Experimental test of Non-Commutative Quantum Gravity by VIP-2 Lead

The SST is based on Lorentz invariance as a fundamental assumption. This means that the PEP is directly related to the fate of the space-time symmetry and structure. Nonetheless, Lorentz Symmetry may be dynamically broken at a very high energy scale, without this phenomenon translating into a fundamental breakdown of the symmetry. In this case the generation of non-renormalizable operators, suppressed as inverse powers of the Lorentz violation scale $\Lambda$, is expected. On the other hand approaches to Quantum Gravity, for which space-time coordinates do non commute close to the Planck scale (about $10^{19}\,{\rm GeV}$), thus deforming the Lorentz algebra at the very fundamental level, were put forward. The idea that the space-time might be non-commutative is usually accredited to W. Heisenberg (see e.g. Ref. Heisenberg ), as an extension of the uncertainty principle, having been later elaborated by H. Snyder and C.N. Yang in Refs. Snyder ; Yang . From a symplectic-geometry approach SymGeo it is possible to unveil the deep relation intertwining space-time symmetries, statistics Arzano:2007gr and the uncertainty principle SymGUP , hence providing concrete path-ways for falsification.

Non-commutativity of space-time is common to several Quantum Gravity frameworks, to which we refer as Non-Commutative Quantum Gravity models (NCQG). The connection of space-time non-commutativity with both String Theory (ST) Frohlich:1993es ; Chamseddine1 ; Frohlich:1995mr ; Connes:1997cr ; Seiberg:1999vs and Loop Quantum Gravity (LQG) AmelinoCamelia:2003xp ; Freidel:2005me ; Cianfrani:2016ogm ; Amelino-Camelia:2016gfx ; NClimLQG ; BRAM ; Brahma:2017yza was extensively studied in literature¹¹1The fact that non-commutativity emerges in both theories may not be a coincidence, but can be conjectured to be instead related to a newly formulated ${\bf H}$-duality — see e.g. Refs. Addazi:2017qwt ; Addazi:2018cyn — since a self-dual LQG formulation can be obtained from topological M-theory. Besides ST and LQG contexts, deformed symmetries may effectively emerge from several other non-perturbative models of quantum geometry..

The two main classes of non-commutative space-time models embedding deformed Poincaré symmetries are characterized by $\kappa$-Poincaré Agostini:2006nc ; 13 ; AmelinoCamelia:2007uy ; Arzano:2016egk and $\theta$-Poincaré AmelinoCamelia:2007wk ; AmelinoCamelia:2007rn ; Addazi:2017bbg ; Addazi:2018jmt ; Addazi:2019ruk ; Addazi:2018ioz symmetries. Among these latter, there exists a sub-class of models which preserves unitarity of the S-matrix in the Standard Model sector AG ; Addazi:2018jmt . From the experimental point of view, the most intriguing prediction of this class of non-commutative models is an energy dependent probability ($\delta^{2}(E)$) for electrons to perform PEP violating atomic transitions. For both $\kappa$ and $\theta$ Poincaré the PEP violation probability turns to be of order one in the deformation parameter, when the probed energy is close to the scale of non-commutativity $\Lambda$ (nonetheless, trivial dimensional arguments are sufficient to show that first order results in $\theta$ Poincaré are suppressed by the square of the energy scale). In the low energy regime, i.e. for energies much smaller than the non-commutativity scale, the PEP violation probability is highly suppressed, accounting for the lack of evidence of PEP violation signals over decades of experimental efforts in this direction.

The experimental tests of the Spin-Statistics connection are based on the search for signals of PEP violating processes, by exploiting different techniques, which may or not respect the Messiah and Greenberg (MG) messiah superselection rule. According to MG, even assuming small mixed symmetry components in a primarily antisymmetric wave function, in a system composed by a fixed number of particles the symmetry of the world Hamiltonian would prevent transitions among two different symmetry states, i.e. electrons or nucleons would not perform PEP forbidden transitions to lower orbits.

PEP tests for electrons, respecting the constraint imposed by MG, are performed by introducing new electrons in a pre-existing system of electrons and constraining the probability that the newly formed state is symmetric. This was accomplished using various methods:

• •

  the estimate of the primordial abundance of anomalous ⁵Li, with three protons in the lowest level, leads to the limit ($\delta^{2}<2\cdot 10^{-28}$) thoma1992 , which is also the strongest constraint inferred from astrophysical and cosmological arguments;

• •

  capture of ¹⁴C $\beta$ rays onto Pb atoms ($\delta^{2}<3\cdot 10^{-2}$) Goldhaber1948 ;

• •

  pair production electrons captured on Ge ($\delta^{2}<1.4\cdot 10^{-3}$) Elliott:2011cx ;

• •

  search for PEP violating atomic transitions in conducting targets, a prototype experiment of this class has been carried out in Ref ramberg1990 following a suggestion of Greenberg and Mohapatra Greenberg:1987aa , it consists in measuring the X-ray emission from a target strip where a direct current is circulated and looking for a difference in the spectra acquired with current on and off (best upper limit $\delta^{2}<8.6\cdot 10^{-31}$) curceanu2017 ; napolitano2022 ;

• •

  a generalized version of the latter experimental technique consists in using as test fermions the free electrons residing in the conduction band of the target, an improved version of the original analysis proposed in Ref. Elliott:2011cx , based on the calculation of the time dependence of the anomalous X-ray emission process, lead to the best upper limit $\delta^{2}<1.53\cdot 10^{-43}$ Piscicchia:2020kze .

NCQG models are not subject to the MG superselection rule. Within this context the prototype Reines-Sobel experiment  reines looked for anomalous transitions in a stable system, i.e. spontaneous PEP-violating transition in a closed system of electrons in an established symmetry state. In this framework strong bounds on the PEP violation probability were set by the DAMA/LIBRA collaboration ($\delta^{2}<1.28\cdot 10^{-47}$) bernabei2009 , searching for K-shell PEP violating transitions in iodine — see also Refs. ejiri . A similar strategy was followed by the MALBEK experiment abgrall , which set the bound $\delta^{2}<2.92\cdot 10^{-47}$ by constraining the rate of K_α PEP violating transitions in Germanium.

This research is based on a different strategy, which is not confined to the evaluation of a specific transition PEP violation probability. We consider the predicted PEP violating atomic transition amplitudes, in the context of the $\theta$ Poincaré model, whose derivation is outlined in Sections III and IV. The deformation of the standard transition probabilities depends on the typical energy scales of the involved reactions, conditioned by the value of the “electric like” components of $\theta_{\mu\nu}$. This enables a fine tuning of the $\theta$ tensor components. The predicted energy dependence of the spectral shape of the whole complex of relevant transitions is accounted for and tested against the measured X-rays distribution, constraining $\Lambda$, for the first time, far above the Planck scale for $\theta_{0i}\neq 0$. Within a similar theoretical framework the DAMA/LIBRA limit on the PEP violating atomic transition probability bernabei2009 was analyzed in Ref. Addazi:2018ioz , and a lower limit on the non-commutativity scale $\Lambda$ was inferred as strong as $\Lambda>5\cdot 10^{16}$ GeV, corresponding to $\Lambda>4\cdot 10^{-3}$ Planck scales.

PEP violating nuclear transitions were also investigated e.g. in Refs. bernabei2009 ; bellini2010 ; suzuki1993 . The strongest bound ($\delta^{2}<7.4\cdot 10^{-60}$) was obtained in Ref. bellini2010 searching for PEP violating transitions of nucleons from the 1P_3/2 shell to the filled 1S_3/2 shell. Based on a parametrization of the PEP-violation probability, in terms of inverse powers of the non-commutativity scale, the impact of these experimental results for Planck scale deformed symmetries was investigated in Ref. Addazi:2017bbg . As a result a class of $\kappa$-Poincaré and $\theta$-Poincaré models was excluded in the hadronic sector. Nonetheless, within the context of NCQG models, tests of PEP in the hadronic and leptonic sectors need to be considered as independent. There is no a priori argument why the standard model fields should propagate in the non-commutative space-time background with the same coupling. As an example non-commutativity emerges in string theory as a by-product of the constant expectation value of the B-field components, which in turn are coupled to strings’ world-sheets with magnitudes that are not fixed a priori.

Underground X-ray surveys, searching for atomic transitions prohibited by the PEP, proved in recent years to provide very strong and promising tests of Spin-Statistics vipepjc2018 ; piscicchiaentropy2020 ; Piscicchia:2020kze and, hence, tests of Quantum Gravity models at unexpectedly high non-commutativity scales. The results of the dedicated VIP-2 Lead experiment, searching for signals of K_α and K_β PEP violating transitions in an ultra radio-pure Roman lead target, are analysed in the framework of NCQG, and summarized in Ref. PRL_version . We show that the experiment sets critical bounds on $\theta$-Poincaré. In particular, VIP-2 Lead rules-out $\theta$-Poincaré up to $2.6\cdot 10^{2}$ Planck scales, if the “electric like” components of the $\theta_{\mu\nu}$ vector are non-null; $\theta$-Poincaré is excluded up to $6.9\cdot 10^{-2}$ Planck scales if they vanish. VIP-2 Lead provides the strongest bound, from atomic levels transitions, on this special type of space-time non-commutativity.

Problems related to a-causality for non-commutative theory were outlined in Ref. Seiberg:2000gc , in which the authors derived the commutator $[\mathcal{O}(x),\mathcal{O}(y)]_{\star}$. Similarly, using perturbation theory and different definitions of time-ordering, it was pointed out in Ref. Bozkaya:2002at that micro-causality and unitarity cannot simultaneously co-exist in non-commutative field theory. A similar result was obtained in Ref. AlvarezGaume:2001ka , where it was found that SO$(3,1)$ micro-causality is violated, but that SO$(1,1)$ micro-causality in the light-cone still holds when perturbative unitarity does. In Ref. Gomis:2000zz it was then pointed out that non-commutativity among space and time coordinates violates unitarity, unless light-like non-commutativity is considered Aharony:2000gz . This problem was supposed to arise because of neglecting massive string states.

A different take on the same problem was developed in Ref. Chaichian:2002vw , where it was argued that for pure space non-commutativity micro-causality is preserved. As a by-product of this analysis, Pauli spin-statistics relation was claimed to remain valid in non-commutative quantum field theories. Nonetheless, as pointed in Ref. Greenberg:2005jq , the special case inspected in Ref. Chaichian:2002vw that involves scalar fields with no time derivatives, only make sense in the light-wedge $x_{0}^{2}-x_{3}^{2}\leq 0$, as already shown in Ref. AlvarezGaume:2001ka . The implications of Ref. Chaichian:2002vw for the spin-statistic theorem contradict the results found in Greenberg:2005jq , namely that the equal time commutator in D space-time dimensions

	$\displaystyle[\mathcal{O}(x),\mathcal{O}(y)]_{\star}=(e^{-\imath p\cdot x-\imath p^{\prime}\cdot y}+e^{-\imath p^{\prime}\cdot x-\imath p\cdot y})\times$
	$\displaystyle\frac{8\imath}{(2\pi)^{2D-1}}\int\frac{d^{D-1}k}{2E_{k}}\,\sin\left(-\frac{1}{2}\theta^{ij}\left(k_{i}(p+p^{\prime})_{j}+p_{i}{p_{j}}^{\prime}\right)\right)\times$
	$\displaystyle e^{\imath\vec{k}\cdot(\vec{x}-\vec{y})}\,\cos(\frac{1}{2}\theta^{ij}k_{i}p_{j})\,\cos(\frac{1}{2}\theta^{ij}k_{i}{p_{j}}^{\prime})$		(3)

does not vanish at space-like separation. This latter result is instead consistent with previous ones derived in the literature – see e.g. Refs. Seiberg:2000gc ; Bozkaya:2002at ; AlvarezGaume:2001ka ; Gomis:2000zz ; Aharony:2000gz , and thus hinges toward the confirmation that the spin-statistic theorem is violated.

The existence of eventual isomorphism has also been considered in order to instantiate a pure equivalence among commutative and non-commutative quantum field theories Fiore:2007vg . Notice that a complete equivalence among the commutative and non-commutative theories, both at the level of the algebra and the co-algebra, would turn non-commutative theory to be un-falsifiable and un-predictive. Consequently, we do not deem as worthy of any phenomenological interest this theoretical approach, since it fails from distinguishing itself from the standard theory and its predictions, and thus decide not to speculate on the implications of this possibility.

The $\star$-product induces deformed commutation and anti-commutation rules:

	$\displaystyle a^{(s_{1})}(p_{1})a^{(s_{2})}(p_{2})=$	$\displaystyle-e^{\imath p_{1}\wedge_{\theta}p_{2}}a^{(s_{2})}(p_{2})a^{(s_{1})}(p_{1})\,,$
	$\displaystyle b^{(s_{1})}(p_{1})b^{(s_{2})}(p_{2})=$	$\displaystyle-e^{\imath p_{1}\wedge_{\theta}p_{2}}b^{(s_{2})}(p_{2})b^{(s_{1})}(p_{1})\,,$
	$\displaystyle\alpha^{(s_{1})}(p_{1})\alpha^{(s_{2})}(p_{2})=$	$\displaystyle+e^{\imath p_{1}\wedge_{\theta}p_{2}}\alpha^{(s_{2})}(p_{2})\alpha^{(s_{1})}(p_{1})\,,$

having introduced the antisymmetric product $a\wedge_{\theta}b\equiv a_{\mu}\theta^{\mu\nu}b_{\nu}$, and

	$\displaystyle a^{(s_{1})}(p_{1})a^{\dagger(s_{2})}(p_{2})=$	$\displaystyle-e^{\imath p_{1}\wedge_{\theta}p_{2}}a^{\dagger(s_{2})}(p_{2})a^{(s_{1})}(p_{1})$
		$\displaystyle+\delta_{\vec{p}_{1},\vec{p}_{2}}\,,$
	$\displaystyle b^{(s_{1})}(p_{1})b^{\dagger(s_{2})}(p_{2})=$	$\displaystyle-e^{\imath p_{1}\wedge_{\theta}p_{2}}b^{\dagger(s_{2})}(p_{2})b^{(s_{1})}(p_{1})$
		$\displaystyle+\delta_{\vec{p}_{1},\vec{p}_{2}}\,,$
	$\displaystyle\alpha^{(s_{1})}(p_{1})\alpha^{\dagger(s_{2})}(p_{2})=$	$\displaystyle+e^{\imath p_{1}\wedge_{\theta}p_{2}}\alpha^{\dagger(s_{2})}(p_{2})\alpha^{(s_{1})}(p_{1})$
		$\displaystyle+\delta_{\vec{p}_{1},\vec{p}_{2}}\,.$

Consistently with these rules, asymptotic in-coming and out-going two-fermion states are represented as

	$\displaystyle|p_{1},s_{1};p_{2},s_{2}\rangle$	$\displaystyle=$	$\displaystyle a^{\dagger(s_{1})}(p_{1})a^{\dagger(s_{2})}(p_{2})|0\rangle$		(5)
		$\displaystyle=$	$\displaystyle e^{\imath p_{1}\wedge_{\theta}p_{2}}c^{\dagger(s_{1})}(p_{1})c^{\dagger(s_{2})}(p_{2})|0\rangle\,,$

and

	$\displaystyle|p_{1}^{\prime},s_{1}^{\prime};p_{2}^{\prime},s_{2}^{\prime}\rangle$	$\displaystyle=$	$\displaystyle a^{\dagger(s_{1}^{\prime})}(p_{1}^{\prime})a^{\dagger(s_{2}^{\prime})}(p_{2}^{\prime})|0\rangle$		(6)
		$\displaystyle=$	$\displaystyle e^{\imath p_{1}^{\prime}\wedge_{\theta}p_{2}^{\prime}}c^{\dagger(s_{1}^{\prime})}(p_{1}^{\prime})c^{\dagger(s_{2}^{\prime})}(p_{2}^{\prime})|0\rangle\,,$

having introduced ladder operators $c$ and $d$ phase-shifted with respect to $a$ and $b$ by

	$\displaystyle a_{\vec{p}}=e^{-\frac{1}{2}\imath p\wedge_{\theta}P}c_{\vec{p}}\,,\qquad b_{\vec{p}}=e^{-\frac{1}{2}\imath p\wedge_{\theta}P}d_{\vec{p}}$
	$\displaystyle{\rm with}\qquad P_{\mu}=\sum_{\vec{p}}\left(c^{\dagger}_{\vec{p}}c_{\vec{p}}+d^{\dagger}_{\vec{p}}d_{\vec{p}}\right)$

and so forth for the hermitian conjugates.

It is useful to consider the deformation of the Feynman amplitude $\mathcal{M}$ of the electron-muon scattering process, as recovered from the interaction vertex derived from Eq. (4)and from the asymptotic states in Eqs. (5)-(6). This reads

	$\displaystyle\mathcal{M}_{\theta}$	$\displaystyle=$	$\displaystyle e^{\frac{\imath}{2}\left(p_{1}\wedge_{\theta}p_{2}-p_{1}^{\prime}\wedge_{\theta}p_{2}^{\prime}\right)}\mathcal{M}_{0}e^{-\frac{\imath}{2}\left(p_{1}^{\prime}\wedge_{\theta}p_{1}+p_{2}^{\prime}\wedge_{\theta}p_{2}\right)}\,,$
	$\displaystyle\mathcal{M}_{0}$	$\displaystyle=$	$\displaystyle\frac{e^{2}}{2q^{2}}\left(\bar{u}^{s_{1}^{\prime}}(p_{1}^{\prime})\gamma^{\mu}u^{(s_{1})}(p_{1})\bar{v}^{s_{2}^{\prime}}(p_{2}^{\prime})\gamma_{\mu}v^{(s_{2})}(p_{2})\right)\,,$

with transferred momentum $q=p_{1}^{\prime}-p_{1}=p_{2}^{\prime}-p_{2}$. Averaging over the initial spins and summing over final ones, the deformation rearranges, in the cross section $\sigma$, into a multiplicative factor

	$\displaystyle\sigma_{\theta}$	$\displaystyle=$	$\displaystyle|\Phi_{\theta}|^{2}\sigma_{0}\,,$
	$\displaystyle\Phi_{\theta}$	$\displaystyle=$	$\displaystyle e^{\imath(p_{1}\wedge_{\theta}p_{2}-p_{1}^{\prime}\wedge_{\theta}p_{2}^{\prime})}e^{-\frac{\imath}{2}(p_{1}^{\prime}\wedge_{\theta}p_{1}-p_{2}^{\prime}\wedge_{\theta}p_{2})}\,.$

The first order expansion in the $\theta$ tensor of $|\Phi_{\theta}|^{2}$ provides the PEP-violating phase $\phi_{\rm PEPV}=\delta^{2}$.

For a generic NCQG model deviations from the PEP in the commutation/anti-commutation relations can be parametrized Addazi:2017bbg as

$$a_{i}a_{j}^{\dagger}-q(E)a_{j}^{\dagger}a_{i}=\delta_{ij}\,.$$

(7)

For $\theta$-Poincaré models²²2In principle, there may exist other models which may have an energy dependence from the electron momentum or mass. It is also interesting to remark that, in general, an angular dependence of the PEPV emission is expected., when two electrons of momenta $p_{i}^{\mu}=(E_{i},\vec{p}_{i})$ (with $i=1,2$) are considered, we may introduce the phase $\phi_{\rm PEPV}$ to parametrize the deformation of the standard transition probability $W_{0}$ into $W_{\theta}=W_{0}\cdot\phi_{\rm PEPV}$. Making explicit $\Lambda$ in the $\theta$-tensor through the relation $\theta_{\mu\nu}=\tilde{\theta}_{\mu\nu}/\Lambda^{2}$, with $\tilde{\theta}_{\mu\nu}$ dimensionless, the energy scale dependence may be introduced: i) either according to

$$\phi_{\rm PEPV}=\delta^{2}\simeq\frac{D}{2}\frac{E_{N}}{\Lambda}\frac{\Delta E}{\Lambda}\,,$$

(8)

where $D=p_{1}^{0}\tilde{\theta}_{0j}p_{2}^{j}+p_{2}^{0}\tilde{\theta}_{0j}p_{1}^{j}$, the quantity $E_{N}\simeq m_{N}\simeq A\,m_{p}$ denotes nuclear energy and $\Delta E=E_{2}-E_{1}$ accounts for the atomic transition energy; ii) or as an energy levels difference, encoding the PEP violating X-ray line energy, namely as in

$$\phi_{\rm PEPV}=\delta^{2}\simeq\frac{C}{2}\frac{\bar{E}_{1}}{\Lambda}\frac{\bar{E}_{2}}{\Lambda}\,,$$

(9)

where $\bar{E}_{1,2}$ are the energy levels occupied by the initial and the final electrons and $C=p_{1}^{i}\tilde{\theta}_{ij}p_{2}^{j}$. The former case, discussed in Eq. (8), amounts to encode non-commutativity among space and time coordinates, namely $\theta_{0i}\neq 0$, while the latter case, in Eq. (9), corresponds to selecting $\theta_{0i}=0$, which ensures unitarity of the $\theta$-Poincaré models AlvarezGaume:2001ka ; Gomis:2000zz . In both cases the factors $D/2$ and $C/2$ can be approximated to unity.

We further notice that Eq. (7) resembles the quon algebra (see e.g. Refs. Greenberg:1987aa ; greenberg1991particles ), in the case of quon fields however the $q$ factor does not show any energy dependence, and it is not related to any quantum gravity model. Thus the q-model necessarily requires a hyper-fine tuning of the $q$ parameter. $q(E)$ is related to the PEP violation probability by

$$q(E)=-1+2\delta^{2}(E).$$

(10)

An experimental bound on the probability that PEP may be violated in atomic transition processes, straightforwardly translates into a constrain to the new physics scale $\Lambda$, consistently with the choice of the $\theta_{0i}$ components.

The VIP-2 lead experimental apparatus is based on a high purity co-axial p-type germanium detector (HPGe), with a diameter of 8.0 cm and a length of 8.0 cm, surrounded by an inactive layer of lithium-doped germanium of 0.075 mm. The active germanium volume of the detector is 375 cm³. The detector is surrounded by a complex of passive and active shieldings (the latter serves as target material). The passive shielding is composed of an outer pure lead layer (30 cm from the bottom and 25 cm from the sides) and an inner electrolytic copper layer (5 cm). The volume of the sample chamber is of about 15 l ((250 × 250 × 240) mm³). A 1 mm thick air-tight steel housing encloses the shielding structure and the cryostat and is flushed with boil-off nitrogen, from a liquid nitrogen storage tank, to minimize the radon contamination. With the aim to reduce the neutron flux impinging the detector this is also surrounded, on the bottom and the sides, by 5 cm thick borated polyethylene plates. The target consists of three cylindrical sections of radio-pure Roman lead, fully surrounding the detector. The thickness of the target is about 5 cm, for a total volume $V\sim 2.17\times 10^{3}\;\mathrm{cm}^{3}$. The inner part of the setup is shown in Fig. 1 of Ref. Piscicchia:2020kze . A more detailed description of the external passive shielding and of the cryogenic and vacuum systems is given in Refs. neder ; heusser2006low . The data acquisition system is a Lynx digital signal analyser controlled via GENIE 2000 personal computer software, both from Canberra-Mirion.

The aim of the measurement is to search for the X-rays signature of PEP-violating K_α and K_β transitions in Pb, when the $1s$ level is already occupied by two electrons. These transitions are shifted, with respect to the standard ones, as a consequence of the shielding effect of the additional electron in the ground state; hence they are distinguishable in precision spectroscopic measurements. Let us notice that the deformation of the algebra preserves, at the first order, the standard atomic transition probabilities, the violating transition probabilities being dumped by factors $\delta^{2}(E)$, hence transitions to the $1s$ level from levels higher then $4p$ will not be considered (see e.g. Ref. krause for a comparison of the atomic transitions intensities in Pb). The energies of the standard $\mathrm{K}_{\alpha}$ and $\mathrm{K}_{\beta}$ transitions in Pb are given in Table 1, where those expected for the corresponding PEP violating ones are also reported. The PEP violating K lines energies are obtained based on a multi configuration Dirac-Fock and General Matrix Elements numerical code indelicato , see also Ref. Elliott:2011cx where the $\mathrm{K}_{\alpha}$ lines are obtained with a similar technique.

The efficiency for the detection of X-rays emitted inside the Pb target, in the energy region of interest (defined in Section VI), was determined by means of Monte Carlo (MC) simulations. To this end the detector was characterised and all of its components have been set into a MC code (Ref. Boswell:2010mr ) based on the GEANT4 software library (Ref. Agostinelli:2002hh ).

We present in Section VI the results of a Bayesian analysis, aimed to extract the probability distribution function (pdf) of the expected number of photons emitted in PEP violating K_α and K_β transitions. The analysed data set corresponds to a total acquisition time $\Delta t\approx 6.1\cdot 10^{6}\mathrm{s}\approx 70~{}\mathrm{d}$, i.e. about twice the statistics used in Ref. Piscicchia:2020kze .

The aim of the analysis is to extract the upper limit $\bar{S}$ of the expected number of total signal counts, generated by PEP violating K_α and K_β transitions in the target. Comparison of $\bar{S}$ with the theoretically expected photons emission, due to PEP violating atomic transitions, provides a limit on the scale $\Lambda$ of the model. The conditional pdf of the expected number of total signal counts $S$ given the measured distribution - called data - is obtained as follows:

$$P(S|data)=\int_{0}^{\infty}\int_{\mathcal{D}_{\mathbf{p}}}P(S,B|data,\mathbf{p})\ d^{m}\mathbf{p}\ dB$$

(11)

where the joint posterior distribution of $S$ and the expected number of total background counts $B$ is given by the Bayes theorem

$$P(S,B|data,\mathbf{p})=$$

$$=\frac{P(data|S,B,\mathbf{p})\cdot f(\mathbf{p})\cdot P_{0}(S)\cdot P_{0}(B)}{\int P(data|S,B,\mathbf{p})\cdot f(\mathbf{p})\cdot P_{0}(S)\cdot P_{0}(B)\ d^{m}\mathbf{p}\ dS\ dB}.$$

(12)

In order to account for the uncertainties in the experimental parameters $\mathbf{p}$, which characterize the measurement and the data analysis, an average likelihood is considered, which is weighted with the joint pdf of $\mathbf{p}$. $\mathcal{D}_{\mathbf{p}}$ represents the domain of the parameters’ space. The likelihood is parametrized as follows

$$P(data|S,B,\mathbf{p})=\prod_{i=1}^{N}\frac{\lambda_{i}(S,B,\mathbf{p})^{n_{i}}\cdot e^{-\lambda_{i}(S,B,\mathbf{p})}}{n_{i}!}\$$

(13)

where $n_{i}$ are the measured bin contents. The number of events in the $i$-th bin fluctuates, according to a Poissonian distribution, around the mean value:

	$\displaystyle\lambda_{i}(S,B,\mathbf{p})$	$\displaystyle=$	$\displaystyle B\cdot\int_{\Delta E_{i}}f_{B}(E,\bm{\alpha})\ dE+$
		$\displaystyle+$	$\displaystyle S\cdot\int_{\Delta E_{i}}f_{S}(E,\bm{\sigma})\ dE\$

$\Delta E_{i}$ is the energy range corresponding to the $i$-th bin; $f_{B}(E,\bm{\alpha})$ and $f_{S}(E,\bm{\sigma})$ represent the shapes of the background and signal distributions normalized to unity over $\Delta E$. Among the experimental uncertainties the only ones which significantly affect $\bar{S}$ are those which characterize the shape of background (parametrized by the vector $\bm{\alpha}$) and the resolutions ($\bm{\sigma}$) at the energies of the violating transitions (the resolutions are reported in Table 2). All the other experimental parameters are affected by relative uncertainties of the order of 1% (or less), which are neglected, hence $\mathbf{p}=(\bm{\alpha},\,\bm{\sigma})$. For the normalized background shape we take a flat distribution, as a result of the fit to the measured distribution described below. In order to obtain $f_{S}(E)$ the expected values of the numbers of PEP violating $K$ transitions, predicted by the model, are to be evaluated.

The most interesting phenomenological feature, emerging from the theory, consists on the PEP violating transition amplitude dependence on the characteristic energy scales involved in the investigated reactions (see e.g. Eqs. (8) and (9)). This, in turn, translates in a dependence on the atomic number of the adopted target, thus opening an intriguing scenario demanding a systematic analysis of the data from ongoing bernabei2009 ; abgrall and forthcoming experiments. Researches in analogy with what presented in Refs. Addazi:2017bbg ; Addazi:2018ioz , addressed on a scan of possible evidences for spin-statistics violation, would supplement the conclusions of this study.

The VIP collaboration is presently implementing an upgraded experimental setup, based on cutting-edge Ge detectors, aiming to probe $\theta$-Poincaré beyond the Planck scale, independently on the particular choice of the $\theta_{\mu\nu}$ electric like components. NCQG models, in a large number of their popular implementations, are not far to be tested and eventually ruled-out. In this sense, contrary to naive expectations NCQG is not only a theoretical attractive mathematical idea but also offers a rich phenomenology, suitable to be tested in high sensitivity X-ray spectroscopic measurements.