Quantum Entanglement in Top Quark Pair Production

Abstract

Top quarks, the most massive particles in the standard model, attract considerable attention since they decay before hadronizing. This presents physicists with a unique opportunity to directly investigate their properties. In this letter, we expand upon the work of G. Iskander, J. Pan, M. Tyler, C. Weber and O. K. Baker to demonstrate that even with the most massive fundamental particle, we see the same manifestation of entanglement observed in both electroweak and electromagnetic interactions. We propose that the thermal component resulting from protons colliding into two top quarks emerges from entanglement within the two-proton wave function. The presence of entanglement implies the coexistence of both thermal and hard scattering components in the transverse momentum distribution. We use published ATLAS and CMS results to show that the data exhibits the expected behavior.

Full text

Quantum Entanglement in Top Quark Pair Production
Mira Varma and Oliver K. Baker∗
Department of Physics, Yale University, New Haven, CT 06520
Abstract
Top quarks, the most massive particles in the standard model, attract considerable attention since
they decay before hadronizing. This presents physicists with a unique opportunity to directly investigate
their properties. In this letter, we expand upon the work of G. Iskander, J. Pan, M. Tyler, C. Weber
and O. K. Baker to demonstrate that even with the most massive fundamental particle, we see the
same manifestation of entanglement observed in both electroweak and electromagnetic interactions. We
propose that the thermal component resulting from protons colliding into two top quarks emerges from
entanglement within the two-proton wave function. The presence of entanglement implies the coexistence
of both thermal and hard scattering components in the transverse momentum distribution. We use
published ATLAS and CMS results to show that the data exhibits the expected behavior.
Key words: Quantum entanglement, Entanglement entropy, Top physics,
Heavy quark production
1 Introduction
Prior literature has established that the transverse momentum distribution of hadrons is best described by
fitting the sum of an exponential and a power law. (See Refs. [1, 2] for more detail). The power law portion
of the fit, which represents hard scattering, is well understood: it arises from the sizable momentum transfer
between the quarks and gluons [1]. The thermal behavior of the transverse momentum distribution, on the
other hand, remains a mystery in particle physics. There have been several competing ideas about why this
behavior is present [3]-[9]. A common belief is that thermalization arises through re-scattering after nuclei
collide [10]. This explanation is limited: it cannot explain the origin of thermalization in proton-proton (pp)
collisions.
A universal explanation for the behavior of the transverse momentum distribution is that it is due to
entanglement between parts of the wave functions of the colliding particles. This idea has been studied for
several interactions. G. Iskander et al. showed that for weak interactions, specifically neutrino scattering,
there is entanglement between the probed and unprobed regions of the nucleon in the collision [11]. This
theory has also been studied for electroweak processes, namely, deep inelastic scattering (DIS). K. Zhang et
al. calculated the von Neumann entropy (interpreted as the entanglement entropy) of the DIS system, which
they proposed was caused by entanglement between the probed and unprobed regions of the proton [12].
As discussed in Ref. [13], when two protons collide, the entire system undergoes a “quench” due to the
sudden presence of a collision and a spectator region. The Hamiltonian is evolved, meaning, H=H0→
H0+V(t), where V(t) is the effect of a pulse of the color field. The uncertainty principle suggests that the
momentum transfer of the collision, Q, and the proper time (time measured in the particle’s rest frame), τ,
are related by: τ∼1/Q [13]. If we approximate a pp collision as a short pulse of a (chromo)electric field,
the effective temperature parameter (arising from thermalization), satisfies the following relation [13]:
Tth ≃(2πτ )−1≃Q
2π.(1)
∗Corresponding author: oliver.baker@yale.edu
1
arXiv:2306.07788v1 [hep-ph] 13 Jun 2023

In Eq. 1, Tth is a parameter of the thermal component of the transverse momentum distribution [13].
(See Refs. [14, 15, 1] for more detail).
In this letter, we extend this idea to the most massive known particle in the standard model, the top quark.
The top quark is particularly interesting due to its short lifetime (i.e. top quarks decay before hadronizing).
We propose that when two protons collide, the thermal part of the transverse momentum distribution is
caused by entanglement in the wave function of the proton-proton system. This entanglement is between
a collision region, where the two protons overlap, which we call A, and a region where the two protons do
not overlap, which we call B. In other words, we are proposing that when entanglement is present, the
transverse momentum distribution has both thermal and hard scattering components. Naturally, therefore,
where there is no entanglement, the thermal component is absent.
2 Background
In the general quantum information formalism, a pure state of two quantum systems, Aand B, is denoted
as,
ρAB =|ψAB ⟩⟨ψAB |,(2)
where ρAB is the reduced density matrix. Physically, when a system is in a pure state, there is complete
information about the wave function. On the other hand, if a system is in a mixed quantum state, there is
incomplete information about the wave function at every point in time. A mixed state is a weighted sum of
pure states. Mathematically, this is denoted as:
|ΨAB ⟩=X
i,j
cij |ψA
i⟩⟨ψB
j|.(3)
The reduced density matrix is:
ρAB =X
i,j
cij |ψA
i⟩⟨ψA
j|⊗|ψB
i⟩⟨ψB
j|.(4)
The reduced density matrix of either one of the subsystems is needed in order to calculate the entanglement
entropy. We choose to use subsystem Abecause it is more interesting. The reduced density matrix of
subsystem Acan be written as:
ρA= TrB(ρAB ).(5)
Once we have ρA, the entanglement entropy of subsystem Ais given by:
SA=−Tr(ρAln ρA).(6)
If Tr(ρA)2= 1, we have a pure state, and there is no entanglement present in |ΨAB⟩. If Tr(ρA)2<1, we
have a mixed state, and |ΨAB⟩is entangled [16].
When two protons collide, both protons are initially in a pure state (see Fig. 1). Once the protons have
collided, two regions are present: an overlap (collision) region, A, and a non overlap (spectator) region,
B. In Fig. 1, A∪Brepresents a pure state, since we are considering the proton-proton system as a whole.
However, when considering Aor Bseparately, they are each in a mixed state, and we expect entanglement
to be present.
3 Results and Analysis
We begin our study by using the transverse momentum distribution of t¯
tpair production in the semi-leptonic
decay channel, focusing on the hadronic decay products, which is described by:
t¯
t→W+bW −b→q¯qb +ℓ¯ν¯
b+ jets.(7)
The process described in Eq. 7 is depicted in Fig. 2. Throughout this letter, the center of mass (pp collision)
energy is √s= 13 TeV. The following relations for the thermal and hard scattering components of the
transverse momentum distribution (Eqs.8 and 11) were originally proposed in Ref. [17] and have been used
2

Figure 1: Diagram depicting a proton-proton collision. (Top) Both protons before they collide. (Bottom)
The two protons during the collision, where region Ais the collision region (region of overlap) and region B
is the spectator region (non-overlap region). Regions Aand Bare entangled.
Figure 2: Top anti-top quark decay in the semi-leptonic channel. The resulting W bosons can decay hadron-
ically, resulting in a quark antiquark pair or leptonically, resulting in a lepton and a neutrino [18].
3

in other studies [1, 11, 19, 13, 20]. The thermal component of the transverse momentum distribution is given
by the following,
1
pT
dσ
dpT
=Ath exp −mT
Tth ,(8)
where pTis the transverse momentum of the system, Ath is a fitting parameter, mTis the transverse mass
of the system, and Tth is the effective (thermal) temperature parameter.
The transverse mass is calculated using the following relation,
m2
T=m2+p2
T,(9)
where mis the mass of the t¯
tsystem, which in this case is the mass of top quark & anti-top-quark together
≈2×(173 GeV/c2). The effective temperature parameter, which was extracted in Ref. [1], is given by,
Tth = 0.098 s
s00.06
GeV,(10)
where √s0is a normalization constant equal to √s0= 1 GeV and √sis the proton-proton collision energy
(which is √s= 13 GeV).
The hard scattering component of the transverse momentum distribution is given by:
1
pT
dσ
dpT
=Ahard
1 + m2
T
T2
hardnn.(11)
In Eq. 11, Ahard is a fitting parameter, Thard is the hard scale parameter, and nis a scaling factor obtained
from the power law fit. The values mT,√s0and √sremain unchanged from their previous definitions.
The hard scale parameter, which was determined in Ref.[1], is defined as:
Thard = 0.409 s
s00.06
GeV.(12)
The CERN ROOT fitting program and the SciPy curve fit function were used to fit Eqs. 8 and 11 to t¯
t
decay data arising from proton-proton collisions at the Large Hadron Collider. The transverse momentum
distribution of t¯
tproduction for the hadronic decay products in the semi-leptonic decay channel (ATLAS)
with an integrated luminosity of 3.2 fb−1, is depicted in Fig.3. As we can see, both a thermal component
(red) and a hard scattering component (green) are needed to properly fit the data, which suggests the
presence of entanglement. The sum of the two fits, which has a reduced chi-squared value of χ2/ndf ≈1.6,
is the blue curve. The error bars are smaller than the size of the data points. Fig. 4 depicts an analogous
fit using CMS data, which yielded a reduced chi-squared value of χ2/ndf ≈1.3. This increase in statistical
precision was expected, since the integrated luminosity of the CMS data was 35.8 fb−1, an order of magnitude
larger than that of the ATLAS data. As the integrated luminosity increases, the number of recorded events
increases as well, which results in a more precise transverse momentum distribution. Again, in Fig. 4, we
can see the necessity of having both a thermal and a hard scattering component in the fit. Fig. 5 depicts
the transverse momentum distribution of the additional leading jet. Since we cannot properly keep track of
the jets, i.e., we lack complete information about their behavior, we cannot have set spectator and collision
regions. Therefore, when studying the transverse momentum distribution of one of the additional leading
jets, we would expect no entanglement due to the lack of information. As we can see in Fig. 5, only the hard
scattering component is needed to fit the data, implying the absence of entanglement, as predicted.
One can quantify the presence of a thermal component in the transverse momentum distribution by
calculating the ratio between the area under the curve (integral) of the hard scattering component (Eq. 11)
and the area under the curve (integral) of the sum of the fits (Eq.8 + Eq. 11). This ratio, R, is defined as,
R=Ip
Ie+Ip
,(13)
where Ipis the area under hard scattering (power law) portion of the curve and Ieis the area under the
thermal (exponential) part of the curve. If there is no thermal component to the fit, Ie= 0. When ATLAS
4

Figure 3: Transverse momentum distribution of top-antitop quark pair production from ATLAS data, with
a center of mass energy of 13 TeV and a luminosity of 3.2 fb−1. The reduced chi-squared fit value is
χ2/ndf ≈24.7/15 = 1.6. Data is taken from [22].
5

Figure 4: Transverse momentum distribution of top-antitop quark pair production from CMS data, with
a center of mass energy of 13 TeV and a luminosity of 35.8 fb−1. The reduced chi-squared fit value is
χ2/ndf ≈10.3/8=1.3. Data is taken from [22].
6

Figure 5: Transverse momentum distribution of top-antitop quark pair production from ATLAS data, with
respect to the additional leading jet. Center of mass energy is 13 TeV and luminosity is 139 fb−1. The
reduced chi-squared fit value is χ2/ndf ≈17.0/13 = 1.3. Data is taken from [23].
7

data was used, Rwas calculated to be 0.19 ±0.03. Using CMS data yielded a slightly different R, which was
0.16 ±0.03. When we examined the transverse momentum with respect to the additional leading jet of the
system, the Rwas found to equal one. This is exactly what we expected, as there was no entanglement present
in this case. The calculated Rvalues are consistent with the ratios computed for other pp collisions, as well
the Rvalues for charged weak interactions [11], [13], [20]. For the process given by ¯νµ+12C→µ++π−+12 C,
no entanglement was expected since the event was diffractive, which implies that the nucleus as a whole was
probed. Therefore, there were no identifiable collision and spectator regions which could be entangled with
one another. This parallels our result for the transverse momentum distribution of the additional leading
jet, since in this case, distinguishable collision and spectator regions were also absent. Table 1 summarizes
the results from previous literature as well as our new results.
RProcess Reference
0.16 ±0.05 pp →charged hadrons [13], [20]
0.15 ±0.05 pp →H→γγ [13], [20]
0.23 ±0.05 pp →H→4l(e, µ) [13], [20]
1.00 ±0.02 pp(γγ)→(µµ)X’X” [13], [20]
0.13 ±0.03 ¯νµ+N→µ++π0+X[11]
1.00 ±0.05 ¯νµ+12C→µ++π−+12 C [11]
0.19 ±0.03 pp →t¯
t→W bW b (ATLAS) current work
0.16 ±0.03 pp →t¯
t→W bW b (CMS) current work
1.00 ±0.05 pp →t¯
t→W bW b →jets (ATLAS) current work
Table 1: Rvalues from prior studies and our current work.
4 Conclusion
In this letter, we have extended upon the ideas in Refs.[1, 11, 19, 13, 20] to show that even in t¯
tcollisions, the
thermal component of the transverse momentum distribution can be attributed to entanglement between
different parts of the wave functions of the colliding particles (in this case, protons). In Ref. [24], Duan
discusses this idea further, introducing a term called “entropy of ignorance.” In his work, Duan agrees that
in proton-proton collisions, there is entanglement between the collision and spectator regions of the proton.
Since an experiment can only measure the collision region, we lack information about the spectator region.
This lack of information is called the “entropy of ignorance.”
Studies of entanglement in t¯
tcollisions can also be used to investigate possibilities of physics beyond the
standard model. In Ref. [25], a term called quantum discord is discussed, which is a fundamental quantity
that measures the “quantumness of correlations” [26]. If the quantum discord is asymmetric, this can hint at
the presence of CP violation. It would be interesting to apply these ideas to new experimental measurements
of t¯
tpair production or to other types of particle collisions.
Acknowledgements
The authors gratefully acknowledge funding support from the Department of Energy Office of Science Award
DE-FG02-92ER40704.
Acknowledgements
The authors gratefully acknowledge funding support from the Department of Energy Office of Science Award
DE-FG02-92ER40704.
8

References
[1] A. A. Bylinkin, D. E. Kharzeev and A. A. Rostovtsev. The origin of thermal component in the transverse
momentum spectra in high energy hadronic processes. Int. J. Mod. Phys. E 23, no.12, 1450083 (2014)
doi:10.1142/S0218301314500839 [arXiv:1407.4087 [hep−ph]].
[2] A. A. Bylinkin and A. A. Rostovtsev. Parametrization of the shape of hadron-production spectra in high-
energy particle interactions. Phys. Atom. Nucl. 75, 999-1005 (2012) doi:10.1134/S1063778812040047
[3] J. Cleymans, H. Satz, E. Suhonen and D. W. von Oertzen. Strangeness Production in Heavy Ion
Collisions at Finite Baryon Number Density. Phys. Lett. B 242, 111-114 (1990) doi:10.1016/0370-
2693(90)91604-A
[4] J. Cleymans and H. Satz. Thermal hadron production in high-energy heavy ion collisions. Z. Phys. C
57, 135-148 (1993) doi:10.1007/BF01555746 [arXiv:hep-ph/9207204 [hep−ph]].
[5] K. Redlich, J. Cleymans, H. Satz and E. Suhonen. Hadronization of quark - gluon plasma. Nucl. Phys.
A566, 391C-394C (1994) doi:10.1016/0375-9474(94)90652-1
[6] P. Braun-Munzinger, J. Stachel, J. P. Wessels and N. Xu. Thermal equilibration and expansion in
nucleus-nucleus collisions at the AGS. Phys. Lett. B 344, 43-48 (1995) doi:10.1016/0370-2693(94)01534-
J [arXiv:nucl-th/9410026 [nucl−th]].
[7] F. Becattini, M. Gazdzicki and J. Sollfrank. On chemical equilibrium in nuclear collisions. Eur. Phys.
J. C 5, 143-153 (1998) doi:10.1007/s100520050256 [arXiv:hep-ph/9710529 [hep−ph]].
[8] F. Becattini, J. Cleymans, A. Keranen, E. Suhonen and K. Redlich. Features of particle multiplicities
and strangeness production in central heavy ion collisions between 1.7A-GeV/c and 158A-GeV/c. Phys.
Rev. C 64, 024901 (2001) doi:10.1103/PhysRevC.64.024901 [arXiv:hep-ph/0002267 [hep−ph]].
[9] P. Braun-Munzinger, K. Redlich and J. Stachel. Particle production in heavy ion collisions.
doi:10.1142/9789812795533 0008 [arXiv:nucl-th/0304013 [nucl−th]].
[10] F. Becattini, P. Castorina, J. Manninen and H. Satz. The Thermal Production of Strange and Non-
Strange Hadrons in e+ e- Collisions. Eur. Phys. J. C 56, 493-510 (2008) doi:10.1140/epjc/s10052-008-
0671-x [arXiv:0805.0964 [hep−ph]].
[11] G. Iskander, J. Pan, M. Tyler, C. Weber, and O. K. Baker,
Quantum Entanglement and Thermal Behavior in Charged-Current Weak Interactions,
Phys. Lett. B 811, 135948 (2020).
doi:10.1016/j.physletb.2020.135948.
arXiv:2010.00709 [hep-ph].
[12] K. Zhang, K. Hao, D. Kharzeev and V. Korepin. Entanglement entropy production in deep inelastic
scattering. Phys. Rev. D 105, no.1, 014002 (2022) doi:10.1103/PhysRevD.105.014002 [arXiv:2110.04881
[quant−ph]].
[13] O. K. Baker and D. E. Kharzeev. Thermal radiation and entanglement in proton-proton collisions
at energies available at the CERN Large Hadron Collider Phys. Rev. D 98, no.5, 054007 (2018)
doi:10.1103/PhysRevD.98.054007 [arXiv:1712.04558 [hep−ph]].
[14] D. Kharzeev and K. Tuchin. From color glass condensate to quark gluon plasma through the event
horizon. Nucl. Phys. A 753, 316-334 (2005) doi:10.1016/j.nuclphysa.2005.03.001 [arXiv:hep-ph/0501234
[hep−ph]].
[15] D. Kharzeev, E. Levin and K. Tuchin. Multi-particle production and thermalization in high-energy QCD.
Phys. Rev. C 75, 044903 (2007) doi:10.1103/PhysRevC.75.044903 [arXiv:hep−ph/0602063 [hep−ph]].
[16] S. J. van Enk. Mixed states and pure states. April 9, 2009, https://pages.uoregon.edu/svanenk/
solutions/Mixed_states.pdf (accessed May 20, 2023)
9

[17] A. A. Bylinkin and A. A. Rostovtsev. Systematic studies of hadron production spectra in collider
experiments. Proceedings of the XL International Symposium on Multiparticle Dynamics (2010) doi:
https://dx.doi.org/10.5689/UA-PROC-2010-09/14 [arXiv:1008.0332 [hep−ph]].
[18] CMS Collaboration. Top Quarks Fast to Arrive at New Energy Frontier. October 24, 2022, https:
//cms.cern/news/top-quarks-fast-arrive-new-energy-frontier (Accessed May 15, 2023)
[19] R. Liu, C. Weber, O. K. Baker and D. E. Kharzeev. Quantum Information Science in pp Collisions at
the Energy Frontier. Meeting of the APS Division of Particles and Fields, Northeastern University, MA
(2019)
[20] C. Weber, O. K. Baker and D. E. Kharzeev. Quantum Entanglement and Proton-Proton Collisions at
the LHC SUSY International Conference, Corpus Christie, TX (2019)
[21] A. A. Bylinkin and A. A. Rostovtsev. Identified hadron production in a two component model. Nucl.
Part. Phys. Proc. 273-275, 2714-2716 (2016) doi:10.1016/j.nuclphysbps.2015.10.040
[22] F. Fabbri [ATLAS and CMS]. Comparative overview of differential measurements at ATLAS and CMS.
[arXiv:1901.10827 [hep−ex]].
[23] V. Khachatryan et al. [CMS]. Measurement of differential cross sections for top quark pair production
using the lepton+jets final state in proton-proton collisions at 13 TeV. Phys. Rev. D 95, no.9, 092001
(2017) doi:10.1103/PhysRevD.95.092001 [arXiv:1610.04191 [hep−ex]].
[24] H. Duan. Entanglement, partial set of measurements, and diagonality of the density matrix in the
parton model. SciPost Phys. Proc. 8, 160 (2022) doi:10.21468/SciPostPhysProc.8.160 [arXiv:2107.10812
[hep−ph]].
[25] Y. Afik and J. R. M. de Nova. Quantum discord and steering in top quarks at the LHC. [arXiv:2209.03969
[quant−ph]].
[26] A. Brodutch et al.. Quantum discord in quantum computation. J. Phys.: Conf. Ser. 306, 012030 (2011)
doi:10.1088/1742-6596/306/1/012030
10