Neutral current neutrino-electron scattering and generalized uncertainty principle

In this article, we study the effect of quantum gravitational corrections on neutrino electron scattering in electroweak theory SU(2) L × U(1) Y . We consider a neutrino that scatters off an electron by an exchange of a massive neutral gauge boson. In the minimal length formalism, starting from the equation of motion for a massive gauge field we obtain the propagator. And currents are obtained from the action for a massless Dirac fermion. Scattering amplitudes and differential cross sections as function of inelasticity variable y are obtained using expressions of the propagator and leptonic current. All usual expressions are recovered when the deformation parameter α tends to zero. We find that quantum gravitational effects become important only for a highly energetic process with a momentum transfer being much greater than the neutral gauge boson mass.


Introduction
The probing of the constituent of matter through scattering has allowed us to further our understanding of natural laws, for instance electron-proton scattering has been utilized to learn about the inner structure of proton.But, more details of the structure require higher scattering energies, in other words, the probing of extremely short distance scales require higher energies which result in nonnegligible quantum gravitational effects, especially for scattering at extremely high energies.Once taken into account, these effects lead to a modification of the usual Heisenberg uncertainty principle which introduces a minimal length scale in natural phenomena.Surprisingly, the minimal length scale is a common feature in most tentative theories of quantum gravity [1][2][3][4][5][6][7][8][9][10].Therefore, to take into account of these gravitational effects in scattering theory we can just reformulate the theory of scattering in the presence of the minimal length.There have been various proposals for the implementation of the minimal length scale [11][12][13][14][15][16][17][18][19][20][21][22].But if we want to preserve the principles of relativity, the minimal length scale can be implemented using the following generators [22], [23]   where is   corresponds the dimensions for a 3D space.We may consider q  and p  as low-and high-energy momentum respectively.The conjugate generators are chosen such that both p  and q  become Poincare invariant (conserved) quantities.
The organization of the article is as follows: In Sec. 2 we derive the propagator for a massive gauge field from the equation of motion with a minimal length and current from the Lagrangian density with minimal length.In Sec. 3 we derive the corrected scattering amplitudes and corrected differential cross sections for the neutral neutrino-electron scattering for both low and high energy neutrinos.Finally, in Sec. 4 we summarize the article.

Massive gauge Field Lagrangian
where 2 .   is defined as d' Alembert operator, and m is just the mass of the gauge field If we want to include quantum gravity corrections, we just apply Eq. ( 1) by replacing the usual d'Alembert operator by the corrected d' Alembert operator, then Eq. ( 2) can be expressed as We can derive the Feynman propagator and S-matrix for the gauge field equation with a minimal length using the usual rules.The propagator which is associated with Eq. ( 3) can be written as follows For a four-momentum q , the Fourier representation for the propagator is If we substitute Eq. ( 5) into Eq.( 4) we obtain It is clearly seen that the Feynman propagator that satisfies both sides of Eq. ( 6) is where     , , 1 q m q q m       .We realize that the propagator depends on  , and the usual propagator is obtained in the limit, 0   .
In accordance with Eq. ( 1) the action for a massless fermion ML  is expressed as For brevity, we will omit the subscript   ML and denote by  the corrected fermion wave-function ML  .In electroweak theory, the covariant derivative can be expressed as [29]   jw D igW ig y B ig A where w g A gW g yB Expanding in terms of ordinary derivatives   , Eq. ( 8) becomes Now, we can write Eq. ( 10) in terms of W  and B  as follows  The interaction Lagrangian for boson exchange up to   where .
The gauge field W  is responsible for the charged current, and the remaining fields are responsible for the neutral current.
The charged current is In a more familiar form, Eq. ( 13) can be written as where 2 Since the neutral weak interaction is mediated by a neutral boson Z and a photon  , and the hypercharge gauge boson field B  is not equivalent to the electromagnetic field A  , then we can choose to express 3 W  and B  in terms of neutral boson field Z  and the photon field A  as follows [29] 3 cos sin sin cos where w  is the weak mixing angle.
Using Eq. ( 12) and Eq. ( 15), the neutral current is In terms of   which is defined in Eq. ( 14), we can write Eq. ( 16) in a more familiar form as follows If we consider a neutrino scattering which is mediated by a weak neutral gauge boson, then Eq. ( 17) becomes We realize that the usual and corrected currents have the same structure under    .This means that the corrected effective Lagrangian is obtained by simply replacing   by   in the usual effective Lagrangian.
The four-momentum is generally dependent of  .But if the initial or final state of   x  is measured in the usual space, then  is explicitly independent of  .In this article, we choose the initial and final four-momentum of   x  to be explicitly independent of  .Now, let us consider the scattering of a muon neutrino off an electron by an exchange of a weak neutral gauge boson.Let k and k be the initial and final four-momentum of a neutrino.Let p and p be the initial and final four-momentum of an electron.For a neutral current, the scattering amplitude can be expressed in terms of Dirac spinors as follows where is defined as the reduced neutral weak fine structure constant, whereas z g is the usual neutral weak fine structure constant.
Since the phase space factor integral does not change due to the invariance of the dispersion relation [22], [23] then the differential cross section is where the quantity   4. pk is the incident flux.Since the initial state of  is explicitly independent of  , then   4. pk is just the usual incident flux.
In terms of Eq. ( 7), Eq. ( 20) becomes If the polarization is not observed, then in terms of trace, Eq. ( 22) leads to In a more compact form, Eq. ( 23) becomes For a straightforward study we can write Eq. ( 24) in a tensor form as follows , , Now, we evaluate the traces in Eq. ( 24).
For a neutrino For an electron In high energy scattering we can still consider a low or a high momentum transfer.In the case where the momentum transfer is much less than the mass of the mediating boson, we can consider,  reduces to 2 m then we can just substitute Eq. ( 26) and Eq. ( 27) into Eq.( 25) and we obtain For 2 0 k  , Eq. ( 28) becomes .
In the case where the momentum transfer is much greater that the gauge boson mass, we can consider, Then Eq. ( 29) can be written as .
Let us define a convenient variable . . . .
In terms of y , Eq. ( 29) is expressed as In terms of y , Eq. (31) becomes To obtain the differential cross section with respect to y , we insert the  function constraint in Eq. ( For the case where the momentum transfer is much less than the boson mass, if we substitute Eq. (35) into Eq.( 37) we obtain the following differential cross section Using Eq. (39) we recover the usual cross section in the limit, 0   .The differential cross section for the low energy case.Since the quantum gravitational effects are significantly small, the graph for usual differential cross section coincides with the graph for the corrected differential cross section For the case where the momentum transfer is much greater than the boson mass, if we substitute Eq. (36) into Eq.(37) we obtain the following differential cross section Using Eq. (40) we recover the usual cross section in the limit, 0   .MeV .For the low energy, quantum gravitational effects are significantly small so that the usual differential cross section and the corrected differential cross section overlap.However, for a highly energetic process with a momentum transfer much greater than the gauge boson mass there is an observable difference between the usual differential cross section and the corrected cross section for small values of y , and for high values of y there is an overlap which possibly indicates that the target becomes transparent due to a high energy transfer because the neutrino beam would tend to lose energy as y increases.

Summary
In this article, we studied quantum gravitational effects on neutrino-electron scattering for low and high momentum transfer for both low and high-energy neutrinos.The appropriate propagator is obtained from the equation of motion for a massive gauge field.The scattering amplitude is expressed in terms of currents which are obtained from the action for a massless fermion.All corrected expressions depend on  and the usual expressions are obtained in the limit 0   .With the assumption that the minimal length scale is near the Planck scale, we found that gravitational effects are not significant for a low energy process, but for a highly energetic process with a momentum transfer much greater than the gauge boson mass gravitational effects become significant.
length, where 0  is a dimensionless deformation parameter, and p is the Planck length.The quantities x  and q  are the index  has values, to time, and i Consider a neutrino scattering from an electron.If   e Jx  is the current of an electron, in the Lorentz gauge,  0Ax    , the equation of motion for the massive gauge field which is exchanged between a neutrino and an electron is[26][27][28]

3 .
Scattering amplitude and total cross section for ee     Let us consider a corrected fermion wave-function of the form

Figure 1 .
Figure 1.The differential cross section for the low energy case.Since the quantum gravitational effects are significantly small, the graph for usual differential cross section coincides with the graph for the corrected differential cross section

Figure 2 .
Figure 2. The differential cross section for the high momentum transfer case.The graph on the right, 0   , corresponds to the usual differential cross section.The graph on the left,

Figures ( 1 )
Figures(1), and(2) show the comparison between the usual differential cross section and the corrected differential cross section for the case where the momentum transfer is less than the gauge boson mass and for the case where the momentum transfer much greater than the boson mass respectively.In each figure, the usual cross section corresponds to 0   and the corrected differential cross section corresponds to 1   .
and integrate.