Weak interaction processes with spectator nucleon in the proto … · 2017. 12. 21. · 1. Motivation Core-collapse supernovae Historical considerations: I 185: Earliest known written

Weak interaction processes withspectator nucleon in the proto-neutron starStephen Frieß

April 1, 2016 | TU Darmstadt, IKP, Theoretical Nuclear Astrophysics Group | Stephen Frieß | 1

Outline

1. Motivation2. Structure functions and opacities

I Description of simple neutrino reactionsI Neutral-current scattering with spectator nucleonI Charged-current processes with spectator nucleon

3. Calculation of matrix elementsI Basic framework for matrix element calculationsI Matrix element for charged-current processes with spectatorI Non-relativistic corrections of the weak interaction

4. Summary and outlook


1. MotivationCore-collapse supernovae

Historical considerations:I 185: Earliest known written record of the

observation of a supernova event.I 1931: Lev Landau speculates about the

existence of (neutron) stars containingmatter at nuclear density.

I 1934: Walter Baade and Fritz Zwicky linkneutron star formation to supernova events.

Modern research:

I Neutron star formation particularlyassociated with type II ’core-collapse’supernovae.

I Requirement: Progenitor mass M > 8M�. From left to right: Lev Landau,Walter Baade and Fritz Zwicky.


1. MotivationCore-collapse supernovae


1. MotivationRole of neutrinos

I Densities are reached at whichneutrinos become effectively trapped byvarious reaction modes.

I Determination of the resulting neutrinospectrum important for shock revivaland nucleosynthesis.

I Shock revival:I Reignition of the stalled shockwave.I Delayed neutrino-driven explosion

mechanism.I Nucleosynthesis:

I R-process in the neutrino-driven wind.I ν process in the outer shells.



Additional reactions of interest:I Neutron decay.I Neutral-current reactions with spectator

nucleon: Prescription developed by Raffelt(2001).

I Charged-current reactions with spectatornucleon: Not yet considered.



Additional reactions of interest:I Neutron decay.I Neutral-current reactions with spectator

nucleon: Prescription developed by Raffelt(2001).

I Charged-current reactions with spectatornucleon: Not yet considered.

Primary goal of this work:Investigation of charged-current processes withspectator nucleon.


Outline






2. Structure functions and opacitiesDescription of simple neutrino reactions

We consider semileptonic processes of the form ν1N2 → l3N4 .

I Using Fermi’s golden rule the opacity or inverse mean free path can becalculated as follows (c.f. Reddy et al., 1998):

χ(E1) = 2∫

d3p2

(2π)3

∫d3p3

(2π)3

∫d3p4

(2π)3 (2π)4δ4(P1 + P2 − P3 − P4)Wif

× f2(E2)(1− f3(E3))(1− f4(E4)).

I Transition rate for non-relativistic and non-interacting nucleons:

Wif ≈ G2F [(F 2

1 + 3F 2A ) + cos(θ13)(F 2

1 − F 2A )]

I Coupling constants F1 and FA dependent on the particular reaction of interest.



I Introduce energy and momentum differences

ω = E1 − E3 and k = |~p1 − ~p3|.

I Define a structure function containing all information about the nucleonicmedium:

S(ω, k ) = 2∫

d3p2

(2π)3

∫d3p4

(2π)3 (2π)4δ(ω+E2−E4)δ(~k +~p2−~p4)f2(E2)(1− f4(E4)).

I The opacity for ν1N2 → l3N4 , may then be rewritten as

χ(E1) = G2F (F 2

1 + 3F 2A )∫

dωE3

E1(1− f3(E3))

∫dk k S(ω, k ).

I Inverse neutron decay ν1N2l3 → N4 can be treated in a similar fashion.




2. Structure functions and opacitiesNeutral-current scattering with spectator

I Given by ν1N2N3 → ν4N5N6 , the structure function can be calculated by

S(ω, k ) ∝ 2∫ ∏

Nucleons j

d3pj

(2π)3 f2f3(1−f5)(1−f6)Wif (2π)4δ(4)(P1+P2+P3−P4−P5−P6),

where ω = E1 − E4 and k = |~k1 − ~k4|.

I Analytic expressions exist in certain limits and on heuristic basis.In the following we use (Raffelt, 2001):

S(ω, k ) =nN

T2

1 + e−ω/T

2γ(ω/T )2 [1− e

π4

(ω/T

γ+κe−γ/κ

)2

] +2√π

κ + γeγ/κe−

(ω/Tκ

)2

with number density nN , effective momentum κ = k

√2/TM and dilution

parameter γ = 1.25 ρ14√

3/(2 + T10).


2. Structure functions and opacitiesNeutral-current scattering with spectator

Validity of the structure function:

I Non-degenerate conditions (µN/T � −1).

I Temperatures of T ≈ 4.9− 9.7 MeV.

Properties of the structure function:

I Dilute limit:

I Detailed balance:

I Normalization:

I f-sum rule:

limγ→0

S(ω, k ; γ) = SνN→Nν (ω, k ).

S(−ω, k ) = e−ωT S(ω, k ).∫ ∞

−∞dωS(ω, k ) = 1.∫ ∞

−∞dω ωS(ω, k) =

k2

2M.


3. Structure functions and opacitiesCharged-current processes with spectator

Utilizing the prescription:

I First step: Modify structure function for usage of charged currents.I Look at the definition of the structure function:

S(ω, k )ν1N2N3→ν4N5N6 ∝∫

d3pi

(2π)3· · · δ(ω + E2 + E3 − E5 − E6)

I The nucleon energies are governed by

Ei =p2

i

2MN+ Mi + Ui .

I Calculate charged currents with substitution: ω → ω +∑

U (i)j − U (f )

j .

I Last step: Choose the final state lepton Fermi-Dirac function and integrationlimits such that we get the non-spectator process in the dilute limit γ → 0.







Problems of the prescription:

I 1. Problem: Prescription only well-defined for a narrow set of conditions.I Extrapolations to degenerate settings are however possible.

I 2. Problem: Detailed balance needs to be reconsidered.I For charged currents, detailed balance is governed by:

Si→f (−ω, k ) = e(∆µif−ω)/T Sf→i (ω, k ).

I Implementation possible with further schemes.

I 3. Problem: Consistency with normalization and f-sum rule.I 4. Problem: Spectator species cannot be chosen explicitely.


Outline






3. Calculation of matrix elementsBasic framework

I In the approximation of contact interactions,the Lagrangian for weak processes can bewritten in the form of a current-currentinteraction:

L =G√

2HµLµ

I The hadron and lepton current are given by

Hµ ≈ ψ4γµ(F1 − FAγ

5)ψ2 and Lµ = ψ3γµ(1− γ5)ψ1.

I The hadron current Hµ is an approximation for low momentum transfers.I F1 and FA are vector and axial coupling constants.I The transition rate Wif is given by

Wif [Mif ] =〈|Mif |2〉24E1E3

with matrix-element Mif =G√

2HµLµ.


3. Calculation of matrix elementsNucleons at rest

I We start from the hadron current as given by:

Hµ ≈ ψ4γµ(F1 − FAγ

5)ψ2.

I The nucleon wavefunctions are further given by the Dirac spinors:

ψN =1√2E

u(~p, s) =

√E + m

2E

(χs~σ~p

E+mχs

)I Where χ1 = (1, 0)T and χ2 = (0, 1)T .I Assuming nucleons at rest, we simply find

ψN ≈(χs

0

).

I The hadron current may then be rewritten as the product of 2d vectors:

Hµ = χ†s4Hµχs2 with Hµ ≈ F1δµ0 − FAσiδµi .


3. Calculation of matrix elementsCharged-current processes withspectator nucleon

I E.g. consider neutron decay with spectator:

I A model for the nuclear interaction is needed.I In the following, we will use one-pion exchange

VOPE = −(

fmπ

)2 (~σ(1)~k )(~σ(2)~k )(~τ (1)~τ (2))k2 + m2

π

.

I Wavefunctions are thus given by: ψNs = χs ⊗ |N〉.





I Example diagram:

I Resulting matrix element:

MAif = − G√

2

(f

mπ

)2

Lµ[χ†3~σ~kχ1(k2 + m2π)−1χ†4(F1δµ0 − FAδµiσi ) ω−1~σ~kχ2]

×〈n|τ j |n〉〈p|τ−τ j |n〉.



I Calculating analogously (B) and (C), we get for the sum (A+B+C):

MABCif = FA

G√2

(f

mπ

)2

ω−1(k2 + m2π)−1Li×

[2kiχ†3~σ~kχ1χ

†4χ2 + 2kjχ

†3(iεijkσk − δij )χ1χ

†4~σ~kχ2].

I The contribution from the (D+E+F) diagrams can be obtained analogously.I The total matrix element is given by adding both contributions incoherently:

Mif = MABCif −MDEF

if

I To utilize the matrix element in further calculations, we need to calculate thespin-summed and squared matrix element:∑

Spins

|Mif |2 =∑Spins

|MABCif −MDEF

if |2 =∑Spins

2×{|MABC

if |2 −MABCif MDEF

if†}

.



I Neglecting terms in the lepton momenta, we get:

∑Spins

|Mif |2 = 768 G2F 2A

(f

mπ

)4 k4

ω2(k2 + m2π)2 EeEνe .

I Comparing this to results from Friman and Maxwell (1979), we find:

∑Spins

|Mif |2 ≡∑Spins

2×|MABCif |2 = 512 G2F 2

A

(f

mπ

)4 k4


I It was obtained by only considering the (A+B+C) contribution.I The (D+E+F) diagrams were accounted for by a multiplicative factor 2.



I The discrepancy arises in our calculation, due the mixed contribution beingnon-vanishing:∑

Spins

|Mif |2 =∑Spins

2×{|MABC

if |2 −MABCif MDEF

if†}

.

I In particular, we find for the mixed contribution:

−2×∑Spins

MABCif MDEF

if† = 256 G2F 2

A

(f

mπ

)4 k4


I We assume in our subsequent calculations a vanishing mixed contribution andconcentrate only on the (A+B+C) contribution.

I The (D+E+F) contribution can be obtained at any time by a simple indexsubstitution.


Outline






3. Calculation of matrix elementsNon-relativistic corrections

I In the following, we will consider the hadron current as given by:

Hµ = vµH − aµH = ψN′ (~p′, s′)

[ΓV − ΓA

]ψN (~p, s)

I With the vector and axial vertex operators

ΓµV = [F1γµ + i

F2

2mσµνqν ]τ± and

ΓµA = [FAγµγ5 +

GP

mγ5qµ]τ±.

I We additionally consider the weak magnetism and pseudoscalar term .I When neglecting them, i.e. F2 ≡ GP ≡ 0, we get the familiar term

ΓµV − ΓµA = γµ(F1 − FAγ5),

which we haved used in previous calculations.



I In the approximation of nucleons at rest, we wrote the hadron current aselement of a 2x2 decomposition of the vertex operator:

Hµ = χ†f Hµχi ≈ χ†f (F1δµ0 − FAδµjσj )χi .

I To extend the 0th order result successively by higher orders, we calculate a2x2 matrix Hµ from the 4x4 vertex operator ΓµV − ΓµA .

I We approximate the hadron current matrix only to first order, to keepsubsequent calculations handable:

Hµ ≈ δµ0

{F1 − FA

~σ(~p′ + ~p)2m

}+δµj

{−FAσj + F1

(~p + ~p′)j

2m+ (F2 − F1)

i(~σ × ~q)j

2m

}I Note that at this order, we get a contribution from the weak magnetism term,

however none from the pseudoscalar term.



I It is useful to reformulate the squared and spin-summed matrix elementM interms of auxiliary tensors.

Hµνa = Tr[Hµa (H†)νa ]

Vµνab = Tr[Hµa ]× Tr[(H†)νb ]

Wµνab = k−2Tr[Hµ

a (~σ~k )(H†)νb (~σ~k )]

Lµν =∑Spins

LµL†ν

I For the (A), (B) and (C) diagrams, we then get:

M = α2 k4 Lµν{8Hµν1 + 2Hµν2 + 2Hµν4 − 2Vµν21 − 2Vµν41

−2Vµν12 − 2Vµν14 + 2Wµν24 + 2Wµν

42 }.

I To obtain non-relativistic corrections, we now just need to calculateexpressions for the auxiliary tensors H, V andW at desired order.



I In 1st order of the nucleon momenta, we get

M(1) =M(0) +

[−128F 2

A

~P1

m− 64

F 2A

m[~k (~P2 + ~P4)]~k

k2

](Ee~pνe + Eνe

~pe)

+ 128FA(F1 − F2)

[(~k × ~q)× ~k

k2m+

2~qm

](Ee~pνe − Eνe

~pe),

where ~Pi = ~pi + ~p′i is the sum of the nucleon momenta before and after a weakinteraction.

I To ensure consistency with the underlying approximation scheme, one mayadditionally assume q/m ≈ 0.



I 2nd order correction terms in the nucleon momenta for q/m ≈ 0:

M(2) =M(1) + 4 (F 21 + F 2

A )8~P2

1 + 2~P22 + 2~P2

4

m2 EeEνe

+ 8 (2 F 21 − F 2

A )(~P2~P4)

m2 EeEνe − 32 F 21

[~P1(~P2 + ~P4)]m2 EeEνe

+ 16 F 2A

(~k~P2)(~k~P4)m2 k2 EeEνe + 4 (F 2

A − F 21 )

8~P21 + 2~P2

2 + 2~P24

m2~pe~pνe

− 8(2F 21 + F 2

A )(~P2~P4)

m2~pe~pνe + 32 F 2

1[~P1(~P2 + ~P4)]

m2~pe~pνe + 16 F 2

A(~k~P2)(~k~P4)

m2k2~pe~pνe

+ 16 F 21

[4(~P1~pe)(~P1~pνe )

m2 +(~P2~pe)(~P2~pνe )

m2 +(~P4~pe)(~P4~pνe )

m2

]


4. Summary and outlookSummary

Structure function studies:

I We extended a prescription for neutral-current scattering with spectatornucleon to charged currents.

I The extension predicts significant enhancement of the opacities for theprocesses νe e−p N → N n and νe p N → N n e+.

I The usage of the prescription however comes with problems.

Matrix element studies:

I We found a discrepancy when trying to verify an approach used to calculatethe spin-sum for νe e−p N → N n with exchanged isospin states.

I A formalism was introduced to systematically calculate non-relativisticcorrections stemming from the weak interaction.

I Further, correction terms up to 2nd order in the nucleon momenta wereexplicitely derived.



Outlook:

I The calculated non-relativistic correction terms to the matrix element can betested in the context of opacity or emissivity calculations.

I In the thesis: We also give a formula to account for the correction termsstemming from the mixed contribution we found.

I A full-relativistic treatment might however become less expansive than theinclusion of higher order correction terms.

I The treatment of the nuclear interaction could be further improved on thebasis of e.g. contributions from boost corrections or chiral EFT.

I Other interaction models could be further investigated.


Thank you for your attention!


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Weak interaction processes with spectator nucleon in the proto … · 2017. 12. 21. · 1. Motivation Core-collapse supernovae Historical considerations: I 185: Earliest known written