**3. Entanglement entropy and thermal behavior in the electroweak interaction**

The material and discussion in Section 2 supporting a picture of thermalization in hadronic physics due to quantum entanglement motivates an investigation of

<sup>1</sup> It is once again emphasized that this does not imply that the Higgs boson is produced thermally, but rather that its transverse momentum distribution is affected by thermal radiation due to entanglement.

#### **Figure 6.**

*(Left side) The depiction of antineutrino scattering from a nucleon via emission of a W boson with an exiting muon in the final state. The W boson samples a partial region of the nucleon, not the entire nucleon, as explained in the text. (Right side) The region of the nucleon sampled by the interacting W boson is denoted as region A. The nucleon spectator region not probed by the boson is region B. Figure from [5].*

whether this same connection is manifested in weak interactions mediated by massive vector bosons. In this section that study, taken mainly from [5], is made using charged-current weak interaction processes such as

$$
\overline{\nu}\_{\mu} + N \to \mu^{+} + \pi^{0} + X \tag{12}
$$

Similar to the partial probing of the nucleon wave function described in Section 2 the vector boson in this investigation probes only a part of the nucleon wave function, again denoted by the region **A** in **Figure 6**. This probed region has a transverse size of approximately *d* ¼ *h=pW*, and a longitudinal size of approximately *<sup>l</sup>* <sup>¼</sup> ð Þ *mx* �<sup>1</sup> [1, 2, 14]. In this analysis, *<sup>h</sup>* is Planck's constant, *pW* is the boson'<sup>s</sup> momentum, *x* is the momentum fraction carried by the struck quark in the interaction (Bjorken-*x*), and *m* is the nucleon mass. Within the struck nucleon, the probed region **A** is complementary to the spectator region **B** that is not probed in the interaction. The entire space within the nucleon (a pure state) is then **A**∪**B**. In this present analysis, as in [14], thermal behavior is attributed to the quantum entanglement between regions **A** and **B** as depicted in in **Figure 6**.

In this current analysis, we test the hypothesis, albeit disfavored by the conventional mechanism of thermalization, that the thermal feature found in the low-*pT* region (corresponding to measurement at late times) of the momentum distribution can instead be attributed to the sub-nucleonic entanglement induced by collisions at high energies. This is the gist of the study using charged-current anti-neutrino interactions at the intensity frontier in particle physics. The claim from the the first two sections of this chapter is further strengthened by the demonstration that when the nucleus as a whole is scattered by the *W* boson so that no sub-nucleonic entanglement is produced, the thermal feature is absent from the spectrum, as expected. And that when quantum entanglement exists in the process, thermalization is present in the momentum distribution.

#### **3.1 Charged current weak interactions: analysis and results**

We begin by considering neutral pion production in charged-current antineutrino interactions with a CH (hydrocarbon scintillator) target; see (Eq. (12)). This *Quantum Information Science in High Energy Physics DOI: http://dx.doi.org/10.5772/intechopen.98577*

experimental data includes the total inclusive charged current weak interaction differential cross sections [39, 40] measurements at 1*:*5 *GeV* <*E<sup>ν</sup>* <10 *GeV* [39] and data at *E<sup>ν</sup>* ¼ 3*:*6 *GeV* [40]. The analysis results from both references, and from [5], are described in this present study. A conversion from pion kinetic energy (*Tπ*) published in [39] to pion momentum published in [40] is made using the expression

$$\frac{d\sigma}{dp\_{\pi}} = \frac{p\_{\pi}c^{2}}{T\_{\pi} + m\_{0,\pi}c^{2}} \frac{d\sigma}{dT\_{\pi}}.\tag{13}$$

The relativistic kinetic energy is related to the pion rest mass, *m*0,*<sup>π</sup>c*2, by

$$T\_{\pi} = (\gamma - \mathbf{1}) m\_{0,\pi} c^2 \tag{14}$$

where *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>v</sup>*<sup>2</sup>*=c*<sup>2</sup> <sup>p</sup> , with *<sup>v</sup>* the pion velocity in this case. We will compare the above results against the inclusive charged-current coherent pion production differential cross sections given in [41].

The normalized differential cross section that is used to describe the thermal behavior from the interaction is given by a very similar formula as in subSection 2.2 but here using

$$\frac{1}{p\_{\pi}} \frac{d\sigma}{p\_{\pi}} = A\_{\text{thermal}} e^{(-E\_{\pi}/T\_{\text{thermal}})} \tag{15}$$

where *<sup>p</sup><sup>π</sup>* (*E<sup>π</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *m*<sup>2</sup> *<sup>π</sup>* þ *p*<sup>2</sup> *π* p ) is the pion momentum (energy) and where the Mandelstam variable *<sup>s</sup>* is approximately equal to *<sup>m</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*Eν<sup>m</sup>* The hard-scattering part of the normalized momentum distribution is given by

$$\frac{1}{p\_{\pi}} \frac{d\sigma}{p\_{\pi}} = A\_{\text{hard}} \left( \mathbf{1} + \frac{m\_{\pi}^2}{T\_{\text{hard}}^2 \cdot n} \right)^{-n} \tag{16}$$

where *n* a power law scaling parameter. These equations are also discussed in [14, 42].

The CERN ROOT fitting program is used to fit these expressions to the MINERvA results. A total of five parameters are used in the fitting procedure: *T*thermal, *T*hard, *n*, *A*hard, and *A*thermal. In each case, the reduced chi-squared statistic and the fitting parameters with their associated uncertainties are recorded.

The results of fitting the thermal and hard scattering components to the distribution in the analysis using data from the MINER*ν*A collaboration [39, 40] are shown in **Figure 7**. As can seen from the fit, there are separate thermal (red-dashed) and hard-scattering (green-full) components in the full momentum distributions. The solid blue curve is the superposition of the exponential and power law fits.

Final state interactions (FSI) are modeled using the GENIE Monte Carlo program [43] in the anayses described in [39, 40]. They show that the larger FSI effects on the data are at low pion momenta. These effects are small compared with the statistical and other systematic uncertainties from the analysis, and did not affect the fits and conclusions drawn in this present study.

Now consider the resulting momentum distribution when the process of antineutrino scattering is from the entire nucleus, and not from a partial region of the nucleon as described above. That is, when the antineutrino scatters from the nucleus coherently, as in

$$
\overline{\nu}\_{\mu} + A \to \mu^{+} + \pi^{-} + A.\tag{17}
$$

In this charged current weak interaction, there is no entanglement between different parts of a struck nucleon, and no thermal component to the momentum distribution of the single produced pion is expected. It is this description of the interaction that is supported by the coherent scattering data from the MINER*ν*A collaboration [41], as shown in **Figure 8**. Only the hard scattering (power law) fit component is needed to describe the momentum distribution. The absence of a thermal (exponential) fit component is due to the absence of entanglement in the proposition presented in this present work.

#### **Figure 7.**

*Antineutrino differential cross section for scattering against ahydrocarbon nuclei with resulting charged current pion production. The dashed (red) line fit to the data is the thermal component fit and the thick solid (green) line shows the hard component fit. The combined thermal and hard scattering thin solid (blue) line best fits to the data. Data taken from [39, 40]. Plot taken from [5].*

#### **Figure 8.**

*Coherent scattering of the antineutrino from the hydrocarbon scintillator nuclei results in the momentum distribution shown here. The differential cross section is well described by a hard-scattering component (solid green line) alone, as expected in the absence of entanglement. The data is from [41]. The figure is from [5].*

*Quantum Information Science in High Energy Physics DOI: http://dx.doi.org/10.5772/intechopen.98577*


#### **Table 1.**

*The ratio R is defined in (Eq. (10)) for different processes as shown. The results listed indicate that the thermal behavior due to entanglement entropy is independent of the interaction (strong or electroweak) but process dependent.*
