Lecture VI: Neutrino propagator and neutrino potential Petr Vogel, - PowerPoint PPT Presentation

Lecture VI: Neutrino propagator and neutrino potential Petr Vogel, Caltech NLDBD school, November 1, 2017

For the case we are considering, i.e. with the exchange of light Majorana neutrinos, the double beta decay nuclear matrix element consists of three parts: GT − M 0 ν M 0 ν = M 0 ν F + M 0 ν T ≡ M 0 ν GT (1 + χ F + χ T ) , g 2 A The Gamow-Teller part M GT is the dominant one. When treated in the closure approximation it is M 0 ν GT = ⟨ f | � lk σ l · σ k τ + l τ + k H ( r lk , ¯ E ) | i ⟩ , ¯ � � The ``neutrino potential” originating from the light neutrino propagator is � ∞ � � h K ( q 2 ) qdq 2 r 12 , E k H K = R f K ( qr 12 ) J π − ( E i + E f ) / 2 . J π π g 2 q + E k � 0 A functions ( ) ( ) and ( ) ( Where f GT (qr) = j 0 (qr) and h GT = g A /(1 + q 2 /M A 2 ) 2 is the nuclear axial current form factor, M A ~ 1 GeV.

As we will see, the neutrino momentum is ~200 MeV so the dependence on the nuclear excitation energy is weak. The potential H(r,E) looks like a Coulomb 1/r radial dependence. Finite size and higher order currents remove the singularity at r=0. fns….nucleon finite size hot…higher order terms in weak currents

Matrix elements M 0 ν evaluated in closure approximation using the QRPA method. Plotted against assumed average excitation energy. Values without the closure approximation indicated by arrows. Closure approximation underestimates M 0 ν by less than 10%. 6 5 M 0 ν cl 4 76 Ge 96 Zr 3 100 Mo 130 Te 2 0 2 4 6 8 10 12 E [MeV]

How does the matrix element M 0 ν GT depend on the distance between the two neutrons that are transformed into two protons ? This is determined by the function C 0 ν GT (r) k δ ( r − r lk ) H ( r lk , ¯ C 0 ν GT ( r ) = ⟨ f | � lk σ l · σ k τ + l τ + E ) | i ⟩ , � ∞ M 0 ν C 0 ν It is normalized by the obvious relation GT = GT ( r ) dr, 0 Thus, if we could somehow determine C(r) we could obtain M 0 ν . In order to obtain C(r) consider first the matrix elements of the operator σ 1 . σ 2 between two neutrons and two protons coupled to the angular momentum J without the neutrino potential: f J = ⟨ p (1) , p ′ (2)( r ); J ∥ σ 1 · σ 2 ∥ n (1) , n ′ (2)( r ); J ⟩ n,n ′ ,p,p ′ ( r ) (1) is introduced where is the relative distance between

Here are few examples for the f 7/2 and f 5/2 orbits. These functions, as expected, typically extend up to the nuclear diameter, peaking near the middle. Some of them, in particular those with J = 0, are asymmetric with larger amplitude at small distances. 0.3 f 7/2 f 7/2 f 7/2 f 7/2 J=0 0.25 f 7/2 f 7/2 f 7/2 f 7/2 J=2 f 7/2 f 7/2 f 7/2 f 7/2 J=4 0.2 f 7/2 f 7/2 f 7/2 f 7/2 J=6 0.15 f 5/2 f 5/2 f 7/2 f 7/2 J=0 f 5/2 f 5/2 f 7/2 f 7/2 J=2 C(r) 0.1 f 5/2 f 5/2 f 7/2 f 7/2 J=4 0.05 0 − 0.05 − 0.1 0 2 4 6 8 10 12 14 r (fm) Figure by G. Martinez

To obtain the matrix element M 0 ν , one has to include the `neutrino potential’ that stresses smaller values of r, and combine the s.p. states based on their || � × ⟨ || × ⟨ J ∥ contributions obtained by solving the corresponding equations of motion. f || [ � In QRPA they are the amplitudes and × ⟨ J π k i || [ c + c n ] J || 0 + × ⟨ 0 + c + c n ′ ] J || J π k f ⟩⟨ p ˜ i ⟩ p ′ ˜ + + � j p � � � ( − 1) j n + j p ′ + J + J √ j n J M K = 2 J + 1 j n ′ j p ′ J J π ,k i ,k f , J pnp ′ n ′ × ⟨ p (1) , p ′ (2); J ∥ ¯ f ( r 12 ) O K ¯ f ( r 12 ) ∥ n (1) , n ′ (2); J ⟩ f || [ � × ⟨ 0 + c + c n ′ ] J || J π k f ⟩⟨ J π k f | J π k i ⟩ p ′ ˜ × ⟨ J π k i || [ c + c n ] J || 0 + p ˜ i ⟩ . (4)

It is instructive to consider the contributions of different angular momenta J to the final result. This is a typical case; J = 0 contributes most, while other J have smaller amplitude but opposite sign; hence a substantial cancellation. Note the qualitative agreement between NSM and QRPA. This is for the 82 Se. The same s.p. space, f 5/2 , p 3/2 , p 1/2 , g 9/2 , is used for both. For QRPA this space is smaller than usual, thus smaller M 0 ν is obtained.

Function C 0 ν (r) evaluated in QRPA. Note the peak at ~ 1fm. There is no contribution from r> 2-3 fm. And the function for different nuclei look very similar, essentially universal. 6 76 Ge 100 Mo -1 ] 4 130 Te C(r) [fm 2 0 -2 7 0 1 2 3 4 5 6 8 9 10 r [fm] From Simkovic et al, Phys. Rev C 77 , 045503 (2008)

Now C(r) evaluated in the nuclear shell model. All relevant features look the same as in QRPA despite the very different way the equations of motion are formulated and solved. 3.5 3 A=48 A=76 2.5 A=82 A=124 2 A=130 C(fm -1 ) 1.5 A=136 1 0.5 0 From Menendez et al, Nucl. Phys. -0.5 A 818 , 130 (2009) 0 1 2 3 4 5 6 7 8 r(fm)

C(r) for the hypothetical 0 νββ decay of 10 He. 10 He 10 Be 0.4 The calculation was performed using the ab initio variational 0.3 Monte-Carlo method. So most of the approximations inherent in NSM or QRPA are avoided. -1 ] 0.2 Yet the C(r) function looks, C(r) [fm at least qualitatively, very similar to the results shown 0.1 before. 0 -0.1 0 2 4 6 0 r [fm] Figure from Pastore et al.,1710.05026

The fact that the resulting C(r) is concentrated at r<~2fm is the result of cancellation between J = 0 and other values of J . We have seen the effect of such cancellation before. It is again common in QRPA and NSM. -2 8 J = 0 -1 ] 6 C(r) [fm 4 2 0 0 -2 -1 ] C(r) [fm -4 -6 J ̸ = 0 -8 0 1 2 3 4 5 6 7 8 9 10 r [fm]

0.025 From C(r) we know that the full 2 νββ operator has a short range 0.020 character. That is also visible in -1 ] the momentum analog C(q). 0.015 C(q) [MeV characteristic momentum is not 0.010 hc/R but hc/r 0 ~ 200 MeV. 0.005 0.000 J =0 0.08 pairing Other J non-pairing This is again the result of cancellation between the 0.04 -1 ] J =0 (pairing) part and the C(q) [MeV 0.00 other J (broken pairs) parts. Note that the lower panel -0.04 has ~ 3 times larger y scale. -0.08 0 100 200 300 400 500 600 q [MeV]

GT-AA with correlations -3 2 × 10 GT-AA without correlations Again, C(q) for the hypothetical 10 He 0 νββ decay, evaluated using -3 2 × 10 the variational Monte Carlo method, with no approximation. -3 C(q) [MeV 1 × 10 The behavior at large values of q (q > 400 MeV) is a bit different. -4 This has to do with the different 8 × 10 treatment of the nucleon finite -1 ] -4 size. 4 × 10 0 -4 -4 × 10 6 0 200 400 600 q [MeV]

The short range character of the 0 νββ operator, revealed by the evaluation of C 0 ν (r) means that the nucleons participating in the decay must be close to each other. That also means that they are mostly in the central region of the nucleus, and less likely near the nuclear surface. The central regions of all nuclei has essentially the same density and thus also Fermi momentum, i.e. it is in the form of nuclear matter. It is thus not surprising that no matter which method is used there is relatively little a dependence in the M 0 ν (Z,A). This is in contrast with the known M 2 ν matrix elements for the 2 νββ decay which show a rather pronounce Z,A variations, 2 νββ decay is low momentum transfer process, while the 0 νββ Is much higher momentum transfer process.

Calculated M 0 ν by different methods (color coded) The spread of the M 0n values for each nucleus is ~ 3. On the other hand, there is relatively little variation from one nucleus to the next. 8 NR-EDF 7 R-EDF QRPA Jy 6 QRPA Tu QRPA CH 5 IBM-2 SM Mi M 0 ν 4 SM St-M,Tk 3 2 1 0 31 48 76 82 96 100 116 124 130 136 150 A Figure from review by Engel and Menendez

The 2 ν matrix elements, unlike the 0 ν ones, exhibit pronounced shell effects. They vary relatively fast as a function of Z or A.

Lets consider once more the GT m.e. for 0 νββ k H ( r lk , ¯ M 0 ν GT = ⟨ f | � lk σ l · σ k τ + l τ + E ) | i ⟩ , If we remove from the operator the neutrino potential ¯ H(r,E) we obtain the matrix element of the double GT operator connecting the ground states of the initial and final nuclei. The same operator would be responsible for the 2 νββ decay if it would be OK to treat it in the closure approximation. M 2 ν cl ≡ ⟨ f | � lk σ l · σ k τ + l τ + k | i ⟩ , 2 ν 2 ν ( ¯ In reality, the closure approximation is not good for the 2 νββ ( decay, but we can still consider the corresponding value if we somehow can guess the correct average energy denominator From the correct expression for M 2 ν ⟨ f || στ + || m ⟩⟨ m || στ + || i ⟩ M 2 ν = � m , E m − ( M i + M f ) / 2 the sumation extends over all 1 + virtual intermediate

We can define the radial function C 2 ν cl (r) the same way as for the genuine M 0 ν matrix element, thus C 2 ν cl ( r ) = ⟨ f | � lk σ l · σ k δ ( r − r lk ) τ + l τ + k | i ⟩ , � ∞ M 2 ν C 2 ν cl = cl ( r ) dr. 0 It is now clear that, at least formally, the following equality holds: 2 � ∞ C 0 ν (r) = H(r,E 0 ) C 2 ν cl (r) while M 0 ν C 0 ν GT ( r ) dr, GT = 0 So, if we can somehow determine the function C 2 ν cl (r) we will be able to determine C 0 ν (r) and thus also the ultimate goal, the M 0 ν .