Universality of Koba-Nielsen-Olesen Scaling in QCD at High Energy and Entanglement

Abstract

Using Mueller’s dipole formalism for deep inelastic scattering in Quantum Chromodynamics (QCD), we formulate and solve the evolution for the generating function for the multiplicities of the produced particles in hadronic processes at high energy. The solution for the multiplicities satisfies Koba-Nielsen-Olesen (KNO) scaling, with good agreement with the recently re-analyzed data from the H1 experiment at HERA (DESY) and the old ALEPH detector data for hadronic $Z$ decay at LEP (CERN). The same scaling function with KNO scaling carries to the hadronic multiplicities from jets in electron-positron annihilation. This agreement is a priori puzzling, since in Mueller’s dipole evolution, one accounts for virtual dipoles in a wave function, whereas in electron-positron annihilation, one describes cross-sections of real particles. We explain the origin of this similarity, pointing at a particular duality between the two processes. Finally, we interpret our results from the point of view of quantum entanglement between slow and fast degrees of freedom in QCD and derive the entanglement entropy pertinent to electron-positron annihilation into hadronic jets.

1. Introduction

Using Mueller’s dipole formalism for deep inelastic scattering in QCD, we formulate and solve the evolution for the generating function for the multiplicities of the produced particles in hadronic processes at high energy. The solution for the multiplicities satisfies Koba-Nielsen-Olesen (KNO) scaling, with good agreement with the recently re-analyzed data from the H1 experiment at HERA (DESY) and the old ALEPH data for hadronic $Z$ decay at LEP (CERN). The same scaling function with KNO scaling carries to the hadronic multiplicities from jets in electron-positron annihilation. This agreement is a priori puzzling, since in Mueller’s dipole evolution, one accounts for virtual dipoles in a wave function, whereas in electron-positron annihilation, one describes cross-sections of real particles. We explain the origin of this similarity, pointing at a particular duality between the two processes. Finally, we interpret our results from the point of view of quantum entanglement between slow and fast degrees of freedom in QCD and derive the entanglement entropy pertinent to electron-positron annihilation into hadronic jets.

Universality is a powerful concept permeating several branches of physics, whereby different physical systems can exhibit similar behavior. This is usually captured by universal exponents, given general assumptions. Perhaps, the best example is the universality of the critical exponents in scaling laws in the vicinity of phase changes. Scaling laws, per se, form an important theoretical corpus in physics. In general, they describe the functional relationship between two physical quantities, that scale with each other over a significant interval.

In the context of high-energy particle physics, the so-called Koba-Nielsen-Olesen scaling (named KNO scaling hereafter), formulated half a century ago, is of paramount importance in the empirical analysis of many high-energy hadronic multiplicities. Yet, it is usually challenging to derive from the first principles in QCD. Historically, KNO scaling was first formulated in two independent theoretical works [1, 2], which suggested that at high energies with large Mandelstam $s$ (squared invariant mass), the probability distribution of producing $n$ particles in a specific collision process, scales as

$$\overline{n}\left(s\right)p_n\left(s\right)=f\left(z\right),$$E1

where $\overline{n}\left(s\right)$ is the average multiplicity at large $s$, and $z\equiv \frac{n}{\overline{n}\left(s\right)}$ is the argument of the scaling function $f\left(z\right)$. Remarkably, the KNO scaling hypothesis precedes the emergence of Quantum Chromodynamics (QCD) and the advent of high energy and luminosity data currently available at colliders.

This contribution is motivated by the recent work in [3], where the deep inelastic scattering (DIS) data from the H1 experiment at DESY were re-analyzed, with interest in an assessment of the quantum entanglement in high energy particle physics. Clearly, the data analyzed, especially for the highest energy range, shows KNO scaling (see Figure 1). Also, the Shannon entropy of the multiplicities presented bears some similarity to the entanglement entropy. However, the explicit form of the scaling function was unknown, and the QCD understanding of the hypothetical entanglement was not specified.

Figure 1.

The numerical (black-solid curve) up to $z=2.0$ and asymptotic (blue-solid curve) scaling of the KNO particle multiplicity $f\left(z\right)$ based on Table 2, compared to the recent data for DIS [3] (red) and the $Z$-decay data at $\sqrt{s}=M_Z$ by ALEPH [4] (gray).

In the first part of this chapter, we discuss the unexpected a priori fact, that identical differential equations, such as the ones we derived recently in [5], yield a scaling function that applies to different settings at high energy, e.g., deep inelastic scattering (DIS) and jets in electron-positron or $e^{+}{e}^{-}$ annihilation. Exploiting the duality [6, 7] between the Banfi-Marchesini-Smye (BMS) construction [8] and the Mueller’s dipole [9] in the conformal limit, we show that the pertinent generating functions for the multiplicity probabilities in the case of DIS and jets, respectively, are mathematically identical. Furthermore, in the double-logarithm approximation (DLA), the duality holds beyond conformal limit by introducing one-loop running in the “minimal” way. Given these, we arrive at the final differential equation for the KNO function, which we then solve using methods based on analyticity. The resulting, parameter-free curve, does not only agree well with DIS data from the H1 experiment at DESY, and the $Z$-decay data from ALEPH at CERN, but also represents a valuable prediction for future DIS experiments, alike electron-ion collider (EIC) or electron-ion-collider in China (EicC).

In sum, this paper consists of three new results: (1) the derivation of the KNO scaling function in QCD for both DIS and jets in the DLA; (2) the use of the KNO scaling function in the DLA, to show the universality of the hadronic multiplicities from current colliders, for both DIS and jets; (3) the explicit derivation of the entanglement entropy for $e^{+}{e}^{-}$ annihilation into hadronic jets, to be measured at collider energies.

2. The BMS equation for infrared (IR) logarithms in $e^{+}{e}^{-}\to$ multi-gluon cross-section

The BMS equation [8, 10, 11], describes the “nonglobal” logarithms in the $e^{+}{e}^{-}$ annihilation process, and is based on the universal features of the soft divergences in the $n$-gluon contribution to the total cross-section ${\sigma }_n$. To leading logarithm accuracy, the “most singular” part of ${\sigma }_n$, can be effectively generated through a Markov process. Defining the directions of the quark-antiquark pair as $p$ and $n$, soft gluons ($k$) are emitted from harder ones ($p$) through a universal eikonal current $\frac{{\mathit{gp}}^{\mu }}{p\cdot k}$, and the emissions are strongly ordered in time and energy. As a result, the emission depends only on the color charges that are already present in the final state but not on the history of how they are emitted. In the large number of colors $N_c$, the generating functional for ${\sigma }_n$,

$$Z\left(\frac{E}{E_0};n,p;\lambda \right)=\sum _{n=0}^{\infty }{\lambda }^n{\sigma }_n,$$E2

satisfies a closed integral equation (4)

$$\begin{array}{lll}Z\left(\frac{E}{E_0};n,p;\lambda \right)& =& S\left(\frac{E}{E_0};\widehat{p},\widehat{n}\right)\\ & & +{\overline{\alpha }}_s\lambda {\int }_{E_0}^E\frac{\mathit{d\omega }}{\omega }S\left(\frac{E}{E_0};\widehat{p},\widehat{n}\right)\int d{\mathrm{\Omega }}_{k}K\left(\widehat{k};\widehat{p},\widehat{n}\right)Z\left(\frac{\omega }{E_0};n,\widehat{k};\lambda \right)Z\left(\frac{\omega }{E_0};p,\widehat{k};\lambda \right),\end{array}$$E3

with the eikonalized gluonic emission kernel $K\left(\widehat{k};\widehat{p},\widehat{n}\right)=\frac{1}{4\pi }\frac{p\cdot n}{\widehat{k}\cdot p\widehat{k}\cdot n}$ and ${\overline{\alpha }}_s=N_c{\alpha }_s/\pi$. The Sudakov factor reads

$$S\left(\frac{E}{E_0};\widehat{p},\widehat{n}\right)=e^{-{\overline{\alpha }}_{s}ln\frac{E}{E_0}\int d{\mathrm{\Omega }}_{k}K\left(\widehat{k};\widehat{p},\widehat{n}\right)}.$$E4

More precisely, the first term is the Sudakov contribution, where all the soft gluons are virtual, and the second term is the contribution where at least one soft gluon is real (3) is the integral form of the BMS equation, which can brought to the standard form discussed in [8, 10, 11], by taking a derivative with respect to $ln\frac{E}{E_0}$ and some re-arrangements.

3. The Mueller’s dipole for small-$x$ logarithms and the BMS-Balitskii-Kovchegov (BK) correspondence

Mueller [9] has shown that the small-$x$ evolution equations such as Balitskii-Fadin-Kuraev-Lipatov (BFKL) [12, 13, 14], and BK [15, 16], are also based on a very similar branching process, where small-$x$ virtual gluons are released into the light-front wave functions (LFWFs). The same reasoning yields an evolution equation, this time for the generating function for the distribution of the virtual dipoles or $Z\left(b,\frac{x}{x_{\min }},\lambda \right)$, which is exactly of the form (3), with the substitution $\int d{\mathrm{\Omega }}_k\to \int d^2{b}_2$:

$$\begin{array}{lll}Z\left(b_{10},\frac{x_0}{x_{\min }},\lambda \right)& =& S\left(b_{10},\frac{x_0}{x_{\min }}\right)\\ & & +\lambda \frac{{\alpha }_s{N}_c}{2{\pi }^2}{\int }_{x_{\min }}^{x_0}\frac{{\mathit{dx}}_1}{x_1}S\left(b_{10},\frac{x_0}{x_1}\right)\int {\mathit{db}}_2^2\frac{b_{10}^2}{b_{12}^2{b}_{20}^2}Z\left(b_{12},\frac{x_1}{x_{\min }},\lambda \right)Z\left(b_{20},\frac{x_1}{x_{\min }},\lambda \right).\end{array}$$E5

Here, the corresponding Sudakov or “soft-factor” for “virtual” emissions is

$$S\left(b_{10},\frac{x_0}{x_1}\right)=exp\left[-\frac{{\alpha }_s{N}_c}{\pi }ln{b}_{10}^2{\mu }^{2}ln\frac{x_0}{x_1}\right].$$E6

In fact, one can show that in the leading order, the BMS and BK equations map onto each through a pertinent conformal transformation [6, 7, 17, 18, 19], where the asymptotic real soft gluons at $t=\infty$, map onto the virtual gluons present at $x^{+}=0$ [5]. In this sense, the mapping is a “virtual-real” duality, in addition to the standard interpretation that it maps rapidity divergence to ultraviolet (UV) divergence [6].¹ In Table 1, we have highlighted this duality through a parallel between the two constructions.

	Dipole	Cusp
Distribution in	Virtual gluon	Real gluon
Large $N_c$	Yes	Yes
Kernel	$\frac{b_{10}^2}{b_{12}^2{b}_{20}^2}$	$\frac{n\cdot p}{\widehat{k}\cdot n\widehat{k}\cdot p}$
Virtual part	TMD soft-factor	Sudakov form factor
Time ordering	In LF time	In center of mass (CM) time
Momentum ordering	Decreasing $k^{+}$	Decreasing energy $\omega$
Virtuality ordering	Increasing	Decreasing
Markov process	Yes	Yes
DLA	$b_{10}\gg b_{12}\gg \dots$	${\theta }_{01}\gg {\theta }_{12}\gg .\dots$

Table 1.

Mueller hierarchy and BMS hierarchy.

4. The DLA limit and the universal DLA KNO scaling function

The BK (BMS) equation resums single logarithms in rapidity (energy). However, in both cases, there are two types of divergences instead of one: in the Mueller’s dipole construction for the LFWFs [9], there are UV divergences when $k_{\perp }$ becomes large, while in the BMS construction with Wilson-line cusps, there are rapidity-divergences when the emissions become collinear to the Wilson-lines. It is natural to resum the double logarithms for both. One can then show that the double-logarithm approximation (DLA) for BK (BMS) is generated from the same branching process, with one more strong ordering in dipole sizes (emitting angles) (see Table 1). The strong ordering in virtuality is preserved by the DLA limit. Clearly, the two DLA and the underlying size (angle) orderings, simply map onto each other under a conformal transformation. The similarity of the branching process follows from the non-Abelian three-gluon coupling and large $N_c$ in QCD, giving rise to a binary tree branching.

A unique feature of the DLA is that the distribution of dipoles (soft gluons) has a nontrivial KNO scaling function $f\left(z\right)$, which coincides with that suggested many years ago for jets in [20, 21, 22]. This is due to the same strong ordering in emitting angles in both cases, with Table 1 making this plausible.

In the DLA, the equation for the generating function (3) simplifies

$$Z\left(\rho \right)=e^{-\rho }+\rho {\int }_0^1\mathit{dx}{\int }_0^1{\mathit{dye}}^{-\rho \left(1-y\right)}Z\left(\mathit{\rho xy}\right)Z\left(\mathit{\rho y}\right),$$E7

where

$$\rho =\frac{2C_F}{{\mathit{\pi \beta }}_0}ylnln\frac{Q^2}{M^2}.$$E8

with $2C_F\sim N_c$ and ${\beta }_0=\frac{11N_c}{12\pi }$ for large $N_c$. Note that we have switched from fixed ${\alpha }_s$ in (3) to one-loop running in (9), with the scale fixed by the emitted small dipoles, hence the emergence of the beta function ${\beta }_0$. A more thorough discussion of the choice of ${\alpha }_s$ at low-x can be found for example in [23], which reduces to the size of the smallest dipole in the presence of strong ordering. In case of $e^{+}{e}^{-}$, the same DLA limit holds when the running was introduced through the energy of the soft gluons. Defining $Z=expW$, and introducing $u=2\sqrt{\rho }$, we arrive at the final equation ${\mathrm{\Delta }}_{2}W=e^W-1$, where ${\mathrm{\Delta }}_2$ is a radial part of the 2-dimensional Laplacian. This equation is reminiscent of the Poisson-Boltzmann equation, if ${\mathrm{\Delta }}_2$ was the full Laplacian. For large energies, it reduces to

$$\frac{d^{2}W}{{\mathit{du}}^2}=e^W-1.$$E9

The solution of this equation, encodes the shape of the KNO scaling function $f\left(z\right)$, through a Fourier-Laplace transform

$$Z\left(t=-e^u\right)={\int }_0^{\infty }\mathit{dz}{e}^{-\mathit{tz}}f\left(z\right).$$E10

A detailed investigation of the above equations can be found in our recent analysis [5]. With the help of complex-analytic method, we are able to unravel the scaling function $f\left(z\right)$ both in the asymptotic region analytically, and throughout numerically. For $z<0.1$, the scaling function can be made explicit

$$f\left(z\right)\sim \frac{\alpha }{z^2}ln\frac{\alpha }{z}exp\left(-\frac{1}{2}{ln}^2\frac{\alpha }{z}ln\frac{\alpha }{z}+\mathcal{O}\left(1\right)\right),$$E11

where $\alpha =1.50972$. For $z>2.0$, the scaling function behaves asymptotically as

$$f\left(z\right)=2r\left(\mathit{rz}-1+\mathcal{O}\left(\frac{lnz}{z}\right)\right)e^{-\mathit{rz}},z\to \infty ,$$E12

where $r=2.55297$. The explicit expression for $r$ can be found in [20, 21, 22], and that for $\alpha$ in [5].

5. Scaling function vs. data

Since this scaling function represents a parameter-free QCD prediction for future experiments, including the EIC and EicC, we record our numerical solution in Table 2, following our analysis in [5]. For completeness, we note that a moment reconstruction of $f\left(z\right)$ using different arguments was used in the context of jets in [21, 22, 24].

z	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
f(z)	0.01	0.21	0.45	0.65	0.77	0.82	0.82	0.78	0.72	0.64
z	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	2.0
f(z)	0.56	0.49	0.42	0.35	0.29	0.24	0.20	0.15	0.12	0.1

Table 2.

Table of the scaling function $f\left(z\right)$.

In Figure 1, we compare our results recorded in Table 2 (black-solid curve) with the H1 data for DIS [3] (red data) and the old ALEPH data for $Z$ decay [4] (gray data). The agreement is very good for both data sets, supporting the universality of our results. For comparison, we also show the exact asymptotic (11) (blue-solid curve), and the KNO particle multiplicity $e^{-z}$ [9], following from the dimensional reduction (diffusive approximation) of Mueller’s dipole wave function evolution (dashed-green curve).

6. KNO scaling and entanglement

The knowledge of the effective reduced density matrix for the virtual dipoles in the LFWF [25], and the shape of the KNO function, allow for the evaluation of the entanglement entropy between fast and slow degrees of freedom [25, 26, 27, 28]. Remarkably, the reduced density matrix for large rapidities is diagonal [25]

$$\rho =\sum _n{p}_n{\rho }_n,$$E13

with $p_n$ the probability for the emission of $n$ dipoles (gluons), and ${\rho }_n$ an effective reduced density matrix with $n$ soft dipoles in-out. Hence, the entanglement entropy

$$S=-\sum _n{p}_{n}ln{p}_n+\sum _n{p}_n{s}_n,$$E14

with $s_n=-\mathrm{tr}{\rho }_{n}ln{\rho }_n$. Since the wave function peaks at $n=\overline{n}$ for large $n$, we assume the same to hold for $s_n$, with the scaling $s\left(\frac{n}{\overline{n}}\right)$, so that

$$S=ln\overline{n}+\int \mathit{dzf}\left(z\right)\left(-lnf\left(z\right)+s\left(z\right)\right).$$E15

For DIS in the DLA with KNO scaling, the result is [5]

$$S_{\mathit{DIS}}\left(y,Q^2\right)\to ln\overline{n}\equiv 2{\left(\frac{2C_F}{{\mathit{\pi \beta }}_0}ylnln\frac{Q^2}{M^2}\right)}^{\frac{1}{2}}.$$E16

a measure of the Sudakov contribution (17) is measurable in the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) regime of DIS. We note that the KNO scaling function in the diffusive (BFKL) regime with $f\left(z\right)=e^{-z}$ [9], leads to maximal decoherence in the entanglement entropy $S_{\mathit{\text{BFKL}}}\sim y$ [25, 26, 27, 28]. In contrast, the unimodal character of the scaling function in the DLA leads to a smaller entanglement entropy $S_{\mathit{DIS}}\sim \sqrt{y}$.

Using the BMS-BK duality, we can readily formulate the entanglement entropy between soft and hard degrees of freedom in the final state of $e^{+}{e}^{-}$ annihilation into hadronic jets

$$S_{e^{+}{e}^{-}}\left(Q^2\right)\equiv 2{\left(\frac{C_F}{{\mathit{\pi \beta }}_0}ln\frac{Q^2}{M^2}lnln\frac{Q^2}{M^2}\right)}^{\frac{1}{2}},$$E17

also a measure of the Sudakov contribution (with no extra 2 in the bracket). The rapidity gap between the quark-antiquark pair $y\to ln\frac{Q^2}{M^2}$, produces another logarithm in $Q^2$. Note that for $e^{+}{e}^{-}$, this contribution dominates the single logarithm resummation in the BFKL contribution at large $Q^2$, which is

$$ln{\overline{n}}_{\text{BFKL}}\sim \frac{2C_{F}ln2}{{\mathit{\pi \beta }}_0}lnln\frac{Q^2}{M^2}.$$E18

The prediction (17) is amenable to experimental verification in high-energy hadronic jet physics.

7. Conclusion

To summarize, in this chapter, we presented the universal KNO scaling function underlying the DLA limit for two different systems: the $e^{+}{e}^{-}\to$ multi-gluon process and the Mueller’s dipole wave function. In both systems, the multiplicity generating function satisfies nonlinear BK type evolution equation with respect to primary evolution variables: energy for the $e^{+}{e}^{-}\to$ multi-gluon process and rapidity for the Mueller’s dipole. They map to each other through the BMS-BK correspondence.

On the other hand, in both systems there also exists another large logarithm: in rapidity for $e^{+}{e}^{-}\to$ multi-gluon process and in energy for the Mueller’s dipole. The second logarithm appears naturally in the multiplicity distribution. When projected to the double-logarithm limit (DLA limit), which resumes both of the two large-logarithms, the evolution equation simplifies considerably, and the corresponding multiplicity distribution exhibits a nontrivial universal KNO scaling function. This scaling function compares well with the measured hadronic multiplicities in DIS and $e^{+}{e}^{-}$ annihilation, especially in the exponential tail. Moreover, the KNO scaling also constraints, in a universal way, in the DLA limit, the asymptotic behavior of the rapidity space or energy-space entanglement entropies of the underlying process.

Acknowledgments

We are grateful to Jacek Wosiek for bringing [20] to our attention and to Yoshitaka Hatta for informing us about [19]. This work is supported by the Office of Science, U.S. Department of Energy under Contract No. DE-FG-88ER40388, and by the Priority Research Areas SciMat and DigiWorld under the program Excellence Initiative—Research University at the Jagiellonian University.