Quantum tomography for collider physics: Illustrations with lepton pair production
QQuantum tomography for collider physics: Illustrations with lepton pairproduction
John C. Martens, ∗ John P. Ralston, and J. D. Tapia Takaki Department of Physics and Astronomy, The University of Kansas, Lawrence, KS 66045, USAAbstract:
Quantum tomography is a method to experimentally extract all that is observable abouta quantum mechanical system. We introduce quantum tomography to collider physics with the il-lustration of the angular distribution of lepton pairs. The tomographic method bypasses much of thefield-theoretic formalism to concentrate on what can be observed with experimental data, and howto characterize the data. We provide a practical, experimentally-driven guide to model-independentanalysis using density matrices at every step. Comparison with traditional methods of analyzing an-gular correlations of inclusive reactions finds many advantages in the tomographic method, whichinclude manifest Lorentz covariance, direct incorporation of positivity constraints, exhaustively com-plete polarization information, and new invariants free from frame conventions. For example, ex-perimental data can determine the entanglement entropy of the production process, which is a model-independent invariant that measures the degree of coherence of the subprocess. We give reproduciblenumerical examples and provide a supplemental standalone computer code that implements the pro-cedure. We also highlight a property of complex positivity that guarantees in a least-squares type fit thata local minimum of a χ statistic will be a global minimum: There are no isolated local minima. Thisproperty with an automated implementation of positivity promises to mitigate issues relating to mul-tiple minima and convention-dependence that have been problematic in previous work on angulardistributions. Keywords: polarization, entanglement, entropy, inclusive production, dilepton, quarkonium, angular distribu-tion, data analysis method, intermediate state, density matrix
I. INTRODUCTION
Tomography builds up higher dimensional objects from lower dimensional projections.
Quantumtomography [1] is a strategy to reconstruct all that can be observed about a quantum physical sys-tem. After becoming a focal point of quantum computing, quantum tomography has recently beenapplied in a variety of domains [2–9].The method of quantum tomography uses a known “probe” to explore an unknown system. Datais related directly to matrix elements, with minimal model dependence and optimal efficiency.Collider physics is conventionally set up in a framework of unobservable and model-dependentscattering amplitudes. In quantum tomography these unobservable features are skipped to dealdirectly with observables. The unknown system is parameterized by a certain density matrix ρ ( X ) ,which is model-independent. The probe is described by a known density matrix ρ ( probe ) . The ∗ [email protected], [email protected] a r X i v : . [ h e p - ph ] J u l matrices are represented by numbers generated and fit to experimental data, not abstract operators. Quantum mechanics predicts an experiment will measure tr ( ρ ( probe ) · ρ ( X )) , where tr is the trace.In many cases ρ ( probe ) is extremely simple: A 3 × What will be observed is strictlylimited by the dimension and symmetries of the probe.
The powerful efficiency of quantum tomographycomes from exploiting the probe’s simplicity in the first steps. The description never involves morevariables than will actually be measured.We illustrate the advantages of quantum tomography with inclusive lepton-pair production. Itis a relatively mature subject chosen for its pedagogical convenience. Despite the maturity of thesubject, we discover new things. For example, the puzzling plethora of plethora of ad hoc invariantquantities is completely cleared up. We also find new ways to assist experimental data analysis.Positivity is a central issue overlooked in the literature, which we show how to control. More-over, the tomography procedure carries over straightforwardly to many final states, including theinclusive production of charmonium, bottomonium, dijets, including boosted tops, HH , W + W − , ZZ [10–32]. Our practical guide to analyzing experimental data uses density matrices at each stepand circumvents the more elaborate traditional theoretical formalism. We concentrate on makingtools available to experimentalists. We give a step-by-step guide where density matrices stand asdefinite arrays of numbers , bypassing unnecessary formalism. II. THE QUANTUM TOMOGRAPHY PROCEDURE APPLIED TO INCLUSIVE LEPTON PAIRPRODUCTION
The tomography procedure reconstructs all that can be observed about a quantum physical sys-tem. For inclusive lepton pair production, what can be observed is the invariant mass distribution,the lepton pair angular distribution dN / d Ω and the polarization of the unknown intermediatestate of the system, contained in ρ ( X ) . In this section we reconstruct dN / d Ω and ρ ( X ) from firstprinciples using tomography. Structure functions and model-dependent assumptions about theintermediate state, common to the traditional formalism [33–37], do not appear.Expert readers, who are accustomed to seeing some of these formulas derived, might note thatthe method of derivation is particularly simple. The particular steps we do not follow are to benoted. That also explains why some of the relations we find seem to have been overlooked in thepast. Polarization and spin are different concepts. The polarization (and density matrix of the unknown state) predicts the spin,while the spin cannot predict the density matrix.
A. Kinematics
Consider inclusive production of a lepton pair with 4-momenta k , k (cid:48) from the collision of twohadrons with 4-momenta P A , P B : P A P B → (cid:96) + ( k ) (cid:96) − ( k (cid:48) ) + X ,where X and the final state lepton spins are unobserved and thus summed over. In the high energylimit k = k (cid:48) = Q = k + k (cid:48) . The azimuthal distribution of total pair momenta inthe lab frame is isotropic. Lepton pair angular distributions are described in the pair rest-framedefined event-by-event. In this frame the pair momenta are back-to-back and equal in magnitude.The frame orientation depends on the beam momenta and the pair total momentum.Defining momentum observables via a Lorentz-covariant frame convention allows calculationsto be done in any frame. In its rest frame the total pair momentum Q µ = ( (cid:112) Q , (cid:126) Q = ) . A set of xyz spatial axes in this frame will be defined by three 4-vectors X µ , Y µ , Z µ , satisfying Q · X = Q · Y = Q · Z =
0. (1)The frame vectors being orthogonal implies X · Y = Y · Z = X · Z = P A =(1, 0, 0, 1), P B =(1, 0, 0, -1) (light-cone ± vectors), a frame satisfying the relations of Eq. 1and Eq. 2 is given by ˜ Z µ = P µ A Q · P B − P µ B Q · P A ;˜ X µ = Q µ − P µ A Q Q · P A − P µ B Q Q · P B ;˜ Y µ = (cid:101) µναβ P A ν P B α Q β .These frame vectors define the Collins-Soper ( CS ) frame. The normalized frame vectors are ( X µ , Y µ , Z µ ) = ( ˜ X µ √− ˜ X · ˜ X , ˜ Y µ (cid:112) − ˜ Y · ˜ Y , ˜ Z µ √− ˜ Z · ˜ Z ) . This is a great advantage compared to making calculations with a complicated (and error prone) sequence of rotationsand boosts. We use (cid:101) =
1. The mirror symmetry of pp collisions also strongly supports a convention where the direction of the Z axis is determined by the sign of the pair rapidity. The formulas shown do not include this detail. These expressions simplify a more complicated convention that included finite mass effects in the original definition
To analyze data for each event labeled J :Compute Q J = k J + k (cid:48) J ; (cid:96) J = k J − k (cid:48) J ; ( X µ J , Y µ J , Z µ J ) ; (cid:126)(cid:96) XYZ , J = ( X J · (cid:96) J , Y J · (cid:96) J , Z J · (cid:96) J ) ;ˆ (cid:96) J = (cid:96) XYZ , J / (cid:113) − (cid:96) XYZ , J · (cid:96) XYZ , J . (3)In fact, ˆ (cid:96) J = ( sin θ cos φ , sin θ sin φ , cos θ ) J , where θ , φ are the polar and azimuthal angles of one(e.g. plus-charge) lepton in the rest frame of Q . The meaning of a ”Lorentz invariant cos θ ” is ascalar Z µ ( k − k (cid:48) ) µ which becomes ˆ z · ˆ ( k − k (cid:48) ) in the rest frame of Q . B. The angular distribution, in terms of the probe and target density matrices
The standard amplitude for inclusive production of a fermion- anti-fermion pair of spin s , s (cid:48) has astring of gamma-matrices contracted with final state spinors v a ( k (cid:48) s (cid:48) ) , ¯ u b ( k , s ) . When the amplitudeis squared, these factors appear bi-linearly, as in u a ( ks ) ¯ u a (cid:48) ( ks ) = ( )[( k / + m )( + γ s / )] aa (cid:48) .Summing over unobserved s and dropping m δ aa (cid:48) , a form of density matrix appears: ∑ s u a ( ks ) ¯ u a (cid:48) ( ks ) → k µ γ µ aa (cid:48) .The Feynman rules for the density matrix of two relativistic final state fermions (or anti-fermions,or any combination) is a factor given by ρ aa (cid:48) , bb (cid:48) ( k , k (cid:48) ) → k / aa (cid:48) k / (cid:48) bb (cid:48) (4)This fundamental equality is not present in pure state quantum systems. There is no spinor corre-sponding to a fermion averaged over initial spins , nor to a fermion summed over final spins.As shown in the Appendix, the rest of the cross section appears in the target density matrix ρ ( X ) ,which must have four indices to contract with the probe indices: d σ ∼ ∑ aa (cid:48) bb (cid:48) ρ aa (cid:48) , bb (cid:48) ( k , k (cid:48) ) ρ aa (cid:48) , bb (cid:48) ( X ) dLIPS = tr (cid:0) ρ ( k , k (cid:48) ) ρ ( X ) (cid:1) dLIPS , (5)where dLIPS is the Lorentz invariant phase space.Note that u a ( ks ) ¯ u a (cid:48) ( ks ) is not positive definite since the Dirac adjoint ¯ u a (cid:48) ( ks ) = ( u † ( ks ) γ ) a has afactor of γ , introduced by convention. Removing it, ∑ s u a ( ks ) u † a (cid:48) ( ks ) becomes positive by inspec-tion. (Any matrix of the form M · M † has positive eigenvalues.) ρ ( k , k (cid:48) ) as written is not normal-ized, because the Feynman rules shuffle spinor normalizations into overall factors. To make thearrow in Eq. 4 into an equality, multiply on the right by γ twice, and standardize the normaliza-tions. The same steps applied to ρ aa (cid:48) , bb (cid:48) ( X ) cancels the γ factors. The result is that the probabilityto find two fermions has the fundamental quantum mechanical form P ( k , k (cid:48) ) = tr ( ρ ( k , k (cid:48) ) ρ ( X )) .The left side of Eq. 5 is d σ ( k , k (cid:48) ) , the same as the joint probability P ( Q , (cid:96) (cid:12)(cid:12) init ) where init are theinitial state variables. The phase space for two leptons converts as k k (cid:48) d σ d kd k (cid:48) = d σ d Qd Ω .We can write P ( Q , (cid:96) (cid:12)(cid:12) init ) = P ( (cid:96) (cid:12)(cid:12) Q , init ) P ( Q (cid:12)(cid:12) init ) .Here P ( Q (cid:12)(cid:12) init ) = d σ / d Q , and P ( (cid:96) (cid:12)(cid:12) Q , init ) = dN / d Ω is the conditional probability to find (cid:96) given Q and the initial state. This factorization is general and unrelated to one-boson exchange, partonmodel, or other considerations. Since P ( (cid:96) (cid:12)(cid:12) Q , init ) is a probability, quantum mechanics predicts itis a trace: dNd Ω = σ d σ d Ω = P ( (cid:96) (cid:12)(cid:12) Q , init ) = π tr ( ρ ( (cid:96) ) ρ ( X )) , (6)where tr indicates the trace, d Ω = dcos θ · d φ , and ρ ( (cid:96) ) , the probe, is a 3 × (cid:96) J . The target hadronic system is representedby ρ ( X ) . Since the probe ρ ( (cid:96) ) is a 3 × ρ ( X ) is a 3 × The description has just been reduced from ρ aa (cid:48) , bb (cid:48) ( k , k (cid:48) ) , a Dirac tensor with 4 possible matrixelements, to a 3 × tr ( ρ ( (cid:96) )) = like-sign and unlike sign pairs, and assumes no model for how the pairs are produced . The Dirac form (andDirac traces) is over-complicated, because describing every possible exclusive reaction for everypossible in and out state is over-achieved in the formalism. C. The probe matrix
The probe matrix ρ ( (cid:96) ) is given by ρ ij ( (cid:96) ) = + a δ ij − a ˆ (cid:96) i ˆ (cid:96) j − ıb (cid:101) ijk ˆ (cid:96) k , (7) We remind the reader that the phase space factors dLIPS originate in further organizational steps computing the quantummechanical transition probability per volume per time, which afterwards restore the phase space factors. This is a more general statement than enumerating “structure functions”. which is derived in the Appendix. The Standard Model predicts only two parameters, a and b . Ifon-shell lepton helicity is conserved (as in lowest order production by a minimally-coupled vectorboson) then a = b = c A c V . The latter is not a prediction but a definition. If the productionis parity-symmetric then c A = The only non-trivial prediction of the Standard Model is the value ofc A c V . Lowest-order production by Z bosons predicts b = sin θ W ∼ itself represents a reduced system that is unknown a priori . Itshould be determined experimentally. Consider the angular distribution of e + e − → µ + µ − . Let ρ ( e ; ˆ z ) describe electrons with parameters a e , b e colliding along the z axis. Let ρ ( µ ; ˆ (cid:96) ) describemuons with parameters a µ , b µ emerging along direction ˆ (cid:96) . A short calculation using Eq. 7 twicegives π tr (cid:16) ρ ( e ; ˆ z ) ρ ( µ ; ˆ (cid:96) ) (cid:17) = π (cid:18) + b e b µ ˆ (cid:96) · ˆ z + a e a µ (( ˆ z · ˆ (cid:96) ) − ) (cid:19) , = π (cid:18) + b e b µ cos θ + a e a µ ( cos θ − ) (cid:19) . (8)Fitting experimental data will give a e a µ and b e b µ . If lepton universality is assumed the probe ρ ( µ ; ˆ (cid:96) ) has measured the probe ρ ( e ; ˆ z ) . D. How tomography works: dN / d Ω as a function of ρ ( (cid:96) ) , ρ ( X ) Let ˆ G (cid:96) be a set of probe operators, with expectation values < G (cid:96) > = tr ( G (cid:96) ρ ( X )) . The tracedefines the Hilbert-Schmidt inner product of operators. The condition for operators (matrices) tobe orthonormal is tr ( G (cid:96) G k ) = δ (cid:96) k orthonormal matrices. (9)There are N − N × N Hermitian operators, not including the identity. When acomplete set of probe operators has been measured, the density matrix is tomographically recon-structed from observables as ρ ( X ) = ∑ (cid:96) G (cid:96) tr ( G (cid:96) ρ ) = ∑ (cid:96) G (cid:96) < G (cid:96) > .For a pure state density matrix, there exists a basis { G (cid:96) } such that only one term appears in thesum over (cid:96) . Then ρ pure = | ψ >< ψ | , and | ψ > is reconstructed as the eigenvector of ρ pure .Each orthogonal probe operator measures the corresponding component of the unknown system,and is classified by its transformation properties. For angular distributions the transformations of This may be a new result, which goes beyond what is known from one-boson exchange with or without radiative correc-tions. The production details can only renormalize the parameters. interest are rotations. ρ ( (cid:96) ) contains tensors transforming like spin-0, spin-1 and spin-2. Each tensorof a given type is orthogonal to the others.Organizing transformation properties simplifies things significantly. Recall the general form of ρ ( (cid:96) ) , from Eq. 7. The most general form for ρ ( X ) that is observable will have the same generalexpansion, with new parameters:Probe: ρ ij ( (cid:96) ) = δ ij + b ˆ (cid:96) · (cid:126) J ij + aU ij ( ˆ (cid:96) ) ; where U ij ( ˆ (cid:96) ) = δ ij − ˆ (cid:96) i ˆ (cid:96) j = U ji ( (cid:96) ) ; tr ( U ( (cid:96) )) = ρ ij ( X ) = δ ij + (cid:126) S · (cid:126) J ij + U ij ( X ) ; where U ( X ) = U T ( X ) ; tr ( U ( X )) =
0. (11)These formulas reiterate Eq. 7 while identifying ( J k ) ij = − ı (cid:101) ijk as the generator of the rotationgroup in the 3 × Upon taking the trace as an inner product, orthogonality selectseach term in ρ ( X ) that matches its counterpart in ρ ( (cid:96) ) . For example (cid:126) J is orthogonal to all the otherterms except the same component of (cid:126) J :12 tr ( J i J k ) = δ ij ;hence 12 tr ( ˆ (cid:96) · (cid:126) J (cid:126) S · (cid:126) J ) = ˆ (cid:96) · (cid:126) S .Orthogonality makes it trivial to predict which density matrix terms can be measured by probematrix terms. We call the matching of terms “the mirror trick.”We now make several relevant comments about Eq. 10 and Eq. 11: • All density matrices can be written as 1 N × N / N to take care of the normalization, plus a trace-less Hermitian part. The unit matrix is the spin-0 part and invariant under rotations. Theonly contribution of the 1 terms is tr ( × ) / N = N . • The textbook density matrix spin vector (cid:126) S consists of those parameters coupled to the angularmomentum operator. This is also called the spin-1 contribution. The quantum mechanicalaverage angular momentum of the system is < (cid:126) J > = tr ( ρ X (cid:126) J ) = (cid:126) S .When the coordinates are rotated, the (cid:126) J matrices transform exactly so that (cid:126) S rotates like avector under proper rotations, and a pseudovector under a change of parity. • The last term of Eq. 10, the spin-2 part, is real, symmetric and traceless. By the mirror trickit can only communicate with a corresponding spin-2 term in ρ ( X ) denoted U ij ( X ) , which is The real Cartesian basis for (cid:126) J is being used because it is more transparent than the J z → m basis that is an alternative. Itwould have complex parameters. real, symmetric and traceless. It can be considered a measure of angular momentum fluctua-tions: < (cid:0) J i J j + J j J i (cid:1) − (cid:126) J δ ij > = U ij ( X ) .A common mistake assumes the quadrupole U should be zero in a pure “spin state.” Actuallya pure state with | (cid:126) S | = ρ pure , ij ( (cid:126) S ) = ( δ ij − ˆ S i ˆ S j ) − ı (cid:101) ijk ˆ S k . (12)For example, when (cid:126) S = ˆ z the density matrix has one circular polarization eigenstate witheigenvalue unity, and two zero eigenvalues. Pure states exist with (cid:126) S =
0: They have realeigenvectors corresponding to linear polarization. From the spectral resolution ρ ( X ) = ∑ α λ α | e a >< e α | , there is no observable distinction between a density matrix and the occur-rence of pure states | e a > with probabilities λ α , which are the density matrix eigenvalues. • As it stands the U ij matrices in Eq. 10 and Eq. 11 have not been expanded in a complete setof symmetric, orthonormal 3 × ρ ( X ) can be fit to data whether or notan expansion is done. The purpose of such work is to complete the classification process toassist with interpreting data. We sketch the steps here. Details are provided in an Appendix.Let E M be a basis of traceless orthonormal matrices where U ( (cid:96) ) = ∑ M tr ( U ( (cid:96) ) E M ) E M . Thisis the tomographic expansion of the probe. Choose E M so the outputs are normalized real-valued spin-2 spherical harmonics Y M ( θ , φ ) . The expansion of the unknown system will be U ( X ) = ∑ M tr ( ρ ( X ) E M ) E M = ∑ M ρ M ( X ) E M . By orthogonality the spin-2 contribution tothe angular distribution will be dNd Ω ∼ tr ( ρ ( (cid:96) ) ρ ( X )) spin − ∼ ∑ M ρ M ( X ) Y M ( θ , φ ) .Writing out the terms gives dNd Ω = π + π S x sin θ cos φ + π S y sin θ sin φ + π S z cos θ + c ρ ( √ − √ θ ) − c ρ sin ( θ ) cos φ + c ρ sin θ cos ( φ )+ c ρ sin θ sin ( φ ) − c ρ sin ( θ ) sin φ . (13)The label X has been dropped in ρ M and c = ( √ π ) . Since E M transform like Y M , thecoefficients ρ M transform under rotations like spin-2. That means ρ M → R ( ) MM (cid:48) ρ M (cid:48) ( X ) , where R ( ) MM (cid:48) is a matrix available from textbooks [38]. The traditional A k , λ k conventions do not useorthogonal functions. Transformations from the traditional conventions to the ρ M conventionare given in an Appendix.Note the transformation properties listed are exact . The systematic and statistical errors of ameasurement appear in fitting ρ ( X ) . E. Fitting ρ ( X ) , dN / d Ω Quantum mechanics requires ρ ( X ) must be positive , which means it has positive eigenvalues.Positivity produces subtle non-linear constraints, similar to unitarity. In the 3 × is a more restrictive set of relations. If density matrices are not used it is quitestraightforward to fit data yielding a positive cross section while violating positivity .Fortunately positivity can be implemented by the Cholesky decomposition of ρ X [39], which isdiscussed in the Appendix. For the 3 × ρ ( X )( m ) = M ( m ) · M † ( m ) ; M ( m ) = (cid:113) ∑ k m k m m + im m + im m m + im m , (14)where the parameters − ≤ m α ≤ ρ ( (cid:96) ) is an array of numbers, and ρ ( X )( m ) is an array of parameters. The resultsare combined to make the J th instance of tr ( ρ J ( (cid:96) ) ρ ( X )( m )) , where ρ ( X )( m ) has been parameterizedin Eq. 14. Fit the m α parameters to the data set. For example, the log likelihood L of the set J = J max is L ( m ) = J max ∑ J log (cid:16) tr (cid:16) ρ ( J ) ( (cid:96) ) · ρ ( X )( m ) (cid:17)(cid:17) + J max log ( π ) . (15)Sample code available online carries out these steps, returning parameters m α . The details of cutsand acceptance appear in fitting the numbers m α using numbers for the lepton matrix ρ ( lep ) (not When tr ( ρ ( X )) = ρ ( X ) exceeds unity, and oneor more goes negative. Then for some vector | e > the quadratic form < e | ρ X | e ><
0, which would appear to providea signal. Yet no such signal might be found in the angular distribution, because tr ( ρ ( (cid:96) ) ρ ( X )) > To help readers appreciate the practical value of these advantages, we constructed standalone analysis code in both ROOTand Mathematica [40]. We expect the code to provide useful cross-checks on code users might write for themselves. Z -boson data we found ρ f it ( X ) = − i − + i + i − i − − i + i Using the Standard Model parameters for ρ ( (cid:96) ) , Eq. 3 and Eq. 7, the trace yields dNd Ω f it ∼ tr ( ρ ( (cid:96) ) ρ ( X )) = + ( φ ) sin ( θ ) + ( φ ) cos ( θ )+ ( φ ) cos ( θ ) + ...where ... indicates several terms there is no need to write out. Integrated over φ , this expressionbecomes dN f it d cos θ = π (cid:16) + θ + θ (cid:17) .A 1 + cos θ distribution is the leading order Drell-Yan prediction for virtual spin-1 boson annihila-tion, while 0.137 cos θ represents a charge asymmetry.It is trivial to go from tr ( ρ ( (cid:96) ) ρ ( X )) to a conventional parameterization of an angular distribu-tion by taking inner products of orthogonal functions. It is also easy to expand ρ ( X ) in a basis oforthonormal matrices with the same results. Note these steps are exact , and much different fromfitting data to trigonometric functions in some convention, which tends to yield multiple solutions,along with violations of positivity, which can introduce pathological convention-dependence. Per-haps struggles with convention-dependence of quarkonium data [41, 42] are related to this. Itwould be interesting to investigate. F. Summary of quantum tomography procedure
To analyze data for each event labeled J : • Compute Q J = k J + k (cid:48) J ; (cid:96) J = k J − k (cid:48) J ; ( X µ J , Y µ J , Z µ J ) ; (cid:126)(cid:96) XYZ , J = ( X J · (cid:96) J , Y J · (cid:96) J , Z J · (cid:96) J ) ;ˆ (cid:96) J = (cid:96) XYZ , J / (cid:112) − (cid:96) XYZ , J · (cid:96) XYZ , J . • Make the lepton density matrix. For Z bosons in the Standard Model it is ρ ij ( (cid:96) ) = ( δ ij − ˆ (cid:96) i ˆ (cid:96) j ) − ı (cid:101) ijk ˆ (cid:96) k . (16) • The results are combined to make the J th instance of tr ( ρ J ( (cid:96) ) ρ ( X )( m )) , where ρ ( X )( m ) has1been parameterized in Eq. 14. Fit the m α parameters to the data set. For example, the loglikelihood L of the set J = J max is L ( m ) = J max ∑ J log (cid:16) tr (cid:16) ρ ( J ) ( (cid:96) ) · ρ ( X )( m ) (cid:17)(cid:17) + J max log ( π ) . (17)Sample code available online (see footnote 10) carries out these steps, returning parameters m α . G. Comments
1. The possible symmetries of ρ ( (cid:96) ) enter here. Suppose c A =
0. Then ρ ( (cid:96) ) is even under parity,real and symmetric. The imaginary antisymmetric elements of ρ ( X ) are orthogonal, and contributenothing to the angular distribution. When known in advance, the redundant parameters of ρ ( X ) can be set to zero while making the fit. (That does not mean unmeasured parameters can be for-gotten when dealing with positivity.) In general a fitting routine will either report a degeneracyfor redundant parameters, or converge to values generated by round-off errors. Degeneracy willalways be detected in the Hessian matrix computed to evaluate uncertainties.2. The normalization condition ∑ k m ( k ) = (cid:112) m α from Eq.14, and subtracting J max log ( ∑ J max k m ( k )) from the log-likelihood (Eq.17). When that is done thefitted density matrix will not be automatically normalized, due to the symmetry ρ ( X ) → λρ ( X ) ofthe modified likelihood. The density matrix becomes normalized by dividing by its trace. Incorpo-rating such tricks improved the speed of the code available online (see footnote 10) by a factor ofabout 100.3. Algorithms are said to compute a “unique” Cholesky decomposition, which would seem topredict m α given ρ ( X ) . The algorithms choose certain signs of m α by a convention making thediagonals of M positive. However that is not quite enough to assure a numerical fit finds a uniquesolution.The fundamental issue is that MM † = ρ ( X ) is solved by M = (cid:112) ρ ( X ) , and the square root isnot unique. There are 2 N arbitrary sign choices possible among N eigenvalues of (cid:112) ρ ( X ) . Forcingthe diagonals of M to be positive reduces the possibilities greatly, and an algorithm exists to forcea unique, canonical form of m α in a data fitting routine. We did not make use of such a routine,since fitting ρ ( X ) is the objective. Depending upon the data fitting method, increasing the numberof ways for M ( m α ) to make a fit sometimes makes convergence faster.4. Let <> exp stand for the expectation value of a quantity in the experimental distribution ofevents. By symmetry < ˆ (cid:96) > exp and < ˆ (cid:96) i ˆ (cid:96) j > exp are vector and tensor estimators, respectively, whichmust depend on the vector and tensor parameters (cid:126) S , U ij ( X ) in the underlying density matrix. A2calculation finds < ˆ (cid:96) > exp = J max ∑ J ˆ (cid:96) J = − (cid:126) S ; < ˆ (cid:96) i ˆ (cid:96) j > exp = J max ∑ J ˆ (cid:96) Ji ˆ (cid:96) Ji = δ ij − Re [ U ij ] .An estimate of ρ ( X ) not needing a parameter search then exists directly from data. However posi-tivity of ρ ( X ) is more demanding, and not automatically maintained by such estimates. III. RESULTSA. Analysis Bonuses of the Quantum Tomography Procedure
1. Convex Optimization
The issue of multiple solutions for ρ ( X ) is different. Multiple minima of χ statistics affects fitsto cross sections parameterized by trigonometric functions. However, quantum tomography usingmaximum likelihood happens to be a problem of convex optimization . In brief, when ρ is positivethen < e | ρ | e > is a positive convex function of | e > . Then tr ( ρ ( (cid:96) ) ρ ( X )) is convex, being equivalentto a positively weighted sum of such terms. The logarithm is a concave function, leading to aconvex optimization problem. That means that when ρ ( X ) is a local maximum of likelihood it is theglobal maximum. Exceptions can only come from degeneracies due to symmetry or an inadequatenumber of data points [43]. Convex optimization is important because without such a property theevaluation of high-dimensional fits by trial and error can be exponentially difficult.
2. Discrete Transformation Properties
Table I lists discrete transformation properties of all terms under parity P , time reversal T , andlepton charge conjugation C (cid:96) . If leptons have different flavors (as in like or unlike sign e µ ) the C (cid:96) operation swaps the particle defining ˆ (cid:96) .When coordinates XYZ are defined the direction of ˆ Y = ˆ Z × ˆ X is even under time reversal andparity, which is exactly the opposite of X and Z . Then (cid:126) S · ˆ Y is T -odd, contributing the sin θ sin φ term. The XY and ZY matrix elements of ρ ( X ) are also odd under T , contributing the termsshown. T -odd terms come from imaginary parts of amplitudes, which are generated by loop cor-rections in perturbative QCD. In a forthcoming study [44] of inclusive lepton pair production near the Z pole, we find interesting, new features in the S y data of Ref. [11]. term origin dN / d Ω C (cid:96) P T C (cid:96)
P PT · (cid:96) · − − − + + · X · · − − + + · Y · · + + + + · Z · · − − + + S x X (cid:96) sin θ cos φ − + + − + S y Y (cid:96) sin θ sin φ − − − + + S z Z (cid:96) cos θ − + + − + ρ XX (cid:96)(cid:96) sin θ cos 2 φ + + + + + ρ XY (cid:96)(cid:96) sin θ sin 2 φ + − − + + ρ XZ (cid:96)(cid:96) sin 2 θ cos φ + + + + + ρ YZ (cid:96)(cid:96) sin θ sin φ + − − + + ρ ZZ (cid:96)(cid:96) √ − √ θ + + + + + TABLE I:
Terms in the angular distribution with their properties under discrete transformations C (cid:96) , P , and T .Here (cid:96) stands for ˆ (cid:96) , X (cid:96) stands for ˆ X · ˆ (cid:96) = − X µ (cid:96) µ , and so on with scalar normalization factors removed. T -odd scattering observables from the imaginary parts of amplitudes generally exist without violatingfundamental T symmetry. See the text for more explanation. Notice that every term in the lepton density matrix (Eq. 7) is automatically symmetric under C (cid:96) P . This is a kinematic fact of the lepton pair probe which does not originate in the StandardModel. As a result the C (cid:96) P transformations of the angular distribution depend on the coupling tothe unknown system. If overall CP symmetry exists the target density matrix will have CP oddterms where C (cid:96) P odd terms are found. In the Standard Model these cos θ and sin θ cos φ termscorrespond to charge asymmetries of leptons correlated with charge asymmetries of the system,namely the beam quark and anti-quark distributions.While weak CP violation is a mainstream topic, P and CP symmetry of the strong interactionsat high energies has not been tested [45]. The gauge sector of QCD is kinematically CP symmetric,because the non-Abelian tr ( (cid:126) E · (cid:126) B ) term is a pure divergence. . Higher order terms in a gauge-covariant derivative expansion are expected to exist, and can violate CP symmetry [45].However, measuring violation of CP or fundamental T symmetry in scattering experiments isinvariably frustrated by the experimental impossibility of preparing time-reversed counterparts.Some ingenuity is needed to devise a signal. It appears that any signal will involve four indepen-dent 4-momenta p J and a quantity of the form Ω = (cid:101) αβλσ p α p β p λ p σ . For example a term going like (cid:96) · Y ∼ (cid:101) αβλσ (cid:96) α Q β P A λ P B σ might possibly originate in fundamental T symmetry violation, and bemistaken for perturbative loop effects. A more creative road to finding CP violation involves twopairs with sum and difference vectors Q , (cid:96) ; Q (cid:48) , (cid:96) (cid:48) , and the scalar (cid:101) αβλσ (cid:96) α Q β Q (cid:48) λ (cid:96) (cid:48) σ , which is evenunder C and odd under P . The pairs need not be leptons (although “double Drell Yan” has longbeen discussed) but might be (say) µ + µ − π + π − . It would be interesting to explore further what atomographic approach to such observables might uncover. Non-perturbative strong CP violation in QCD by a surface term has been proposed. Tests have been dominated by theneutron dipole moment, while calculations of non-perturbative effects are problematic.
3. Density Matrix Invariants
We mentioned that scattering planes, trig functions, boosts and rotations could be avoided, andthe examples show how. Once a frame convention is defined the lepton “coordinates” ( X J · (cid:96) J , Y J · (cid:96) J , Z J · (cid:96) J ) are actually Lorentz scalars. However they depend on the convention for XYZ , whichis arbitrary. At least four different conventions compete for attention. Moreover, once a frame ischosen, at least two naming schemes (the “ A k ” and “ λ k ” schemes) exist to describe the angulardistribution in terms of trigonometric polynomials..Well-constructed invariants can reduce the confusion associated with convention-dependentquantities [44, 46–48]. Since (cid:126) S transforms like a vector its magnitude-squared (cid:126) S is rotationallyinvariant. The spin-1 part of ρ ( X ) does not mix with the real symmetric part under rotations. Sinceit is traceless, the real symmetric (spin-2) part has two independent eigenvalues, which are rota-tionally invariant. Finally the dot-products of three eigenvectors ˆ e J of the spin-2 part with (cid:126) S arerotationally invariant. Then ( ˆ e j · (cid:126) S ) are three invariants not depending on the sign of eigenvectors.That suggests six possible invariants, but ∑ j ( ˆ e j · (cid:126) S ) = (cid:126) S makes the (cid:126) S invariants dependent, leav-ing five independent rotational invariants. That is consistent with counting 8 real parameters in a3 × • The degree of polarization d is a standard measure of the deviation from the unpolarized case.It comes from the sum of the squares of the eigenvalues of ρ minus 1/3, normalized to themaximum possible: d = (cid:113) ( tr ( ρ X ) − ) /2,where 0 ≤ d ≤ d = d = • The entanglement entropy S is the quantum mechanical measure of order. The formula is S = − tr ( ρ X log ( ρ X )) In terms of eigenvalues ρ α , S = − ∑ α ρ α log ( ρ α ) . When ρ → N × N / N the system is unpo- Work by Faccioli and collaborators [49, 50] attempted to construct invariants by inspecting the transformation propertiesof ratios of sums of angular distribution coefficients upon making rotation about the conventional Y axis. The methodcannot identify a true invariant unless Y happens to be an eigenvector of the matrix. By the same method the group alsoidentified (cid:126) S as a “parity violating invariant,” while (cid:126) S is actually even under parity. Parity violation is not required tomeasure (cid:126) S with polarized beams. b (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) a FIG. 1:
Contours of constant entropy S of the lepton density matrix ρ ( (cid:96) ) (Eq. 7) in the plane of parameters ( a , b ) . Contours are separated by 1/10 unit with S = b = sin θ W . Annihilation with on-shell helicityconservation is indicated by the vertical dashed line a = larized, and S = log ( N ) . That is the maximum possible entropy, and minimum possibleinformation. When S = e S as the “effective dimension” of the system. For example theeigenvalues ( + b , 1/2 − b , 0 ) occur in the density matrix of on-shell fermion annihilationwith helicity conservation. One zero-eigenvalue describes an elliptical disk-shaped object.The entropy ranges from S =
0, ( e S = b = S = log ( ) , ( e S = e S → ρ ( (cid:96) ) (Eq. 7) in the plane of parameters6 FIG. 2:
Boundary of the positivity region of a density matrix depending on three parameters a , b , c describedin the text. The two-dimensional region cut by the plane c = ( a , b ) . The matrix eigenvalues are ( − a /3, 1/3 + a /3 − b , 1/3 + a /3 + b ) . The trian-gular boundaries are the positivity bounds on these parameters, outside of which the entropyhas an imaginary part. The corners of the triangle are pure states. The left corner representsa purely longitudinal polarization, ρ L = | L >< L | where | L > = (
0, 0, 1 ) in a coordinate sys-tem where ˆ (cid:96) = ˆ z . The two right corners are purely circular polarizations, ρ ± = | (cid:101) ± >< (cid:101) ± | ,where in the same coordinates (cid:101) ± = ( ± ı , 0 ) / √
2. The interior lines a = ± b , b = a and b are actually unrestricted in all directions, so long as they lie within thebounding curves.The Standard Model leptons from lowest order s -channel Z production have a = b = sin θ W , which is shown in Fig. 1 as a dot. The edge a = − sin θ W , 1/2 + sin θ W . The a , b parameters of It can be shown that the eigenvalues λ k = + ( d /3 ) cos ( θ k ) , where d is the degree of polarization and θ k = cos − ( det (( ρ ( X ) − × ) / d ) /2 + π k /3 ) . b = sin θ W just touches the b = − a line, which happens at sin θ W = θ W one cannot rule out a deeper connection.It is tempting but incorrect to assume the bounds discussed would apply to the same termsof a more general density matrix. For example, add − c ˆ n i ˆ n j to the expression in Eq. 7, whereˆ n · ˆ (cid:96) = a , b , c isshown in Figure 2, which also shows the plane c = c = ± ( a , b ) parameters shrinks to single points.The matrix for ρ ( X ) computed earlier is an example where all terms in any standard con-vention happen to occur. By inspection this system (mostly quark-antiquark annihilation) issuperficially much like the lepton one. The entropy of is 0.68 and e S = (cid:126) S , and its magnitude. IV. DISCUSSION
The quantum tomography procedure offers at least seven significant advantages over standardmethods of analyzing the angular correlations of inclusive reactions: • Simplicity and Efficiency.
Tomography exploits a structured order of analysis. By construction,unobservable elements never appear. • Covariance.
Physical quantities are expressed covariantly every step of the way. That is notalways the case with quantities like angular distributions. • Complete polarization information.
The unknown density matrix ρ ( X ) contains all possible in-formation, ready for classification under symmetry groups. • Model-independence.
No theoretical planning, nor processing, nor assumptions are made aboutthe unknown state. The process of defining general structure functions has been completelybypassed. It is not even necessary to assume anything about the spin of s - or t − channelintermediates. The observable target structures is always a mirror of the probe structure. The“mirror trick” is universal as described in Section II D. • Manifest positivity.
A pattern of misconceptions in the literature misidentifies positivity as be-ing equivalent to positive cross sections. It is not difficult to fit data to an angular distributionand violate positivity. In fact, an angular distribution expressed in terms of expansion coefficientsactually lacks the quantum mechanical information to enforce positivity. • Convex optimization.
The positive character of the density matrix leads to convex optimiza-tion procedures to fit experimental data. This provides a powerful analysis tool that ensuresconvergence.. • Frame independence.
Once the unknown density matrix has been reconstructed, rotationallyinvariant quantities can be made by straightforward methods. This is illustrated in SectionIII A 3, which includes a discussion of the entanglement entropy.Quantum tomography has already yielded significant results. Our tomographic analysis [44] ofa recent ATLAS study of Drell-Yan lepton pairs with invariant mass near the Z pole [11] discoveredsurprising features in the density matrix eigenvalues and entanglement entropy. By way of adver-tising, we have also gained insight into the mysterious Lam-Tung relation [51], including why itholds at NLO but fails at NNLO. These topics will be presented in separate papers. ACKNOWLEDGMENTS
The authors thank the organizers of the INT17-65W workshop ”Probing QCD in Photon-NucleusInteractions at RHIC and LHC: the Path to EIC” for the opportunity to present this work. We alsothank workshop participants for useful comments.
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V. APPENDIXA. Cross section in terms of density matrices
The density matrix approach stands on its own as an efficient tool kof quantum mechanics. How-ever we see the need to relate it to more traditional scattering formalism.Consider the cross section for the inclusive production of two final-state particles of spin s , s (cid:48) from particle intermediates: d σ ∼ ∑ s , s (cid:48) | ∑ J M ( χ J → f s , s (cid:48) ) | · d Π LIPS ,where ∑ J M ( χ J → f ) = ∑ J < f | T | χ J > δ ( ∑ p f − ∑ p i ) , T is a transfer matrix, and d Π LIPS is theLorentz-invariant phase space.Then, d σ ∼ ∑ s , s (cid:48) [( ∑ J < f s , s (cid:48) | T | χ J > ) · ( ∑ K < χ K | T † | f s , s (cid:48) > )] · d Π LIPS , (18) = tr [( ∑ s , s (cid:48) T † | f s , s (cid:48) >< f s , s (cid:48) | T ) · ( ∑ J , K | χ J >< χ K | )] · d Π LIPS ,where ∑ J , K | χ J >< χ K | accounts for any interference between intermediate states. We identify thequantity in the first set of parentheses on the last line of Eq. 18 with the probe density matrix ρ probe and the quantity in the second set of parentheses with the density matrix for the unknown particleintermediates, ρ X . The T , T † in ρ probe ensure the overlap of ρ X with ρ probe , inside the trace, is takenat ’equal times’.Rewriting d σ in terms of the density matrices, we find, d σ ∼ tr ( ρ probe · ρ X ) · d Π LIPS ,1where d Π LIPS ∼ d Ω · d q and q is the sum of the final-state pair momentum. Then, for given pairmomentum q , the angular distribution is, d σ d Ω ( θ , φ | q ) ∼ tr ( ρ probe · ρ X ) ,where the proportionality suppresses an overall normalization. The conditional probability givenq captures the event-by-event character of angular correlations. The explicit q dependence mightsuggest we assumed an s -channel boson intermediate state, but we have not.If we know d σ / d Ω and ρ probe for a given event, we can reconstruct ρ X for that event. This is theessence of the quantum tomography procedure. The probe is what is known, and it determineswhat can be discovered. B. Probe matrix
Figure 3 shows the diagram for the simplest lepton-pair probe. It is completely non-specificabout the process creating a lepton with momentum k and Dirac density matrix polarization αα (cid:48) ,and an anti-lepton with momentum k and polarization ββ . (cid:48) From the Feynman rules ρ ( lep ) ββ (cid:48) αα (cid:48) ∼ ( k / ) αα (cid:48) ( k / ) ββ (cid:48) . (19)The symbol ∼ indicates the high energy limit and ignoring a trivial overall normalization. FIG. 3:
The lepton pair density matrix ρ ααββ (cid:48) ( lep ) , in black, coupled to the colliding system density matrix ρ ( X ) . The matrix labels on legs are diagonal in momenta k , k . Off-diagonal polarization (Dirac) indices areexplicitly shown. The Feynman rules are the same as for ordinary diagrams. Continuing, ρ ( X ) is something of vast complexity, which can only couple to ρ ( lep ) via the indicesshown. The Dirac structure of ρ ( X ) can be expanded over several complete sets. However therelevant (observable) part of ρ ( X ) ββ (cid:48) αα (cid:48) will be its projection onto the subspace coupled to this particular probe , just as the general analysis prescribed. It is ideal to classify only what will be observed. Thatsector is predetermined by the very limited Dirac structure of the probe. Thus d σ ∼ k µ k ν ρ µν ( X ) ,where ρ µν ( X ) = tr (cid:0) ( γ µ ⊗ γ ν ) ρ ( X ) (cid:1) .With this probe massless fermion pairs produce no combinations of k µ k ν or k µ k ν or anything else.The operations show how the tr symbol comes to be used repeatedly with different meanings im-plied by the context.We now use Hermiticity, which makes two predictions: ρ ( lep ) = ρ µν ( lep ) γ µ γ µ ; ρ µν ( lep ) = ρ νµ ( lep ) ∗ ,plus the same relation for ρ ( X ) . All k µ k ν factors must be strictly bilinear, and occur in a real sym-metric plus ı × antisymmetric combinations. The most general possibility is ρ µν ( lep ) = α k · k η µν + β ( k µ k ν + k ν k µ ) + ı γ ( k µ k ν − k ν k µ ) + ı δ(cid:101) µναβ k α k β , (20)where η µν is the Minkowski metric. By algebra ρ µν ( lep ) = α q η µν + β q µ q ν − β (cid:96) µ (cid:96) ν − ı γ ( q µ (cid:96) ν − q ν (cid:96) µ ) − ı δ (cid:101) µναβ q α (cid:96) β . (21)Event by event there exists a preferred, oriented pair rest frame where q µ = ( q , (cid:126) ) and (cid:96) µ =( q ˆ (cid:96) ) . In that frame Eq. 21 predicts a normalized 3 × ρ jk ( lep ) = δ jk + a J jkp (cid:96) p − b U jk ; (22) U jk ( ˆ (cid:96) ) = ˆ (cid:96) j ˆ (cid:96) k − δ jk a and b are real, and J p are the spin-1 rotation generators in Cartesian coordinates: J jkp = − ı (cid:101) pjk .Equation 22 has been decomposed into tensors transforming under rotations like spin-0, spin-1 andspin-2. The expansion above is kinematic and not a consequence of any special theory.The approach has the virtue of maintaining strict control of how outputs depend on assumptions.We have made few assumptions, yet we have a result, which is that ρ jk ( lep ) only depends on two scalar a , b . The only scalar available from ρ ( lep ; (cid:96) , q ) is q , hence a = a ( q ) , b = b ( q ) . Theenormous body of field theory and the Standard Model only predicts only two parameters of Eq.322. If and when the lepton pair originates from an intermediate boson with vertex c V γ µ + c A γ µ γ ,then a = c A c V ; b = q , namely vertex form factors, hasemerged on its own. Notice that in the rest frame oriented naturally along the lepton momen-tum ρ ( lep ) is not diagonal. The diagonal elements are interpretable as probabilities, even classical probabilities. The off-diagonal elements convey the information about entanglement. C. Positivity
There is a positivity issue in fitting angular distribution data. Represent ρ = MM † , and then ρ >
0. Any M = HU , where H = H † and UU † =
1. We can make M self-adjoint since U cancels out. To parameterize N × N matrices M use SU ( N ) representations G a , normalized to G a G b = ( ) δ ab . For N = M = m × / √ + √ m a G a . (24)Compute MM † = m /3 + (cid:114) m m a G a + m a m b G a G b .The symmetric product is G a G b + G b G a = δ ab /3 + f abg G g .Check the trace of both sides for the normalization of δ ab . Everything else must be traceless andspanned by G g . Then MM † = ( m + ∑ a m a ) + (cid:114) m m g G g + f abg m a m b G g .The normalization tr ( ρ ) = ∑ µ = m µ m µ =
1. This requires each 0 < m µ <
1, while it ismore restrictive.There is a degeneracy issue in the nonlinear relation of m µ to a straight expansion ρ = + c g G g , c g = (cid:114) √ − m · m m g + f abg m a m b (25)4Notice [ ρ , M ] = [ M , M ] = M and ρ have the same eigenvectors. The eigenvalues of ρ are those of M , squared. For N eigenvalues of ρ there are 2 N possible M ’s. If ρ is positive definite, however, there is only one M with strictly positive diagonal entries. In that sense, the M satisfying ρ = MM † can be said to beunique.The positivity problem is often solved with the Cholesky decomposition: ρ = LL † , where L is a lower-triangular matrix with real entries on the diagonal. L is related to M by a similaritytransform. There are N + N ( N − ) /2 = N free real parameters in a lower-triangular matrixwith real diagonals, which is just right for Hermiticity. As before, the condition tr ( ρ ) = ∑ m µ m µ =
1. The Cholesky decomposition is unique, in the sense above, when ρ is positive defi-nite. D. Collected conventions
As a consequence of consistent definitions, our ρ M and Y M transform under rotations like realrepresentations of spin-2. Other conventions have long existed. Table II shows the relations ofthe ρ M parameters compared to the ad-hoc conventions known as A k and λ k . The ρ M are self-explanatory because they correspond to orthonormal harmonics and transform like spin-2 repre-sentations. The arbitrary normalizations and conventions relating different basis functions havebeen a barrier to interpretation, needlessly complicated transformations between angular frameconventions.16 π /3 ( √ A /2 − √ ) A A /2 A A π / ( + λ θ ) − λ θ / ( √ + √ λ θ ) λ θφ / ( + λ θ ) λ φ / ( + λ θ ) λ ⊥ φ / ( + λ θ ) λ ⊥ θφ / ( + λ θ ) c ρ ρ ρ ρ ρ ( π ) √ − √ ( θ ) sin ( θ ) cos ( φ ) sin ( θ ) cos ( φ ) sin ( θ ) sin ( φ ) sin ( θ ) sin ( φ ) TABLE II: