# Presentation: 803637

## Theory Overview on Double Beta Decay 

- Petr Vogel, Caltech

- PANIC05 Neutrino Satellite Meeting:
-       Future of Neutrino Physics
-         Santa Fe, Oct. 29, 2005

## Thanks to the discoveries of the recent past  we know a lot. But, there are still many things we do not know and would like to know:

- Are neutrinos Majorana particles?
- What is the pattern of neutrino masses?
- What is the absolute mass scale?
- Is CP invariance violated in the lepton sector?
- Is there a relation between all of this and the baryon
-     excess in the Universe?

- Double beta decay is a process that can help in
-          answering some of these questions

## APS Joint Study on the Future ofNeutrino Physics (2004)
     (physics/0411216)

We recommend, as a high priority, a phased program
of sensitive searches for neutrinoless double beta
decay (first on the list of recommendations)  

The answer to the question  whether neutrinos are 
their own antiparticles is of central importance, not
only to our understanding of neutrinos, but also to
our understanding of the origin of mass. 

**     ****(physics/0411216)**
- We recommend, as a high priority, a phased program
- of sensitive searches for neutrinoless double beta
- decay (first on the list of recommendations)
- The answer to the question  whether neutrinos are
- their own antiparticles is of central importance, not
- only to our understanding of neutrinos, but also to
- our understanding of the origin of mass.

## Whatever processes cause 0, its observation would imply the existence of a Majorana mass term:

- Whatever processes cause 0, its observation would imply the existence of a Majorana mass term:

- Schechter and Valle

## What are the questions that we should

-           ask and try to answer?

- What is the physics of the fundamental dd -> uuee
-      lepton number violating process? How can we tell?
- How to relate the  decay rate to the fundamental
-      parameters? In other words, how can we evaluate the
-      nuclear matrix elements? And how uncertain their
-      values are?
- What is the relation of the deduced fundamental
-      parameters and the neutrino mixing matrix? Or,
-      in other words, what is the relation between the
-       decay rate and the absolute neutrino mass.

## Light Majorana neutrino,

- only Standard Model
- weak interactions
- ~ <m>2

- Light or heavy Majorana
- neutrino. Model extended
- to include right-handed WR.
- Mixing extended between
- the left and right-handed
- neutrinos.
- 

- Heavy Majorana neutrino
- interacting with WR.
- Model extended to include
- right-handed current
- interactions.
- 

- Supersymmetry with
- R-parity violation.
- Many new particles
- invoked. Light
- Majorana neutrinos exist
- also.
- 

## The relative size of the heavy (AH) vs. light particle (AL)
 exchange to the decay amplitude is (a crude estimate)

  AL ~ GF2 m/<k2>,        AH ~ GF2 MW4/5 , or AH ~ 

where  is the heavy scale, is unknown coupling constant,
and k ~ 50 MeV is the virtual neutrino momentum.

For ~ 1 TeV and m ~ 0.1 – 0.5 eV  AL/AH ~ 1, hence both
mechanisms contribute equally.

Thus if LNV characterized by ~ TeV exists we will not
know whether we can deduce the magnitude of <m> 
from the lifetime or not. 

-  exchange to the decay amplitude is (a crude estimate)
-   AL ~ GF2 m/<k2>,        AH ~ GF2 MW4/5 , or AH ~ 
- where  is the heavy scale, is unknown coupling constant,
- and *k ~ *50 MeV is the virtual neutrino momentum.
- For ~ 1 TeV and m ~ 0.1 – 0.5 eV  AL/AH ~ 1, hence both
- mechanisms contribute equally.
- Thus if LNV characterized by ~ TeV exists we will not
- know whether we can deduce the magnitude of <m>
- from the lifetime or not.

## As long as only limits on the  rate exist, there is no
reason to worry much. One can deduce limits on both <m> 
and  since it is unlikely that the two amplitudes exactly
cancel each other. 
However, if decay is observed, we would like to know
what physics is responsible.
In most cases it is impossible to decide, based on observations
of the decay, which mechanism is responsible for 0 since
the electron spectra, angular distributions, polarizations, etc.
are independent of it (the left-right current interference is a
famous exception).

- reason to worry much. One can deduce limits on both <m>
- and  since it is unlikely that the two amplitudes exactly
- cancel each other.
- However, if decay is observed, we would like to know
- what physics is responsible.
- In most cases it is impossible to decide, based on observations
- of the decay, which mechanism is responsible for 0 since
- the electron spectra, angular distributions, polarizations, etc.
- are independent of it (the left-right current interference is a
- famous exception).

##  It turns out that Lepton Flavor Violation (LFV) involving charged leptons provides a “diagnostic tool” for establishingthe mechanism of   decay.

- See, “Neutrinoless double beta decay and lepton flavor violation”
- Phys. Rev. Lett. 93 (2004) 231802, V. Cirigliano, A. Kurylov,
- M.J.Ramsey-Musolf, and P.V.

- The two processes, LFV and LNV are, generally, governed
- by different mass scales. However, one can establish
- (with some ``fine tuning” exceptions) a relation between
- these scales.

##  it is possible that SM extensions with low ( TeV) scale

-  it is possible that SM extensions with low ( TeV) scale
-  LNV exist.

-  SM extensions with high (GUT) scale LNV,

- ~ *O(1) >> *10-2

- If

- then

- On the other hand if

- then

- are essentially the only possibility.

- Consider the well studied LFV processes:

## Assume that indeed  ~ <m>2. What is the 

- relation between this quantity and the absolute
- neutrino mass?

- <m> =|  Uei2 mi | .  This quatity depends on the oscillation
- parameters, on the _unknown phases_ of Uei, and on the
- _unknown _absolute neutrino mass. The dependence on the
- Majorana phases can be eliminated by considering the
- maximum and minimum values of <m>. Only the dependence
- on the absolute mass remains.

## <m> vs. the

- from  decay

- from observational
- cosmology,
- M = m1+m2+m3

- blue shading:
- normal hierarchy,
- m231 > 0.
- red shading:
- inverted hierarchy
- m231 < 0
- shading:best fit
- parameters, lines
- 95% CL errors.

- minimum mass,
- not observable

- _<m____> vs. the_
- _absolute mass scale_

- Thanks to A. Piepke

## The degenerate mass region can be explored by ways

- Katrin
- sensitivity

- Planck +SDSS
- sensitivity

- independent on the Majorana nature of neutrinos.

## Nuclear matrix elements

- hep-ph/0403167 ,
- spread of published
- values of squared
- nuclear matrix element for 76Ge

- from Bahcall et al

**Nuclear matrix elements**

- A provocative question: Do we know at all how large the matrix elements really are? Or, in other words, why there is so much variation among the published results?

- This suggests an
- uncertainty of as much as
- a factor of 5. _**Is it really**_
_**so bad?**_

## In contrast, Rodin et al, nucl-th/0503063 suggest that the

- uncertainty is much less, perhaps only ~ 30%. So, who is right?

## More on nuclear matrix elements 

-   In order to relate decay rate to the effective mass <m> we have to
-    know the corresponding nuclear matrix elements. Any error in them is
-    directly reflected as a like size error in <m>.
-    The most important operator is     h(rij)[i  j – (gV/gA)2] where
-     the sum is over all nucleon pairs, and rij is the distance between the
-     nucleons. h(r) is the `neutrino potential’, the Fourier transform of the
-     neutrino propagator h(r)~e-1.5Er/r. (Often Rxh(r) is used to make the
-     m.e. dimensionless, with R being the nuclear radius.)
-     Tests show that  it is OK to treat this two-body operator in closure;
-      the result depends only weakly on the assumed average nuclear
-      excitation energy. It is customary to expand the operators
-      in terms of multipole states of the intermediate odd-odd nucleus. As a
-      consequence of the high momentum of the virtual neutrino many multipoles
-      contribute significantly, unlike for the case of the  decay where only
-      1+ contributes.

## Many multipoles

- contribute in each
- case. Most of them,
- with the exception
- of 1+, have the same
- sign.
- This is from Rodin et al,
- other calculations get
- a similar pattern.

## Here the contribution of

- the 1+ multipole, and of
- all other ones is plotted
- against the parameter gpp
- that signifies the strength
- of the neutron-proton
- interaction. The nominal
- value is gpp=1.0. The dots
- denote the adjusted value
- that reproduces the 
- decay rate in each nucleus.
- Note the steep slope of the
- 1+  curve and the relatively
- gentle slope of the dashed
- lines.
- _This suggests that adjusting_
- _properly the contribution_
- _of the 1_+_ is crucial._

## Nuclear matrix elements for the 2 

- Nuclear matrix elements for the 2
- decay deduced from measured halflives.
- Note the pronounced shell dependence.

- 1/T1/2 = G(E,Z) (MGT

- easily calculable
- phase space factor

## The  matrix element contains products of the - (right leg) and

- The  matrix element contains products of the - (right leg) and
- EC (left leg) amplitudes, weighed by the energy denominators.
- Single state dominance is often invoked.

- Energies, and squares of the amplitudes (but not signs) of the 1+
- states could be determined by charge exchange reactions, testing
- the SSD idea.

- However, for contribution of the 1+ multipole to 0m.e. the energy
- denominator represents a wrong weight. It looks more like the
- closure m.e. 

## QRPA calculation for 100Mo. The closure m.e. and Mare not

- M(x3.5)

- Mclosure

- M(1+)

- gpp

- m.e.

- QRPA calculation for 100Mo. The closure m.e. and Mare not
- proportional to each other. But Mclosure and M(1+) essentially are.

- adjusted
- gpp

## QRPA calculation for 76Ge. The closure m.e. and Mare not really

- QRPA calculation for 76Ge. The closure m.e. and Mare not really
- proportional to each other. But Mclosure and M(1+) essentially are.

- M(x7.5)

- Mclosure

- M(1+)

- gpp

- m.e.

- adjusted
- gpp

## Everybody agrees that within QRPA, the most popular

- method of evaluating the  matrix elements, the
- crucial parameter is gpp the strength of the particle-
- particle (neutron-proton) interaction. (There are other,
- implicit, parameters within QRPA that affect the results
- as well, more on them later)
- Consequently, the choice of gpp is important. Since the
- 1+ channel is the most sensitive one to the variation
- of gpp it is almost obvious that it should be used  for
- the adjustment. The contribution of all other multipoles
- to m.e. is less sensitive to the exact value of gpp,
- Hence that contribution is likely to be correctly reproduced
- by QRPA. There is no agreement on the proper way of gpp
- adjustment. Rodin et al. use the M matrix element.

## SM 96

**SM 96**
**Caurier**

- QRPA
- Rodin
- 2003

- Bobyk 2001
- 2 wrong
-    20x

- Tomoda 86
- proj.m.f.
- 2 wrong
- 6x

- The outliers predict wrong 2 halflife. The matrix elements of SM
- and Rodin et al. are quite close.

## Comparison of M of Rodin et al. (RQRPA) and

- Nowacki et al. (SM, private comm., preliminary 2004)
- and older published (Caurier et al. 1996)
- Nucleus                            RQRPA                          SM
- 76Ge                                   2.3-2.4                          1.6
- 82Se                                                   1.9-2.1                           1.7
- 96Zr                                    0.3-0.4                         0.4
- 100Mo                                  1.1-1.2                           0.3
- 116Cd                                   1.2-1.4                           1.9
- 130Te                                   1.3                                2.0  (1.0)
- 136Xe                                   0.6-1.0                          1.6  (0.6)
*Except for **100**Mo the agreement between these very different *
*calculations is reasonably good.*

## Rodin et al. Phys. Rev. C68, 044302 (2003),
and  Rodin et al. nucl-th/0503063.
have shown that their method of adjustment
results in very little sensitivity to:

a)   how many s.p. states are included?
b)   which parametrization of the G-matrix is used?
c)    whether the gA is quenched? 
d)    whether  one of the generalization of QRPA is used?

It is likely that most, if not all, of the spread in the QRPA
results is caused by different choices a) - d) without
corresponding renormalization.
There is a heated debate currently whether this is indeed
the answer to the problem.  

- and  Rodin et al. nucl-th/0503063.
- have shown that their method of adjustment
- results in very little sensitivity to:
- a)   how many s.p. states are included?
- b)   which parametrization of the G-matrix is used?
- c)    whether the gA is quenched?
- d)    whether  one of the generalization of QRPA is used?
- It is likely that most, if not all, of the spread in the QRPA
- results is caused by different choices a) - d) without
- corresponding renormalization.
_**There is a heated debate currently whether this is indeed**_
_**the answer to the problem.**_

**Notes:**
Make a comment on short range correlations 
and induced pseudoscalar

## 0half-lives for <m> = 50 meV based on the matrix elements of Rodin et al. 

- 0half-lives for <m> = 50 meV based on the matrix elements of Rodin et al.

- 76Ge         (2.1 - 2.6) x 1027 y
- 82Se         (6.0 - 8.7) x 1026 y
- 100Mo        (1.1 - 1.7) x 1027 y
- 130Te         (0.7 - 1.7) x 1027 y
- 136Xe         (1.5 - 5.6) x 1027 Y (no 2observed yet)

- Note: Calculated matrix elements decrease with
- increasing A, but the phase-space factors usually
- increase, particularly the Coulomb factor, hence
- relatively little variation of T1/2 with A.

- Note: The sensitivity to <m> scales as 1/(T1/2)1/2

## Summary and/or Conclusions

-  Study of  decay entered a new era. No longer is the aim just to
-  push the sensitivity higher and the background lower, but to explore
-  specific regions of the <m> values.
- In agreement with the `phased’ program the plan is to explore the
- `degenerate’ region (0.1-1 eV) first, with ~100 kg sources, and
- prepare the study of `inverted hierarchy’ (0.01-0.1eV) region
- with ~ ton sources that should follow later.
- In this context it is important to keep in mind the questions I discussed:
- Relation of <m> and the absolute mass (rather clear already, becoming less uncertain with better oscillation results).
- Mechanism of the decay (exploring LFV, models of LNV, running of
-        LHC to explore the ~TeV mass particles).
- Nuclear matrix elements (exploring better, and agreeing on, the
-       reasons for the spread of calculated values, and deciding on the
-       optimum way of performing the calculations, while pursuing vigorously
-       also the application of the shell model).

## 76Ge

- 76Ge

- 100Mo

- From G. Gratta

- Two modes of  decay, one allowed and another
-                       forbidden in SM.

## F.Nowacki, preliminary, private comm.