[PPT] - Matrix Elements for Double-Beta Decay I. Overview A. Introduction PowerPoint Presentation

SLIDE 1

Matrix Elements for Double-Beta Decay

I. Overview
A. Introduction
J. Engel

November 1, 2017

SLIDE 2

A Little on the Standard Mechanism

!" n n p p e e W W

x

Here mνM ≪ me.

SLIDE 3

How Effective Mass Gets into Rate

[T0ν

1/2]−1 =

spins
|Z0ν|2δ(Ee1 + Ee2 − Q)d3p1

2π3 d3p2 2π3 Z0ν contains lepton part

k

e(x)γµ(1 − γ5)Uekνk(x) νc

k(y)γν(1 + γ5)Uekec(y) ,

where ν’s are Majorana mass eigenstates. Contraction gives neutrino propagator:

k

e(x)γµ(1 − γ5) qργρ + mk q2 − m2

k

γν(1 + γ5)ec(y) U2

ek ,

The qργρ part vanishes in trace, leaving a factor meff ≡

k

mkU2

ek.

SLIDE 4

What About Hadronic Part?

Integral over times produces a factor

n

f|Jµ

L (

x)|nn|Jν

L (

y)|i q0(En + q0 + Ee2 − Ei) + ( x, µ ↔ y, ν) , with q0 the virtual-neutrino energy and the J the weak current.

SLIDE 5

What About Hadronic Part?

Integral over times produces a factor

n

f|Jµ

L (

x)|nn|Jν

L (

y)|i q0(En + q0 + Ee2 − Ei) + ( x, µ ↔ y, ν) , with q0 the virtual-neutrino energy and the J the weak current. In impulse approximation: p|Jµ(x)|p′ = eiqxu(p)

gV(q2)γµ − gA(q2)γ5γµ

− igM(q2)σµν 2mp qν + gP(q2)γ5qµ

u(p′) .

SLIDE 6

What About Hadronic Part?

Integral over times produces a factor

n

f|Jµ

L (

x)|nn|Jν

L (

y)|i q0(En + q0 + Ee2 − Ei) + ( x, µ ↔ y, ν) , with q0 the virtual-neutrino energy and the J the weak current. In impulse approximation: p|Jµ(x)|p′ = eiqxu(p)

gV(q2)γµ − gA(q2)γ5γµ

− igM(q2)σµν 2mp qν + gP(q2)γ5qµ

u(p′) .

May not be adequate.

SLIDE 7

What About Hadronic Part?

Integral over times produces a factor

n

f|Jµ

L (

x)|nn|Jν

L (

y)|i q0(En + q0 + Ee2 − Ei) + ( x, µ ↔ y, ν) , with q0 the virtual-neutrino energy and the J the weak current. In impulse approximation: p|Jµ(x)|p′ = eiqxu(p)

gV(q2)γµ − gA(q2)γ5γµ

− igM(q2)σµν 2mp qν + gP(q2)γ5qµ

u(p′) .

May not be adequate.

q0 typically of order inverse inter-nucleon distance, 100 MeV, so denominator can be taken constant and sum done in closure.

SLIDE 8

Final Form of Nuclear Part

M0ν = MGT

0ν − g2 V

g2

A

MF

0ν + . . .

with MGT

0ν = F| |

i, j

H(rij) σi · σj τ+

i τ+ j

|I + . . . MF

0ν =F |

i, j

H(rij) τ+

i τ+ j

|I + . . . H(r) ≈ 2R πr ∞ dq sin qr q + E − (Ei + Ef)/2 roughly ∝ 1/r Contribution to integral peaks at q ≈ 100 MeV inside nucleus. Corrections are from “forbidden” terms, weak nucleon form factors, many-body currents ...

SLIDE 9

Matrix Elements for Double-Beta Decay

I. Overview
B. Basic Ideas of Nuclear Structure

SLIDE 10

Traditional Nucleon-Nucleon Potential

From E. Ormand, http://www.phy.ornl.gov/npss03/ormand2.ppt

SLIDE 11

Shell Model of Nucleus

Nucleons occupy orbitals like electrons in atoms. Central force on nucleon comes from averaging forces produced by other nucleons.

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/shell.html

Reasonable potentials give magic numbers at 2, 8, 20, 28, 50, 126

SLIDE 12

An Example

← − d3/2 ← − s1/2 ← − d5/2

SLIDE 13

Simple Model Can’t Explain Collective Rotation...

From Booth and Combey, http://www.shef.ac.uk/physics/teaching/phy303/phy303-3.html and

E. Ormand, http://www.phy.ornl.gov/npss03/ormand1.ppt

Collective rotation between magic numbers

SLIDE 14

Or Collective Vibrations

Two vibrational ”phonons” with angular momentum 2 give states with angular momentum 0, 2, 4.

From Booth and Combey, http:///www.shef.ac.uk/physics/teaching/phy303/phy303-3.html and http://www.fen.bilkent.edu.tr/˜aydinli/Collective%20Model.ppt

SLIDE 15

Alternative Early View: “Liquid Drop” Model

Protons and neutrons move together; volume is conserved, surface changes shape. Ansatz for surface: R(θ, φ) = R0

1 +
m

αmY2,m(θ, φ)

SLIDE 16

Alternative Early View: “Liquid Drop” Model

Protons and neutrons move together; volume is conserved, surface changes shape. Ansatz for surface: R(θ, φ) = R0

1 +
m

αmY2,m(θ, φ)

The 5 α’s are collective variables. For vibrations, Hamiltonian
btained e.g. from classical fluid model:

H ≈ 1/2B

m | ˙

αm|2 + 1/2C

m |αm|2

with B ≈ ρR5 2 = 3 8πmAR2

0 ,

C ≈ aSA2/3 π − 3e2Z2 10πR0 , ω =

C/B

ω is roughly the right size, but real life is more complicated, with frequencies depending on how nearly magic the nucleus is.

SLIDE 17

Deformation in Liquid Drop Model

If Coulomb effects overcome surface tension, C is negative and nucleus deforms. 5 “intrinsic-frame” α’s replaced by 3 Euler angles, and: β ≡

α2

0 + 2α2 2 ,

γ ≡ tan−1[ √ 2α2/α0] so that Ψ(θ, ϕ, ψ) ≈ DJ∗

MK(θ, ϕ, ψ)Φint.(β, γ).

deformed spherical V β

−0.3 −0.2 −0.1 0.1 0.2 0.3

SLIDE 18

Deformation in Liquid Drop Model

If Coulomb effects overcome surface tension, C is negative and nucleus deforms. 5 “intrinsic-frame” α’s replaced by 3 Euler angles, and: β ≡

α2

0 + 2α2 2 ,

γ ≡ tan−1[ √ 2α2/α0] so that Ψ(θ, ϕ, ψ) ≈ DJ∗

MK(θ, ϕ, ψ)Φint.(β, γ).

deformed spherical V β

−0.3 −0.2 −0.1 0.1 0.2 0.3

. . . . . . .

2+ 4+ 6+ 2+ 4+ 4+ 3+ 2

β γ

+ + +

.

Low-lying states

1. Rotations of deformed nucleus
2. Surface vibrations along or against

symmetry axis

SLIDE 19

Density Oscillations

Photoabsorption cross section proportional to “isovector” dipole

strength. Resonance lies at higher energy than surface modes.

Ikeda et al., arXiv:1007.2474 [nucl-th] Szpunar et al., Nucl. Inst. Meth. Phys. A 729, 41 (2013)

Giant dipole resonance

SLIDE 20

Development Since the First Models

SLIDE 21

Modern Shell-Model Basic Wave Functions

Nucleus is usually taken to reside in a confining harmonic oscillator. Eigenstates of oscillator part are localized Slater determinants, the simplest many-body states: ψ(

r1 · · ·

rn) =

φi(
r1)

φj(

r1)

· · · φl(

r1)

φi(

r2)

φj(

r2)

· · · φl(

r1)

. . . . . . . . . . . . φi(

rn)

φj(

rn)

· · · φl(

rn)
−

→ a†

i a† j · · · a† l |0

They make a convenient basis for diagonalization of the real internucleon Hamiltonian. To get a complete set just put distribute the A particles, one in each oscillator state, in all possible ways.

SLIDE 22

Truncation Scheme of the Modern Shell-Model

Core is inert; particles can’t move

ut.

Particles outside core confined to limited set of valence shells. Can’t use basic nucleon-nucleon interaction as Hamiltonian because of truncation, which excludes significant configurations. Most Hamiltonians to date are in good part phenomenological, with fitting to many nuclear energy levels and transition rates. All

perators need to be

“renormalized” as well.

We’ll return to this problem later.

Example: 20Ne

core valence

0s 1p 0f 1p 0s 1d

SLIDE 23

What the Shell Model Can Handle

From W. Nazarewicz, http://www-highspin.phys.utk.edu/˜witek/

All these are easy now. But more than one oscillator shell still usually impossible.

SLIDE 24

Level of Accuracy (When Good)

48Ca 48Sc

From A. Poves, J. Phys. G: Nucl. Part. Phys. 25 (1999) 589 597.

SLIDE 25

Shell Model Calculations of 0νββ Decay

M0ν with shell-model ground states |48Ca and |48Ti Effects of varying the phenomenological Hamiltonian

Problem with shell model: Experimental energy levels tell us, roughly, how to “renormalize” Hamiltonians to account for orbitals

mitted from the shell-model space. But what about the ββ
perator? How is it changed? Most calculations use “bare” operator.

SLIDE 26

The Beginning of Nuclear DFT: Mean-Field Theory

For a long time the best that could be done in a large single-particle space. Call the Hamiltonian H (not the “bare” NN interaction itself). The Hartree-Fock ground state is the Slater determinant with the lowest expectation value H.

SLIDE 27

Variational Procedure

Find best Slater det. |ψ by minimizing H ≡ ψ′| H |φ′ / φ′| ψ′: In coordinate space, resulting equations are −∇2 2mφa(

r) +

   

d

r′V(|

r −

r′|)

jF

φ∗

j (

r′)φj( r′)

ρ(

r′)

    φa(

r)

−

jF
d

r′V(|

r −

r′|)φ∗

j (

r′)φa( r′)

φj(
r) = ǫaφa(
r) .

SLIDE 28

Variational Procedure

Find best Slater det. |ψ by minimizing H ≡ ψ′| H |φ′ / φ′| ψ′: In coordinate space, resulting equations are −∇2 2mφa(

r) +

   

d

r′V(|

r −

r′|)

jF

φ∗

j (

r′)φj( r′)

ρ(

r′)

    φa(

r)

−

jF
d

r′V(|

r −

r′|)φ∗

j (

r′)φa( r′)

φj(
r) = ǫaφa(
r) .

First potential term involves the “direct” (intuitive) potential Ud(

r) ≡
d

r′V(|

r −

r′|)ρ( r′) .

SLIDE 29

Variational Procedure

Find best Slater det. |ψ by minimizing H ≡ ψ′| H |φ′ / φ′| ψ′: In coordinate space, resulting equations are −∇2 2mφa(

r) +

   

d

r′V(|

r −

r′|)

jF

φ∗

j (

r′)φj( r′)

ρ(

r′)

    φa(

r)

−

jF
d

r′V(|

r −

r′|)φ∗

j (

r′)φa( r′)

φj(
r) = ǫaφa(
r) .

First potential term involves the “direct” (intuitive) potential Ud(

r) ≡
d

r′V(|

r −

r′|)ρ( r′) . Second term contains the nonlocal “exchange potential” Ue(

r,

r′) ≡

jF

V(|

r −

r′|)φ∗

j (

r′)φj(

r) .

SLIDE 30

Self Consistency

Note that in potential-energy terms Ud and Ue depend on all the

ccupied levels. So do the eigenvalues ǫa, therefore, and Solutions

are “self-consistent” To solve equations:

1. Start with a set of basis states φa, φb, φc...and calculate

construct Ud and Ue.

2. Solve the HF Schr¨
dinger equation to obtain a new set of

basis states φa′, φb′ . . .

3. Repeat steps 1 and 2 until you get essentially the same basis
ut of step 2 as you put into step 1.

SLIDE 31

Second-Quantization Version

Theorem (Thouless)

Suppose |φ ≡ a†

1 · · · a† F |0 is a Slater determinant. The most general

Slater determinant not orthogonal to |φ can be written as |φ′ = exp(

m>F,i<F

Cmia†

mai) |φ = [1 +

m,i

Cmia†

mai + O(C2)] |φ

Minimizing E = ψ| H |ψ: ∂H ∂Cnj = φ| Ha†

naj |φ = 0

∀ n > F, j F = ⇒ hnj ≡ Tnj +

k<F

Vjk,nk = 0 ∀ n > F, j F where Tab = a| p2

2m |b and Vab,cd = ab| V12 |cd − ab| V12 |dc. This

will be true if ∃ a single particle basis in which h is diagonal, hab ≡ Tab +

kF

Vak,bk = δabǫa ∀ a, b . Another version of the HF equations.

SLIDE 32

Brief History of Mean-Field Theory

1. Big problem early: Doesn’t work with realistic NN potentials

because of hard core, which causes strong correlations.

SLIDE 33

Brief History of Mean-Field Theory

1. Big problem early: Doesn’t work with realistic NN potentials

because of hard core, which causes strong correlations.

2. Hard core included implicitly through effective interaction:

Brueckner G matrix Still didn’t work perfectly; three-body interations neglected.

G

=

V

+

V V

+ + . . .

SLIDE 34

Brief History of Mean-Field Theory

1. Big problem early: Doesn’t work with realistic NN potentials

because of hard core, which causes strong correlations.

2. Hard core included implicitly through effective interaction:

Brueckner G matrix Still didn’t work perfectly; three-body interations neglected.

G

=

V

+

V V

+ + . . .

3. Three-body interaction included approximately as

density-dependent two-body interaction, in the same way as two-body interaction is approximated by density-dependent mean field. Results better, and a convenient “zero-range” approximation used.

SLIDE 35

Brief History (Cont.)

4. Phenomenology successfully evolved toward zero-range

density-dependent (Skyrme) interactions, with H = t0 (1 + x0^ Pσ) δ(

r1 −

r2) + 1 2t1 (1 + x1^ Pσ)

(

∇1 − ∇2)2δ(

r1 −

r2) + h.c.

+ t2 (1 + x2^

Pσ) ( ∇1 − ∇2) · δ(

r1 −

r2)( ∇1 − ∇2) + 1 6t3 (1 + x3^ Pσ) δ(

r1 −

r2)ρα([

r1 +

r2]/2) + iW0 ( σ1 + σ2) · ( ∇1 − ∇2) × δ(

r1 −

r2)( ∇1 − ∇2) , where ^ Pσ = 1 + σ1 · σ2 2 , and ti, xi, W0, and α are adjustable parameters.

SLIDE 36

Brief History (Cont.)

4. Phenomenology successfully evolved toward zero-range

density-dependent (Skyrme) interactions, with H = t0 (1 + x0^ Pσ) δ(

r1 −

r2) + 1 2t1 (1 + x1^ Pσ)

(

∇1 − ∇2)2δ(

r1 −

r2) + h.c.

+ t2 (1 + x2^

Pσ) ( ∇1 − ∇2) · δ(

r1 −

r2)( ∇1 − ∇2) + 1 6t3 (1 + x3^ Pσ) δ(

r1 −

r2)ρα([

r1 +

r2]/2) + iW0 ( σ1 + σ2) · ( ∇1 − ∇2) × δ(

r1 −

r2)( ∇1 − ∇2) , where ^ Pσ = 1 + σ1 · σ2 2 , and ti, xi, W0, and α are adjustable parameters. Abandoning first principles leads to still better accuracy.

SLIDE 37

Brief History (Cont.)

5. Convenient because exchange potential is local; easy to solve.

Also, variational principal can be reformulated in terms of a local energy-density functional.

SLIDE 38

Brief History (Cont.)

5. Convenient because exchange potential is local; easy to solve.

Also, variational principal can be reformulated in terms of a local energy-density functional. Defining ρab =

iF

b| φiφi |a , ρ(

r) =
iF,s

|φi(

r, s)|2

τ(

r) =
iF,s

|∇φi(

r, s)|2 ,
J(
r) = −i
iF,s,s′

φi(

r, s)[∇φi(
r, s′) ×

σss′] and E = φ| H |φ =

d

r[

h2

2nτ + 3 8t0ρ2 + 1 16ρ3 + 1 16(3t1 + 5t2)ρτ + 1 64(9t1 − 5t2)(∇ρ)2 + 3 4W0ρ ∇ · J + 1 32(t1 − t2)

J2]

SLIDE 39

Brief History (Cont.)

5. Convenient because exchange potential is local; easy to solve.

Also, variational principal can be reformulated in terms of a local energy-density functional. Defining ρab =

iF

b| φiφi |a , ρ(

r) =
iF,s

|φi(

r, s)|2

τ(

r) =
iF,s

|∇φi(

r, s)|2 ,
J(
r) = −i
iF,s,s′

φi(

r, s)[∇φi(
r, s′) ×

σss′] and E = φ| H |φ =

d

r[

h2

2nτ + 3 8t0ρ2 + 1 16ρ3 + 1 16(3t1 + 5t2)ρτ + 1 64(9t1 − 5t2)(∇ρ)2 + 3 4W0ρ ∇ · J + 1 32(t1 − t2)

J2]

you find ∂ (E −

i ǫiρii)

∂ρab = hab − ǫaδab = 0, ∀a, b i.e. the Hartree-Fock equations, with Hamiltonian hab = ∂E/∂ρab.

SLIDE 40

Brief History (Cont.)

6. “Shoot, we can include more correlations, get back to first

principles, if we mess with the density functional via:”

SLIDE 41

Brief History (Cont.)

6. “Shoot, we can include more correlations, get back to first

principles, if we mess with the density functional via:”

Theorem (Hohenberg-Kohn and Kohn-Sham, vulgarized)

∃ universal functional of the density that, together with a simple one depending only on external potentials, gives the exact ground-state energy and density when minimized through Hartree-like equations. (Finding the functional is up to you!) There is some work to construct functionals form first principles, but they are determined largely by fitting Skyrme parameters. Results are pretty good, but it’s hard to quantify systematic error.

SLIDE 42

Densities Near Drip Lines

This and next 2 slides from J. Dobacewski

100Sn

0.00 0.05 0.10 2 4 6 8

Particle density (fm

3)

(p) (n)

100Zn

2 4 6 8 10

R (fm)

(p) (n)

SLIDE 43

Two-Neutron Separation Energies

Experiment Theory

SLIDE 44

Deformation

SLIDE 45

Collective Excited States

Can do time-dependent Hartree-Fock in an external potential f(

r, t) = f(
r)e−iωt + f†(
r)eiωt. TDHF equation is:

−idρab dt = ∂E[ρ] ∂ρab + fab(t)

SLIDE 46

Collective Excited States

Can do time-dependent Hartree-Fock in an external potential f(

r, t) = f(
r)e−iωt + f†(
r)eiωt. TDHF equation is:

−idρab dt = ∂E[ρ] ∂ρab + fab(t) Assuming small amplitude oscillations ρ = ρ0 + δρe−iωt + δρ†e−iωt gives iωδρmi =

n>F,jF

∂hmi ∂ρnj δρnj + ∂hmi ∂ρjn δρjn + fmi

SLIDE 47

Collective Excited States

Can do time-dependent Hartree-Fock in an external potential f(

r, t) = f(
r)e−iωt + f†(
r)eiωt. TDHF equation is:

−idρab dt = ∂E[ρ] ∂ρab + fab(t) Assuming small amplitude oscillations ρ = ρ0 + δρe−iωt + δρ†e−iωt gives iωδρmi =

n>F,jF

∂hmi ∂ρnj δρnj + ∂hmi ∂ρjn δρjn + fmi Resulting δρ(ω), is the transition density. “Transition strength” to excited state with energy E = hω is roughly R =

mi fmiδρmi(ω).

This is the “random phase approximation” (RPA).

SLIDE 48

Isovector Dipole in RPA

Strength Distribution Transition Densities

' & $ % Evolution

f

the IV dip

le

strength IV dip

le

strength in Sn isotop es

5 10 15 20 25 30

E [MeV]

2 4 6 8

R [e

2fm 2]

0.20

0.00 0.20

0.10

0.00 0.10 neutrons protons 2 4 6 8 10 12

r [fm]

0.20

0.00 0.20 132Sn

r

2

δ ρ [ f m

1

]

E=14.04 MeV E=11.71 MeV E=7.60 MeV

14.04 MeV 11.71 MeV 7.60 MeV

SLIDE 49

Generalization to Include Pairing

HFB (Hartree-Fock-Bogoliubov) is the most general “mean-field” theory in these kinds of operators: αa =

c
U∗

acac + V∗ aca† c

,

α†

a =

c
Uaca†

c + Vacac

,

Ground state is the “vacuum” for these operators. In addition to having density matrix ρab, one also has “pairing density:” κab ≡ 0| abaa |0 , κ(

r) =
ab

ϕa(

r)ϕb(
r)κab .

Quasiparticle vacuum violates particle-number conservation, but includes physics of correlated pairs. Energy functional E[ρ] replaced by E[ρ, κ]. Minimizing leads to HFB equations for U and V. Generalization to linear response is called the quasiparticle random phase approximation (QRPA).

SLIDE 50

Gamow-Teller Strength

Transition operators are those that generate allowed β decay: στ±. Gamow-Teller strength from 208Pb

2 4 6 8 10 5 10 15 20 25 30 B (GT) EQRPA B(GT +) B(GT -)

SLIDE 51

QRPA Calculations of 0νββ Decay

These very different in spirit from shell-model calculations, which involve many Slater determinants restricted to a few single-particle shells. QRPA involves small oscillations around a single determinant, but can involve many shells (20 or more). Recall that the 0ν operator has terms that look like ^ M =

ij

H(rij)σi · σj . where i and j label the particles. QRPA evaluates this by expanding in multipoles, and inserting set of intermediate-nucleus states: F| ^ M |I =

ij,JM,N

F| ^ Oi,JM |N N| ^ Oj,JM |I , and uses calculated transition densities to evaluate the matrix elements.

SLIDE 52

Matrix Elements for Double-Beta Decay

II. Hadronic Physics

SLIDE 57

e e ~ ?

p n

p p n n

p n

u u d d u d

Relating Theory to Experiment

Many-body methods ab-initio methods LQCD

SLIDE 58

e- e-

W − W − ¯ ν ¯ ν

d d u u

How do we get to nuclear scales?

SLIDE 59

e-

d u

e-

d u

e- e-

W − W − ¯ ν ¯ ν

d d u u

Λ≪ΛBSM

SLIDE 60

Prezeau, Ramsey-Musolf, Vogel (2003)

e-

d u

e-

d u

e- e-

W − W − ¯ ν ¯ ν

d d u u

Λ≪ΛBSM

SLIDE 61

e-

d u

e-

d u

n

p p

n n

p

n

p

e- e- π- π+

n

p

n

p

e- e- π+

Prezeau, Ramsey-Musolf, Vogel (2003)

e-

d u

e-

d u Λ≪ΛQCD Λ≪ΛQCD Λ≪ΛQCD

SLIDE 62

e-

d u

e-

d u

n

p p

n n

p

n

p

e- e- π- π+

n

p

n

p

e- e- π+

Prezeau, Ramsey-Musolf, Vogel (2003)

𝜓PT: NNLO 𝜓PT: NNLO 𝜓PT: LO

e-

d u

e-

d u Λ≪ΛQCD Λ≪ΛQCD Λ≪ΛQCD

SLIDE 63

n

p

n

p

e- e- π- π+

SLIDE 64

n

p

n

p

e- e-

~gA

π- π+

SLIDE 65

n

p

n

p

e- e-

~gA

π- π+ e-

d u

e-

d u _ _ π- π+

This is the matrix element we need to calculate using LQCD

SLIDE 66

What Do You Do With These Amplitudes?

Chiral effective field theory! In QCD vacuum

q=u,d

mq¯ qq = 0 which spontaneously breaks a chiral (left-right) symmetry. Like spontaneous magnetization, which gives rise to massless “magnons” (spin waves). Pions are the analog of magnons for chiral

symmetry. If the u and d quarks had the same mass, pions would

be massless. In the real world they have mass, but much less than

ther hadronic objects.

Chiral perturbation theory is the “effective theory” for interacting

pions. It has infinitely many parameters but only a finite number at

each order of λχ = q/Λ or mπ/Λ, the expansion parameter (q is a typical momentum and Λ is the scale at which other hadrons can exist, about 1 GeV.) The theory breaks down if λχ gets close to Λ.

SLIDE 67

Chiral Effective Field Theory with Nucleons

Here you try to add nucleons to the mix. There is no problem with adding a single nucleon, but with two or more, things get a little

tricky. Proceeding naively, the terms in the nucler interaction have

effect only at increasingly large powers of λχ.

+... +... +... +...

2N Force 3N Force 4N Force

LO (Q/Λχ)0 NLO (Q/Λχ)2 NNLO (Q/Λχ)3 N3LO (Q/Λχ)4

SLIDE 68

Chiral Effective Field Theory with Nucleons

Here you try to add nucleons to the mix. There is no problem with adding a single nucleon, but with two or more, things get a little

tricky. Proceeding naively, the terms in the nucler interaction have

effect only at increasingly large powers of λχ.

+... +... +... +...

2N Force 3N Force 4N Force

LO (Q/Λχ)0 NLO (Q/Λχ)2 NNLO (Q/Λχ)3 N3LO (Q/Λχ)4

SLIDE 69

Comes with Consistent Weak Current

Pions are axial, just like the part of the weak current important for ββ decay. The leading piece of the axial current is π c , c just the usual one-body current, more or less. At next order, you get

π c3, c4 cD

with the constants fixed by the three-body interaction:

SLIDE 70

Operators for Heavy Particle Exchange

Leading diagrams for heavy particle exchange

n p e π π e n p

Subleading diagrams

n π e e p p n e e n n p p

SLIDE 71

How Useful?

In principle, this is exactly what you’d need for a controlled calculation of weak processes with controlled error bars. In practice...

1. Extension of “power counting” to nonperturbative

nuclear-structure calculations not fully rigorous.

2. Arguments about how best to determine parameters
3. You also need a many-body calculation with quantifiable

errors (we’ll get to that next).

SLIDE 72

Matrix Elements for Double-Beta Decay

III. Ab Initio Nuclear Structure

SLIDE 73

Ab Initio Shell Model

Partition of Full Hilbert Space ^ PH^ P ^ PH^ Q ^ QH^ P ^ QH^ Q P Q P Q Shell model done here. P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with Heff in P reproducing d most important eigenvalues.

SLIDE 74

Ab Initio Shell Model

Partition of Full Hilbert Space Heff Heff-Q P Q P Q Shell model done here. P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with Heff in P reproducing d most important eigenvalues.

SLIDE 75

Ab Initio Shell Model

Partition of Full Hilbert Space Heff Heff-Q P Q P Q Shell model done here. P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with Heff in P reproducing d most important eigenvalues. For transition operator ^ M, must apply same transformation to get ^ Meff.

SLIDE 76

Ab Initio Shell Model

Partition of Full Hilbert Space Heff Heff-Q P Q P Q Shell model done here. P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with Heff in P reproducing d most important eigenvalues. For transition operator ^ M, must apply same transformation to get ^ Meff. As difficult as solving full problem. But idea is that N-body effective operators may not be important for N >2 or 3.

SLIDE 77

Method 1: Coupled-Cluster Theory

Ground state in closed-shell nucleus: |Ψ0 = eT |ϕ0 T =

i,m

tm

i a† mai +

ij,mn

1 4tmn

ij a† ma† naiaj + . . . m,n>F i,j<F

States in closed-shell + a few constructed in similar way.

Slater determinant

SLIDE 78

Method 1: Coupled-Cluster Theory

Ground state in closed-shell nucleus: |Ψ0 = eT |ϕ0 T =

i,m

tm

i a† mai +

ij,mn

1 4tmn

ij a† ma† naiaj + . . . m,n>F i,j<F

States in closed-shell + a few constructed in similar way. Construction of Unitary Transformation to Shell Model for 76Ge:

1. Calculate low-lying spectra of 56Ni + 1 and 2 nucleons (and 3

nucleons in some approximation), where full calculation feasible.

2. Do Lee-Suzuki mapping of lowest eigenstates onto f5/2pg9/2 shell,

determine effective Hamiltonian and decay operator.

Lee-Suzuki maps d lowest eigenvectors to orthogonal vectors in shell model space in way that minimizes difference between mapped and original vectors.

3. Use these operators in shell-model calculation of matrix element for

76Ge (with analogous plans for other elements). Slater determinant

SLIDE 79

Option 2: In-Medium Similarity Renormalization Group

Flow equation for effective Hamiltonian. Asymptotically decouples shell-model space. d dsH(s) = [η(s), H(s)] , η(s) = [Hd(s), Hod(s)] , H(∞) = Heff

V [ MeV fm3] 10 5

5
10
15
20

hh pp

✛ ✲

hh pp

❄ ✻

s = 0.0 s = 1.2 s = 2.0 s = 18.3

Hergert et al.

Trick is to keep all 1- and 2-body terms in H at each step after normal ordering. Like truncation of coupled-clusters expansion. If shell-model space contains just a single state, approach yields ground-state energy. If it is a typical valence space, result is effective interaction and operators.

SLIDE 80

Ab Initio Calculations of Spectra

1 2 3 4 5 6 7 8

+

2

+

3

+ +

2

+

4

+ +

2

+

3

+ +

2

+

4

+ +

2

+ +

3

+

2

+

4

+

4

+ +

2

+

3

+

(0

+ )

(2

+ )

(4

+ )

1 2 3 4 5 6

1 2 + 5 2 + 3 2 + 1 2 + 5 2 + 3 2 + 1 2 + 5 2 + 3 2 + 1 2 +

(5

2 + )

(3

2 + )

1 2 3 4 5 6 7

+

2

+

1

+ +

2

+

1

+ +

2

+

1

+ +

2

+

1

+

22O 23O 24O

Ex [ MeV]

C C E I I M

S

R G U S D B E x p . C C E I I M

S

R G U S D B E x p . C C E I I M

S

R G U S D B E x p .

Neutron-rich

xygen isotopes

Deformed nuclei

1 2 3 4 5 6 7 8 9 10 11 12 13 14

+

2

+

4

+

6

+

8

+ +

2

+

4

+

6

+

8

+ +

2

+

4

+

6

+

8

+ +

2

+

4

+

6

+

8

+ +

2

+

4

+

6

+

8

+ +

2

+

4

+

6

+

8

+ +

2

+

4

+

6

+

8

+ +

2

+

4

+

6

+

Ex [ MeV]

C C E I I M

S

R G U S D B E x p . C C E I I M

S

R G U S D B E x p . 20Ne 24Mg

SLIDE 81

Coupled Cluster Test in Shell-Model Space: 48Ca − →48Ti

No Shell-Model Mapping From G. Hagen 0+ 0+ 2+ 2+ 2+ 2+ 4+ 4+ 3+ 3+ 4+ 4+ 2+ 2+ 1+ 1+ 2+ 2+ 3+ 3+ 4+ 4+ 0+ 0+ EOM CCSDT-1 Exact

48Ti Spectrum

P r e l i m i n a r y

SLIDE 82

Coupled Cluster Test in Shell-Model Space: 48Ca − →48Ti

No Shell-Model Mapping From G. Hagen 0+ 0+ 2+ 2+ 2+ 2+ 4+ 4+ 3+ 3+ 4+ 4+ 2+ 2+ 1+ 1+ 2+ 2+ 3+ 3+ 4+ 4+ 0+ 0+ EOM CCSDT-1 Exact

48Ti Spectrum

ββ0ν Matrix Element GT F T Exact .85 .15

.06

CCSDT-1 .86 .17

.08

P r e l i m i n a r y

SLIDE 83

Full Chiral NN + NNN Calculation (Preliminary)

From G. Hagen

Method E3max M0ν CC-EOM (2p2h) 1.23 CC-EOM (3p3h) 10 0.33 CC-EOM (3p3h) 12 0.45 CC-EOM (3p3h) 14 0.37 CC-EOM (3p3h) 16 0.36 SDPFMU-DB

1.12

SDPFMU

1.00

P r e l i m i n a r y

Last two are two-shell shell-model calculations with effective interactions.

SLIDE 84

Complementary Ideas: Density Functionals and GCM

Construct set of mean fields by constraining coordinate(s), e.g. quadrupole moment Q0. Then diagonalize H in space of symmetry-restored quasiparticle vacua with different Q0. β2 = deformation

Robledo et al.: Minima at β2 ≈ ±.15

Collective wave functions

(b)

0.6

0.4
0.4
0.2

0.2 0.4 0.6

76Ge (0i +) 76Se (0f +)

β2

Rodriguez and Martinez-Pinedo: Wave functions peaked at β2 ≈ ±.2

SLIDE 85

Complementary Ideas: Density Functionals and GCM

Construct set of mean fields by constraining coordinate(s), e.g. quadrupole moment Q0. Then diagonalize H in space of symmetry-restored quasiparticle vacua with different Q0. β2 = deformation

Robledo et al.: Minima at β2 ≈ ±.15

Collective wave functions

(b)

0.6

0.4
0.4
0.2

0.2 0.4 0.6

76Ge (0i +) 76Se (0f +)

β2

Rodriguez and Martinez-Pinedo: Wave functions peaked at β2 ≈ ±.2

We’re now including crucial isoscalar pairing amplitude as collective coordinate...

SLIDE 86

Capturing Collectivity with Generator Coordinates

How Important are Collective Degrees of Freedom? Can extract collective separable interaction —— monopole + pairing + isoscalar pairing + spin-isospin + quadrupole —— from shell model interaction, see how well it mimics full interaction for ββ matrix elements in light pf-shell nuclei.

0.5 1 1.5 2 2.5 3 3.5 4 22 24 26 28 30 32 34 36 38 40 Ca → Ti MGT Nmother KB3G Hcoll.

Hcoll. (no T = 0 pairing)

SLIDE 87

Capturing Collectivity with Generator Coordinates

How Important are Collective Degrees of Freedom? Can extract collective separable interaction —— monopole + pairing + isoscalar pairing + spin-isospin + quadrupole —— from shell model interaction, see how well it mimics full interaction for ββ matrix elements in light pf-shell nuclei.

0.5 1 1.5 2 2.5 3 3.5 4 22 24 26 28 30 32 34 36 38 40 Ca → Ti MGT Nmother KB3G Hcoll.

Hcoll. (no T = 0 pairing)

Good news for collective models!

SLIDE 88

GCM Example: Proton-Neutron (pn) Pairing

Can build possibility of pn correlations into mean field. They are frozen out in mean-field minimum, but included in GCM.

0νββ matrix element

−5 5 10 15 0.5 1 1.5 2 2.5 3

M 0ν gpn

pn-GCM Ordinary GCM

Collective pn-pairing wave functions

0.1 0.2

76Ge

0.1 0.2 2 4 6 8 10

|Ψ(φ)|2 φ = pn pairing amplitude

76Se

Proton-neutron pairing significantly reduces matrix element.

gpp

SLIDE 89

GCM in Shell-Model Spaces

1 2 3 4

GCM Exp. 76Se

+ 3 + 3 + 3 + 3 + 2 + 2 + 2 + 2

2

+ 1

2

+ 1

2

+ 1

2

+ 1 + 1 + 1 + 1

Excitation energy (MeV)

+ 1

76Ge Exp. GCM

GCM Spectrum in 2 Shells ββ Matrix Elements in 1 and 2 Shells

SLIDE 90

Combining DFT-like and Ab Initio Methods

GCM incorporates some correlations that are hard to capture automatically (e.g. shape coexistence). So use it to construct initial “reference” state, let IMSRG, do the rest.

Test in single shell for “simple” nucleus.

In progress: Improving GCM-based flow. Coding IMSRG-evolved ββ transition operator. To do: applying with DFT-based GCM.

SLIDE 91

Improving RPA/QRPA

RPA produces states in intermediate nucleus, but form is restricted to 1p-1h excitations of ground

state. Second RPA adds

2p-2h states.

16O

0.02 0.04 0.02 0.04

Fraction E0 EWSR/MeV 5 10 15 20 25 30 35 40 E (MeV)

0.02 0.04 (a) (b) (c) RPA SSRPA_F Exp

DFT-Corrected Second RPA

SLIDE 92

Issue Facing All Models: “gA”

40-Year-Old Problem: Effective gA needed for single-beta and two-neutrino double-beta decay in shell model and QRPA.

from experimental τ1 2 ISM gA,eff

ISM

1.269A 0.12 from experimental τ1 2 IBM 2 CA SSD gA,eff

IBM 2 1.269A 0.18

Ca Ge Se ZrMo Cd Te Xe Nd 40 60 80 100 120 140 160 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Mass number gA, eff

from F. Iachello

If 0ν matrix elements quenched by same amount as 2ν matrix elements, ex- periments will be much less sensitive; rates go like fourth power of gA.

SLIDE 93

Arguments Suggesting Strong Quenching of 0ν

Both β and 2νββ rates are strongly quenched, by consistent factors. Forbidden (2−) decay among low-lying states appears to exhibit similar quenching. Quenching due to correlations shows weak momentum dependence in low-order perturbation theory.

SLIDE 94

Arguments Suggesting Weak Quenching of 0ν

Many-body currents seem to suppress 2ν more than 0ν. Enlarging shell model space to include some effects of high-j spin-orbit partners reduces 2ν more than 0ν. Neutron-proton pairing, related to spin-orbit partners and investigated pretty carefully, suppresses 2ν more than 0ν.

1 2 3 Ca → Ti 1 2 3 22 24 26 28 30 32 34 36

2ν 0ν

MGT Nmother full no T = 0 pairing

4 8 12 r (fm)

4
2

2 P(r) (fm

1)

gpp=0.0 gpp=0.8 gpp=1.0 gpp=1.2

Large r contributes more to 2ν.

SLIDE 95

Effects of Closure on Quenching

Two-level model: Initial |0I |1I Intermediate |0M |1M Final |0F |1F

Shell-model space

E0 E1 Assume Lower levels: 0M| β |0I = 0F| β |0M ≡ Mβ Upper levels: 1M| β |1I = 1F| β |1M = −α Mβ Operator doesn’t connect lower and upper levels. “Shell-model” calculation gets Mββ = M2

β

E0 Mcl

ββ = M2 β

SLIDE 96

Effects of Closure on Quenching (Cont.)

β = Mβ(cos2 θ − α sin2 θ)2

M′

2ν = M′ β 2

1

E0 + (α + 1)2 sin2 θ cos2 θ E1

M′

2ν cl = M′ β 2

1 + (α + 1)2 sin2 θ cos2 θ

SLIDE 97

Effects of Closure on Quenching (Cont.)

β = Mβ(cos2 θ − α sin2 θ)2

< Mβ M′

2ν = M′ β 2

1

E0 + (α + 1)2 sin2 θ cos2 θ E1

M′

2ν cl = M′ β 2

1 + (α + 1)2 sin2 θ cos2 θ

SLIDE 98

Effects of Closure on Quenching (Cont.)

β = Mβ(cos2 θ − α sin2 θ)2

< Mβ M′

2ν = M′ β 2

1

E0 + (α + 1)2 sin2 θ cos2 θ E1

≈

M′

β 2

E0 M′

2ν cl = M′ β 2

1 + (α + 1)2 sin2 θ cos2 θ

E0 ≪ E1

SLIDE 99

Effects of Closure on Quenching (Cont.)

β = Mβ(cos2 θ − α sin2 θ)2

< Mβ M′

2ν = M′ β 2

1

E0 + (α + 1)2 sin2 θ cos2 θ E1

≈

M′

β 2

E0 M′

2ν cl = M′ β 2

1 + (α + 1)2 sin2 θ cos2 θ

>

M′

β 2

= Mcl

2ν,

α = 1

E0 ≪ E1

So if α = 1, the closure matrix element is not suppressed at all.

SLIDE 100

Effects of Closure on Quenching (Cont.)

β = Mβ(cos2 θ − α sin2 θ)2

< Mβ M′

2ν = M′ β 2

1

E0 + (α + 1)2 sin2 θ cos2 θ E1

≈

M′

β 2

E0 M′

2ν cl = M′ β 2

1 + (α + 1)2 sin2 θ cos2 θ

>

M′

β 2

= Mcl

2ν,

α = 1

E0 ≪ E1

So if α = 1, the closure matrix element is not suppressed at all. If α = 0, it’s suppressed as much as the single-β matrix element, but still less than the non-closure ββ matrix element.

SLIDE 101

We Hope to Resolve the Issue Soon

Problem must be due to some combination of:

1. Truncation of model space.

Should be fixable in ab-initio shell model, which compensates effects of truncation via effective operators.

2. Many-body weak currents.

Size still not clear, particularly for 0νββ decay, where current is needed at finite momentum transfer q. Leading terms in chiral EFT for finite q only recently worked

ut. Careful fits and use in decay computations will happen in

next year or two.

SLIDE 102

Benchmarking and Error Estimation

Systematic Error:

1. Calculate and benchmark spectra and transition rates (including β

decay) with all good methods.

2. Calculate β, 2νββ and 0νββ matrix elements in light nuclei — 6He,

8He, 22O, 24O — with methods discussed here plus no-core shell

model and quantum Monte Carlo.

3. Do the same in 48Ca.
4. Test effects of “next order” in EFT Hamilton, coupled-cluster

truncation, restrictions to N-body operators, etc.

5. Benchmark methods against spectra and electromagnetic transitions

in A = 76, 82, 100, 130, 136, 150.

SLIDE 103

Benchmarking and Error Estimation

Systematic Error:

1. Calculate and benchmark spectra and transition rates (including β

decay) with all good methods.

2. Calculate β, 2νββ and 0νββ matrix elements in light nuclei — 6He,

8He, 22O, 24O — with methods discussed here plus no-core shell

model and quantum Monte Carlo.

3. Do the same in 48Ca.
4. Test effects of “next order” in EFT Hamilton, coupled-cluster

truncation, restrictions to N-body operators, etc.

5. Benchmark methods against spectra and electromagnetic transitions

in A = 76, 82, 100, 130, 136, 150. Statistical Error: Chiral-EFT Hamiltonians contain many parameters, fit to data. Posterior distributions (for Bayesian analysis) or covariance matrices (for linear re- gression) developed to quantify statistical errors for ββ matrix elements.

SLIDE 104

Finally...

Existence of topical collaboration will speed progress in next few years.

Or else I’m in big trouble.

Goal is accurate matrix elements with quantified uncertainty by end of collaboration (5 years from now).

SLIDE 105

Finally...

Existence of topical collaboration will speed progress in next few years.

Or else I’m in big trouble.

Goal is accurate matrix elements with quantified uncertainty by end of collaboration (5 years from now).

Matrix Elements for Double-Beta Decay

Matrix Elements for Double-Beta Decay

← − d3/2 ← − s1/2 ← − d5/2

. . . . . . .

.

Matrix Elements for Double-Beta Decay

Relating Theory to Experiment

How do we get to nuclear scales?

n

p p

n n

p

n

p

n

p

n

p

n

p p

n n

p

n

p

n

p

n

p

n

p

n

p

n

p

n

p

n

p

n

p

Matrix Elements for Double-Beta Decay

P r e l i m i n a r y

P r e l i m i n a r y

P r e l i m i n a r y

That’s all; thanks for listening.