2.27) and (2.39) are essentially Minkowskian and are non-time ordered. This means that we will be dealing with Wightman correlation functions. As we have shown in the previous section, they define the charge flow correlations in the detector limit (see Fig. 5).
In this section we explain how the Wightman correlation functions in (2.18), (2.27) and (2.39) can be obtained from their Euclidean counterparts by analytic continuation. The Euclidean correlation functions have singularities at x 2 ij = 0, corresponding to short-distance separation between the operators, x i → x j . In Minkowski space, additional singularities appear when the operators become light-like separated. In this case the analytic properties of the correlation function crucially depend on the ordering of the operators (time ordering versus Wightman). Therefore, performing the analytic continuation of the correlation functions from Euclid to Minkowski we have to pay special attention to their analytic properties.

Lorentz covariant definition of the detectors
In Section 2.2 we have defined the flow operators (2.14), (2.19) and (2.28) in the rest frame of the source q µ = (q 0 , 0). To make use of the conformal symmetry of the correlation functions, we have to restore the Lorentz covariance and extend the definitions to any reference frame.
We recall that the time integral in the definition of the flow operators (2.14), (2.19) and (2.28) has the interpretation of the working time of the detector located at the position r n relative to the collision point. The space-time coordinate of the detector x µ = (t, r n) can be decomposed in the basis of two light-like vectors, with x + = 1 2 (t + r) = (xn)/2 and x − = 1 2 (t − r) = (xn)/2. We can restore manifest Lorentz covariance by rescaling each light-like vectors by an arbitrary positive scale, with ρ, ρ ′ > 0. This lifts the restriction that the time component of n µ is equal to 1. Then the covariant definition of the light-cone coordinates in (4.1) looks as Notice, however, that covariance can only be maintained if all the expressions we encounter are homogeneous under such local rescalings, allowing us to go back to the original form of vectors n andn in (4.1). By 'local' we mean that the coordinates of each flow operator should rescale with their own, independent parameter ρ. The next step is to reformulate the detector limit, r → ∞ and 0 ≤ t < ∞, in terms of the light-cone variables x ± . We illustrate the correct procedure relying on the example of the energy flow. If we just take r → ∞ but keep t fixed, as in the original definition (2.14), we would have x ± → ±∞ which is clearly too strong. We need to keep one of these variables fixed while taking the other one to infinity. In physical terms, the flow operator admits the following interpretation. We can think of a massless particle captured by the detector as of a wave front propagating in the direction n µ and spreading along the directionn µ . This suggests to first send the detector to future infinity along, say, the light-cone direction n µ and then integrate over the position of the massless particles on the wave front along the directionn µ . In terms of the light-cone coordinates this means that we first take the limit x + → ∞, whereas x − remains finite. The time integral in (2.14) then becomes an integral over −∞ < x − < ∞. This brings us to the new definition where J + (x) ≡n µ J µ (x)/(nn) is the light-cone component of the R−current. They have rescaling weights (−2) and (−1), respectively. The expressions in the right-hand sides of (4.4) and (4.5) involve the two light-like vectors n andn, but the dependence on the latter is redundant. To illustrate this point, it is sufficient to rewrite these operators in terms of creation and annihilation operators. For instance, we can start with (4.4) and repeat the calculation of (2.15) to find, with the help of (A.1), so that the dependence onn µ drops out. Notice that the annihilation/creation operators in (4.6) depend on the momentum k µ = n µ τ , collinear with the light-like vector defining the space-time orientation of the detector. The charge and scalar flow operators admit similar representations (see Eq. (A.2) for the latter). An important difference is, however, that the corresponding τ −integral measure becomes dτ τ and dτ , respectively. Relation (4.6) (together with the analogous relations for the scalar and charge flow operators) confirms the transformation properties of the flow operators under the rescaling (4.2) derived above. Changing the integration variable τ → τ /ρ, we find where the factors of ρ are related to the power of τ in the corresponding integral representation.

Symmetries
Let us introduce a generic notation D i (n) for all flow operators (scalar, charge and energy). We are interested in the symmetry properties of their correlations Here we introduced the subscript q on the left-hand side to indicate the dependence on the total momentum transferred (see Fig. 5). By construction, the correlation function (4.8) is invariant under permutations of any pair of detectors. In addition, D 1 (n 1 ) . . . D(n ℓ ) q is invariant under Lorentz transformations of the total momentum q µ and of the light-like vectors n µ i defining the detector orientation, (4.9) The correlations (4.8) have two additional symmetries related to the independent rescaling of n µ i and q µ . The flow operators are defined in terms of the operators O 20 ′ , J µ and T µν carrying Lorentz spins 0, 1 and 2, respectively. According to (4.7), the spin controls their transformation properties under the rescaling n µ → ρ n µ , leading to where s O = 0, s Q = 1 and s E = 2. The other scaling property is related to the rescaling q µ → λq µ . Since the vectors n µ i are dimensionless, this transformation can be compensated by the dilatations x µ → λ −1 x µ of the coordinates, By virtue of (4.9), the single-detector correlation D(n; s) depends on the two Lorentz invariants (qn) and q 2 . The dependence on the former is controlled by the spin s, i.e., by the required degree of homogeneity under the rescalings (4.10). Then the q 2 −dependence is fixed by the scaling dimension ∆, resulting in (4.13) For the charge and energy flow correlations, we can fix the normalization constant c D by making use of the fact that the flow operator in both cases is determined by a conserved current. As a consequence, integrating over the position of the detector on the sphere n µ = (1, n) with n 2 = 1 in the rest frame of the source q µ = (q 0 , 0), we should find the corresponding conserved quantity -the total energy dΩ n E( n) = q 0 and the total charge dΩ n Q( n) = 4 Q of the state. 13 In this way we get (4.14) Let us now consider the double-detector correlation D(n; s)D(n ′ ; s ′ ) . The main difference, compared to the previous case, is that it is not fixed anymore by symmetries. The reason for this is that we can define a ratio invariant under the transformations (4.9) -(4.11), . Here we introduced the factor of 2 to ensure that 0 ≤ z ≤ 1 for a time-like vector q and light-like vectors n and n ′ . In particular, in the rest frame of the source q µ = (q 0 , 0), the variable z is related to the angle between the two detectors, z = (1 − cos θ)/2. This variable already appeared in the amplitude context, see (3.17). Thus, all the symmetries mentioned above allow us to determine the double-detector correlation up to an arbitrary function of the invariant variable z, 14 In what follows we shall refer to F ss ′ (z) as the event shape function. The symmetry of the correlator in the left-hand side of (4.16) under the exchange of the flow operators leads to the crossing symmetry F ss ′ (z) = (2z) s ′ −s F s ′ s (z). Equation (4.16) is consistent with the fact that D(n; s)D(n ′ ; s ′ ) is related to the four-point correlation function which is fixed by conformal symmetry up to a function depending on two conformally invariant cross-ratios. In the detector 13 Equivalently, the three-point functions of two scalars with a current or a stress-energy satisfy a Ward identity which relates their normalization to that of the two-point functions.
14 For certain detectors D the function F ss ′ will also depend on the auxiliary 'isotopic' variables, see below.
limit, i.e., after sending r → ∞ and the subsequent time integration, this function effectively depends on a single variable. It is straightforward to extend (4.16) to the multi-detector correlations (4.8) involving an arbitrary number of flow operators. In that case, D 1 (n 1 ) . . . D ℓ (n ℓ ) is given by the product of single-detector correlations (4.13) times an arbitrary function depending on the relative angles between the detectors. We present below a list of double-detector correlations involving the three types of detectors considered in this paper. Combining together (4.16) and (4.12), we get where F (z) are arbitrary functions of z. Matching these relations with the amplitude results (3.18), (3.21) and (3.23), we find to the lowest order in the coupling and similarly for the mixed correlations from (3.25), Notice that F OO (z) agrees (up to an overall factor) with the result presented in [26]. We observe that the lowest order corrections to all event shape functions (except F QQ (z)) involve the same function z ln(1 − z)/(1 − z). We come back to this issue in Section 6.

Single-detector correlations
As an illustration of the general considerations above, in this subsection we revisit the calculation of the detector correlations with a single detector insertion, O(n) , Q(n) and E(n) , by using their relation to the three-point correlation functions of the scalar half-BPS operators O 20 ′ , the R−symmetry current and the energy-momentum tensor. The latter was discussed previously in Ref. [13]. Here we illustrate the main steps needed to obtain the cross section from the correlation function: analytic continuation from Euclidean to Wightman functions, taking the detector limit, performing the time integrals and the Fourier transform.

Single scalar detector
According to (2.39), the single scalar detector correlation is given by We do this in three steps: (i) start with the expression for the Euclidean three-point correlation function of the half-BPS operators O IJ 20 ′ (x) and project the SO(6) indices according to (4.21); (ii) perform an analytic continuation to obtain the Wightman function in Minkowski spacetime; (iii) take the detector limit (by sending x + → ∞ and integrating over −∞ < x − < ∞) and make the Fourier transform with respect to the position of the source to introduce the total momentum q µ .
The correlation functions of the half-BPS operators O IJ 20 ′ have a number of remarkable properties in N = 4 SYM. First of all, the conformal weight of O IJ 20 ′ is protected from quantum corrections and, as a consequence, their three-point correlation function in Euclidean space, is fixed by conformal symmetry up to an overall normalization factor. Furthermore, this factor also does not receive quantum corrections, meaning that the three-point correlation function (4.22) is given by its Born-level expression. Since O(x i , Y i ) is bilinear in the scalar fields, the latter expression reduces to the product of three free scalar propagators D E (x) ∼ 1/x 2 : . (4.24) Here D W (x) is given by the two-point (non-time ordered!) correlation function of free scalar fields in Minkowski space-time We would like to emphasize that the analytic continuation in (4.24) relied on the simplicity of the three-point correlation function (4.23). Later in the paper we shall consider the four-point correlation function of half-BPS operators G E (1, 2, 3, 4). It is not protected anymore and the perturbative corrections to G E (1, 2, 3, 4) are described by a complicated function of the conformal cross-ratios. It then becomes a non-trivial task to find its analytic continuation G W (1, 2, 3, 4). The Wightman function (4.24) depends on three auxiliary isotopic null vectors Y i (with i = 1, 2, 3). In the case of the scalar flow correlation (4.20) the choice of the analogous auxiliary variables was dictated by the requirement for O(n) to have real values. This is why we associate the complex null vector Y with the source and its conjugate Y with the sink, while the detector matrix S IJ is chosen real. In the case of the correlation function, the variables Y i are holomorphic coordinates of the half-BPS operators in the internal space. However, what really matters is that we can always remove the auxiliary variables Y i to reveal the R−tensor structure of the correlation function, and then project it with the new set of auxiliary variables Y, Y , S appearing in (4.20). In the case of the Wightman function (4.24), this amounts to the substitution rule for the isotopic variables For a more detailed discussion of the two sets of auxiliary isotopic variables and their matching see Appendix D. The next step is to take the detector limit of the Wightman function G W (1, 2, 3). To match the correlation function on the right-hand side of (4.21), we put x 3 = 0 and identify x µ 2 = x 2+ n µ + x 2−n µ to be the detector coordinate. Then, for x 2+ → ∞ we make use of the relations to find from (4.24) and (4.23) .

(4.27)
According to (4.21), we have to integrate this expression over the detector light-cone coordinate x 2− . The expression on the right-hand side of (4.27) has two poles in x 2− located on the opposite sides of the real axis. Closing the integration contour in, say, the upper half-plane, we get .

(4.28)
Notice that, as announced in Sect. 4.1, the second light-like vectorn has dropped out of the right-hand side. Finally, we perform the Fourier transform of (4.28) and collect various factors to obtain . (4.29) In the rest frame of the source q µ = (q 0 , 0) this relation coincides with (3.12) and agrees with the general form (4.13).

Single charge detector
Let us repeat the above analysis for the single-detector correlation of the charge flow operator (2.27), Replacing Q A B (n) with its definition (4.5), we find that it is related to the three-point Wightman correlation function of two half-BPS operators O 20 ′ and the R−symmetry current J, upon an appropriate identification of the coordinates. As before, we start with the Euclidean version of three-point function (4.31). This is is another example of a protected correlation function, hence it is given by the Born level result where Γ IJ is the generator of the fundamental representation of SU(4) defined in (B.4) and {IJ} denotes the traceless symmetrization of the pairs of indices I 1 J 1 and I 3 J 3 , as required by the index structure of the operators O 20 ′ . It is easy to check that the expression in the right-hand side of (4.32) satisfies the current conservation condition at point 2 and is conformally covariant with the relevant scaling weight at each point. The expression for Q(n) in (4.30) involves a projection of the SU(4) tensor structure in (4.32) with Y where tr(yQȳ) = y AB Q B Cȳ CA and the variables y AB andȳ AB are defined in (B.1). As in the previous case, the Wightman correlation function (4.31) can be obtained from the Euclidean one in (4.32) by replacing the intervals (see footnote 16 below) as x 2 ij → −(x 2 ij −iǫx 0 ij ) (for i < j) in the right-hand side of (4.32). Further, we specify the coordinates of the operators as x 3 = 0 and x µ 2 = x 2+ n µ + x 2−n µ and obtain in the detector limit .

(4.34)
Notice that this expression is proportional to the light-like vector n µ determining the orientation of the detector. Next, the integral over the detector time produces .

(4.35)
Finally, performing the Fourier integral and reinstating the y/Q−structure from (4.33), we obtain where Q was defined in (3.10). In the rest frame of the source, this relation coincides with (3.9) and agrees with (4.14).

Single energy detector
Here we essentially repeat the calculation from [13]. The single-detector correlation of the energyflow operator (4.4) takes the form As follows from (4.4) and (4.37), the energy correlator E(n) is determined by the three-point function of two half-BPS operators and one energy-momentum tensor. In Euclidean space it is given by x µ Since the energy-momentum tensor is an SO(6) singlet, the R−symmetry tensor structure in (4.38) is much simpler than before. Moreover, the isotopic factor in (4.38) is canceled by that of the total transition probability σ tot in the right-hand side of (4.37). Repeating the steps outlined above, analytic continuation to the Wightman function, detector limit and time integration (see (4.4)), followed by a Fourier transform, we obtain For q µ = (q 0 , 0), we reproduce the result of the amplitude calculation (3.7), in accord with (4.14).

Double-detector correlations from four-point correlation functions
As was explained in the previous section, starting with two (or more) detector insertions we have to deal with Wightman correlation functions involving four (or more) operators -two half-BPS operators serving as the source/sink, and more than one flow operators serving as detectors. Unlike the single-detector case discussed above, such functions are not protected from quantum corrections and we expect them to have a complicated form order-by-order in the coupling constant. As a consequence, the question about the analytic continuation of the Wightman functions from their Euclidean counterparts becomes very non-trivial. In this section we explain the procedure on the example of the double-detector correlations.

Energy-momentum tensor supermultiplet
We recall that the flow operators (4.4) and (4.5) are built from three protected operators -the half-BPS scalar operator, the R−current and the energy-momentum tensor. It is well known that in N = 4 SYM these operators belong to the same supermultiplet T (x, Y, θ,θ), This is the starting point for the analysis of the scalar-scalar flow correlation. It is known [27,28] that N = 4 superconformal symmetry allows one to reconstruct the complete super-correlation function (5.2) in terms of its lowest component (5.3). Each term in the (θ,θ) expansion of (5.2) is given by a particular differential operator acting on the coordinates x i of the scalar operators in (5.3). Most importantly, these differential operators do not depend on the coupling constant. Their explicit form for the cases of interest (the components involving one or two R−currents and/or energy-momentum tensors) will be worked out in a forthcoming paper. Thus, to compute the quantum corrections to the super-correlation function (5.2) it suffices to know the correlation function of the four half-BPS operators (5.3).

Four-point correlation function of half-BPS operators
The Euclidean four-point function of the half-BPS operators O 20 ′ has been studied extensively in [29,30,31,32,33,34]. It is convenient to split it into a sum of two terms, G E = G Here Φ E (u, v) depends on the two conformal cross-ratios and admits a perturbative expansion, The functions Φ ℓ (u, v) are currently known in terms of Euclidean scalar Feynman integrals up to six loops (ℓ = 6) [35]. The one-loop correction is given by the so-called one-loop box integral In Euclidean space, the function Φ E (u, v) has logarithmic singularities in the limit x 2 ij → 0, that corresponds to short-distance separations between the operators.
Let us apply (5.4) and (5.5) to obtain the correlation of two scalar flow operators O(n)O(n ′ ) , Eq. (2.39), from the correlation function G E (1, 2, 3, 4). At the first step, we associate the operators at points 4 and 1 with the source O(0, Y ) (we set x 4 = 0) and the conjugate sink O(x 1 , Y ), respectively. Further, the operators at points 2 and 3 are associated with the two detectors, i.e., the scalar flow operators O(n, S) and O(n ′ , S ′ ), involving the real detector matrices S IJ and S ′ IJ . The description of the correlation function in (5.4) and (5.5) is holomorphic in the auxiliary null vectors Y i at all four points. So, to match the isotopic structures of the two objects, we have to make the substitutions Next, we recall that the scalar flow operators O(n, S) and O(n ′ , S ′ ) do not commute for generic detector matrices S and S ′ because of a possible cross-talk between the detectors. To avoid this unwanted cross-talk, when defining the scalar detector correlations we had to impose the additional condition (2.38) on the detector matrices. Making the substitution S IJ → Y 2I Y 2J and S ′ IJ → Y 3I Y 3J described above, we find that the condition (2.38) on the scalar flow detectors is translated into the orthogonality condition Here we dressed the symbols in the left-hand side of these equations with hats to indicate that both expressions have been evaluated for y 2 23 = 0. Comparing these relations with the general expressions (5.4) and (5.5), we observe that the suppression of the cross-talk between the detectors has a simple interpretation in terms of the correlation function. Namely, the condition y 2 23 = 0 eliminates the most singular terms in (5.4) and (5.5) in the short-distance limit x 2 23 → 0. To conclude the discussion of how to match the isotopic structures of the correlation function of four half-BPS operators in (5.8) and (5.9) and that of the double scalar flow correlation, we give the translation table between the SO(6) invariant y− and Y /S−structures, after imposing the condition y 2 23 = 0: 15 The next step is to use relations (5.8) and (5.9) to obtain the Wightman correlation function of four half-BPS operators.

Analytic continuation
Let us first examine the contribution to O(n)O(n ′ ) coming from the Born level approximation to the correlation function (5.8).
A distinguishing feature of G (Born) E is that, like the three-point function (4.23), it is a rational function of the distances x 2 ij . This suggests that we can obtain the Wightman function G   (1, 2, 3, 4). Since we assigned the points 4 and 1 to the source and to its complex conjugated image (sink), respectively, and the points 2 and 3 to the detectors, the above rule should be applied to all factors of 1/x 2 ij with i < j except 1/x 2 23 . Notice that, in virtue of the commutativity of the flow operators, the Wightman function should be insensitive to the ordering of the operators located at points x 2 and x 3 . Indeed, it is easy to see from (5.8) that G (Born) W is regular at x 2 23 = 0. The same is true for the expression in the square brackets in (5.9). To compute O(n)O(n ′ ) , we convert (5.8) into a Wightman function and go to the detector limit by putting x 4 = 0 and identifying the coordinates of points 2 and 3 as x 2 = x 2+ n+x 2−n and Then, for x 2+ , x 3+ → ∞, we notice that the second term on the right-hand side of (5.8) gives rise to an expression containing double poles in x 2− and x 3− . These poles are located in the same half-plane and, therefore, vanish upon integration over x 2− and x 3− . In this way, with the help of (5.10) we obtain , (5.11) where in the second relation (3.13). The Fourier integral in the last relation reduces to . It is easy to see that, in the rest frame of the source, for q µ = (q 0 , 0), n µ = (1, n) and n ′µ = (1, n ′ ), it vanishes unless n = − n ′ . The latter case corresponds to detecting particles moving back-to-back and it produces a contribution proportional to δ(1 − z). This result is in agreement with the calculation of O(n)O(n ′ ) (Born) based on the amplitudes, Eq. (3.14). Namely, at Born level, for q µ = (q 0 , 0), the final state in O 20 ′ → everything consists of two scalar particles moving back-to-back. They can be detected only for n = − n ′ , corresponding to z = 1 in (4.15). Thus, for z < 1, the correlation function contributes to the event shape function F O (z) defined in (4.17) only starting from order O(a).
Let us now repeat the same analysis for the Euclidean correlation function (5.9). The main difficulty in this case is that it involves a complicated function of the two cross-ratios. How to convert it to the Wightman function Φ W (u, v)? The answer to this question was proposed in [36] and a particularly elegant formulation was given in Ref. [37]. It relies on the Mellin representation of the correlation function, where u, v > 0 in Euclidean space and M(j 1 , j 2 ) is a meromorphic function of the Mellin parameters. At weak coupling, the Mellin amplitude is given by a perturbative expansion in a = g 2 /(4π 2 ), For our purposes here, we will only need the lowest-order term, which is obtained by rewriting the one-loop box integral in (5.7) in the Mellin form [35]. At strong coupling in planar N = 4 SYM, the Mellin amplitude was computed in [38,39,40] via the AdS/CFT correspondence, The invariance of the correlation function G (1, 2, 3, 4) under the exchange of any pair of points (x i , Y i ) ↔ (x j , Y j ) leads to the crossing symmetry relations which translate into relations for the Mellin amplitude, It is easy to check that the Mellin amplitudes (5.14) and (5.15) verify these relations. According to [37], the Mellin amplitude M(j 1 , j 2 ) is a universal function, depending neither on the space-time signature, nor on the ordering of the operators in the four-point correlation function. Extending (5.12) to Minkowski space, we have to specify the analytic continuation of (x 2 ik ) j to negative x 2 ik for arbitrary complex j. The choice of the prescription is determined by the ordering of the operators. In the special case of the Wightman function G W (1, 2, 3, 4), the prescription is ( 16 In this way, we obtain from (5.12) Notice that Φ W is locally conformally invariant. It coincides with Φ E (u, v) for all x 2 ij < 0 and differs otherwise.
For our purposes we will only need the Wightman function (5.18) in the detector limit. As before, we assign points 4 and 1 to the source and sink, respectively, and points 2 and 3 to the detectors. We put x 4 = 0, x 2 = x 2+ n+x 2−n , x 3 = x 3+ n ′ +x 3−n ′ and take the limit x 2+ , x 3+ → ∞ to get 17 where the notation was introduced for the function In what follows the dependence of f (j 1 , j 2 ) on the coordinates x 1 , x 2− and x 3− is tacitly assumed.
Viewed as a function of the detector times x 2− and x 3− , the expression in the right-hand side of (5.20) has singularities located on both sides of the real axis. This ensures that the integrals over the detector times x 2− and x 3− do not vanish.

Master formulas
We are now ready to perform the detector limit of the four-point Wightman correlation function. Combining together (5.9), (5.10) and (5.19), we find where σ tot is given by (2.10). Here we see three independent R−symmetry structures, Each of them is accompanied by the function f (j 1 , j 2 ) with shifted arguments. Finally, to obtain the correlation O(n)O(n ′ ) we have to integrate both sides of (5.21) over the detectors times, x 2− and x 3− , and perform the Fourier transform with respect to x 1 . Assuming that the order of integrations can be exchanged, we find from (5.21) where we introduced a notation for Changing the integration variables, x 2− → x 2− /(nn) and x 3− → x 3− /(n ′n′ ), we verify using (5.20) that K(j 1 , j 2 ; z) is dimensionless and invariant under (independent) rescalings of the lightlike vectors n and n ′ . Therefore, K(j 1 , j 2 ; z) can depend on the four-dimensional vectors only through the scaling variable z defined in (4.15). Going through the calculation of (5.24) we find (see Appendix E for details) where 0 < z < 1. The function K(j 1 , j 2 ; z) characterizes the scalar detectors. It is a symmetric function of j 1 and j 2 and, most importantly, it is independent of the coupling constant. In what follows we refer to this functions as the detector kernel.
Equation (5.23) agrees with the general expression for the scalar detector correlation in (4.17). It leads to the following remarkable master formula for the event shape function F OO (z) where F ± OO (z) are given by the Mellin integrals These relations establish the correspondence between the four-point Euclidean correlation function of half-BPS operators and the double-scalar detector correlation. Given the correlation function G E (1, 2, 3, 4) in the form (5.5), we start by extracting the Mellin amplitude M(j 1 , j 2 ; a) from (5.12). Then we obtain the event shape function F O (z) by integrating the Mellin amplitude in the Mellin space with the kernel K(j 1 , j 2 ; z), which is uniquely fixed by the choice of the detectors. In this representation, the dependence on the coupling constant resides in the Mellin amplitude M(j 1 , j 2 ; a), whereas the z−dependence comes from the detector kernel K(j 1 , j 2 ; z).

One-loop check
To illustrate the power of the master formula (5.26), let us use the known one-loop expression for the Mellin amplitude (5.14) to compute the event shape functions (5.27) to the lowest order in the coupling. Taking into account (5.25), we find To perform the Mellin integration in (5.27) it is useful to shift the integration variable as j 2 → j 2 − j 1 . In this way, we obtain where the j 2 −integration contour goes to the left from the one over j 1 . The integral can be easily computed by residue method after closing the j 1 −integration contour in the right half-plane and the j 2 −integration contour in the left half-plane. Then, the integral is given by the residues at j 1 = 0 and j 2 = −k (with k = 1, 2, . . . ) Going through the same steps for (5.28) we find where the integration contours are the same as in (5.29). The main difference, however, is that the integrand has only a double pole at j 1 = 0 and, therefore, the j 1 −integral vanishes, Substituting (5.30) and (5.32) into the master formula (5.26) we notice that, firstly, the one-loop correction to the event shape function does not receive a contribution proportional to S, S ′ and, secondly, the coefficient of S S ′ coincides with the one-loop result (4.18) based on the amplitude calculation.

Generalization of the master formulas
In this section we extend our previous analysis to the general case of double-detector correlations (4.16) and (4.17) involving scalar, charge and energy flow operators. As was explained in Sect. 4.2, these correlations are uniquely specified by the event shape functions depending on the scaling variable 0 < z < 1.
To compute the event shape functions (4.16) and (4.17), we can follow the same procedure as before. Namely, we start with the four-point correlation functions involving two half-BPS operators (for the source and sink) and various components of the stress-tensor supermultiplet (5.1) (for the two detectors), analytically continue them to get the corresponding Wightman functions, and finally go to the detector limit to compute the double-detector correlations. We recall that the various four-point functions of this type appear as particular components in the expansion of the super correlation function G E (1, 2, 3, 4), Eq. (5.2), in powers of the Grassmann variables.
According to Sect. 5.1, the super correlation function can be obtained from its lowest component G E (1, 2, 3, 4) by applying a (complicated) differential operator. This operator does not depend on the coupling constant, so the perturbative corrections to G E (1, 2, 3, 4) are described by the unique scalar function Φ(u, v) from (5.5) and by its derivatives. Therefore, in order to get the Wightman function G W (1, 2, 3, 4) it suffices to replace Φ(u, v) by its Wightman counterpart Φ W (u, v) defined in (5.18). Then, using the Mellin representation (5.19) and going through the same steps as in the previous subsection, we obtain the following general representation for the event shape function (4.17) 18 Here the subscripts A, B = O, Q, E denote the type of detectors (scalar, charge or energy) and M(j 1 , j 2 ; a) is the universal (detector independent) Mellin amplitude defined in (5.12) and (5.13). 18 Here we give a short summary of the results, the details will be presented elsewhere. In obtaining them we have used the Mathematica package xAct [41], especially the application package Spinors [42].
For the scalar and charge detectors, the event shape function (6.1) also depends on the detector matrices S IJ and Q B A , respectively. This dependence enters into (6.1) through the R−symmetry invariant factors ω R built from the matrices S, Q and from the auxiliary variables Y, Y defining the source and the sink (see Eqs. (3.24) and (5.22)). The set of such factors is in one-to-one correspondence with the set of R−symmetry structures that appear in the four-point correlation function 0|O(1, Y )A(2)B(3)O(4, Y )|0 , which we use to compute the event shape function (6.1). Denoting the R−symmetry representations of the two detectors by A and B, respectively, (20 ′ for scalar, 15 for charge and 1 for energy flow), we can identify the above mentioned structures with the overlap of irreducible representations in the tensor products A × B and 20 ′ × 20 ′ . They are labelled by the index R on the right-hand side of (6.1). In the detector limit, each irreducible component R of the four-point correlation function gives rise to the detector kernel K AB;R (j 1 , j 2 ; z). As was already explained, this kernel depends on the Mellin variables j 1 and j 2 and the scaling variable z but does not depend on the coupling constant.
Let us first consider the kernels K EE , K EQ and K EO involving the energy flow. Since the energy flow is an R−symmetry singlet, the overlap of the two tensor products 1×B and 20 ′ ×20 ′ consists of a single irreducible structure and, therefore, the sum in (6.1) reduces to a single term involving ω EE 1 , ω EQ 15 and ω EO 20 ′ (see (D.13)). Quite remarkably, in these three cases the detector kernels turn out to be equal to the scalar-scalar kernel K defined in (5.25): As a consequence of these relations, the corresponding event shape functions F EE (z), F EQ (z) and F EO (z) are proportional to each other for any coupling constant. We interpret this as a corollary of N = 4 superconformal symmetry which uniquely fixes the correlation functions OT T O , OT JO and OT OO , starting from OOOO . The remaining kernels K OO , K QO and K QQ have a more complicated structure. We found however that, similarly to (6.2), they can be expressed in terms of the function K, Eq. The kernel for each channel can be written as a differential operator acting on K, where P OO R are polynomials in t 1,2 listed in Table 1 and t i = e ∂ j i (for i = 1, 2) act as shifts of the arguments of the kernel, j i → j i + 1.
Several comments are in order regarding this table. Each polynomial P OO R (t 1 , t 2 ) corresponds to a particular irreducible representation R of SU(4) appearing in the decomposition (6.3). Comparing these polynomials with those listed in (D.4), we notice that P OO R (t 1 , t 2 ) coincide with the eigenfunctions of the quadratic Casimir of SU(4) in the 'mirror' representation In particular, we have P OO 105 = Y 1 = 1, with the corollary K OO;105 = K(j 1 , j 2 ; z) . Table 1: Scalar-scalar correlations. The first column lists the various channels R in the tensor product (6.3). The second column gives the polynomial of shift operators which acts on the kernel K(j 1 , j 2 ; z), Eq. (5.25), according to (6.4). The third and fourth columns list the event shape functions F OO;R at one loop and at strong coupling, respectively. They were obtained from (6.1) using (5.14) and (5.15).
Secondly, from Table 1 we see that the event shape functions F OO (z) in the channels 15 and 175 vanish identically. This is due to the antisymmetry of the kernels K OO;15 and K OO;175 under the exchange of j 1 ↔ j 2 , which is incompatible with the symmetries of M(j 1 , j 2 ), as can be seen in (5.17).
Thirdly, Table 1 lists the contributions to F OO (z) in two regimes, at weak coupling (one loop) and at strong coupling 19 . We observe the presence of divergences, at one loop in the singlet channel, and at strong coupling in all non-vanishing channels but the 105. The appearance of divergences at weak coupling is not surprising and is related to the cross-talk between scalar detectors, as explained in Sect. 2.2.3. We recall that the cross-talk can be eliminated by imposing the additional condition on the detector matrices, [S, S ′ ] = 0, Eq. (2.38). As explained in Sect. 5.2 (see also Eq. (D.5)), this condition has the effect of suppressing half of the y−structures in (5.4) and (5.5), see (5.8) and (5.9). Equivalently, half of the irreps on the right-hand side of (6.3) also drop out, namely 1, 15 and 20 ′ . So, we can reinterpret [S, S ′ ] = 0 as a condition for removing some of the divergences in the Mellin integrals. In the absence of this condition the channel 20 ′ remains finite at one loop, but diverges at strong coupling 20 . The two remaining channels 84 and 105 give finite contributions at one loop, but the former diverges at strong coupling. We can avoid this divergence if we impose a stronger condition, Y 2 = Y 3 (see (D.6)). Its effect is to suppress all channels but the 105, which turns out to be finite both at one loop and at strong coupling.
Our next choice of detectors is A = Q and B = O. In this case, the sum in (6.1) runs over the overlap of irreducible representations in the tensor product with those in (6.3), i.e. R = 15, 20 ′ , 175. We can represent the results in a manner similar to 19 The correlation function and its Mellin amplitude at strong coupling were computed in [38,39,40]. More details about the event shapes at strong coupling will be presented elsewhere. 20 A mechanism to tame them is discussed at length in Refs. [13,14]. Table 2: Charge-scalar correlations. The channel 20 ′ vanishes identically for symmetry reasons. The channel 15 is removed by the condition (D.6).
the scalar case discussed above, introducing  Table 3. An interesting new feature is that P QQ 1 and P QQ 20 ′ explicitly depend on z. At one loop we need no condition since all five channels are finite (the two channels 15 s and 15 a give identically vanishing contributions for symmetry reasons). However, at strong coupling we again observe a divergence in the singlet channel. The difference with the previous case is that now the singlet can only be removed by imposing a stronger condition Y 2 = Y 3 . Its effect is to suppress all channels but the 84, which is finite both at one loop and at strong coupling.
Summarizing the various tables above, we remark the following interesting property, generalizing (6.2) and (6.6). For all choices of the two detectors, the kernels corresponding to the R−symmetry channel with the highest value of the quadratic Casimir of SU(4), are identical: This is a non-obvious corollary of N = 4 superconformal symmetry, which will be investigated in our future work [43]. Finally, putting together the one-loop results for the event shape functions (6.1) from the different tables, we exactly reproduce the one-loop results of the amplitude calculations (without with ω QQ R given by (D.12).

Conclusions
In this paper we studied the weighted cross sections describing the angular distribution of various global charges (energy, R−charge) in the final states of N = 4 SYM created from the vacuum by a source. We applied the approach developed in the companion paper [14] and computed them starting from Euclidean correlation functions both at weak and strong coupling. The starting point of our analysis was a relation between the weighted cross sections and Wightman correlation functions involving various flow operators. We defined three different types of the flow operators based on certain components of the N = 4 stress-tensor multiplet: the half-BPS scalar operators, the R−symmetry currents and the energy-momentum tensor. In addition, we used the half-BPS scalar operator as a source that creates the physical state out of the vacuum. Then, the weighted cross section is determined by the Wightman correlation function involving different components of the N = 4 stress-energy multiplet in a particular detector limit which includes sending some of the operators to the null infinity and subsequently integrating over their light-cone coordinates.
The Euclidean version of such correlation functions have a number of remarkable properties in N = 4 SYM. In particular, three-and four-point correlators of operators belonging to the N = 4 stress-energy multiplet are uniquely determined by analogous correlation functions of half-BPS scalar operators. The corresponding relations do not depend on the coupling constant and will be explained in more detail in an upcoming publication. The next step consisted in the analytical continuation of the Euclidean correlation functions to their Wightman counterparts following the Lüscher-Mack procedure [36]. We demonstrated that the most efficient and convenient way to do this is via the Mellin representation for the correlation functions.
In this way, we derived a master formula which yields an all-loop expression for the weighted cross sections as a convolution of the Mellin amplitude, defined by the Euclidean correlation function, with a coupling-independent 'detector kernel' determined by the choice of the flow operators. We performed thorough checks for single-and double-detector correlations to leading order at weak coupling and found perfect agreement with the conventional amplitude calculations of the corresponding observables in N = 4 SYM. We would like to emphasize that our approach is applicable to all orders in the weak coupling expansion as well as at strong coupling. In the latter case, we made use of the prediction for the four-point correlation functions of half-BPS operators, obtained via the AdS/CFT correspondence, to compute the double correlations at strong coupling.
There are several directions in which the construction presented in this paper can be extended even further. Notice that the Mellin amplitude for the four-point correlation function of half-BPS operators is known to two loops and it should be possible to extend it beyond using the recent progress in computing the correlation function up to six loops [44]. This opens up a possibility to compute the flow correlations at higher orders of perturbative expansion and confront them with techniques based on Keldysh-Schwinger diagram technique [45]. We will address this question in a forthcoming publication.
Another important issue that has to be developed further concerns remarkable relations between the detector kernels corresponding to different flow operators and in different R−symmetry channels, Eqs. (6.2), (6.6) and (6.11). They lead to analogous relations between the weighted cross sections. We expect that there should exist a supersymmetric Ward identity that directly relates various weighted cross section studied in this paper. Finally, infrared finiteness of the charge flow correlations at weak coupling, the origin of the divergences at strong coupling in some R−symmetry channels require a more detailed study. These and other issues will be addressed elsewhere.
should vanish. This means that the commutator of the flow operators, say [O( n; S), O( n ′ ; S ′ )], can receive a contribution from particles with zero momenta only. To investigate this, we use the representation of the scalar flow operator (2.33) in terms of annihilation/creation operators of scalars combined with the identity 21 with n µ = (1, n) being a light-like vector, n 2 = 0. Thus, we obtain an equivalent representation for the flow operator (for ℓ = 0), where (φ i a(nτ )) ≡ 6 I=1 φ i I a I (nτ ). Notice that each additional factor of k 0 on the right-hand side of (A.1) is translated into a factor of τ in the integral on the left-hand side. This implies that the charge and energy flow operators, Q( n; Q) and E( n), respectively, admit a representation similar to (A.2) with the important difference that the τ −integral involves additional factors of τ and τ 2 , respectively (see Eq. (4.6)).
Making use of the commutation relation where φ i , φ ′j and S, S ′ define the detector matrices (2.32) and (φ i φ ′j ) = 6 1 φ i I φ ′ I j . The delta function localizes the τ −integral at τ 1 = τ 2 = 0 but leads to an ambiguous expression of the form 0 × δ(0). To carefully evaluate it we approximate the delta function and examine the following test integral involving an arbitrary test function f (τ 1 , τ 2 ) where in the second relation we rescaled the integration variables as τ i → ǫ 1/2 τ i . Applying this identity to an appropriately chosen function f (τ 1 , τ 2 ) we find from (A.  4 a †I (0)(SS ′ − S ′ S) IJ a J (0) 1 − ( n n ′ ) , (A.6) 21 To verify this identity it suffices to integrate both sides with a test function.
where in the second relation we applied (2.32). We conclude from (A.6) that, for general detector matrices S and S ′ , the scalar flow operators do not commute. Moreover, in agreement with our expectations, the commutator receives a contribution from particles with zero momentum only. We recall that for the charge and energy flow operator, the integral representation analogous to (A.2) contains additional factors of τ . Calculating [O( n; S), Q( n ′ ; Q)] and [O( n; S), E( n ′ )] as in (A.6), we find that the additional factor of τ makes both commutators vanish. The same applies to all remaining commutators involving charge and energy flow operators.
B SU (4) versus SO (6) To simplify the R−symmetry structure of the correlation functions, throughout the paper we use two different but equivalent sets of isotopic (or harmonic) variables, Y I and (y AB ,ỹ AB ). The former defines a complex SO(6) null vector 6 I=1 Y I Y I = 0, whereas the latter carries a pair of SU(4) indices A, B = 1, . . . , 4 and satisfies the relations The variables Y and y are related to each other as follows: where (Σ I ) AB are the (chiral) Dirac matrices for SO (6). They satisfy the relations and can be expressed in terms of the 't Hooft symbols Σ AB I = (η AB I , iη AB I ) [46]. Combining (Σ I ) AB with their complex conjugates (Σ I ) AB = (Σ I ) AB , we obtain the generators of the fundamental representation of SU(4): We then use (B.2) and (B.3) to get the following identity In the special case Y I 1 = Y I 2 , or equivalently (y 1 ) AB = (y 2 ) AB , it follows from (Y 1 Y 1 ) = 0 that 1 2 ǫ ABCD (y 1 ) AB (y 1 ) CD = (y 1 ) AB (ỹ 1 ) AB = 0 . (B.6) Making use of (B.1) and (B.2) we can obtain two equivalent representations for various operators in terms of SO(6) and/or SU(4) harmonic variables.
In particular, the projected scalar field admits two representations, where Φ I and φ AB are related to each other as and the scalar field satisfies the additional reality conditionφ AB = (φ AB ) * . Notice that we do not impose a similar condition on y AB and treat (y AB ) * as independent variables where (Y Y ) = 1 2 y A 1 B 1ȳ A 1 B 1 and D(x) denotes the propagator of a free scalar field, The explicit expression for D(x) contains a prescription, for Feynman (time-ordered) and Wightman correlation functions, respectively.

C Scalar, charge and energy correlations at one loop
In this Appendix, we compute scalar-scalar, charge-charge and energy-energy correlations to lowest order in the coupling, using amplitude techniques. contribution from the state |sλλ and it comes from the diagrams shown in Figs. 6(c) and (d).

Converting the structures (D.2) into correlation function ones, we get
) and The polynomials Y R coincide with the eigenfunctions Y nm (t 1 , t 2 ) of the quadratic Casimir operator of SU(4) for the irreps with Dynkin labels [n − m, 2m, n − m], as listed in Appendix B in [47]. Further, the condition (2.38) for absence of cross-talk between the two scalar detectors, [S, This condition has two solutions, the weaker constraint (Y 2 Y 3 ) = 0 or the stronger Y 2 = Y 3 22 . Inserting these constraints back into (D.2), we find that the weaker one implies Here QS = Y I Q IJ S KL Y L , and we have rewritten the detector matrix Q IJ = −Q JI = (Γ IJ ) B A Q A B in SO(6) notation, with the help of the SO(6) gamma matrices defined in (B.4).