aa r X i v : . [ h e p - ph ] D ec Observables of QCD Di ff raction Mikael Mieskolainen and Risto Orava Department of Physics, University of Helsinki and Helsinki Institute of Physics, P.O. Box 64, FI-00014, Finland CERN, CH-1211 Geneva 23, Switzerland a) Corresponding author: [email protected] b) [email protected] Abstract.
A new combinatorial vector space measurement model is introduced for soft QCD di ff raction. The model independentmathematical construction resolves experimental complications; the theoretical framework of the approach includes the Good-Walker view of di ff raction, Regge phenomenology together with AGK cutting rules and random fluctuations. INTRODUCTION
Soft di ff raction bases theoretically on (soft) Pomeron exchange, the vacuum singularity of Regge theory. In QCD,this is described as a non-local (long-wavelength) gluonic color singlet ladder based object. However, a completetheoretical description of the multi-Pomeron exchanges and interactions, the exact nature of Reggeon exchanges andrandom fluctuations in hadronization process are still lacking. All these processes can generate large rapidity gaps(LRGs) of several units. An alternative approach is to treat high energy hadronic di ff raction as a coherent process,where the relativistic wave function, and its components, undergo unitary scatterings and absorptions. This well knownGood-Walker picture [1] is usually implemented by using multichannel eikonal models to account for the complicatedproton structure and its coherent fluctuations [2], albeit in an integrated way.Experimentally, soft di ff raction is traditionally equated with registering large rapidity gap events. In practice,a number of approximations are required for defining and measuring the rapidity gaps as pseudorapidity intervalsvoid of particles. There are several limitations in defining the rapidity gaps as experimental observables. First of all,electrically neutral particles often remain unaccounted for, even if their presence can be partially inferred using themeasured secondaries. Low mass di ff ractive systems require very forward instrumentation which, in general, coversonly part of the small angle scattering at the LHC. Theoretically and experimentally, the low mass resonance region ofsingle and double di ff raction, and high mass asymptotic are poorly understood. Counting events with widely varyingdefinitions of rapidity gaps and relatively large E ⊥ / p ⊥ thresholds cannot give a complete view on the subject of QCDdi ff raction.In the following, binary vector spaces over the number field F = { , } are defined for the chosen observables,then a probabilistic extraction of di ff ractive cross sections and, finally, new combinatorial approaches are introducedfor probing issues like the famous AGK cutting rules [3]. In all this, the experimental limitations together with theabove theoretical motivations are used as a guidance. DEFINING THE OBSERVABLES
Now instead of merely counting rapidity gaps, the partial cross sections σ ppinel ≡ σ + σ + σ + . . . + σ n are con-sidered, where each σ k corresponds to one particular final state ”topology class” over a finite d interval discretizedpseudorapidity axis, integrated over the transverse ϕ -plane. These span k = , . . . , d − = n non-zero binary vectorsin F d , where each vector component Bernoulli random variable X i ∈ { , } , 1 ≤ i ≤ d , encodes whether at least onefinal state in the given rapidity (detector) slice is observed. Simplified, these components correspond to pseudorapid-ity intervals of [ η min i , η max i ] which represent geometric projection boundaries. Experimentally, these intervals mayverlap or remain distinct from each others. In the limit d → ∞ the problem turns into track counting, and at d = ff erent process cross sections is lost. Depending on the exact definition, the number of distinct”topology classes” is not necessarily n , the number of binary combinations. The corresponding abstract final statevector space F d is illustrated in Figure 1. FIGURE 1.
Binary vector space as a d -hypercube graph or lattice. Each vertex represents one vector in F d , here d =
6. Thesevertices and weights associated with them are interpreted directly as di ff erent final state configurations, corresponding to the partialcross sections σ k . Events with varying rapidity gap combinations are contained in this space. In practice, due to finite statisticsand finite discretization, there is both Poissonian and discretization uncertainty influence the exact gap sizes. It ispossible to use also a di ff erent number field, such as the real valued vector space R d as in [4], where it was shown thatlarge rapidity gaps are obtained as a particular limit of the multivariate space. In principle, this allows event-by-eventutilization of p ⊥ or multiplicity degrees of freedom. Real valued distributions can be constructed also ”after” the binarysubspace projection, as a hybrid approach. The main benefit in using the binary vector spaces, is to allow concisealgebraic representations of the measurement itself, and to factorize the model parameter extractions in addition.Visible or fiducial partial cross sections σ ( vis ) k are defined in terms of ( η, p ⊥ ) acceptance and e ffi ciency functionsof charged and (or) neutral particles for each i -th discretized pseudorapidity interval. Crucial experimental issue is theseparation of e ffi ciency corrections on visible part versus the pure extrapolation to outside of the fiducial acceptanceregion. This is di ffi cult in the forward domain, due to limited granularity of the calorimetry, tracking and intensefluxes of secondaries from the interactions in the beam pipe and surrounding material. Low p ⊥ -thresholds and high- | η | coverage are of utmost importance. Matrix unfolding procedure is necessary in order to turn the visible detector levelpartial cross sections to the particle level cross sections U : { σ ( vis ) k } 7→ { σ k } . Detector ine ffi ciencies tend to createartificial rapidity gaps and this is to be taken into account in the unfolding process.When defining the measurement in terms of the invariant mass of the di ff ractive system M X (or ξ = − p ′ z / p z as ξ ≃ M X / s ), no direct geometrical fiducial definition exists. Unless the pseudorapidities and true rapidities areassumed to be approximately equal, η ≃ y , and the average kinematical relations are used for rapidity gaps in singledi ff raction: h ∆ y i S D ≃ − ln (cid:16) M X / s (cid:17) and in double di ff raction: h ∆ y i DD ≃ − ln (cid:16) M X M Y / ( m p s ) (cid:17) , h y i ≃ ln (cid:16) M X / M Y (cid:17) .In practise, Monte Carlo chain definitions of di ff ractive mass acceptances are used, based on varying hadronizationmodel assumptions of the di ff ractive systems. This includes, for example, non-trivial final state p ⊥ behaviour of softprocesses, which is not understood from the first principles. The invariant masses of the SD or DD systems are notdirectly measured at the LHC, even if the leading proton longitudinal momentum is known, thereby allowing the useof 4-momentum conservation. Figure 2 shows the phase space of single and double di ff raction at the LHC. High massdouble di ff raction is seen to be kinematically constrained to be high- | t | process, while single di ff raction is not. EXTRACTING DIFFRACTIVE CROSS SECTIONS
The following extraction of di ff ractive cross sections is based on probabilistic inversion, density estimation or multi-dimensional fitting procedure. Concerning the earlier work by the authors, with real valued multivariate approaches,see [4, 5]. The term extraction is used here, because the measurement of inclusive di ff ractive cross sections is always X (GeV) R a p i d i t y √ s = 13 TeV h y -span i of M X h ∆ y i limit of √ s h ∆ y i ≃ M (GeV) √ − t ( G e V ) -2 Ω Ω √ s = 13 TeV √ − t max √ − t min FIGURE 2.
On left: The high mass ”coherence condition” ξ < . ∼ x F > . ∼ h ∆ y i > → t -channel with variable invariant masses for 2 outgoing legs. Ω is for SD and Ω for DD with M X = M Y . The general case of DD ( M X , M Y ) is between Ω and Ω . model dependent at the LHC. The chosen framework aims at making the model dependence explicit and as trans-parent as possible. Each scattering process class C j , such as single or double di ff raction, is described in terms of a d -dimensional probability density or likelihood function p ( x | θ j , λ ) with x ∈ F d . These densities are from MC sim-ulations or possibly from simple parametrizations. That is, they give us likelihood of observing a binary vector finalstate x originating from the class j . In addition to likelihood functions, a priori probability distributions p ( θ j , λ | α )are constructed for physics model parameters θ j and detector simulation nuisance parameters λ , which can both bevectors. Variable α denotes a generic hyperparameter, a parameter of the parameter distribution. Many of these stepsare often implicitly included in traditional large rapidity gap event analyses; here these are explicitly accounted for.In a fully Bayesian treatment, the posteriori probability distributions p ( θ j , λ | x , α ) ∝ p ( x | θ j , λ ) p ( θ j , λ | α ) areobtained for each process class by first posing prior distributions for each process and their parameters, and thenproceeding with Monte Carlo sampling of the parameter space. This can be a computationally heavy process de-pending on the number of free parameters, distribution shapes and correlations. Point estimates and credibility in-tervals are then obtained directly. For the process cross sections or fractions, a frequentist fitting via MaximumMarginal Likelihood is especially straightforward when the iterative Expectation-Maximization (EM) algorithm [6]is used. This approach maximizes the denominator or ”evidence” in the Bayes formula with respect to the fractions:arg max { f j } Q Ni = P | C | j = p ( x i | C j ) f j , over a sample of N events. After iterating, integrated process fractions are obtainedas ˆ f j ≡ h p ( C j | x ) i x with P j ˆ f j =
1. These can be scaled to physical cross sections with a van der Meer scan.Multidimensional fitting allows also estimation of parameters such as the e ff ective Pomeron intercept α P (0) = + ∆ P . The intercept is an interesting model parameter, not only because it controls asymptotic energy behaviour ofcross sections, but also due to its controlling role of the di ff erential mass distribution in the triple-Pomeron ( PPP )high mass limit and at t → d σ S D / dM X ∝ / ( M X ) +∆ P . To emphasize, the purity or background corrections areautomatically taken into account here, because all major inelastic processes are simultaneously fitted. Since no explicitlarge rapidity gaps are required, di ff ractive cross sections can be extracted at the actual high mass limit. ALGEBRAIC REPRESENTATIONS
All the di ff erent r -subspaces, 0 ≤ r ≤ d , contained in our binary vector space of final states are encapsulated inthe Grassmannian manifold Gr ( r , d , F ), in the object describing all possible r -dimensional subspaces in F d . This is avery rich object of algebraic geometry with variety of applications from coding theory to mathematical physics. Themanifold is defined as Gr ( r , d ) = { r × d matrices with rank r }\ row operations, with dim Gr ( r , d ) = r ( d − r ). Usingthe Grasmannian subspaces allows to probe the Regge (vertex) factorization of type d σ DD dM X dM Y dt ∼ d σ SD dM X dt d σ SD dM Y dt / d σ EL dt bycomparing specific partial cross sections σ k with a simple algorithm. This is experimentally feasible to do at the LHC,iven the lacking di ff erential measurement capabilities of M X and t . By comparing the di ff erent subspace combinationrates, information about multi-Regge type of factorization is gained.In practise, an algebraic representation of the binary data is needed. The most direct representation amountsto just counting the relative rates of 2 d di ff erent (or 2 d − p . However, the components of ordinary moments m k and the components of central moments δ k ,are also easily defined using the Kronecker (tensor) products as [7] p = * −
10 1 ! ⊗ d X d ! ⊗ X d − ! ⊗ · · · ⊗ X !+ , (1) m k = DQ di = X k i i E = * X d ! ⊗ X d − ! ⊗ · · · ⊗ X !+ k , (2) δ k = DQ di = ( X i − h X i i ) k i E = * X d − h X d i ! ⊗ X d − − h X d − i ! ⊗ · · · ⊗ X − h X i !+ k , (3)where k = + P di = k i i − (little endian binary expansion), 1 ≤ k ≤ d and k i ∈ F are used. The central momentsdescribe the correlations ( d − d −
1) between any 2 or more subspaces (rapidity intervals). X i are the correspondingBernoulli random variables.The probability distributions of rapidity gaps ∆ y for simplified Pomeron and Reggeon exchanges and randomfluctuations are expected to be approximately [8] P P ( ∆ y ) = c P exp( ∆ y ( α P − , α P ∼ .
08 (soft) . . . . P R ( ∆ y ) = c R exp( ∆ y (2 α R − α P − , α R ∼ / P F ( ∆ y ) = ℓ F exp( − ∆ y /ℓ F ) , (6)which give the short range correlation length for a Reggeon with ℓ R = − / (2 α R − α P − ∼
1, the long range cor-relation length for a Pomeron with ℓ P = / ( α P − ∼
10 and for the fluctuations at Tevatron ℓ F ∼ . − .
75 [8].These examples motivate the present combinatorial construction which goes beyond the multidimensional fitting andextraction of cross sections presented earlier, and is now compatible with discussion about multigaps, gap destruc-tion and rescattering and short / long range y -correlations. The algebraic representation chosen here is motivated bysimple arguments that di ff ractive dissociation should represent the statistical dispersion h F i − h F i in the absorptionprobabilities of the di ff ractive eigenstates such as in [2], by the Good-Walker view.The approach intrinsically includes the combinatorial AGK rules [3], that is, the density of particles over rapidityintervals with cut Pomerons. Basis for an experimental algorithm could be obtained, for example, by histogrammingthe multiplicity or charge information over n di ff erent combinations. As an example of the AGK cutting rules: thetotal cross section for exchange of µ Pomerons, σ tot µ , partial cross section σ ( ν ) µ of a final state with a number of ν cutPomerons and their ratio as given in [9] σ ( ν ) µ σ tot µ = ( − µ − ν µ ! ν !( µ − ν )! (2 µ − − δ ν ) . (7)Substituting for example µ = v = , ,
2, the usual alternating AGK factors of 1, − µ one needs an explicit model, such as eikonal probabilities for the number of Pomerons being exchanged. COMBINATORIAL INVERSION
The novel approach presented here also meets an interesting combinatorial challenge, which is the statistical inversionof ”pileup” final states. These can be simultanenous proton-proton interactions at the LHC, but in principle any Poissonprocess which superimposes independent interactions, such as in the classic Miettinen-Pumplin model of wee partons[10]. The ”direct model” equation is a convolution between Poisson and multinomial distributions as y i = e − µ − e − µ ∞ X k = µ k k ! X Ω ik k ! Q nj = x j ! n Y j = p x j j , (8)here y i is the pileup diluted or enhanced probability of observing i -th binary final state, i = , . . . , d − = n and µ isthe Poisson mean. The multinomial term in brackets and its values of x j ∈ N are evaluated over all valid combinationsgenerating the i -th final state c i ∈ F d at Poisson order k from the set of n -tuples Ω ik . Those which are allowed bypartially ordered set (poset) combinatorics Ω ik = { ( x , . . . , x j , . . . , x n ) | _ j x j c j = c i and X j x j = k } , (9)where the Boolean W operator takes care of ”summing” the binary vectors c j of multiplicity x j and thus evaluatingthe pileup compositions. µ -3 -2 -1 K S -4 -3 -2 -1 d = 8 FIGURE 3.
Inversion performance in solving p under Kolmogorov-Smirnov error (KS) as a function of Poisson µ . Dashed lineswithout inversion. N = , (black), 10 , (blue), 10 (red) events. Performance is fundamentally limited by √ N statistics andsaturation at high µ . The basic idea is that the probabilities y are measured, and it is p to be solved by inverting Equation 8. Analternating sign solution similar to AGK rules can be obtained using the so-called principle of inclusion-exclusion(PIE) which is the M¨obius inversion for subsets in the combinatorial incidence algebra context [11], also utilized forexample in mathematical physics in [12]. Exact details of the present combinatorial inversion will be discussed andpresented elsewhere. Finally, a performance demonstration of the inversion algorithm is shown in Figure 3. ACKNOWLEDGEMENTS
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