Mueller hierarchy and BMS hierarchy.
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
Keywords
- QCD
- DIS
- entanglement
- multiplicity
- KNO scaling
- perturbation theory
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
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,
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
where
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.
In the first part of this chapter, we discuss the unexpected
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
2. The BMS equation for infrared (IR) logarithms in e + e − → multi-gluon cross-section
The BMS equation [8, 10, 11], describes the “nonglobal” logarithms in the
satisfies a closed integral equation (4)
with the eikonalized gluonic emission kernel
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
3. The Mueller’s dipole for small-x logarithms and the BMS-Balitskii-Kovchegov (BK) correspondence
Mueller [9] has shown that the small-
Here, the corresponding Sudakov or “soft-factor” for “virtual” emissions is
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
Dipole | Cusp | |
---|---|---|
Distribution in | ||
Large | Yes | Yes |
Kernel | ||
Virtual part | TMD soft-factor | Sudakov form factor |
Time ordering | In LF time | In center of mass (CM) time |
Momentum ordering | Decreasing | Decreasing energy |
Virtuality ordering | Increasing | Decreasing |
Markov process | Yes | Yes |
DLA |
4. The DLA limit and the universal DLA KNO scaling function
The BK (BMS) equation resums
A unique feature of the DLA is that the distribution of dipoles (soft gluons) has a nontrivial KNO scaling function
In the DLA, the equation for the generating function (3) simplifies
where
with
The solution of this equation, encodes the shape of the KNO scaling function
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
where
where
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
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 |
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
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]
with
with
For DIS in the DLA with KNO scaling, the result is [5]
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
Using the BMS-BK duality, we can readily formulate the entanglement entropy between soft and hard degrees of freedom in the final state of
also a measure of the Sudakov contribution (with no extra 2 in the bracket). The rapidity gap between the quark-antiquark pair
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
On the other hand, in both systems there also exists another large logarithm: in rapidity for
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.
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Notes
- In nonconformal theory, the exact mapping breaks at two-loop already. But for the virtual part, it can be generalized to all orders.