I

20.3

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LOS ALAMOS SCIENTIFIC LABORATORY

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SOME LOW ENERGY NE UTRINO CROSS SECTIONS

—
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I

LAMS-3013

UC-34, PHYSICS

TKD-4500 (26thEd.)

LOS ALAMOS SCIENTIFIC LABORATORY

OF THE UNIVERSITYOF CALIFORNIA LOS ALAMOS NEW MEXICO

REPORT WRITTEN December 1, 1963

REPORT DISTRIBUTED: March 17, 1964

SOME LOW ENERGY NEUTRINO CROSS SECTIONS

by

Leon Heller

—

I

ContractW-’74O5-ENG.36 withthe U. S. Atomic Energy Commission

All LAW reporte are informal documents, usually prepared for a speokl purpdae and primarily prepared for use withfn the Labcmtory rather than for

general distribution. Tbia repert he not been edited, reviewed, or verified

for accuraoy. All LAMS reporte express the views of the authore M of tbe

time they were written and do net necessarily reflect the opfnione of tbe Los

Mamos Scientific Laboratory or the final opinion of the authors on the eubject.

I

1

ABSTRACT

The cross sections for elastic neutrino and antineutrino scattering

on electrons, and anti.neutrinoabsorption on protons (giving a neutron

and a positron)> are calculated using the V-A four-fermion interaction,

Recoil electron spectra are presented for the elastic scattering, and

the angular distribution of positrons is given for the absorption process,

in addition to total cross sections for all these processes. Results are

given analytically and in graphical.form.

The steps necessary to perform these calculations are explained in

detail.

For the elastic scattering the results are expected to be valid for

center of mass energies much less than the mass of the intermediate boson

(if it exists), and also for energies too smELU to probe the structure of

the neutrino and/or electron (if there is any). Therefore the fornni!se

should be reliable up to at least Xl MeV neutrinos and antineutrinos.

For the absorption process the anomalous magnetic moment of the nucleons

presents the first correction (with increasing energy) to the calculation

presented here which is accurate only below 10 MeV antineutrino energy.

3

An Appendix reproduces some lecture notes from 1957 on nucle~ beta

decay. Although a parity-conserv~ interaction was used h those notes,

the formulae are still valid for all those results involving initially

unpolarized nuclei and in which all final polarizations are summed over.

CONTENTS

1.

II.

III.

me

v.

VI*

VII.

Abstract

Introduction

Notation

Interaction

Kinematics

Density of States

Cross Sections

Examples A. ~+e~~+e

B. v+e-+v+e

CO X+p~n+e+

References

Page

3

7

8

9

16

18

20

22

25

26

29

33

5

I. Introduction

Calculations of cross sections involving neutrinos have appeared in

the literature, and none of the results presented in this report are new

although we do point out some errors in the literature. It was felt

desirable, however, to collect in a single place the detailed steps involved in performing these calculations.

We use the four-fermion interaction as given by Feynman and GeUmannl, slightly generalized to sUow for unequal vector and axial vector

coupling constants. This interaction

nor an “induced” pseudoscalar3 term.

volving nucleons becomes increasingly

does not include %eak magnetism”2

The former term for processes unimportant as the energy increases,

and represents, already at Ml MeV, a correction of-~ to our calculation

OfV+p-n+e+. The pseudoscalar term is”only significant for processes

involving y-mesons. A complete cross section formula containing contributions

fermion

If

here of

only up

mass.

from aid.possible Lorentz invariant interactions of the four-

%Ype is given in reference k.

there is an intermediate boson, then the calculations presented

neutrino and

to center of

antineutrino scattering on electrons wilJ be valid

mass energies which are much smaller than the boson

7
After a brief section on notation, Section III folkws in detail

the calculation of the square of the matrix element, summed over all

polarizations of all four fermions. The utility of this quantity, with

special reference to the role of neutrinos, is discussed. Sections IV

andVs ummarize the needed kinematics and density of states formulae,

and in Section VI all the results are collected in a cross section for-

-e Section VII applies these results toy + e ~~ + e, v + e +V + e,

and? + p +n + e+, and numerical cross sections are presented in graphical.form. Included in this section

of neutral symmetric currents on the

Some lecture notes from 1957 on

is a discussion of

elastic scattering

nuclear beta decay

the possible role

processes.

are included as

an Appendix since they contain

operators which are needed for

in the Appendix which refer to

many of the kesults about projection

the present calculations. AU the results

initially unpolarized nuclei, and final

states in which aU polarizations are sumned over, are valid even though

a parity conseting interaction was used since no pseudoscalar quantities

can be formed from just the two lepton momenta.

II. Notation

Greek indices run from zero to three; Roman indices from one to

three. Three-dimensional vectors are written with arrws, ~; fourdimensional vectors with a wavy underline,~ Dot products are defined

where the metric tensor g has been introduced for later convenience:

pv

8
-

01000

4

1 0-1 0 0

g

=

2 0 0-1 0

3 0 0 0-1

The y-matrices which shaU be used are the folkwing:

4

where I and a are the two-dimensional identity and Pauli spin matrices,

respectively. Some properties of these matrices are that

Y~ = 7.; ;+ = -% YPYV + YVYV = 2&3 t

WV; YVYO = 707V; 757P =

The dagger superscript denotes the Hermitean conjugate.

- 7P75; (i75)2 = 1.

h and c will often be set equal to unity, but all cross sections

will finally be expressed in c&. The symbol v or; without a subscript

refers to the electron’s neutrino or antineutrino; the ~-meson’s neutrino

is designated VW.

III. Interaction

We shall use a slightly generalized version of the Feynman, GellMann interaction> which allows for unequal vector and axial-vector

strengths. This is not the most general possible Lorentz invariant

9
interaction, but is valid at low momentum transfers.

A and B are leptons, but C and D need not be. The Vts are the field

operators (containing spinors) for the corresponding particles> with

v = @70a /( )

G = 1,41.X 10-49 erg-cm= = 1.01 x 10-5 (~c)3 MPC2 2 with

Mp the mass of the protonIn first order perturbation theory, the matrix element M of this

Hamiltonian looks just like H except that the ~’s are now just the

spinors appropriate to the particles$

where a ~ 1 - i7=, b = f - gi75* Note that af = a, b~ E b (assuming f

and g are real-)$ a2 = 2a$ b2 = 2cd where c ~ (~ + 5)/2~ =d

d –l--& ‘g 175. =

Also, (f - r3i7J(f+gi7J =F -f.

The quantity needed to calculate cross sections is

using the properties of the y-matrices listed previously.

10
Define

Then,

To evaluate this expression one could insert explicit spinors which

tell whether A is a particle or antiparticle and what its momentum and

polarization are; do the same for B, C, and D; and carry out the matrix

multiplication. This would be the simplest thing to do if all the particle

polarizations were actually measured. If some of the initial state

particles are Unpolarizec$then [Ml= must be evaluated separately for each

tnitial polarization state and an average taken. Similarly, if some of

the polarizations of the final state particles are not measured, then

IM12 must be calculated separately for each final polarization state and

the sum taken. If IM12 must be recalculated for several different polarizations states, the use of

alternative procedure which

were measured, but which is

it avoids all the work just

ators which only permit the

for particles, negative for

explicit spinors becomes very

could be used even if all the

especially useful if they are

tedious. An

polarizations

not (because

mentioned), is the use of projection operdesired energy states to contribute (positive

antiparticles). Also> if desired, one can

11
pick out particular polarization states. After inserting these projection operators one can sum IM12 over all.four spinors of a given momentum (two different energies and two different polarizations), and this

eliminates the spinors via the closure relation5. We shall work out

only the case where the polarizations are not of

A special word about neutrinos is required.

interaction whichwe are using is such that only

(and right-handed antineutrinos) can take part.

site polarization (even if they did exist) would

interest6.

The Feynman, Gell-Mann

left-handed neutrinos

Neutrinos of the oppogive rise to a zero

value for IM12. Any neutrino which is present in the final state will

be completely polarized. There is no harm, however, in formally summing

IM12 over the final polarization states of this neutrino since the

“wrong” polarization state automatically contributes zero to the sum.

Assuming that any neutrino present in the initial state is completely

polarized the “correct”way, there is again no harm in summing over its

polarization states, but now this must be just a sum and not an average.

To Summarize, we shall sum IM12 over the polarizations of alJ four

fermions and insert a factor of 1/2 for each unpolarized particle in the

initial state. The results wi12 be valid provided no final state polarizations are measured (except neutrinos) and no initial state particles

are polarized

For each

tion operator

4

(again excepting neutrinos).

particle of momentum;, mass m, a positive energy projecA+(~) must be used, and for each antiparticle of momentum

+

p} mass m, a negative energy projection operator A-(-p) must be Useds.

M?
-)4

A-(-;) = ‘oy”p: 7@ + E ,

-0

where E and p are the physical values of the energy and momentum of the

particle (or antiparticle).

-;O; +m

A+(P)70 = 7oE ~ ; A-(-P)70 =

We sha13.write

A70=~.Q ,

where the + sign is used for a particle and the - sign for an aritiparticle.

The result of inserting these projection operators and using the

closure relation iss

x Ilfjz = $ E Tr(SwV)Tr(TVv)gVVgW J

aU 11,v

polarizations

where S

Wv = 7p~7v*AJ TVV = 7Vb~yVbQC, andTr denotes the trace. Now

13
and hereafter summations over Greek indices are understood

zero to three.

Commuting the b which is to the right in TVV towards

the following sequence of steps

to run from

the left via

yvb= (f +giy5)yV

~yvb = ~ [(f -gi7J?J~D* (f+8i7.)~]7v

b~7Vb = ~

[

2C(1 - 1

hiy5)~”yD+ 2% ‘v

1

7pb~7Vb = —

2%[ 1

2C(1 + hi75)7v@l?D+ 2y7p y~

finally yields

T =—

‘v 4EC{[ 1( C(1+hi&p~~D*~7w 7V P??C * mC

)/

9

where we have put h =fg/c=2fg/(~ +r), ande= (~ - f)/2. SVV

has

c+

the same fo?nnas T with the following replacements:

llv

C-A,D-B,

l,h41,ande-0.

Most of the formulas needed for evaluating the trace of T are WV

given in the App_. In addition there are the results which involve

75: 75 multiplied by zero, one, two, or three y-matrices has zero trace;

( )

and Tr 757a7P7V7v = ‘4~a~pv , where ~ is a completely antisymmetric

tensor. & vanishes if any two subscripts are equal; &0123 = ‘1; au

even permutations of (0123) have ~ = +1; if the permutation of the s~-

14
scripts is odd$ ~ = -1.

ECl%Tr(TWV) =+2mC~eg

pv

-WC z p#c#Y.c&x3 (%gvfl -5-JW+%$%

a,B

)

where now the plus sign in the first term is to be used if C and D are

both particles or both antiparticles, and the minus sign is to be used

if one is a particle and the other an antiparticle. The sutumationsin

the second term can be simply performed,

ECEDTr(TVv) =

With the same substitutions as before,

‘A%Tr (s~v)= 2(pI@v

Using the definitions of g and ~, one gets

15
(“0$ (@)]

- (kli)(%”pd] ‘

where we have used the fact that Z g g C

P,v w Vv Wtiellkw = 2(%%, - %A%a)

to simplify the last term.

The result of this section can be summarized in the following formula, where c and

stants f and g.

h have been reexpressed in terms of the coupling conpolarizations

The minus sign in the last term is used if C and D are

both antiparticles; otherwise, the plus sign is used.

pression in brackets is a Lorentz invariant quantity.

* -J

both particles or

Note that the exmO Kinematics

Given two particles in the final state, labeled 1

-4

momenturn-energyp, E,

16

and 2, with total
a

-+ -0-$

P= P=+P2

the energy-angle relation for

E= E1+E2

particle 1 cam

2

be found by eliminating

;= and~.

(;-;)’=g

which can be written as

~ - ppl Cos OA,=

angle between ~= and ~ (the total momentum). &= is the invariant square

of the total energy-momentum four vector. Squaring and rearranging

yields a quadratic

(F - p’

equation for E=

Co+e ) & - aml 1

+A2+m2$cos20=0 ,

1

which can be solved to give El as a function of e,

E(&2+TL+*P]cos e! [&2-(x@@]2 - 4nf(ngh?’ dJ12e )

E= =

2(F -pa Cos’e )

●

The maximum and minimum values of E= can be found from this by putting

sin e =

If

initial

In

00

the total energy

state, labeled 3

and

and

momentum is due

4, then 6 = ~

the laboratory system, with particle

17

to two particles in the

+ #4 + 2p3@p ●

.- A

4 originally at rest,
V. Densitv of States

Given a system of total momentum; and total energy E which disintegrates into two particles (~ = ~= + ~2; E = E= + E2), one can get

expressions for the nuniberof states per unit total energy in different

forms depending upon what physical variables are chosen. Willisms7

gives the derivation for the case in which the density of states p is

expressed in terms of the element of solid angle of one of the particles.

F

That derivation is poor in that it fails to take account of the fact

that (for fixed values of p and 8=) E may decrease as pl increases.

However, the result of that calculation is acceptable provided absolute

values are Ix&en:

p;dsll

P= (:)3 ‘% lEP~-E150fi=l

If one chooses to express p in terms

one of the particles, then this smbiguity

18

of the element of energy of

is not present since for fixed

v
p and El, E is a monotonically increasing function of Ql (just because

pz is a monotonically increasing function of 9=).

The idea behind the WilJiams~ derivation, as applied to the vari.

able El, is to find the total number of states with energy less than

+

or equal to E, with fixed El and p (and cpl),and then differentiate

this expression with respect to E to find the number of states per unit

total energy.

[Jml P=* (~)cl

1

— ~ dpl d(-cos e=) ,

where the range of integration is from e= = O to that value of 91 which

makes (El + ~) equal to E, i.e., the physical value of Cl=for the variables p and E.

19
%%

P=——

(:)” p ‘l@l “

Of course this result can be obtained directly from the formula in

terms of solid angle by using the relation (which can be shply derived

from Section IV)

VT. Cross Sections

The cross section for a reaction induced by neutrinos or antineutrinos on an unpolarized target, as explained in Section III, is

all

polarizations

20
where we have

this, observe

two puwers of

still to insert the necessary factors of h and co TO do

that IM12p has the dimension of [Energyl*J Length16 with

energy coming from & and two powers coming from particle

energies. Multiplying by (Ac)-* will give the proper dimensions for the

cross section. Defini~

0

2 ~ (m@)2 [1 ~_ 2(fic)(m#) 2

01-r .W=; (loolx lo-’) —

(hf# )2

= 8e53 x lo-e d ,

where m is the electron mass, yields the result

where all masses are exq?ressedin units of m,

mc, and all energies in units of mc’. E= and

title energies, and we have chosen to express

of one of the final particle energies rather than per unit solid angle●

The minus sign in the last term is used if C

or both antiparticles; the plus sign is wed

J

mm

all momenta in units of

~ are the two final parthe cross section per unit

and D are both particles

otherwise.

21
VII o ExamPles

A. Antineutrino-Electron Scattering

If the four-fermion interaction arises from the interaction of a

current with itselfl, Z g J J+, where one of the terms in J iS due to

pP#PP P

an electron-neutrino combination> ~v7W(l - i75)*eS then ~tine~rinoelectron scattering and neutrino-electron scattering wi13 both occur.

If there is a charged intermediate boson they must occur. (We are now

talking about the neutrino which is associated with the electron Ve$ not

the neutrino which is associated with the v-meson.) We first consider

the reaction

y’ i-

?%-’ %+e~ ‘

where the subscripts are used for identification purposes. The first

step is to properly identify the four particles taking part in the reaction with the four labels

and D=vo‘~ (One could, in

with C and B withD.) This

;a + ~+yP+e@

I t I )

A,B,C, ~dD: A=~a,B=~$C=ep~

this example, simultaneously interchange A

means that the particles are paired as shown

●

In this reaction the form factors f and g should bothbe put equal

to unity. From the general result of Section VI}

22
where all masses, energies, and momenta are in units of

respectively.

To

System,

3 = v=,

are the

faci~tate the specialization of this result to

make use of the labeling system from Section IV:

andk=

%“ We have already made use of the fact

two final particles; particle 3 is the one which

m, mc’ and mc,

the laboratory

l=eB,2=v5,

that 1 and 2

is incident in

the laboratory and particle 4 is initially at rest. From Section IV

P=Pa=P ZE =E4=m

‘a %

&2=fi+~ ,A=mm+E

‘a () ‘a

P ePe = P~ “Pl

(

=mm+ E-E

‘a e -~

M- ‘a ‘~)

Pe “P = p4°P= =m

(

=mm+E. E

a % -- ‘B ‘a )

‘P ‘ ~-

where conservation of energy

Introducing T = E - m, the

‘P

putting together the results

where E ~ E , the incident

v

‘a

1.67 x 1.0-44 & .. Mev-le

gives the last

kinetic energy

form of the last equation.

of the recoil electron, and

just obtained gives

antineutrino energy, and (oo/mc2) =

23
The next

values of T.

step is to determine the maxinm and minimum possible

From Section IV these are found to be

with the mtibuxn (maximum) T coming from

(forward) in the center of mass system.

Figure I. The cutoff at T- is sharp.

by integrating over all recoil energies,

electrons which go baclanwd

()

do

=*

is plotted vs T on

The total cross section is found

Note that this result agrees with that obtained in reference 4, but is

exactly twice as large as the value quoted in footnote 17 of reference

1. The total cross section is

linearity at small energies is

One point which stiIl has

of neutral.symmetric currents,

plotted onFigure 11.

hardly visible on this

to be discussed is the

terms such as TV7V(1 -

The depsrture from

scale.

possible existence

Te7v(l - i75)ve0 These terms wi~ certainly be present if there is a

neutral intermediate boson. If they are present, then the scattering

which we have been studying can also proceed via this type of coupling

[ 1

v+e~~+e.

.

L 1

24
It is shown in footnote 7 of reference 1 that the matrix element for this

coup~ng has exactly the ssme magnitude as the matrix element for the

other coupling scheme (the one we have used), but that the relative

sign is not certain. This sign is positive if the neutrino field operator and the electron field operator anticommute; it is negative if they

commute. Since the matrix elements must be added together (if both

couplings are present) there is the possibility of obtaining, in the

cross section, a factor of zero or a factor of four. This assumes that

the coupling constant is the ssme for both types of currents, and there

is no a priori reason for this to be the case.

An example of a reaction which can proceed onJy if neutral currents

efistis~p+e+~ +e. The observation of this reaction with the

v

cross section computed above would be very strong etidence for neutral

currents with the ssme coupling constant as that for the charged currentsIn reference 8 it is demonstrated that; - e scattering can take place

v

even if there are only charged intermediate bosons, but that the process

is of higher order and is expected to have a cross section very much

smaller than if there were neutral currents.

B. Neutrino-Electron Scattering

% + %Gll+‘6 “

The coupling is as shown; i.e., make the identifications A = eP = 1,

B =va=3>c=v =2, andD= $ %

=4. Againf=g=l.

25
and from Section IV both these products are equal to mEv~ (LAB). From

w

the general result of Section VI

()0

0

‘%m== dT ,

where again T is the kinetic energy of the recoil electron.

matics is exactly the same as in Example A. The spectrum of

The kinerecoils is

flat fromT =

/(

OtoT=fiv m#+2Ev

)

,and is plotted on Figure 1. The

total cross section is therefore

which is plotted on Figure

is again twice as large as

Just as in Example A,

these results.

110 This result

the value quoted

the existence of

C. Antineutrino Absorption on Protons

agrees with reference 4, but

in reference 1.

neutral currents would alter

v+p4n+e+ .

. &

The identification required here is A = V$ B = e, C = n} and D = p. The

coupling constants which fit nuclear beta-decay are f = 1, g = 1.2.,

26
At energies much smaller than the nucleon rest mass, the recoil energy

of the neutron is negligible, and Ee ~ E~-(Mn- Mp) (all energies

are in the laboratory system). Since the center of mass system is identical with the laboratory system, it is necessary to use the solid angle

form of the density of states

The total energy E

second term in the

is approximately the nucleon mass as is En, so the

denominator is negligible (above Ev = 2 MeV), and

an P=—(2rr)3 ‘epe

with

Pe=p ●

pA“PC =p.p =EM

*-22 vn

P~●PD =p.p =EM

,_e P ep *W

pA “PB =pv.p =EE

e ve

- p$e Cos e

mm .%%%-

pAOPD =p.p =EM

2P VP --

P~ “Pc = peop = EeMn }

n

M* W*

where 6 is the angle between the positron’s direction and that of the

27
incoming antineutrinoO Using the formula for IM12 at the end

III gives

Dodsl

‘rA.B== [

1.2#-#e +

where again the energies

to mc2 and mc.

( )

OO~pe Ee - Pe cos e + O.01 EePe 1

and manenta are dimensionless}

1.3200dsl

*lLAB= m (>-2.53) J-@zJ=l.

Note that

The cos 6

simply

x

being

of Section

>

referred

●

there is already an anisotropy of 7* with 2.5 MeV antineutrinos.

Cos 0

term does not contribute to the total cross section, which is

The threshold is at Evs (Mn - Mp) + m= 1.@5 MeV. The totsJ.cross

section is plotted on Figure III.

28
REFERENCES

1.3

20

30

4.

5.

6.

Feynman and Gell-Mann, Phys~ RevO, 109, 193 (1958).

GelJ_-Mann,Phys. Rev., Ill, 362 (1958); Lee, Mo, andWu, Phys. Rev.

Letters, 10, 253 (1963). —

GoIdberger and Treiman, Phys. Revs, ill, 354 (1958).

Azimov and Shekhter, Soviet Phys. JEIT, ~, 424 (1962).

See Appendix for details about the closure relation and energy

projection operators.

See “An Introduction to Relativistic Quantum Field Theory” by So Se

Schweber, 1961, pp. 498-5oo, for projection operators which single

out a given polarization.

See, for example, “An Introduction to Elementary Particles” by

W. S. C. Williams, 1961, Appendix B.

Fei.nbergand Pais, Phys. Rev., 131, 2’724(1963).

29
I III

ml m*

u)

(0h .h

%-l ?

a) H

30
—

-\

—

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

I

I

I

I

1I

1

I

1

k

1I

I

(9 u)

I

*m

N

\\\\\\\\\\\

\

\

\

\

\

\

\

\

\

\

‘\b

\\\

\\\

Nm*

rco

w

E+

%-l

!4

aJ

31
I I 1 I I I I I I I I I i

N (9 m w m Cu

w

1%

+!!I

aJ
APPENDIX

Lectures on Beta-Decay Theory

Los Alsmos Scientific Laboratory, Los Alamos, New Mexico

FalJ 1957

II The Energy-Angle Distribution in Old-Fashioned Beta-Decay

L. Helder

33
In the following, the Coulomb effect on the electron wi~ be omitted.

The main modification due to the Coulomb field is to put a factor F(Z,W)

into the spectrum. This factor cuts down on the number of positrons, and

increases the number of electrons, especia~y at low energies. F(Z,W)

can be found on p. 2@ of the review srticle by Rose.*

Using the notation of K. Ford’s lectures,w the transition probability per unit the is written:

~o is the initial state of the nucleus, $f the final state, and j goes

over ti the neutrons (for negative electron decay) in the nucleus. In

the absence of the Coulomb field we can write

Beta and Gamma-Rsy Spectroscopy, ed. K. Siegbahn,

*

M. E. Rose 1955,

chap. lx.

+(+6

K. Ford 1957, Lectures on Beta-Decay Theory, Los Alamos Scientific

Laboratory, 1. “Old Fashioned Beta-Decay Theory”.

34
since the negative energy state with

emitted neutrino having momentum %.

sides of these equations are spinors

momentum -~ corresponds to the

The functions cpon the right-hand

which are independent of positionO

Now the light particle functions can be separated from the nuclear

functions.

We wilJ.use the notation~f(x)pOn to represent

-i(fim)o=w~ch is where f(x) is that part of e

time.

the nuclear matrix element>

under consideration at the

M =x Cn

n

These matrix elements are to be taken between partic~ spin states of

the particles involved.

q represents a one-column, four-row spinor, andq* represents that

one-row, four-column spinor which has as entries the complex conjugates

of the entries of V.

Bywel.1-known matrix rules,

35
+

where On

()

*

qpn~v = q$qkpe ,

is the Hermitean conjugate of On. P, in our notation is

Hermitean. FinalLy

Introduce the notation (S, -~)m for the quantity in parentheses.

To compute IM12, these operators have to be taken two at a time,

(ego, Om = ~, On = P; or Om = i3,On = ax, etc.), put in whatever spinors

one is interested in, carry out the matrix products, and sum over all

values of m and n. This last step involves the nuclear matrix elements.

A tremendous simplification results if one is not interesti in the spin

of the light particles, i.e., if one sums IM12 over alJ.possible spins

of electron and neutrino. We can do this for each pair (m,n) and then

sum over m and n. To see how this comes about, we can put in the explicit

spinors and carry out the operations The work canbe greatly simplified

by using projection operators. We will make a

this topic.

PROJECTION OPERATORS

The reason

closure theorem

that

slight digression onto

projection operators are useful can be seen from the

for complete orthonormal functions. This theorem says

36
&;)uJ*(;) =I

J=l

J+ where u (p) is one of the four independent normalized, orthogonal spinors

of momentum ~ (two have positive energy, and two have negative energy).

The sum goes over the four states of momentum~. Note that us(~), which

j* *

is a column vector comes first, and u (p), a row vector,follows. The

result of the matrix multiplication for given index j, is a 4 X 4 matrix.

The theorem says that the sum of the four matrices (j = 1,...4) is the

identity. The proof fo120ws exactly along the lines given by Schif@,

and is repeated here. An arbitrary spinor of momentums can be written

as a linesr combination of the four basic spinors uj(~)

4

Multiply both sides on the left by ui*(~)

Ui*(3)$ (s) = fAj(fi)ui*(@(s) o

j-l

From the orthonormality of the u’s, we have

*

L. 1. Schiff 1949, Quantum Mechanics, First Edition, p. 46.

37
and therefore

Ui*(~)$(~) =Ai(~) ●

Put this back into the expression for $(~):

4

j=l

If this is to be true for arbitrary $(~), we must have

4 4

~u’(p)u’*(p)=I,or~uJ(3)au’*(p), =6aB ●

Now we

combination

j=l j=l

have in the middle of the light particle portion of IMl= the

to sum over the final states of

the neutrino. If this sum had been over all four of the neutrino spin

states having

the identity.

it must be in

Hamiltonian).

momentum -~, we could use the theorem just proven and get

However, not all four states are available to the neutrino;

a negative energy state (the way we have set up the

Similarly, the electron must be in a positive energy state.

We want to carry out the following sum

38
which involves

xq++a$(-?i) ●

Neg En

for ;

Let us look for operators A+(fi)andA-(s), with the following properties:

I

A+(6)w+(;)= q+(;)

A+(;)c@ = O

(

A-(i’)w+(;)= O

where q+(~) and q-(~) are positive and negative

tively, of momentum;.

A+(5) operating on a positive energy state

gives zero on a

has the reverse

that

A+(;)

negative energy

properties. It

$?+&n+E+

= =

2E+

energy states, respecgives it back, and it

state (all of the same momentum). A-(s)

is easily seen, using the Dirac equation,

H+E+

~

39
have the desired properties, where E+ is the absolute value of the

energy. (H iS the Dirac Hsmiltonian: H = s~”p+$lm)

We can rewrite our sum~ ~ (~,-~ as

+ ‘-

Pos En Pos a -

for & Neg En

for V

since A- gives zero on the positive energy states and just gives back

the negative energy states.

Now the closure theorem can

goes over all.four spin states.

be used on the neutri.no,since the sum

We get

z z @Fa)m ‘q [@h30#wao:?4@] ●

@L+ il. ‘+

We can do the same trick on the electron by introducing a positive

energy projection operator for it.

and

&-

40
By grouping the matrices, this really means

which can be rewritten as

Again, using the closure relation gives

which finally equals

E[!30mL(-~)o;f%(i)

1

>

u au

which by definition is

[ 1

Tr f30mA-(-3)O~PA+(fi) ●

41
Putting au this together, we get

of electron

and neutrino

ALLOWEDTRAN9~IONS

This means two things: f(x) = 1,

must be used.

1. Pure Scalar

Om=I

Cm=c

s

Since the nucleon is

,n=I=,+

n

Cn = Cs C>m =

non-relativistic,

and the

ICS12

non-relativistic

.

+C?)$(YT’-9] ●

Using the properties of

(most of the terms have

operators

the Dirac matrices, and their traces, this becomes

zero trace)
1 = 3 “Ii

-q

The complete contribution from the pure

z

IM12 = ICJ2

pure scalar

Final

.

scalar term is

‘%’2 t-pc~s’) ‘

Syi.ns

of leptons

where e is the angle between the electron and the neutrino. Note that

2=;.

w

For this pure scalar case, the electron and neutrino tend to

come off in opposite directions.

II. Pure Vector

om=13 on= f3=o: ●

Result:

z

IM12

( c~se)”

pure vector = IfJvlz11’qz l+P

Final SpinS

of leptons

43
III . Pure Tensor

There are two types of terms

(

Om=Oi, on =Oi)

and

(

Om= CJij )

On=d .

Consider:

om = Oi On=oi=o:

Tr

44
This term iS

of the three terms of this type (x, y, and z) is

II 2

CT l%?J&@-j+7jx12.’P’+lfy~.’Py +Jzrz”P’)’)

where the Gamow-Teller matrix element is defined by

I“G,12 =(j*%dv)*o (j;%av)

If all the

complete result

T

IM12

pure

lep on tensor

spins

=11111 CT2 MW2

z

terms in which

for the tensor

Om= (p, on = OS = O: are added on, the

interaction is

In this form, the fact that the angular distribution is independent of

the coordinate system is clearly demonstrated.

45
If the initial nucleus

is summed over, then, since

——

is unpolarized and the final.nuclear spin

there is no preferred direction in space

Ifxl’”w‘H’‘;lMm12

[J”X]* [fy]=O, et.,

where the bar denotes averaging over initial nuclear polarizations

summing over final nuclear polarizations.

z

IM12

pure

lepton SP s;

‘1’T121Mm12 (l’@ “

tensor

ffial nu~lear spin;

average over initial

This

just

nuclear spin

result seems to

considered. If

be more generally

the basic nuclear

valid than the

wave functions

and J., then the off-diagonal terms must be zero*.

&

Iv. Pure Axis3.Vector

Similar to tensor

and

unpolarized case

have definite #

*

See Jackson, Treiman, and Wyld, Phys. Rev. ~s 517 (1957)for more on

polarization.

46
1,,&J“’kl=

spins vector

lepton spins;

final nuclear spin;

average over initial

nuclear spin

v. Scalar-Vector

Om=I

H?++&[.fq’[jj] (k’+JPJ)I

!M12

PWe ‘ICA121MGT12

axial

vector

INTERFERENCE

On= fl=o:

$;”AhTw

m )

1 ;“~ -—— )

3 EVW “

m =-0

w

Adding the case where I and 9 are interchanged gives another similar

term.

spills

since W is measured in units of me?.

47
VI. Scalar-Tensor Interference

Om=I On=o =()+

Ln

“ +Xx

rs

(gr’gks-grkg’s) ff’ps

i

(

~jpk-qkpj

‘~ )

J,’,4 CyCliC .

This complete term is

&wz!’f7[p]*(.’p’.q’)’) .

v

After summing over the three components of spin, we obtain

48
If we include the case when Om and On are interchanged, the result

is

x

IM12 a

scalarlepton tensor

spins

If there is no polarization, and final nuclear spins are summed over,

this term wil.1.be zero. If time reversal is satisfied, this term wil.1

be zero. The Vector-Axial.Vector Interference term is the negative of

the Scalar-Tensor term. The other Fermi-Gamow-Teller interferences are

identically zero.

VII . Tensor-Axial-Vector Interference

Om=o i

on = BuJ =0+

n

Adding together aXL terms of this type, as well as the ones in which the

role of tensor and axial-vector is interchanged, gives

z lMj2

= (CTC: + CAC; )1 I

21 tensor- ‘GT ~ .

lepton axial

spins vector

49
Adding together all terms considered produces the formula given on

pages 2i’9-2&3of the Rose article and reproduced below. The probability

per unit time that an electron and neutrino are emitted into the relative

solid angle ~ = 2TTsin e de, with the electron having energy between W

md W + dW~ is (for the alkwed transitions)

() 14w,0dwdQ = $ & F(+Z,W) ~’1 W(WO-W)2 x

x[(,%,2+,%,=),%,2+ (,%,2+,%,2)%f] x

x

(pI+a cose+b

w )

#WsinEld9 .

The original coupling constants (unprimed) have the dimensions energy X

volume. The primed coupling constants in this expression are dtiensionless numbers equal to the unprimed constants divided by the natural.units

of energy and volume, e.g., C;= c,/(me+}. Now drop the primes.

The upper and lower signs refer to negative and positive (charge)

electrons, respectively. The main Coulomb effect, the factor F(+Z,W)

has been written in, even though it hasn’t been considered explicitly

thus far. All energies are measured in units of me=. One factor of 21T

m

has come in from the ~ in the perturbation theory formula. Another

factor of 2rTcomes from &l = ~ sin O de, and a factor of k comes from

50
integrating over the electron’s directions (only the relative electronneutrino angle remains).

as $(ICT12 -ICA12 )I“GT12 - (ICS12 -ICV12)1%12

0CT12 +ICJ2) I“GT12 +(ICS12 +lcd2)P“42

( ~s)h12+(c;cA+c:c~ )1MGT2I

;;: ~c~’) hi2+(lcd2+ICA12 )I”GT12

y = + - (aZ)= .

The term+ b/W is the Fierz interference term. It doesnotappear

in a theory which treats electrons and positrons qymnetrically (in the

absence of coulomb effects). All terms arising from Fermi-Gamow-Tell.er

interference have been omitted from the above expression. The GamowTeller part of the coefficient (a), has been written for the no-polarization case.

If one integrates the energy-angle transition probability over the

electron-neutrinorelative angle, the term in cos e contributes nothing,

and the result is the allowed spectrum

N.#w)dw= y -.+ F(*Z,W)W d~” (Wo-W)2

‘[(lcd2 +ICJ2) PF12+(ICT12 +1%12)1%12] (+)dw51
Except

of the

for the interference term b/W, the spectrum shape is independent

coupling.

FORBIDDEN TRANSITIONS

Two effects were neglected in the allowed transitions: The relativistic operators a‘, ~, 75 and 1375;=d retardation% i=e”~ the higher

terms in the expansion of e

-i(i%-a)”~

. This latter expansion can be

written

e

-i(;+-m! = z &-i)n[(xi)”qn ●

The nth

term in this expansion has angular momentum components n, n-2,

()

1

n-4,.o.o . The reason every other angular momentum occurs is that

the nth

term has definite parity (-l)n. The general selection rules for

an arbitrary transition can be obtained by combhing these facts with

4

the intrinsic properties of the operators. 0 and;, being vectors,

carry one unit of angular momentum along with them; P> IS ~d Y5 ~ve

no angular momentum. +~$ ~, and I do not change the parity of a wave

function, whereas & and 75 do. For the allowed transitions, the result

is no parity change for Fermi and Gamow-TelJ-er,AJ = O f’orFermi, and

AJ=+ 1,0 for Gamow-TelJ_er(no 0~0, however).

The first forbidden transitions arise from’two types of terms:

the non-relativistic operators multipliedby

[

-i(~ + a) ● ?1and the

relativistic operators multiplied by unity. All first forbidden transitions have the selection rule that the parity must change; in fact,

52
for the n‘h forbidden transitions the parity change is (-l)n. The

angular momentum selection rules follow directly from the above remarks

except in that part of the tensor and axial vector interactions which

involves a product of a spin component a

i

with a position component x .*

J

These nine terms 0 x form a tensor which can be broken down into ij

three parts: (1) a scalar, 3.; (one component); (2) a vector ~ X ~

(three components); =d (3)a smetric tracelesstensor(five tidependent components). The scalar produces no change in angular momentum;

the vector has the selection rule AJ = * 1,0 (no O A O); and the symmetric, traceless tensor, which carries two units of angular momentum,

has the rule AJ = *2,*1, O (no O ‘O, ~~~, O* o. The exceptions

follow directly from the vector model rule for the addition of angular

momenta.

In each order of forbiddenness n, there will be a tensor coming

from the Gamow-Teller coupling, which carries n+l units of angular momentum

th This is the maximum possible angular momentum change in an n forbidden

transition. This is called the “unique” forbidden transition.

One calculates

the same way as for

effects, the lepton

the electron-neutrino

allowed transitions.

portion of the matrix

energy-angle distribution in

In the absence of coulomb

element separates from the

nuclear part. The sum over lepton spins proceeds as before; there are

*

E. Konopinski 1955,Beta and Gamma Ray Spectroscopy, ed. K. Siegbhan,

chap. X, p. 294.

53
some new traces to be evaluated, however, coming from the relativistic

-$

operators a, y=. The traces involving; are similar to those involving

~, and the traces involving 75 are s~lar to those which conta~ the

identity. Additional angular dependencies arise from the factors of

$+;”

For the n-times forbidden spectrum integrated over all angles, one

writes

()m+

N(W)dW= ~ -$ F(Z,W) ~- W(Wo-W)2Sn(W)W ,

where Sri(W)is cadled the shape factor. The shape factors are quite

involved and depend upon the coulomb effect rather strongly.

so(w)=

[(lCsI

2 + lc~12) 1~12+ (ICT12 +ICA12) IMGT12] (1+}) ●

EXPERIMENTAL SITUATION IN BETA DECAY (before P~itY)

Experimental spectra are generally displayed in the form of Kurie

plots:

(N(w)/F,z,w,wd= versus W.

For the most carefully measured allowed spectra, C&4, T#s, Ss5, this

plot is a straight line, which indicates that the Fierz interference term

is negligible. A conservative upper limit on the coefficient ‘b’ is

b< (3.2. Since b involves products of the two Fermi coupling constants,

34
and products of the two Gamow-Teller

that only one of the Fermi couplings

constants, the assumption was made

and only one of the Gamow-Tel.ler

couplings is present.*

The conclusion that Fermi and

essary followed from the existence

which has AJ = 1. This is allowed

Gamow-Teller couplings are both necof allowed transitions such as He6-Li6

under Gamow-Teller, but forbidden by

Fermi selection rules. Similarly C13%S34 is an allowed O 4 0 transition

which can take place under Fermi, but is forbidden under Gamow-Teller

selection rules. From a study of several decays (with some reasonable

approximations for the nuclear matrix elements) the conclusion was reached

that the Fermi and Gamow-Teller couplings are approximately equal in

strength.

The decision that the correct

and not the axial-vector, resulted

Gsmow-Teller coupling is the tensor,

from the electron-nuclear recoil angular

correlation experiment in the decay of He6. Using conservation of linear

momentum, one can convert the expressions above, which are written in

terms of the electron.neutrtio angle, to expressions involving the electronnucleus angle. The results of this difficult experiment agree rather well

with the prediction of the tensor coupling, and disagree with axialvector coupling. The shapes of once forbidden spectra give information

about the proper choice of Fermi coupling. These are more difficult to

*

c. s.Wu 1955,Beta and Gamma Ray Spectroscopy,

55

ed. K. Siegbhan, Chap. XI.
unravel than allowed spectra and indicated that the scalar coupling is

the correct one.

There seems to be some evidence that the pseudoscal.arcoupling constant is not zero.

An approximate value for the Fermi coupling constant is CF = 1.4-1X

10-49 ergs-c&. This means that the dimensionless Fermi coupling constant is c:- 3.00x IXI-12.This shows howwee.k the P-decay coupling

is ● Using the proton-mass instead of the electron as the unit of mass

gives C~(k$) = 1.01 X I.O&.

56
MMEADDITIONALPROPERTIES OF THE GAMMA MATRICES

(same definitions as h K. Ford’s lectures)

Greek indices W, v,=.. go from zero to three, and Roman indices j,k,...

go from one to three (x to z)

y~yv+yvy~ = 2&’

0+

7°=@=7 Yk =@k= -7&

w

i

-i

E?v=l

2

O-l

0 0-1

OO

0

31000-1

()01

75 = -i = 7°@P73

10

1. Relation between 7‘s and a‘s

7J7k . ~ jak

757k = @k

(j,k,~ cyclic permutation of x,y,z)

~k7J=

-7jok

(j # k)

kk

~7 = 7%k
0% = @ck

~ j7k

= iy~ (j,k,L cyclic)

II. Traces*

Tr(A) =~A = sum of the diagonal elements of A. The trace of

p w

the product of an odd number of 7 matrices vanishes.

In particular:

Tr(~) = O

Tr(Ok~) = O (aa ismade up of twoy ’s)

Tr(yvyv) = Tr(yv#) =~Tr(#yv+yv#) = #vTr(I) = %?” .

Tr(#7ti) . @Q’

In particular:

Tr(p2) = 4

Tr(~) = -4

Tr(yvyv) =0 (v + v)

Tr(Ok) = O

*

Schweber, Bethe, and de Hoffmann 1955,Mesons and Fields 1, Chapter 1,

section 7d.

58