Binary vector spaces and Diraction

For the algebraic construction of Bernoulli observables, we need a vector space F^N2

over the nite Galois eld GF(2) with addition as a component wise Boolean OR (∨) and multiplication as a component wise Boolean AND (∧). As the inner product we can use ha|bi = W^N_i=1ai ∧ b_i ∈ F2, where a, b ∈ F^N2 . The vector space is spanned by N standard basis vectors as span(F^N₂ ) = {e_k}^N_k=1 and the number of elements in an N-dimensional binary vector space is |F^N₂ | = 2^N, illustrated in Figure 6.2. The total number of subspaces is given by a sum over q-binomial coecients, given in Appendix C.1.

The most useful property concerning the physics involved here is that the sub-spaces form an orthomodular poset (partially ordered set). Another aspect is that Boolean operators which are diagonal in one basis are diagonal in all bases. Projec-tion operators are always diagonal, which is semi-intuitive if one thinks about the N-dimensional unit-hypercube structure. In [158] it was pointed out that the mul-tivariate Bernoulli can be considered as a certain generalization of the Ising model.

However, here we apply these to high energy physics.

Bernoulli random variables

Every component observable of vectors in F^N₂ is treated as a Bernoulli random vari-able Bi with 1 ≤ i ≤ N. The random variables Bi follow the distribution

Ber(Bi|m_i) = P_B(B_i= b|m_i) = m^b_i(1 − m_i)^1−b, (6.1) where b ∈ {0, 1}. The mean is mi and variance Var[Bi] = m_i(1 − m_i). Together, these can be encapsulated with a multivariate Bernoulli distribution P (B1= b₁, B₂= b2, . . . , BN = bN|θ). The full distribution is either described directly with 2^N − 1 parameters (−1 from normalization) associated with every 2^N elements of the binary vector space, in what we may call the fundamental or natural representation, or with parameters describing the expectation values and multipoint correlations between dierent Bernoulli variables, what we may call the correlation representations. We use the notation and representations constructed by Teugels [159].

Example N = 3: A centralized correlation representation is

mi = hBii, i = 1, 2, 3 (6.2)

θij =Cov[Bi, Bj] = h(Bi− m_i)(Bj − m_j)i (6.3)

= hB_iB_ji − hB_iihB_ji, (i, j) ∈ {(1, 2), (1, 3), (2, 3)} (6.4) θ₁₂₃ = h(B₁− m₁)(B₂− m₂)(B₃− m₃)i, (6.5)

where the moments or correlation functions are centralized, that is, the mean is subtracted. The structure is only slightly more complicated for larger values of N, and is described in the following chapters. One interesting property is that for the multivariate Bernoulli distribution component variables which are pairwise uncor-related Cov[Bi, Bj] = 0 are also always independent, that is, the density factorizes P (B_i, B_j) = P (B_i)P (B_j) [158]. This does not hold in general for multivariate prob-ability distributions.

Probabilities and cross sections

In the fundamental representation, a probability vector p encapsulates the probabil-ities of mutually exclusive combinations of observables in the binary vector space

p = (P_b0, P_b1, . . . , P_b

2N −1)^T, X

c

pc= 1 (6.6)

where indices go through all the elements of the binary vector space. The one-to-one correspondence is given uniquely by the binary expansion¹

c =

N

X

i=1

b_i2ⁱ⁻¹ ⇔ b^c= (b₁, b₂, . . . , b_N), (6.7) between the index 0 ≤ c ≤ 2^N − 1and the binary vector b^c∈ F^N2 .

Now turning into physical observables, the dierential cross section element is given by the usual

dσn⁰ = 1 F

1

S_n⁰ dΠn⁰|M_i→n⁰|², (6.8) where the matrix element squared |Mi→n⁰|² is the probability amplitude for the transition from the initial state to the n⁰-particle nal state, encapsulating all dy-namics. The Sn⁰ is a QFT combinatorial symmetry factor for identical nal states, F = 4p(p₁· p₂)²− m²₁m²₂ −→ 2s(high energy limit) is the incoming invariant Möller

ux and the standard Lorentz invariant phase space element is

dΠ_n⁰ = (2π)⁴

n⁰

Y

j=1

d³kj

(2π)³2k_j⁰. (6.9)

1Here we remark on the dierence between left and right ordered bit strings. We use the con-vention of most signicant `bit' (MSB) on right. In the most signicant `bit' on left representation, this expansion would be c = P^Ni=1bi2^d−i.

We now divide the ducial phase space seen by the detector into N elements.

Using the combinatorial incidence algebra notation, then the so-called 2^N partial cross sections can be expressed with

σ = 1 F

X

n⁰

1 S_n⁰

Z

dΠ_n⁰δ⁽⁴⁾ p₁+ p₂−X

n⁰

k_j

!

|M_2→n⁰|²1 −1

0 1

⊗ N

 1

A{{k_j}; Ξ_N}



⊗

 1

A{{k_j}; Ξ_{N −1}}



⊗ · · · ⊗

 1

A{{k_j}; Ξ₁}



. (6.10)

The expression above is a 2^N-vector, where the component σ0 denotes the null part outside the total ducial phase space. The acceptance (indicator) functions A : {kj} → {0, 1}encoding the Bernoulli trials are

A_i({k_j}; Ξ_i) =

(1, if ∃kj ∈ Ξ_i

0, otherwise. (6.11)

The acceptance function returns 1 if any of the nal state particles was inside the phase-space domain Ξi dened for Bi and returns 0, if none were inside the domain.

This function can be constructed with simple phase space cuts suitable for analytical calculations. In practice, they are most easily calculated with soft QCD Monte Carlo event generators, where arbitrary ducial phase space cuts are possible. We abstract out the detailed singularity and theory structure connections with the phase space cuts out at this point, they are theory specic, but certainly collinear and infrared singularities may be used to design a specic structure for the acceptance functions.

Experimental eciency and resolution functions work as a mixing map σc7→ σ_c⁰, which is a highly complicated stochastic non-linear operator modulating the pure acceptance indicator function. For the expectation values, it can be represented with a nite linear folding operator acting on the nal states partial cross section vector as F : σ → ˜σ, where σ denotes the partial cross sections at the ducial phase space particle level and ˜σ denotes the visible partial cross sections at the detector level. Both can have almost arbitrary denitions, depending on the problem. The hierarchy goes as follow

dO

P

n0

R

Ωn0

−−−−−−→

∀k σ−−−−−−→^F

−σ_E,+σL

σ,˜ (6.12)

where the rst map is the `theory' or `generator' and the second one is the `detec-tor', to illustrate. The zero-th component σ0 of σ denotes the invisible ducial cross section in the ducial phase space division (geometrically invisible), only accessible theoretically or via indirect measurements. The inverse operator F⁻¹ is the so-called unfolding process, which can be naively solved with a 2^N − 1matrix inversion, ob-tained via detector simulation. The σE denotes the integrated eciency loss, which is not the same as pure acceptance loss σ0. Integrated leakage from outside the du-cial to inside the visible is denoted with σL, which may happen due to e.g. material re-scattering  this can be a non-negligible eect with very forward detectors. We emphasize here that the unfolding is for eciency and smearing induced corrections, not to extrapolate the detector geometry.

As an example of the indicator functions one could consider a simple divi-sion of the (pseudo)rapidity into N intervals with suitable pt thresholds per in-terval. Experimentally, the intervals may overlap over pseudorapidity such that [η_i,min, η_i,max] ∪ [η_j,min, η_j,max] 6= {0}for a detector pair (ij), and thus making them trivially correlated. It is possible, in principle, via suitable non-overlapping ducial denition to take this into account in the unfolding procedure which de-correlates the data. Another option is simply to take care that no overlap happens physically. Also, the detectors may have gaps over η such that [ηi,min, η_i,max] ∪ [η_j,min, η_j,max] = {0}. However, it is worth pointing out that the formalism presented here does not assume statistical independence of any kind, either implicit or explicit.

Abramovski-Gribov-Kancheli cutting rules

We collect here some illustrating results of the AGK cutting rule calculus [25]. By no means this is a complete description, but rather of an introductory nature. These rules are somewhat similar to more well known Cutkosky-Landau cutting rules based on the S-matrix unitarity, the rules which treat scattering amplitude discontinuities.

The AGK rules were introduced in the Regge theory context, but they are actually

exible in terms of the underlying theory. What we need here is the reggeization, which happens in eld theories under Regge limit but also in string theory. The AGK calculus results here are collected from a stringy black hole context [160] and from QCD papers [161,162]. We assume the usual unitary S-matrix and T matrix related with

S = 1 + iT . (6.13)

Let us have the elastic scattering amplitude TAB(s, t) with a Sommereld-Watson integral transform representation in the complex angular momentum space as

T_AB(s, t) = Z dω

2iξ(ω)s^1+ωF (ω, t) with ξ(ω) = τ − e^−iπω

sin πω , (6.14) where F(ω, t) is the corresponding partial wave, ω = J − 1 ∈ C being the conjugate variable to s together with the signature factor ξ for the positive or negative signature τ = ±1. The positive signature case, such as Pomeron exchange, can be written more compactly as ξ(ω) = i + tan(^π₂ω). Now assume an exchange of n Reggeons, that is, exchange of Regge trajectories.

The n-cut discontinuity over ω of F(ω, t) can be written in terms of vertex func-tions V_n^a, V_n^b and integrating over transverse momentum phase space as [162]

DISC⁽ⁿ⁾ω [F (ω, t)] = 2πi for the soft Pomeron often used parametrization is α(t) ' 1.08+0.25t. The factor n!

comes from all possible exchange diagram planar and crossed non-planar orderings.

Now the amplitude for the n-cut exhange can be written using the Sommereld-Watson transform and the cut discontinuity as [161]

T_AB^n−cut(s, t) = Next we write down the Abramovski-Gribov-Kancheli sign alternating combina-torial factors [162]

Gⁿ_k =

(2ⁿ⁻¹(−1)ⁿ+ Π({β_j}), for k = 0 cut Pomerons,

2ⁿ⁻¹(−1)^n−k_(n−k)!k!^n! , for 0 < k ≤ n, (6.20)

n, k 0 1 2 3 4 5 6

1 0 1 0 0 0 0 0

2 1 -4 2 0 0 0 0

3 -3 12 -12 4 0 0 0

4 7 -32 48 -32 8 0 0

5 -15 80 -160 160 -80 16 0

6 31 -192 480 -640 480 -192 32

Table 6.1: AGK factors Gⁿ_k for the even signature (Pomeron) case with n exchanges and k of them cut.

which we tabulate for illustration in Table 6.1.

Let us now point some integer sequence properties of the AGK series, which might be illuminating in our hyperspace context. The series |Gⁿ_k=1| is the number of edges in an n-hypercube. The series |Gⁿ_k=2| is the number of diagonals in an n-hypercube of length √

2. The series |Gⁿ_k=3|is the number of 3-cycles in the halved n-cube graph. These suggest to us directly that in order to be (experimentally) sensitive to n Pomeron exchanges, the number of rapidity slices N should be at least n + 1. Now the use of AGK cutting factors is that they allow us to represent the total discontinuity over s-space scattering amplitude as a weighted sum over all k-cut Pomeron contributions [162]

Aⁿ(s, t) ≡DISCs[T_AB^n−cut(s, t)] =

n

X

k=0

aⁿ_k(s, t), where

aⁿ_k = 2πi Gⁿ_k

Z dΩ_n

n! s^1+PiβiV_n^a({k_j,X

i

β_i})V_n^b({k_j,X

i

β_i}). (6.21) The last step in our short discussion on this topic is the impact parameter b space picture and eikonalization. The eikonal opacity function in the impact param-eter space is obtained using the (inverse) Fourier transform of a single discontinuity integrated over ω [162]

Ω(s, b) ≡ 1 πi

Z d²q (2π)²e^−ib·q

Z

dωDISCω[F (ω, −q²)] s^ω (6.22) with the amplitude recovered in the s-space by (forward) Fourier transform of the n-times exponentiated amplitude, by assuming factorization of each exchange as

10 20 30 40 50 60 70 80 10

20

30

40

50

60

70

80

10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

80

Figure 6.1: A tensor product chain M^{⊗ 4} on the left and a tensor sum chain ¹₄M^{⊕ 4} on the right. Black to red to white denote values in [0,1].

V_n^a = [V1]ⁿ, giving

Aⁿ_k(s, t) = 4is (−1)^n−k k!(n − k)!

Z

d²be^ib·q[Ω(s, b)]ⁿ (6.23) with summation n ≥ k resulting in

A_k(s, t) = 4is Z

d²be^ib·qP (k, s, b). (6.24) Where the probability is a sum over the S-matrix elements [162]

P (k, s, b) ≡ X

n≥k

[S(s, b)S(s, b)^†]ⁿ_k = [Ω(s, b)]^k

k! e^−Ω(s,b), (6.25) which gives a Poisson distribution with hk(s, b)i = Var[k(s, b)] = Ω(s, b). That is, the number of k-cut Pomerons at given energy squared s and impact parameter b is Poisson distributed in this picture. Now, in typical Monte Carlo implementations, one needs also the parton densities to pick the longitudinal momentum fractions from vertex factors V₁^a,b for each exchanged Pomeron and at least triple-Pomeron interactions with diractive cuts, in addition (hard) QCD matrix elements, parton shower and fragmentation.

As an example, we shall refer to the description of QGSJet-II event generator for details [163], which implements advanced Reggeon calculus for soft and hard inter-actions. The scope of this paper is to provide new algebraic methods for measuring the observables. We use the Poissonian AGK distribution as our motivation for the later construction. We shall also note here that a left stochastic matrix devised by Ryskin and Bartels [161], describes the `non-diagonal' cut transitions necessary for rapidity gap congurations

M =





0 0 ¹₂ 0 1 0 1 0 ¹₂



, (6.26)

which has x = [1, −4, 2]^T as its eigenvector, the rst non-trivial row from Table 6.1. That is, for every rapidity interval, these ratios hold, and the transition matrix describes the stochastic probability of transition from dierent congurations per chain. In Figure 6.1 we illustrate the nature of this matrix under the Kronecker product A ⊗ B and the Kronecker sum dened as A ⊕ B = A ⊗ IA+ IB⊗ B, where I is an identity matrix. The product matrix belongs to the class of inclusion-exclusion matrices whereas the sum matrix resembles a hypercube adjacency (connectivity) matrix. Later we learn how these concepts appear in the incidence algebra.

Rapidity gaps as a spacing distribution

It is typical that `non-diractive events' have an exponentially decreasing maximum rapidity gap distribution ∼ exp(−`c∆y), where the correlation length `c∼ 1depends on the average multiplicity density. This can be derived assuming n independent nal state emissions over rapidity. Diractive events are those which have `anomalously' large gaps. The rapidity gap itself is a kinematic consequence given the excited (dissociative) proton system mass is small enough and the initial state boost large enough, however, the M² distribution itself is driven dynamically. The standard triple Pomeron limit distributions obtained via generalized multibody unitarity to-gether with Regge asymptotics give dσ/dM² ∼ 1/(M²)^αP(0), where the αP(0)is the eective Pomeron intercept αP(0) ≡ 1 + ∆_P ∼ 1.08. Then by standard kinematic² change of a variable φ⁻¹(∆y) ≡ s exp(−∆y) = M², we obtain the single diractive

2Beam rapidity is y^±= ± ln(√

s/mp)and the diractive system spans yspan= ln(M²/m²_p).

rapidity gap distribution Above, we neglected Mandelstam t-dependence of the triple Regge expressions by taking t → 0, but this is not relevant for this discussion. Now Equation 6.27 is a slowly rising exponential, instead of decreasing such as for the random non-diractive processes. In the case of a double diractive process, the equiva-lent derivation goes via the Jacobian determinant det J(∆y, y0) = sm²_pe^−∆y, from φ⁻¹_i (∆y, y₀) = (sm²_pe^−∆y±2y0)^1/2 = M_i² with i = 1, 2 for the two systems and

where the dependence on y0vanishes and the functional dependence is the same as in the single diractive case. The boundary conditions are dierent, because the both edges of the gap are moving in the case of double diractive scattering. In general, the gap may be taken with respect to the detector geometric boundary or with re-spect to the minimum or maximum of the rapidity interval (type I), or we can count distances between particles (type II). One must remember that any unitarization scheme may modify these results. That is, the triple Pomeron expressions do not satisfy unitarity under s → ∞, which is a long standing problem, with numerous par-tially plausible solutions (eikonalization, triple Pomeron decoupling, ...). Currently, there is no detailed data enough to gain new insights on this problem neither the non-perturbative QCD techniques are advanced enough to solve it. However, new LHC data may change the situation.

At this point, we refer the reader to mcplots.cern.ch for the measured rapidity gap distributions at the LHC with dierent minimum ptthresholds and Monte Carlo event generator comparisons. The large (diractive) gaps spanning several units of rapidity originate from charge-neutral exchanges, such as exchange of color neutral gluon systems (pomerons) or photons. The interaction rate of gap events may be

screened by additional elastic or inelastic interactions, this is known as the gap survival discussion. An important property is that measuring the nal states down to an arbitrary infrared cuto, not experimentally accessible, modies the observed rates of rapidity gap events. This corresponds e.g. to extra emission of soft gluons.

Thus, the rapidity gap events are not clearly `infrared safe' without an explicit pt

threshold denitions. Later in this work, in Figure 6.6, we show with a toy Monte Carlo simulation how the combinatorial gap rates run as a function of ptthresholds for four dierent particle densities per rapidity slice. We see that it is possible, given dierent pt thresholds and particle densities, to run the relative event rates of dierent nal state topologies to wildly dierent values.

For the non-diractive events, now let Y0, Y1, . . . , Yn be n + 1 rapidity intervals spanned by an n particle nal state proceed e.g. via `string breaking process'. To calculate statistics of these, we may use the Darling's contour integral technique from mathematical statistics [164] via the Laplace transform

E[f0(Y0)f1(Y1) . . . fn(Yn)] = n!

2πi

Z c+i∞

c−i∞

e^z

n

Y

j=0

Z ∞ 0

dz drje^−rjzfj(rj). (6.30) The path of the integral is along the straight line Re{z} = c and fj(x)are arbitrary real valued functions. Using this expectation integral, one can derive numerous results regarding the spacing statistics. For example:

MaxGap: The probability that all rapidity gaps are smaller than β is P (Y_j < β ∀j) = X

0≤j<1/β

n + 1 j



(−1)^j(1 − βj)ⁿ. (6.31)

MinGap: The probability that all rapidity gaps are larger than α is

P (Y_j > α ∀j) = (1 − (n + 1)α)ⁿ s.t. α < 1/(n + 1). (6.32) Both of these are derived in Darling's paper [164] under the uniform distribution assumptions. We see that given uniformly distributed rapidities, the number of nal states gives physical boundaries for the probabilities, visible in the inequalities in Equation6.31and6.32. Now clearly, one can consider higher-order spacing statistics such as distributions of the largest and the second largest gap and so on. Clearly, our incidence algebra construction is an integrating embedding of these taking also into account the translation (boost) position on the rapidity axis.

0 8 16 24 32 40 48 56 0

8

16

24

32

40

48

56

Figure 6.2: The binary vector space illustrated as an N = 6 hypercube graph with its corresponding adjacency matrix (black is 0, white is 1).

Our construction includes, in the limit N → ∞, exactly the `traditional' rapidity gap description. That is, one can recover the one dimensional rapidity gap distribu-tion from the N-dimensional combinatorial distribudistribu-tion. Moreover, in practice, N needs not to be innite due to the random correlation length over rapidity which smears out useful information at very small gap values.

Hypergraph and topology

In Figure6.2 we demonstrate the vector space as an equivalent hypercube G(E, V ).

Each vertex in set E corresponds to one particular nal state conguration and partial cross section component. The number of edges in the graph is N2^{N −1}, also counting the number of non-zero phase space elements (bits). The total number of subspaces is 2825, for N = 6. That is, it is a radically richer space than a single dimensional projection of rapidity gap distributions. In the following sections, a complete algebraic construction for any nite N will be given.

We point out that our denition of `event topology' is reasonably well dened even if slightly abstract at rst sight. Regarding the topological invariants of the hypercube, the Euler characteristic gives us V − E + F = 2 − 2γ(GN) where F is the number of faces and the minimum genus of N-cube for N ≥ 2 is [165]

γ(G_N) = (N − 4)2^{N −3}+ 1. (6.33)

That is, if N ≤ 3, then γ(GN) = 0and we can embed the planar graph without edge crossings on a sphere, N = 4 requires a one hole torus and then exponentially more complicated topologies are generated. The connection between nal state geometry (kinematics) and topology is now well dened  more complicated `event topologies' require larger N. An interesting construction would be the limit N → ∞, that is, from nite dimensional hypercubes to an innite dimensional construction. In practice, approximation schemes such as neural networks could be useful proxies for that.

In document Spinful Algorithmization of High Energy Diffraction (Page 154-166)