J.E.N. 186-DF/l 57
Beta and muon decays A. Galindo, P. Pascual
Madrid, 1967
Toda correspondencia en relación con este trabajo debe dirigirse al Servicio de Documentación Biblioteca y Publicaciones, Junta de Energía Nuclear, Ciudad Universitaria, Madrid-3, ESPAÑA.
Las solicitudes de ejemplares deben dirigirse a este mismo Servicio.
Las publicaciones señaladas con la signatura / i pertenecen a la categoría a, "Memorias Científicas Ori ginales"; las señaladas con la signatura /N pertenecen a la categoría b\, "Publicaciones Provisionales o Notas Iniciales"; y lo si señalados con la signaturas /C, /CM, / B , /Conf pertenecen a la categoría c, "Estudios Reca p i t u l a t i v o s '"L"de^cxtcTrxixD^xroT3r-l-erT^crcrriiend.a-ci 011 G C / V I I \ /RES/150 del OIEA, y la UNESCO/NS/177.
Se autoriza la reproducción de los resúmenes analíticos que aparecen en esta publicación.
Este trabajo se ha recibido para su publicación en Noviembre de 1966.
Depósito legal n° M. 4928-1967.
COMTENTS CHAPTER 1.1.
I.GENERALITIES.
líotation
1
1.2. Properties of the Dirac Matrices ......... 2 1.3. I.H. 1.5. 1.6.
Dirac Eq_uat ion, 7 Solutions of Dirac Equation .........11 Second Quant ization , 30 Transition Probabilities and Cross Sections ........37 I . 7 . Invariant Phase Space Integráis.. 44 1.8, Final State Intecactions
52
CHAPTER II.FQUR FERUIONS POIIiT INTERACTIOH: GENERALITIES 11.1. Lorentz Invariant Interaction.
...57
11.2. Difficulties with the Punctual Four Fernion Interaction. ............. .......63 CHAPTER 111. BETA DECAY 111.1, Introduction,,,,t ,..,,,..,,,...........65 111.2. Electron-Antineutrino Correlation5Elec» tron Ehergv Spectrun and Lifetimes ...... 73 I I I . 3 . AsyiT : Trietry Polarized
of
llectrons
Enitted
froír,
N u c l e i . . . . . . . . . . . . . . . . . , , . . » . 84
III.U. Electron and tieutrino Polari zat ion . . . . . 91 III.5. Tine Reversal Invariance in Beta Decay.9'5 CHAPTER IV.r-'UOn DECAY IV . 1 . Introduction
99
IV .2. Decay of Polaíized Muons without Radia•= tive Corrections«.í t .«••»«.. ....... .,..101 IV.3. Rédiative Corrections to the Electron Spectrun of Unpolarized Huons in the V-A Theory . .'
,. ,
, ,120
CHARTER V.AHALOGY BETWEEN ELECTRONS AND HUONS. V . 1. Generalities.... V.2. Weak Magnetisn
140 ....144
APPENDICES. A .R.elat ions Involving y ^ ^ r i c e s , , ......155 B.Traces of Products of Y-matrices............. 156 C.The Phase Space -Integral I,..,.., ..161 vi
D . Electr on-iiucleus Final State Interaction ..... 16 3 E.Some Integráis of Trequent Use ,.,..,,....... .175 F.Calculation of V^( p „ .p, ) . . ,,....180 c r¿" 1 G.Calculationsof Z-ÍAkhiezer-Berestetskii, 1965).. 190
rOFCV/ÜRD
These notes represent a serie of lectures aelive red bv the authors in the Junta de Energía Luclear ,Jtiií , M
adrid,durinj; the Spring terr; of 19C5.They v:ere devoted
to gradúate students interested in the Theory of Elementa ry Partióles .Special ercphasis was focussecl ir;$o the coinputational r-roblens arising in teta and iruon decay. The authors v.'ish to acknowlecge the financial su.j. port of the JLü and speciallv the kinc interest of Prof. C.Sánchez del Río,
June 19 6 6
A . Ga 1i nd o F . ?ascual
CHAPTER I GENERALITIES
1. NOTATION Although all symbols and notations are explained when first introduced, some general remarks are giva here, The greek índices run from 0 to 3 and the latin in dices from 1 to 3. The relativistic metric tensor is chosen as
=
= -i
+
n)
= 2/m A £ í/3^
/> e -íl
Consequently when rr.^O A_+(p) = -^- • A +_(p)
(1.4-6)
are tv; o complemen-
tary orthogonal projectors. It can be checkeá that the ima ge domains of the four-dimensional spinor space
underA+(p)
are two-dimensional subspaces , which are complementary for - 12 -
and coincide when n\ = 0. The image domain of \ + ( p ) consist of all the solutions of (1.4-4). Henee it is obvious that \\> (p) has the form ^l P )= Where
h ( y1-- mv1-) X+. tp) je tp)
Y (p) is an arbitrary four-spinor
valued function.
Then the most general solution , ^ ( x ) , of the Dirac equation can be written
Integrating over p° we obtain
1
¿ £(?) where E(p) is defined
If u r (p) and v r ( p ) (r= +,-) are two sets of two linearly independent solutions of
(^ + ^^ /vrr Cp ) = O then we can writte:
; ^ -- v^n típ Á - ^ (?) *a(p J - 13 -
where . b ("p) and d (p) are c-number functions» Henee the most general•solution of the Dirac equation can be written as
(1.4-10)
I/-fas
or
_ ex)
Therefore v;e have divided , in a Lorentz covariant way, the most general solution, ty (x), in the sum of two terms: ^ ( x ) with only positive frequencies and ^L'(x) with only negative ones. That the decomposition is Lotentz invariant follows from the procedure we have followed to ob tain it. In the space of Solutions of the Dirac eqnation.. in momentum space we can introduce a Lorentz invariant sea -lar product inducedby the noria
íp)
(I .4-12)
which is clearly definite positi-ve. From the relation vj) cr^> *f C P )
( I. H -
we may write, for
(1.4-14)
which shows the Lorentz invariance of (I.-4-12) in the case m^O. For m = 0 the Lcrentz invariance can be proved either directly from the initial expression (1.4-12) or it will be asimple conseauence of the equations given below. One soraetimes finds that the noria is given by II H-'li1" --
cf.cr^ Í W Y ^ + ^
(I.H-15)
when d )* To clasify their meaning let us note that ,Z. p/lpj'v - Y" r when acting on the positive energies solutions of
¿U?(p)=0 and therefore for zero
mass partiólesthe helicity projection operators are
^i
= ¿ ( ' + A Tr)
(1.4-27)
As a final case let us consider the case in which th§ particle is at rest, i.e. p=(m,0,Q , 0 ) 3 then if (p.s)=0 we have s=(0 9 s) and we get
But we have Y-°W\I since P_ {x
l0
t
\vp .}s\_ 1 fi + \ — ^ p
> ~ 2. i
Is I J
acts A acts on J~i
» than
(particle at rest)
(1.4.28)
which are the spín projectors in the direction s*. - 20 -
Until now we nave specified the four-spinors u (p) and v (j?) up to a normalization factor. To fix the of this factor we choose the following Lorentz invariant normalization
=. ¿ E l pp ) (1.4-29)
v C (f) ÍUV (f ) = 2- E íf) Henee and taking into account the equation (1.4-90) as well as (1.4-20) it is easy to prove (Schweber 1961) that
íf ) =• From here we get X
immediatly the important relations
U*lp)Hvíp)= /{^
(1.4-32)
Somejtimes it is interesting to have explicit ex pressions for the plañe wave four-spinors. These expressions are obviously representation dependent and furthermore it is necessary to fix the 51*~ ,VJe are going to assume that s /"* is given by (1.4-22) and then we get
- 21 -
Representation A
i?'
E(p) ( I .4-33A)
+IYTV
'V- cpj = M_
Representation B
/m
* N
iI.4-33B)
Etp) + =H
4
i
v
fm.
-
22 -
where - / w , -t- L/n-¡_
(I.^-
1 — m-
m From (1.4-34) it is easy to check th'at
(1.4-35)
i?1 Sometimes it is necessary to consider the very high .or v.ery low energy limit of the above given spinors or some related quantities. As usual we defined the
IOVJ
ener
gy limit as v « c and the high energy likit as v—$» c. i) In the low energy limit we have
(RepresentationA)(l.4-J6A)
I* M , O J
^vj
N,
I y )- H 4
Cf) -
(Representation B) (I .4-36B)
H
.- 23 -
ii) In the very high énergy limit
(Representation A) (I .4-37A)
ip ) c M,
O
U- tf)= NI (Representation B) (I .U-37B) +J
1H Y
From eq(I.4-36) and (£-4-37) it is easy to rea lize that in the lov? (high) energy limit the conputations. will be simplified-if the representation A(B) is used, iii) We would like to study the very high energy limit of the operator jf.pVip). From (1.4-25) and (1.433) it is easy to check in a direct way that in both representations and in the very high energy limit L
Ipl
ip]
henee the spín projectors (1.4-22) at very high energies are (1.4-38)
i.e. the helicity proijectors . iv) As we have said above in the low energy limit the representation A is mure useful than the representation B. It -is interesting to study in this representation the limit of the bilinear quantities at low energies. These expressions will beused in the stu dy of p-decay. Let üs considerthe matrixA, whose- 2x2 matrix elements are A..: A», A -A 2.1 Then from (I.4--29A) it can be obtained in a straighforwards way
U.r('pl)Auslp-
N
(f)
a. 5 +
o" •
A
¿••m
Vrr (?') A AT¿
?1)
M
b l
b
(I . 4 - 3 9 )
-M* Ms cxv ( f ) c 'X_¿ t f ) c s. + A ,,.
A
P
O". P
Z
A d -= -A,
-
25
From here it is easy to construct the table I 9 in which all the interesting limits aré considered. TABLE I
Cu-
A
+ r
I
b
-r
«TV."
r
i
r°
ra*
«
•
.
.¿ícf.po
2/m
cr""
fe
Yr
r
ai
1
-
-i
^
+- X
2,/m
- 26 -
5. TRAHSFORMATION LAWS , We would like to know the transformation law of ^ (p) under an inhomogéneas Lorentz transformation. We have
1
J •SL^mi)
(1.4-40)
Under an inhomoganeous Lorentz transformation x—$> x" = = Ax+a
we havethe spinor transformation law (1.3-3) and
we can write
"J Using the éa (1.3-3) v.'e have
which can be written as
Un) 1 J
L +S
T+ LA)
u;_ t-p) e
&
J
and introducing the new integration variable p -?/\~p' we get - 27 -
e. Comparing t h i s
^
e
,-i'Pb-~\
J
e x p r e s s i o n wáth( I . 4-41) we g e t f o r ^ í t í )
(>=+) t h e transfbrmation y (/V*p)
i-i"?*'
e
law
[U(A,cJ) 4>+](p}3 ^ lp> = e
S(A)lf+(A-'p-) (1.4-42)
q, (_A-p)
^ [ U I A , u.) vp.]i-p) = iplt-pí= ^~ l p
s Wi
íf.t-^'W
T-he discrete transformations , T
. -* _ -•
~» '^
r — ti t)
(Space reflexión) (Time reversal)
are given by the operators
[ £ y A ](p) r C S fA (l s p) (1.4-44)
where P and T are, respecfively , unitary and antinnitary operators. The^' end are natriceswhich in order for P 9 T to leave invariant the Dirac equation they must sa-tisfy:
(1.4-45)
It is a simple exercise to check the transfor mation properties under P of the bilinear quantities gi ven before, and thus justify the ñames given therein. Furthermore we can defined (GalindqPascual ,1965) - 28 -
a new operation, C, which relates the positive energy solutions of the Dirac equation with the negative ene£_ gy ones. There exists a unique (necessarily antiunitary, since the representations of the Poincaré group in h
and fl ~
are non
equivalent) C, which makes
commutative the following diagram
U(aA)
This operator C is given explicitly by the expression
where (o is a 4-x4 matrix
which satisfies the equation
=- r If we define (pi
s
|_ c-p)
i l t OV)
(1.4,1+8)
the spinor *P (p) satisfies the equation (|5 + m) u? o°(p) = 0 and it is called the charge conjúgate spinor of U> (p)
- 29 -
5. SECOND QÜ-AHTIZATION
Up to this moment we have consider'ed the wa ve funtion of a free Dirac partióle and we have shown that the most general solution of the Dirac equation is give by (1.4-10) where b (p) and d (p) aré c-fflámber functions, To introduce the second quantization we conside-r that the free Dirac Field operator, ty (x)s satisfies the Dirac equation and therefore t ( x ) can be written as
Here we assume that b (p) and d (p) areannihilation ope rators which satisfy the following anticommutacion relations
' •
+
L M r > , el,, tp')J. - L M p >
-, i
(1.5-2)
, d-c. IP')]^» O
Since üie -Dirac equation must be invariant un der inhomogeneus proper Lorentz tranformations the • • transformation law of the spinor field can be written (Schweber, 1961) ^00
a, /\_ -
S-CM ^
\
,
\x + o-
S (A) = A^v Y^
(i,5_3) - ' a. £ •• j
Furthermore we are going to assumed that b 1 vp) is the operator creating a partióle momentum p and spín projection along p equal to rl/z, and d + (p) is the one creating an antiparticle with momentum tp1 jection along tp
equal to r i/2..
and apín pro
The b (p) and d (p) are
the corresponding anhihilation operators, We shallshow that with these . assumptions the four-spionor u (p) and «=» 3 0 •=»
v (p) are defined through eouations (1,4-9) and (1.4-23) i Taking into account (1,5-1) and the anticommu tation relations it is easy to prove that
tp
(2n) 3/2 -
M
ECp)
If R-, is the operator corresponding to an active rotation of angle 0 along p we obtain from the physical me_ aning of b p (p) that -t
R We have
t'p
o>
therefore
(1.5.4)
T q? n o>-Furthermore , since
iO> = iO> and
_i
—*
(1.5-5)
Z we get lo> =
Un)^ J Performing the change of the integration variable p-"*-» ^ R rf
and taking into account the invariance of "/ V2E(p>-) under the rotations 9 we get - 31 -
+ Comparing the last expression with (1.5-4-) we get
' -LT í.
±. 1-1
Since 0 is an arbitrary angle we obtain
Z . "p —•
^
JU-r ( p ) =. V
^ XLv I p )
I?' In a completely similar way we can prove the other parts of the initial statement, We are going to study with some more detail the transformation properties of the Dirac field under inhomogeneus Lotentz transformation, If the DiraE field is wat ten as iñ (1,4-80 0 with ^\(p) considered as ope rators and-we take into account (1.5-3) we arrive to the following transformation low (in a similar way as(l.4-42 was obtained from (1.3-3)). - U ( a , A ) Vpx (A-'p) V ' U , A) = S^' (A) e
fA(p)
(1.5-6)
The usual way to study the transíormatioasPs C and T ( in the second quantization formalista, is to assume the transformation laws of the creation andanmihiia tion operators under P9 C., and T and ask for local trans formations of the fields considered above, The transformation of the annihilation operators are de.fined ( Man di, 1960) - 32 -
-I
T
bT (f ) T"1 = y\_ kv l-'j> )
T C
x - 35 -
(p¿ ) --X rc( p ) • If X is antiunitary ' a similar reasening, using the fact that now r (p ) =)£• r*( p) „ gives the transformation law
X^i»X" = £ f('xx]
(1.5-14)
Equations (I . 5-13 ) (I . 5-14) conveniently spe cialized show the possibilily of defining local transíorination for the Dirac field. The operation C is not comprised in --this scheme but its study is ' iminediate. Before going on we would like to consider the possible definitions of time reversal.In the lit._rature are defined frequently two different. time reversal operations %' and ~Z " •>
In the Hilbert space of the state vectors an anti
linear operator T is defined through the equarions
T | C L ) 3 la') (1.5-15) la.)
T" —?
A1 *
T A T-' (1,5.17)
"= T
A+T" - 36 -
Henee A'=A" if and only if A=A , i.e.for hermitian
operators we get the same results.
6.TRANSITION PROBABILITIES AND CROSS SECTIOHS. To calcúlate transition probabilities and cross sections it is useful to introduce the S-matrix formalism . As it is well known (Schweber , 19 61)the S-matrix is given by a** HT
ÓO 1
(1.6-1)
where H T ( X ) is the interaction Hamiltonian density and P is the time ordering operator, To study the transition amplitude from the jhitial state I i) to the final state )f) we introduce the Tmatrix, whose elements are related to those of the S-matrix by the equation (1.6-2)
- C
shere P^: and P^ are the four-momentum veiztors oí the initial and final state respectively. N is a normalization factor chosen in the following way
N
—^—n^n^ • Í2n)
c«i
fu
(I>6 3)
'
o
is the energy of the i-particle and we have assumed - 37 -
that we havé n. p a r t i c l e s the f i n a l s t a t e . With rentz invariant
place
ponding
infinite
this d e f i n i t i o n
transition
and
future
transition
proisability
is given by
interested. in the t r a n s i t i o n volume which
the
(fISI
covariant probability
is defined
is L_o
amplitude
¡f) i I i )
all time from
in our
and n j in
the T m a t r i x
s
past
. The
corres-
is not
a mea-
f o r m a l i s m we 9
V
from
taking
infinite
i) 2
p r o b a b i l i t y „ l(flSli)|
ningful. quantity. since
ce-time
state
below.
li) to a f i n a l state
over all space
to the
initial
as we shall p r o v e ThP
an i n i t i a l state
in the
9
are
per u n i t spa.
as
l'Ul ^ i O I* 1
where
I C-V l s 1 ¿. ) \\r
is
I (>t I s u ) |
l
calculated
for a finite space-time volumen VT 0 We zemember that the energy-momentum conservation ¿-function in (1.6-2) is the result of an integration over all space-time9 Since we assume |f) ¿li) we can writte (1,6-2) in the following way
I s 1 c) = - i N (4
and
ITU)
I ¿S e.
henee
U i s1O
J
Therefore
P
1
= NI1 1 (J |T)¿->|2 ¿üm. '
— — í/T
VT
- 38 -
Let us now compute the last factor. This can be obtained as generalization of QT I 2 "
S.Vn X
-.'Ot L/YTV
~
e
a. t
Therefore the transition pr.obability per unit space-time volume is given by
(1.6-4)
r, =
In a decay process we have only one partble in the initial state and the meaningful quantity is the transition probability per unit-space-time volume divi_ ded by the number of decaying centers per unit volume.With —3 our normaliza!ion this is (2n) and therefore we get as decay rate
=. Un)
su.--
Sometimes the initial state is not a puré •state but a mixture, and a suitable average on the initial states must carried out. Simdslarly,, we are interested in the transition to a set of possible final states and then a summation on the final states must be introduced. In the general case we have as decay rate
r = s.s
Site-
(1.6-5)
-
39
-
The generalized summation,, S s and averaging 9 S 9 symbols indícate integrations over momenta and summations over spín and polarizations s depending of the type of process considered. The partial lifetime is defined as
£
=
_i_
(1.6-6)
Let us now consider the particular case when the initial state contains only two particles of masses m a and n, and four momenta p a and p, respectively and the final state has n particles with fourmomenta Pj_9P2»«»Pn and masses m i S m2 S ..»m
respectively. If the initial sys-
tem is in a pcre state we can compute the total cross section for this process dividing the transition probability (1.6-4) by the incoming relative flux and integrating and summing the obtained quantity over the momenta and polarizations of the final particles. Since the incoming flux is computed as the product of the relative velocity s v.
5
of
the incoming particles and the number of particles per unit volume, (2n)
4-3-n.
t
we get for the total cross section
d P l( d
"> |
^^,¿J " '' ^J'^ M P " P t > "^ P O i Cl'5-7) z.
cross From its physical meaning it is obvious that the total sec tion must be a Lorentz invariant quantity. Furthermore if the state is a mixture of different spín states the total cross section is obtained from (1,6-7) by an averaging procedure . Let us now consider the term 4 v ¿ n E E,. A straightforward calculation shows us that this term is Lorentz invariant and therefore can be expressed in terms of invariant quantities. We will denote the quantities refer red to the reference system in which the b particle is at rest with a superindex zero. We have - 4-0 -
Furthermare we can introduce the invariant quantity c S -
t o C x 3 y s z ) defined as
Then (S, mt,
.Henee (1.6-7) can be wíitten as
—
¿c ? ^ ? b -Z *>.
ci.6-10)
From where in particular the Lorentz invariance of the Tmatrix elements shows-up. This expresión can be written in a more obvious invariant way transíorming the integration over the three momenta to integrations over. the f our-motv.e nta.This can be done by the usual way and we get- for the total cross section the expression. - 41 -
(1.6-11)
We have shown that CflTiij is an invariant quantity and therefore can be written in terms of the inva_ riant quantities s(square of the total center of mass ener_ gy) and t. . (momentum transfer variables) which and defi-ned in the usual way, i.e. S-
C?c -r f b ) X
(1.6-12)
tCj -- t pe - pj >*" We have asaumed that. the indexes i and j in (1.6-12) can run from 1 to n and also over a and b s i.e. the labels of the partióles. We can defined, following KHllén
(KMllén,
1964-), a differential cross section with respect to the variables t.. by artificially introducing a S-function in the
right-aand side of ( 1 . 6 - 1 1 ) .
The invariant
differential
cross section is CÍO-
(*»)
, J
, '""
m
i"! kin1K».
•
(1.6-13)
Integrating this expression over the variable t.. we recover the total cross section (1,6-11). From these invariant differential cross. section one easily gets the differential cross section which is usually defined referrédto the center of mass system. - H2 -
Let us consider the case of two partióles 2 in the final state. We define t as t=(p a -p 1 ) and 6 as the angle betwwen p and p 1 in the center of mass system. In this CoM.S the differentÉl cross section is I el O" \
'
l
^c..M.
" ~2^"
ci-íi-1
n
cLc dio»a)
As
we get
From (1.6-8) we obtain easily
-, mi)
Henee
The quantity (1,6-1) can be often developed in terms of powers of the coupling cons tant and if. only the first order term is kept? then S= L- C ja"x H X UJ -- 1 - c jdfc w x l^
(1.6-15)
where H v (t) is the three-dimensional space integral of H(x), Then following a process similar to the oner given above we get as the expression corresponding to (1.6-4) the follo-wing one - 43 -
r =
where
*" •• ¿ I E ¿ - E.) K 4 I M | ¿ ) | 1 + En
(f I M I i )
is
related
with
the
(1.6-16)
S matrix
element
by t J
7.
l S
U)
,
-ú
i i£w -
ev)
C ^ M I O
(1.6-17)
IHVARIANT PHASE SPACE IKTEGRALS. Sometimes it is useful to consider that
the T-matrix elements which appear in (1.6-5) anc (1.6-8) are independent of the spín and momentunv of the final -ta te partióles, then the transition probabilities and cross sections are proportional to Í1 Upí -**;) & épc) dCP¿
(1.7-1)
where O is an arbitrary time-iike fourvector. We are going to study the properties of these integráis. Non-invariant phase space integral can also be studied (Koch, 19 6 M-) but we are not interested in them. Another way to writte I (Q) is
I m CCi) * I R - ^ - & C Q - Í/>J
(1.7-2)
which is obtained from (1.7-1) by integrating over the zero components of the four-momentum vectors of the final -state particles.
From (1.7-1) we can derive inmediatly ai? equation relating I
with I n * • We have
Henee (1.7-3) One of the must used phase space integráis is the corresponding t> two partióles in the final state. Since I 2 (Q) is Lorentz invaiiant we can perform its calculation in the reference systeía Q = (Q°O). Then
Henee
And going to sphericál coordinates and perf orming the inte grations we get
.
I, (Q'.o) = -r— From Lorentz invardance 1 7 (Qr mf, «í) = •£— « M Q 1 "r>?~ f1?i & ÍO) © D-^" tfm,-i-'miJJ!"J (1.7-4) Let us now consider the oase of three fi nal state partióles, i.e. I 3 ( Q ) . This quantity can be computed taking into account
(1.7-3):
2 This can be written ( (Q-p q )
hío-.m^mt) \?i
=
PJ ^
^
^ f i ^ "^
S CQ
"
and from here
2a,
Taking into account (1.7-4) this can be expressed as:
(1.7-5)
The last integration leads to an elliptic integral. By a simple generalization of a method in troduced by KMllén (KMllén, 1964) we make a transíormation of the variable of integration and introduce an angle U) defined by
This transformation is useful for the last Qumeñoal integration. A straighforward calculation (see Appendix C) gi ves
i
,
t^
M
K I\idf
-
- JL_ © (Q) é [QZ-
.
z=
8
(1.7-7)
¿y A x O2" ( C2" -
M=
(c1 +
B3- f
( B'1 -f cx - ZÁ1) U 0a-- a1" - ^ >
c = c? D= Q
6,,
In the usual cases fcl3 fe „ and M are small compared to 1 and the integrand can be expanded in powers of costó » The series is quickly covergent and the integral can be Performed easily. If the assume m =m =m we get from(I.7-7)
((i) © L Q 1 -
UTO
i I [O -
W
^
(1.7-8)
( Cí
k - [ (Q- «n-a? - ^ ^ (Q -
fe, = + 4 (Q
(Q-
which is the result
/J^J2- + "+ 0< b IUl *Í)
(1.8-7)
The expression just obtained is completly rigo rous añd no approximations have been made.Usually the chan - 53 -
neis a and b have different partióles and the interaction U is such that
T
= o
= < 4; I V I 4¿ )
.Then :
(I.8-8)
Let us now consider the case in which the particle 1 and 2 interact through the potential U in the final state and the other partióles are not affect by the existence of U. Let us now introduce a complete system of momentum eigenvectors. If th ; e
are n parti-
óles in the final state we can use the system ..p
Po...P o - n are fixed and we can writte
I $b ) - jd N * W • °1 i PJ P3 — ?-n )
(1.8-8)
and therefore
T,
°
, a
(a) denotes the transition amülinade when *
the final state interaction is not taken into account, If the equation (1.8-9) is written in the position representation we get
e.
where
^>~ W
is the relative rnotion wave function of the
particlesl and 2. Henee
a
(2-nT
and substituting this expression into (1.8-10) we get
(i
If the assumption is doúe that
h
J,
c.
'
b ja . _
o-
then
l
—
~p
I d -* ^
(1.8-12)
'-^
J T.ie brackit in (J.,8-13 ) is % (r) and therefore
In the applications
'^b-% _ ^"^^
^-s approxima-
tely indepeiident of o up to sorae valué at which it starts te decrease quickly to zero. In this case the bracketed term in (1-8-12) Is not exastly a Dirac delta function, but is extraordinarely peaked at the origin. We get b -*eu ~
Where
\-
^ ^ u
is a small average
distance
- 55 -
The result (1.8-13) is extraordinarely important and it means that the transition probabilities in the case that a final state interaction bstween two partióles is present can be computed as if the final sta te interaction did not exist and multiply the obtained result by
I ip~ ("Co^Ol
where
U>~ (V)
is the relative rao
tion wave function 'óf the interact'ing partióles and Y^^. a small averaee distance.
- 56 -
CHAPTER II FOUR FERMIONS POINT INTERACTION.; GENERALITIES
I.LORENTZ INVARIANT INTERACTION The prototype of a weak decay process is A¿ —> A, + A3 -+ !"„
' (II .1-1)
Where A. (i=l 9 ...4) are four partióles of spín 1/2 and they are described by the Dirac 'field'S >f; (xi ( i = l , , . , 4 j , Following Fermi's idea (Fermi 9 193U) w. hall assume that the most general interaction Lagrangi .:. to ex plain (II-l-l) is a Lagrangian invariant under proper Lorentz transf ormations consisting of ,. punctual i eraction between the four spinor field. There are some difficulties related with this assumption but they will be considered later on. From (1.3-11) it is obvious that the r.ost general interaction of the considsed cless is
Z
C¿ L¿ (¿ZIH)
+
h.c.
where the subindex i runs over the set (SSV ,T 5 A S P) . the quattities C. and C*. are complex coupling constants and the quantities L^ and L1 are defined by the relations
L T íii-it.» •= -j [ + i « S account
the
0
—?> 0 '
transi--
. . . . , Considering the
1963) , (Durand 9
co?fections
the
due t o Cou-
1963)
(I.ÍJI-7O -+ O- 0 0 22. ) lO*** erg.ow 5
( I I I . 2-21)
' -
exis-
80 -
ii) Let us noíi -: consider the neutrón decay, In this case "^ Cx)
and x^ fx)
are solutions of the
free Dirac equation. In the allowed transitions if k is the neutrón momentum then k is also de protón momentum and we have •~h ~2
-TI ~°
Henee if M denotes the nucleón mass
f •4 p,X
2 -u>,U)J = —
~ =3
where the low energy limit has been used. Henee forra (III.2-10) and (III.2-18)
I Cpf ^3IC tT r= i ^ l - i ~
(III.2-22)
From this equation and (III. 2-20)
Taking into account the radiative ccitections and using the data from many ^-decaying elements we get_, (Ericson, 1961!-) (Tolhoek, 1963). : -
i L2 -i O. os-
(III.2-23)
- 81 -
The information contained in eq 0 (III.2-16, 22 5 23) is all that can be obtained from the electrón spectrua shape.and from life-time measurements. Nowvs are going to study the information which can be obtained from the stij dy of electrón- neutrino angular correlation» Prior to 1957 the data on electron-neutrino angular correlationsveré scarcé and soné of them (Alien, 1953) completely wrong. We are going to discuss this kind of éxperiment with more recent data. Our starting equation will be (III,2-11) where we hall neglect the' small term with a" and where we shall put b=0 o From an inspection of (III,2-11) and the mea ning of a it is clear that for vector and tensor couplin'gs the electrons and nsutrino are emitted prefenrentica.lly in the same direction and for axial and scalar couplings in opposite direction,, To study the angular correlation it proves useful to introduce the Scott diagram, which is obtained when we plot the quantiíy a of (III.2-10)
O. =
I HF fCtfw - tfss ) + ±- I Mc-x I* UTT - dM) •
2
(III.2-2UA)
versus | H F | z ( c( + c* r' r*> • *o. " * o.OÍ.
C CA CA J •+ ¿Ka.. [Ov(-ñ •*• Cv €„ )
(Burgys1958)
y*
More detailed information on this kind of experiments can be found in (Lundy s 1960),
- 90 -
4. ELECTRON. AND NEUTRINO POLARIZATION Another kind of experiments which gii*e some insigbt into the beta decay Hamiltonian are those of elec_ tron and neutrino polarization,. If N
is the number of electrón emitted
with their spín parallel to its momentum and N_ the number of éLectrons with the spín antiparallel to its momentui, we define the polarization in the direction of the momentum as
P -
N+
"
N
-
Víe are going tp compute this quantity
(III.4-1)"
s
using the V,A s theo-
ry. Our starting point is the equation (III.3-1). We are not interested in the neutrino polarization and in the initial and final nuclear polarizarions . Taking into account (III.2 -3) and (III.2-6) we get
A
^Z r-
Using (1.4-27) and taking into account that we are not interested in the direction of the neutrino which is to be in tegrated over at the end, the terms linear in q can be ¿rpg ped. The only important part of A is A = l M F l ' ü , (f) ^
(cv + Tc Cl) y" ( c* - y* c'v* ) Y1" tuAíp).E- +
f Let us now copute the valué of A when
X is parallel to
From eq.(13-23)
- 91 -
Henee
— _,
J. I Mt T P
But
and similarly for the other trace. Therefore
A.
=
i
j
-i and similarly
Integrating" over the neutrino momentum and the angle of emission of the electrón we finally get 9 (including final state interactions). N + (p) = —~ ^
p E (Emay. E)Z F(Z, E) f ( I £• Av) ....*..
(III.4-3) '
and therefore the polarization (III.4-1) is given by
r=
¿RÍ.(CVC?) IM,.!1-* a^cc.c^llH^i 1 — — — _—^^j- .
(III.4-
- 92 -
The first experiment of this kin'd was carried out by Frauenfelder.(Frauenfeider„ 1957) and aftefc.this many decays have been studied using different techniques (Lundy s l'9 6-l) s( Kof oed-Hansen , 1962). We are going to analize a few of these experiments» Let.üs consider the Co — » Ni + e~ + v 9 ín this case J. = 5 and J.f=4- and therefore is an allowed puré Gamow-Teller transition.In this case
Some experimental results are
+ 4.
(Frauenf eiderS1957)
+o.^*6.,,
(Cavanagh,1957)
+o HU to ot. Bienlein, 1958)
If we assume :that this quantity is really +1 the only.solution of (III, 4-5) isC.=C',. All the experiments of this kind are compatible (Lundy9 1960) with the as sumption C A « + el
both real
(IIIo4_6)
Now we are in the position to study the relative sign of C and C^. Let us writte CA = _ X C V From (III.2-26) we get IXI =
4.4S ± O-os-
(III-.«+-7)
(III.1+-8)
Let us return now to the experiment on electrón asymmetry
- 93 -
from polarized neutrons, discussed in the §4 of this chapter. Some results were quoted in (III.3-22). From (III.4-7) we get
Ke) = 4 -
— i, -v u» & U3X
Taking
into
account
jf 2 >. ( X -+ i ^ _ . ^ ^
I -i- "= X
Therefore
l
(III.5-8) O.19 r + I. I? ± O.OJ
lead
to
(III, 4-9)
This valué has been confirraed from the analysis of more da •
ta-
All the experiments about beta decay lead to the following interaction Lagragian
^
6 1
^ ( I I I . 4-10) 3
t-
/IC
(7
0 2 ¿ ) W
X -+-L.IS ± O. OS"
Since C v and C have opposite sign, according to (III.4-7) this is known as V-A theory. Obviously more experiments in beta decay can be performed and up to this moment there is no evidence — ?ainst the interaction (III. 4-10). Here we are going to discuss only the neutrino polarization experiments, In the same way that we have computed the -equation for the electrón polarization we can compute the Jbrmula corresponding to the neutrino polarization and we get _ 94
-
P = +1
(III.4-11)
i.e. the spín of the antineutrino emitted in ^-decay is always in the direction of motion. Nevertheless this re_ sult can be obtained im.mediatly from (III. 4-10) since (1+ y )/2. is the spín projector along -p* for zero mass partióles and this factor appears in front of vj^, „ To confirm this prediction Goldhaber et al.(Goldhaber 9 1958) studiedthe reaction 152.
IS"Z*
l52
where the spín of E¿*52 ±s o~ „ thaj of Sin 152 " is 1" and the corresponding to Sm
is 0'. The beta decay is the-
refore an allowed Gamow-Teller transition and the electro_ magnetic is a electric dipole one, Let us now take as our z axis the knov/n direction of propagation of the neutrino. The orbital an guiar momentum along z is zero since all partióles raoYs' along this line . It is found experimentally that the spín 15 2 of the Y along the z axis is +1. The spín of Eu and 152 Sm
i s .zero and therefore +1 needs to be compensated
by the pesitron and neutrino spíns.This is only possible ff the z-component. of. the neutrino sp£n is -1/.2, i.e.The neutrino spín is direcified oppositely tó its direction of propagation and the theoretical prediction is confirmed.
5.TIME REVERSAL IHVARIANCE IN BETA DECAY From the preceeding analysis it is clear that the beta-decay interaction Lagrangian is not inva-riant under partide-antiparticle conjugation and space reflection. We would like to study if time reversal ihva - 95 -
riance is also violated in beta decay. If time reversal is violated then the coupling constants would be com-plex. Some of the experiments discussed above pointed out that the coupldng constants were real and we have accepte'd this fact. Now we would like to consider again this prolilem and analyze up to what point the couplonh constants are real. Thfere are four vector quantiti^s •. which conceivably could be measured in a beta-decay experiment: i) ^ J > ii) T iii) p iv) q
:the polarization of the decaying nncleus . : the polarization direction of the electron. : the electrón Tnomentum : the neutrino momantum
Since all these four vectors change sign.under time reversal, the triple product of any three of them gives a term invariant under rotations but not invariant under time reversal. Henee the detection of such a term in a beta-decay experiment would indícate non-invariance under time reversal. It is possible to perform 12 experiments of this kind but we shall deal in detail with only one . He are going to study the distribution function for allowed beta decay of oriented nuclei in which both the electrón and the neutrino momenta are observed, Our starting formula is (III.3-2). The traces appearing in this expressions are computed in (B-12). From these equations we get many different kinds of terms . From
e} fJcf
- 96 -
From i.
fiAv
^
^ p,
II
we get terms
like
v A
n
n
ii
..
A
ii
11
ii
"
We are interested only in the térra
f •< ^ ^ ,
( p * CJ > < 3 ;>
cj.
com-
pared with the term 1; by keeping only these terms in (III-I+-2) we obtain
Furthermore since the final nuclear polarization is not measured vre must sum over Mf . Taking into account the results obtained in the electrón asymmetry study \¡e get
A >. By integrating over the modulus of neutrino moinentuni, we get
— ^ f> E +
2 Ion [ ^Av ^,-T
Usually the initial state is not completely polarized. If a M i
is the population of the state M. and the a., X i
are normalized in such a way that • 2L CL~H¡. -i.
5
in
the above formula we must substitute M . bv 2- cuu M.- = - < ~3 s1 . Furthermore the t e r n ^ ^ (fx ^) tuded < j > fp'x '^)
must be substi»
when the polarizaiion of the initial
nucleus is .considered in an arbitrary direction. Finally we shall assume that the time reversal invariance holds in strong interactions. Our final result is - 97 -
-
2. Ion
LCACV
+
'r'*l
CA Cv
u
u
J .íK.^ cU
For the neutrón we know that
/
T
¿
é,-,.
I «XF I7" = ^-
(III.5-1)
and ÍiXtTl = 3 .
We can take C y real without any restriction and we writte CA s -
X Cv e ( I I I . 5-2)
c ; . cA
c t -- c v
Henee
*
l+3>*
(HI.5-J)
If time reversal invariance holds 0=0 and we get D=0. If time teversal invariance is violated in -a máximum way 0= +_n /2 and we get D= +_ 0.1+51 if A =1.18. The experimental valué (Burgy, 1958) is D= +0.04 +_ 0.07 which implies • + 7^ ± ^ (III. 5-,) ifA=1.18. This is the more aecurate experiment to study invariance under time reversal for beta decay experiments, The result (III.6-4) points out that time reversal invariance either holds or is violated in a very small quantííy.
-
98 -
CHAPTER IV MUON DECAY
1. INTRODUCTION
'
.
The M-mesons were first discovered by Anderson and Neddermayer (Anderson, 1936) in the cosmic radiation and they were thought to be the Yukawa partióles. It was found laiter , though:, that íhese mesons do* not interact appreciably with nucleons and behave like heavy electrons in many respects. Muons are charged, spín 1/2 partióles with
mass fflip * Í O ? . é í i ' and
gyromagnetic
í
0.002
HJU-.
factor
^f*. - 2 [ 1 •+ 14.. 16 2 ± O oes:) i c f 1 ] This last valué agrees with the theeretical predictions of Quantum Electrodunamics . See (KSllen9 1964^ Feinberg, 1963) for details 'on-, . these and other related subjeets; the valúes 6£ masses, lifetimes and so on are of Rosenfeld et al. (Rosenfeld^ , 1965) . The yw-particle decays with a lifetime t.;
¿.lOO I ± O O O O S
pi*J-.
giving normally rise to an electrón plus neutral partióles. As the electr.on spe.ctrum is continuous and its end point energy is cióse to m^/2, the decay is at least a 3-body decay and the neutral produets must have negligible masses. There is experimental evidente (Hincks ,194-8) that none of these partióles are pbotons (at least iñ the major channel) and henee the simplest assumption is to suppose that the muon principal decay mode proceeds as y—» e + (two mass^JLess fermions , say V. 9v^_
- 99 -
actually, though, just an upper limit ,£7 u& has been obtaifled for the sum of the neutral partióle masses. Other extremely rare decay modes have been observad , as p _> 4. + v¡ i. v^ ^ y
,
p -* e -*• u, + 'v,. T e-~ -t- £+
Neutrinoless decays as M — e+ v*. s r e theoretically expec — ted to occur with a branching ratio (Feinberg, 1958)
R -
P ( n -, e + y ) ~ r(.|^-»e-i-^»-7¿J
s
_¿y lo
provided that »^- and e have all quantum numbers in common. Exper imentally , hov/ever, it has never been observecl and an upper limit R é 10
has been found (Bartlétt, 1962). The sim
plest explanation is to assume that v. and v
differ in so-
me quantum number, so as to make the virtual recombination Q. + v, t- P¿ —r> €-
strictly forbidden. The existence of two kinds
of neutrinos has been brilliantly canfirmed
(Danby, 1962)
and the principal yf-decay mode is nowadays believed to be
where w
'Stands for the y5-decay (or electrón) neutrino , and
Uu (¿= ve ) no accompanying
denotes the muon neutriiio, i.e. that neutriu
in the n,» -decay
(' - Ys- X,) Cfl + w o ] (S P) 3 -£-
(w) r = ±
(AA/% JL
- 10"+ -
And similarly 9
(sp)
S 4-
TY
(
= ±
Tr L
)
£
y
( t b. V
let
[
J. Tr t U r r t
= ± -fv [ I « VA +
s
± 7V í ^ T T r ^ - T y r ) ^
where the sy.ntbol
indicates
under th.e exc&ange
As
. Tv[(ol¿w +
p3 p3)(p¿ PH) - ÍJ>, pH>(p-t
sIj.¿)(pJ p4) - 2[(p.tPj) (s,
(£S)(pS) r -
---Z
PVA
/Tn
'
By s u b s t i t u t i n g these e x p r e s s i o n s in ( I V . 2 - 1 1 ) we finally get
v
- (
(p,
. 2-1 2)
- 107
-
where X
has been obtained
e-s dictated
Electron
bv
from X
spcctrum
and
We are now
an arbitrary thus
the
(IV. 242) into
momentum
electrón
decay
parameters:
in position
p* of the decaying
spectrum
to calcúlate
d P_
( I V - 2 - 8 ) . This can be done muons
for muons decaying
will r s t r i c t our considerations, i.e.
m.
(IV.2-10).
momentum
by introducing
by the replacement
and in
for
obtain
flight.We
however, to muons
at
rest,
=
E.. TÍI,. „
As for any E)
time-life
'
vector
q(see
Appendix
.
(IV.2-13) —
, dr3 dt: q
i"
^ (cj - p-¿ - pc ooZKu- ; (IV.-2-24
yields. C -- ( l ^ i S tó.ooor) lo'^Lc^3 •a
(IV.2-15)
.
- 110 -
The error quoted in (IV.2-25) is only significant when the second term víithin the brackets in the r.h.s. of ClV.2-22) Q
is^lO
9
as it happens s for insfcance5 in the.V-A theory s
where it vanishes exactly. Otherwise, if nothing is known about the coupling constants C. 3 C1
s
the muon total rate
allows a determination of C only with an accurancy of a few percent. The, valué of C given (IV.2-25) is often ex — pressed in natural units in térras of the protón roass : C =(1.0 2.3 8 i Oooo¿)lo"r//«if
(IV .2-26)
There is a striking similarity between C s as given by (IV. 2-25) sand the valué of the Fermi coupling constant C_ for ^-decay 5 given in (III O 2-21). That remarkable. closeness of the two valúes will be the starting point of the conserves vector current hypothesis , as explained in the next chapter .
Micheln parameter: For an unpolarizea• sample of muons the normalized electrón spectrum depends only on the momentum of the emitted electrón inrough the expression
P ko dx ^ -, b *
-
obtained from (lV.2-16) by integrating over all posible angles SL and multiplying by the normalization factorp
.As
^ tQp/m-, is either nuil or small s the electrura shape essentially depends on the so-called Michel parameter . The.lowenergy shape parameter
, being determined primarily by the
- 111 -
less probable low-momentum electrons , has not been d i rectly éstimated because of insufficient statistics it is usually neglected for it has a negligible effect on th-e spectrum.
.
and
over-all
"
A comparison of the experimental results with the theoretical prediction yieldsp = 0.780+_ 0.025
( P l a n o , 1960)
•.
0.751 +_ 0.034
(Block et al.,1962)
0.747 + 0.005
(Bardon et al. ,1965)
(IV.2-28)
when both.the radiative corrections to(lV.2-27) and/the resolution of- the experimental set-up are prpperly
taken
in.to account.
The asymmetry parameter.
s:
Careful-measurements of the asymmetry pa rameter ^ for thé decay of polarized muons yield the r e - sults '\t\ > 0.975 +_ 0.054 ^ 0.94
+_ 0.07
(Bardon et a l . i . 9 5 9 ) (Plano
,1960) (IV.2-29)
where the inequality comes out from the possibility of a small depolarizing effect of the stopping target (bromo--' form or liquid h y d r o g e n ) on the arredted m u o n s . Since the sign of $ depends on the muon helicity for muons from f"1 decay.s 5 -and it has been experimentally established A*.- ( JA
+
)
that
from n~ (nV decay is right (left) handed
(Alik-
hanov \ 1960; D a c k e w s t o s s , 1 9 6 1 ; Bardon , 1961 ) , it follows from the above quoted u - d e c a y
experiments that "§ in p o s i t i -
ve . - 112 -
The oparameter: The momentum dependence of the asymmetry. has been measured s and the best valué is % = 0.78 ± O. OS"
(Plano, 1960)
(IV.2-30)
From the general expressions of j s "? s & in terms of the coupling constants the inequlity I ^ %\ é § ned 3 which combined with Plano's results
is easily obtai(IV-2-28 s 29 9 30)
leads to the conclusión that "í- ~ 1 .
Two-component neutrinos: It is clear that the experimental knowledge of .the five real parameters A5 o s "$* s ~§ s h does not lead to a unique determination of the 19 real oonstants which essentially fix the interaction Lagrangian
(IV.2-3)(ten
complex coupling constans C. 5 CT one of which may arbitrarily be taken a real). Even if the electrón polarization is considered there still remains a large freedom of choice of C. 5 C' (Kinoshita, 1957). The neutrino emitted in |3-decay has been established to have a definite helicity, which is + 1 in p T -decay. If we assume that also theneutrinos produced in muon decay have definite helicities then a simple argument shows that only either the V,A terms or the S,P,T terms of (IV,2-4) survive s depending on whether or opposite helicities 9 respectively
v
s
u
have equal
: in fact, in the firs'
(resp„second) case one has
(IV.2-31)
- 113 -
and as
O" * ^
* o
(IV.2-32)
ti * Y O =
(i * r>->
the contention obviously, follows. But the second possibility (namely 9 twocomponent neutrinos v , v„ of opposite helicities) is experimentally ruled out s for several measurements of the polarization of the fast eléctrons (positrons) from JA+-decay reveal a 100% left (right) handed polarization (Biihler, 1963); B±>om, 1963) , and therefore the above hypothesis would imply that the electrón (positrón) are preferentially emitted in the direction (in the opposite direction) of the -muon spín (see Fig.l), in contradiction to what is experi-
r „ with equal helicities. Without an.y loss of generality we will assume that both v . B \¿ 2 are left hnnded neutrinós¿ for in case that they had positive helieities it would suffice to coilsider instesd of the basic fields tyv 3 i/>y the charge cobjugate fields ^ , = r-
.,-_
'
with
-j=. & r -
Cv r
~ C^
(IV. 2-43)
and because of the (1+ Y 5 ) matrix wjithin the first parenthesis in (IV.2-42) the electrón (positrón) wave am plitude is projected when v-^ 1 onto a longitudinally po larized amplitude against (along) the direction of motion (more explicitly, a simple computation shows that the probability, d i~L ,per unit time for an unpolari1 zed Y * to decay at rest with eirsission of an e^ of helicity é. (either +1 or a -1) and momentum in the range (x,x + dx) (see (IV.2-14)), Írrespective of the electrón direction and the momenta of the accomDanying neutrinoss and calculated by using (IV.2-41) as the decay Lagraagian density3 is given by
-2^
6¿sJj2 -
(IVo2-44)
with
v
(IV.2-45)
C
) are aeglected 9
p.w:n¿
=
Í x — - ; iv
(iv.2-H7) .
whence it follows that the fast (i.e. v —» 1) electrons (positrons) of f-^-decay are 100% percent polarized against (along) their. direction of motion(if and only if "g"= + l, i.e. Cy= -C A ) . All experimental results so far available from muon deaay can be explained by using (IV.2-4-2) as the weak interaction Lagrangian and because of (IV.243) that four-field point interaction is known as the V-A theory of muon decay. It is properly a V-A interaction in charge retention order , for both the local operators ^l») TtJ i|- u) and ¡J^i*) T tJ ify L>.) "carry" no electric charge, i.e. they commute with the electric charge operator. But using (II.1-7) the general Lagrangian density (IV.2-8) can also be written in charge-exchange order j
M
r"J (? v + ti A
YV) %^ü
^- . This Yr-invariance of the V-A interaction is a characteristic feature of it in the sense that if one reauires the most general Lagrangian (IV.2-3) or (IV.2-48)to remain invariant under each replacement (IV.2-52) then the following relations are obtained: /
Cc = C¿
A
y\ I
Ce - Ce
(IV.2-53) cs = cP - c T ' -c¿ = cP =• c r - o'
i.e. the Y 5 ~ i n v a r i a n c e implies the interaction Lagrangian to be V-A.
Conclusions: Even " though the experimental data from
- 119 -
muon decay do not point definitely to a specific form of the phenomenological Lagrangian density«¿(x) it is nevertheless extremely satisfactory that all such data can be fitted by using (IV.2-54) w
íx;
thc
' £ £--, fr í'+rr>^ J ~ in cióse analogy with the V - Á A theory of/3-decay. V/e shall not enter here into the discussion of the effects of introducing an intermedíate vector ie-sson and in the next section we shall analyze the radiative corrections to *¿ (x) just to illustrate some t^pical calculations which arise in this corr.ection.
3. RADIATIVE CORRECTIONS TO THE ELECTROIi SPECTRUM OF UMPOLARIZED MUONS I1J THE V-A THEORY. Two of the four leptons entering the muon decay are charged fermions and henee interact electromagnetically. The calculation of the radiative corrections to the decay of unpolarized muons was' first carried.out in 1956 (Behrends , 1956) and it turned' out that these corrections are quite largeB mainly in the low energy part of the electrón spectrumr a being very important their knowledge in order to reconcile the experimental results with the predictions of the V-A theory.A mistake in the treatment of low energy photons íinner brerass-, trahlung) was corrected by Bermañ (Bermari 9 1958) and the electromagnetic corrections to the decay if polarized muons has been carried out by Kinoshita-Sirlin (Kinoshita, 1959a).
- 120 -
:
¥hen the electromagnetic interactions of the electrón and muon fields are taken into account, the Lagrangian density «£, (x) (IV. 2-54) or (IV.2-42) nvust be replaced by
fr .
i- h c.J-
t, (x)
with *,--Mr; %?
«k , vf3i V
/
^3^^
the pboton f ield , e the bare electrón charge and .the last two térros ofo¿(x) are the mass renormalization counterterms. The weak decay four field punctual vértex
(o)
which describes the u-decay with the phenomenolgical c¿ (x) is therefore to be replaced by
(0)
(1)
(2
(3)
-¿/Bt,
(3-)
- 121 -
when only lowest order electromagnetic corrections are considerad. As we just want to keep térras of order non higher thansí (o( being the fine structure constant) 9 we shall need to consider only the contribution of (0) and the interferences of (0) with each of (1) ,(2) , (7')9(3 ) s (3') and add them coheeently s for there is no physical way for isolating their individual contributions. Even t h o u g h each (1 )s ( 2 )s ( 2' ) , ( 3 )s ( 3' ) presents ultraviolet disergences, these divergences will exactly cancel out when added U D together, Only the infrared divergences will stay and their elimitnation takes place as in puré Quantum Electrodynamics 9 naicely, by adding incoherently to the above diagrams thoses graphs (4)+(5) describing inner brerasstrahlung
(4)
(5)
In general, it can be shown (Sirlin 3 1962; Berman 9 1962) that for an arbitrary Lagrangian of the type (IV,2-21) the electromagnetic corrections are finite to all orders in c(.
'
•
The transition.matrix element (finalITi linitial)
computed from (0) for the M.~-decay contains the
factor ü 2 y H (1+f 5 )u 1 (see (IV.2-5)and specialize it).The effect ofiincluding the other diagrams (1 ) + ( 2 ) + ( 2" ) + ( 3 ) + (3^) is simply to replace {L 1|"^(l + v' )u. = ü "/^u by u o P' u , vrhere the new vértex matrix V^~t when calculated 2 ' up t o trón
first charge)
order is
in
giiren
0 and le te. -^ +0 at the end, We have adopted the gauge in vrhich the free photon propagator has -the Feynman form F
á
(Bjorken-Drel.1,1965) .
>"
F
(IV.3-6)
•
To handle properly the nltraviolet and infrarred divergences of the integráis appearing
in(IV.
3-3,5) we shal regularize the photon propagaton(Feynman 5 1 9 4 9 ) , naraely, we shall replace (k 2 + i e ) ~
by Cík1) (k'+re)'' s
where - 123 -
Cikz)
V
=
L
1
z - \ ,
and we shall denote by V
c
d
•^ \ K
-
~
^
t- "•
(IV.3-7)
/
( pL, p,) , t c (p(/2.)
the
expressions obtained from (IV.3-3,5) by the replacement (kx + C£ )"'
> C (hl) íkx -+ fer)"' . At the final stage of the computation,
wewilllet
>-^oo
, >-m —> O
It is clear that Aacts as. a sort cut-off for hio-h virtual momenta and A °
Drevents the m *
infrared divergence. When ^-*o^>^the propagator . ( kl +iér)~'
of •
C Lh7-).
corresponds to photons with a non-vanis-
hing small mass A „ m
C(l 2) If Z 9 5 stands for the valué of
(1 ?) v 5 Z calculáred replacing E( p that „) in 3. -4)of byo E . ( p „) , weby have (seeapp.G) up (IV. to terms er o< order o< and neglecting the terms o( A
)
(
o( 1/>»•),when
, l/A
and
similarly
(IV.3-9)
(see App.F for the derivationof this expression and for the notation).
. From (IV ..3-8) it follows that
(Z
' Z°^) *~ = 1 - — R
(iv.3feio2_
-
124 -
and henee c
h> ?t
_¥_ +
(IV. 3-11)
The ultraviolet divergences, contained orily in R, have cancelled out. There still remains the infrared divergence in the S-function.
Radiative corrections arising from virtual photons(Behrends , 1956) : These are evidently obtained when is üsed instead of
P'
v^f.The corresponding transition matrix
element will b e , for the case of M -decay, -!f ( ) + fs-i-'J^ - o , v;here
(iv.3-12)
c¡ = p, - p¿ = p 3 ••- p M
, the
above expression simplifies t'o
(IV.3-13)
Shere
i, = ^ j i ^ ^L (S - i T . -±-~ - Z ) J
(IV.3-
- 125 -
(see App 0 F for the definition of S,T , 9 9 F~ s F_ ) . Likswise, it can be eaály checked that the virtual
photon radiative corrections to the M. —áe_
cay amount to substitute
njj y^ ( i -+• f¿)-V¿. by (IV.3-15)
( h a d we n o t
assumed
G,_ r e a l ,
ue s h o u l d
replace
G^_
by
f. 5 g • entering (IV . 3 -15 ) ) . i
i
*-
To compute tlie differential spectrum íK
'-
for the
e+
from t¿ + -decay at rest, when mei-
therneutrinos ñor the polarizations of .. the inite 1 and final states are observad we need (in analogy to.the case treated in the orevious scction) to kñov;
y
r S. 2 i I(£ + ,vz . , ^>¿ i T ! l^+ ) ^
( I V . 3-1 5 )
v; here 2L. means a sum o ver final spln, the average over the initiál spín and symmetrization in p , p u , For then
V
.\ ( o. _ ». _ o. - P.A dr, dtü
But
+ T, T T [ ( ^ T gz
Ts)
Cff,
+
nnO (-J* +
RJTTi
-
126
-
(IV . 3 -1 7 )
I -Jz. I* > IgJ2" will not be considerad 2 they a r e of o r d e r o< , henee
The ter.rr.s involving for
Y_ -
Ur. Z ( I g , I7" - l ^ l
2
Cp. ( I V . 3 - 1 8 )
-
- í p. potpi and
since
(IV,2-13)
and APDS . E and
F
fe) (IV.3-19)
f.4 p^) = ¿n -mf/mt
(p, Pi"i
ir.,
i — up to order
it easily
r
V. ph
T -
(ÍV.3-20)
£- s V, A
where -i- U J iiin>i u ; — —
é
the
upper
and where
(lower) o
ci. /_
differential
s igr, c o r r e s p o í i d i n g
(IV.3-2Í)
t o t h es u b j c i r i p t
V(A),
denotes the ün.pertuíbbed trahsition
probability
- 127 -
(IV.3-22)
(IV.3-23)
.
i
ct
©
csrrespondiiig , respective lv s to puré V ( i e -¡,-¿fc ^
~ U ~ «d,-o)
or puré A ( i . e . ^r J££ ^ , - ^ = 6^0) coupling. Because of the smallness of the electrón ir.ass relatively
to the rrraon inass, the expression (IV.3-21) can be
simplified by introducing the variable X d V . 2 - 1 4 ) ana neglec ting those terir.s Kíiich are- comparatively of order « -^ } (2¿) or highsr,
9
V7hen- ver no divergence is introduced by this appro
ximatlon in '-.;.. •.. high-energy part of the spectrun, This approximation •'ÍII Le apóllesele except for the loxí-cnergy part of the electrón spectral d is tribution . By using the following approxiinaxe expressions Co>K & —
one easily F, es
\
6-
obtains
L U Í- L U )
-
í IV. 3-24)
whence
S a ínx [ t-n ~
- ^] -
-O (IV . 3 - 2 5)
128 -
i
r
and
.
(IV.3-25)
therefore In -—
-u;I-
L- OO •+ L. ( . • * ) •+ (tu + -É.n>. - 0 ( 1 -u; -+ ¿ U _ O
I - x + i ix i 2 '2 Since terms of the form o< (ir. /m ) are negj.ectéd we may writte, by using the . explicó.t expressions for L(x),L(l):
ci.rV"ph-«-
where
a -
f i + — tavi) ^ r m>-
av ~
1- cu - ^ ÜT _ 3 C u . 2 . -
a
is
"* i :
-
T
given
x 1 (3- Z ^ d x
(IV.3-26)
by
>* v I" * >-* , _ 1 , £ tu - u;^ ^ ¿ « . ^ n - r - ¿ c J + 2 _ u , (IV.3-27) •^ - 2 . x
The functiin a(x) contains a divergent térra, because of the appearance of U-^ , coming from the infrared divergence. It is well known that to cancel it one should add to ct P_r"' the differential spectrtuiT!
ci-P_
real photons are also emitted:
of the electrons when WÍ"1*-^
e + + u , ,-h ü. , +. y
.Actúa
lly s given a certair. experimental set-up and its corresponding detection effidency, it is impossible to discrimínate between the non-rad iací i ve process radiactive éecay
u -» &. * u -r ?
and the.
j
^u_» e -r ¡_ -i- v -t f in the soft-pnoton limit.
Itturns out, as we shall see, that like in puré Quantum Electrodynamics
the emission of soft photons is more pro-
bable the softer the photons, becoming infinite with increa- 129 -
sing wave-length. This infrared divergence in the inner brem^trahlung exactly compensates the divergence arising from the soft-photon virtual corrections,
Inner bremsstEahlung (rsehrends, 1956; Berman 8195 8 ; Kinoshita, 1959a) : The contribution of diagrams (H) + (5 ) to the matrix element (e." ^-v, [ I T 1 f v
t
must be kept.
On the other hand, the part of the first trace in which m
enters vanishes, since
for any ^t-xU matrix X . Consequently s the first trace in (IV.3-29) reduces to
= Z Tv j (i- Y.-HT^ 1 "--^^^)^
(IV.3-31)
> wherein one na y further drop the feria m
as i± multiplies
Finall\r 9 we need only
the trace of an odd number of y "*s.
con.sider the symnietric part of the above trace under M «%—5> u
,for it is to be contracted with the seconds
trace of (IV.3-29) which was to be symmetrized in the índices /•')>•''5 as v;e saic above. On the other hand, the
y ,.
can be ODÍtted in (IV,3-31) for the contraction of the tvjo traces in (IV. 3-29) must be a real number s and the second trace, when symnetri zea im M? marks
2
¡s , is real. These re-
, and the fact taat
Tr(x¿y¿z)=-
Tr [xr*yr,z)
- ~
Trixttypz)
when we sum over the three polarizations of the massive photons , notably sinplify the corcputstiovjs víhen the photon polarizationsare not observed.
- 131 -
If X^" = 1/LJ times the first trace in(IV-3-29) symmetrized in u. s V and . summed over the photon three polarizations», one easily gets
= 2 gG? (p,X) [ kH p/
(IV.3-32)
where
Jl -
^>
/ e-Fz &F« ^ (— — ~ TT J
(IV.3-33)
and terms of order 0( X w ) after carrying out the- E, > K
K
0 one obtains the transiiion
o
•
_
rate for a M-+ to de cay according to H- ** e f H i f ^ i with the emission of a photon of energy larger than E, , The result follttws (Kinóshita, 1959i) that about 4.9% of ft-decays are accompanied by photons of energy > 2mo , and 1.2% by photons with E, > 10 MeV ; this' last prediction is in agreement with the experimental results (1.3+0.1%), (Sirlins1962). Caltulation of
d. P
tota
^ x ):
Since V
Ph-
T.ph
- 136 -
it follows from (IV.3-26) and (ÍV.3,39) that fIV.3-42) 3, Zs,
with
(IV.3-43)
í (£
-f
-hU3-i)+s*LJ¿ J
where (IV.3-44)
We see therefore that no infrared divergence appears in cL p_ "b^ \ only the logarithmic divergence at x = l remains which should be taken care of by the method previously outlined. The singularities of f(x) at x=0 are due to the approximation ('mi/wifüH í/oit/mi) a ü and (IV.3-4-2) is certainly aplicable for x ¿ 0 .1 . The differential spectrum (IV.3-42) diverges point-wise when iñ2 —^ 0 because of the appearance of fes.We shall see however that this mass-singularity vahishes for the integrated spectrum, and although the correction to the differential spectrum are quite large5 the corrections to the lifetime are very small. We list the valué of
¡O2 -2L J (x) 2n r
for the some
•valúes of x: ...X
0.1
0.2
0.3
0.4
0.5
0.6
0.7
25.2
13.3
7.6
4.0
1.5
-0.4
0,8
0,9
8,95
-3.7
-5.0
102.
•*-£fe): 2n
6S.2
-2.0
137 -
Radiative correction to the lifetime: The integration of d, P_ Cx) over all x can be carried out analyticaliy and presents no difficulty. By using the following relations
one easily géts
(IV.3-45)
and since
—
( nz -
~cr--rfe _
— j ~ ^. /ü
«1 -/O' 2
s
we can writte (IV.3-
n
where
- 138
We see therefore that the radiative corrections suppress the muon decay rate rather than enhance it.A phy_ sical explanation of this strange fact is not known. Taking into account the experimental valué C = 2.2001 +_ 0.0008 ^sec. (IV. 3-45) leads to & - C fl f —
fn1- i£) U (I.M3HS- ± O. QQOS) lo" er,om3 (IV.3-47)
where C is given (IV.2-25) and can be considered as the incorrected or renorroalized G whereas G^ , as given by (IV.3-4-7), represents the bare coupling constant extering (IV.3-1).
Some final remarks: Because of the difference in mass of the muon and electrón it has been suggested that the cancellation of di vergenees in the analysis of radiative correction to muon decay is ambiguous. The regulator- mass for- the photon propagator has been arbitrarily chosen the same for the diagrams (1),(2) and (3). But Berman and Sirlin (Berman, 1962) have proved that this is the only possible choice compatible with gauge-invariance if the bare coupling constanf G K is a given gaüge-independent quantity. (as required for the universality to be meaningful). We leave the discussion at this point and refer the reader to the original papers for a detailed account.
-• 1 3 9-.-
CHAPTER V ANALOGY BETWEÉN ELECTRONS AND MUONS
I. GENERALITIES In Chapter III we have cnnsidered in detail the interaction Lagrangian. that is needed to expláin the ^-ée_ cay. This turas - out to be
L C=
(-Í.M17O i
D 0022.) lo"^ í.va c/vn2
(¥.1-1)
X ; + 1.1S ¿ ü UT In Chapter IV we havs found that the results of muon decay can be explained if the interaction Lagrangian is
v¿
P
' \• s
_ o u e>
(V.l-2) ~«g-
or, more accurately, when the electromagnetic
corrections
to the above Lagrangian are considered, G=(l .¿+345+_0.0005)x 10~" erg.cm 3
(seeIV.3-47)
Surprisingly one finds that the cpupling cons-tants so
determined are nearly equal a'nd this difference
is reduced wher. Cábibbo£ 1963) theory of weak interactions is taken into account. There is furhter evidente from leptonic decays of hyperons that the weak interactions of the muon and the electrón are identical The equality of Cand G
is in excellent agreement
with the notion of universality except póssible for an im-
portant point. The muon seems to possess no strong coupling while the nucleón is strongly coupléd to pions and strange partióles. If the universality applies, as we used to think it does9 to the bare .partióles,, then we would expect a considerable nenormalization of C in the beta decay and C¿Gu. . Since C = G/Liwe must try to get a satisfactory explanation for it or abandon.the idea of universality. Let us remember that we have a similar situation in the electromagnetic theory. Leptons have no strong interactions while the baryon and raesons have it. In spite of this difference the u and e have the same electric charge as P ! n , K , 2 ..... . This means that in the electromagnetic theory the equálity of the bare charges implies the equality of renormalized charges. Let us studythe reason for this remarkable fact, As it is well known the interaction Lagrangian of the strongly coupled partióles and an electromagnetic field is
*¿ = ec Ar ex) T^tx) (V=l-3) where A^_(x) is the potential four-vector and f-€o J~f'(*)) is the so-called electromagnetic current:
where the isual spín isotopic notation has been used. As it is well known this current satisties the conservation eauation. = o In terms of this current the charge operator can be written as
(V.l-7) - 141 -
The physical chargs of one of the partióles which eppear¿ - in (V.l-4) 9 in a state 30>,is given by I cu¡0 14>> and (V a 1-7) proves that the quantity is independent of time,, Thersfore the physical charge is the same at a finite time t as in t= - «»o But at time t= -aa all the partióles ha ve same charge because of Ward's identity and the asumption that the haré charges are equsl (Berman- , 1962). Feynwan and Gell-Mann (Fey.ni3an3 1958) have used the fact that a conservad current implies no renormaliza-tion of the associated charge9 to explain the equality of C and G«,.The vector J^can be divided in two parts, one with the transíormation properties of a scalar in the isotopic spín space and the other part is the third component of the isovector ,-«-, u.
-.
c
C,
,
i^ ^
M
^
vL /—A
which satisfies5 when electromagnetic interactions are neglected, _s
To explain the absence of renormalization effects in the vector part of the beta decay in_teraction, Feynman and Gell-Mann9 suggested that the vector quantity which is coupled with the lepton current
is not merely ^, 60 yf tym ¿x) but a conserved quantity of which this is no more that one íern, They ásame that the eo rrect vector coupling is %
- trr -C^U) 1+ U)
1^1*
I." f-x) ^ ¿ i T i-x)
(V.l-li;
This implies the equality of G^and C. .One may think about the possibility of conserved axial vector current, i.e, that the axial vector --quantity that is coupled with l^(x) is not merely --E íz) yP rT $m Lx) , but a conserved quaníity of which this is oñly one ternu Let us assume that this is so and that the correct axial vector coupling is
where
A
ÍX)
is some axial vector current which sa--
tisfies (V.l-13) In this case the same argument gives us that the constant A of (V.l-1) must be equal to 1. The fact that A¿1 seems to point out the existence of renormalization' effects which cannnot be interpreted if (V.l-13) holds. Furthermore it is very difficult, but no imposible, to construct explicit ly such A^" (x) . Another strong argument againsr (V.l-13) is provided by the fact that if (V.l-13) holds then the decay of the pión in two leptons is forbidden against all the expe= rimental evidente. That this decay is forbidden can be -seen easily starting from
where P is the four-momentum operator, Let us nov; consider the expectation valué between the vacuum., iO) 9 and one pión state, | n) , with momentum k. If (V.l-13) holds we get
ky. (o I A ^ W I n ) = o
(V.l-15)
The transition amplitude of the consider.ed decay is proportional to (V.l-15) and therefore vanishes identically0
- 143
Let us now return to the study of the Feynman and Gell-Mann theorys which is usually called the conserved vector current theory ( C . V . C ) . We would like-to see if it is possible to find some direct evidente for There ara several predictions of C.V.C.which can be confirmed experimentally. The more intere.sting-prediction of C.V.C. theory is the so called weak magngtism and this will be the anly studied here.
2. WEAK MAGHETISM Let us now study the weak magnetism phenomenon, The vector part of the electromagnetic interaction Lagran gian for a nuclear system can be written as
and the vector part of the beta decay Lagrangian for a nuclear system is
4 / -r 2
O."xc 1+
(V.2-2)
Comparing (V.2-1) and (V.2-2) we see that !-(x) plays in the beta decay the same role as A (x) plays in the electromagnetic interactions . This relation can be made more c¿e.a~r by means of Feynman diagrams, noting that for each diagram in which the ph'oton interacts with a real nucleón there is an analogous diagram with two leptons. One example is - -14% -
W
As it is well known in the ele-ctromagnetic theo ry the effect of the pion-nucleon coupling is to change the bare vector matrix elenent in the following way
VJhere
a1-- Cp-p'.)^ and
F,v Cq}
and
F/ i.c¡)
are the v e c -
tor electromagnetic form factors which are real if time reversal invariance holds. Finally ¿u-_ . is the an-.o malous magnetic moment of the i-particle. The valué of K
-/'a
is (Wapstra, 1955) ^aFfV,,
(^
i 0. OooC) f*0
(V.2-4)
The form factors are nornalized in the following way
F,,Y (o) (o)== F^ i o) = -i
(V.2-5)
The effect of the • pion-nucleon interaction on the weak interactions is to change the bare vector matrix e.lement in the following way; (V.2-6)
Taking into account (V.2-3) and (V.2-60 we have as consequence of C.V.D. theory that (V.2-7)
F,v
Therefore a test of the valué of Ftw C
= ^ L Pj
-
159
-
JL.Tr
(B-13)
-
160
-
APPEKDIX C THE PHASE SPACE INTEGRAL I .
We want to study ftow how the eq (1.7-5) changes when the transformation (1.7-6) is performed. Let us introduce A=
Tn
2~Tnl
B=m 2 +m 1
C = Q-m3
D=Q+m3
. (C-l) -
with this notation the transformation (1.7-6).- can. be wr-ítteii ^
"*"
(C-2--)?
if
Henee
We introduce X, - ^ X ( a , m>?-, *nt) It is easy to p'rove that
and
X ^ 4 X ( 0¿, a, *n-f)
Introducing the q u a n t i t i e s :
one obtains
Ther'ef ore -±~
~
P?" C¿->tf lí -f-' (.S, + Si") R iu.^ip + (.RTi +.pNT,.t..S,'
- 161 -
I t can be pi*óved
-
T, Tx.
Heneé . 1 * T.T,.) «o3-y» 4 T.ft. = -
-I'Y
-
f ts.rOP-mif- T,T¿J
Theft
(c-6)
üsing the results obtained above^
¿4 = (.C'4 B 1 ) + ft ^
VT ' r 2 . ~
which Can be w-rittén explicitly as
(C-7)
a n c {í.t-f)
ís
tíbtained
- 162 -
APPENDIX
D
ELECTRON-NUCLEUS FINAL STATE INTERACTION In th-is appendix we are going to obtain the factor F(Z S E) which modifies the shape of the electrón spectrum due to the electron-nucleus final state interaction, From what we have said in (1.8) we must sol ve the Dirac equation for an extermal Coulomb field» We are going to follow the method outlined in (Akhiezers 1965). As it is well known (Schwebers 1961) the Dirac equation in an external electromagnetic field is
[ (^ - e. # ) - «* 1 + U) * o
(D-l)
where A' is the potential fbur-vector, We are going to stu dy the case in which the external field is timeindepen-dent and therefore the time dependence of the wave func-tion can be separatéd in the usual way. The time independent solution satisfies the Hamiltonian equation M ij- tí) r E ^ c*} H= ^ ( . P* - e A ) -i- e At + ^/m where
Let us consider the case of a Dirac particle in an electromagnetic field described by t=0s
A =A ( r ) s
i.e. a central field. Then the Hamiltanian equation is i-j lf (.^ i = E vf Í-?) (D-3) where Vír)=e A Q . Since we are dealing with a cemtral field p?oblem it will be useful to introduce spherLi cal coordinates. To this end let us introduce the angular momentur¿ operator - 163 -
(D-4)
We are going to need the following expressions J.t
* - ¿ (3.r)(*.f) i- C (? p) ¿
— -a
• • - » ' - «
P tv) ; n ( | ) C4)
£v'¿^> y'-' n X (.?)
(D-ll)
165
Thérefore the simultaneous eigenstates of J
JM n
9 Ja
and P aré
(D-12)
n « cThe functions a(r) and *-(r) must be determined in such a way that "4T ,
is an eigenstáte of H. We are going to
use the representation (I.2-10A), Then (D-7) can be written as-
(D-13)
Let.us now prove the following equation M >'=
Proof: We can writte
Y
v
• '
TL
*
„
,,
But since (o\r) is a ps eudo-scalar duces to
Tu
then this equations ré
TU
- 166 -
And using the equation (6.21) of Rose's book
(Rose,1957)
we get
But i t i s well known (Rose,1957i pág.154) that (L'HTHuVs - CCL'-LL; O O O )
í'
Furthermore
C (T-'/i.; -L, T* Va ; O O ©) * - j -
Henee
and equation (D-1H) follows Taking this into account(D-13) can be written as
r Let us now introduce x. » = x
v
- C T + "/O
J • U + »t
^.tj + i/o
y.
L-
vi.
Then we can writte the considered equations as
- 167 -
r ) J ú Cr) -C . y
-'
. p
Where we have
^F
assumed VW = -
^^
(D-19)
X"
If a new independent variable o =2irp is in troduced we get
- 168 -
p?
=
and satisfy the equations
(D-23)
From here it is easy to check that G 2 satisfies the following second order equation
The solution of this equation which is finite at the ori"gin is the coofluent hypergeometric function and consequent' iy
C(^)TF
( f z
2
r-> >^pr)
(D_24)
- 169 -
Where >;¿
7 7 ^ , «,
(D-25)
Furthermore [
-i-ij b ; 5) - PCo-, b;y)3
(D-26)
Using the second equation of (D-20) 9 (D-24) and (D-26) we get (D.27)
X -h C
p We can introduce a new quantity rt through the equation
and it is easy to check that vj is real, Introducing a new constant, factor, C 5 (D.24) and (D 6 27) can be rewritten as F,W = C -e
(y+ LV) (Lpv V F (y + i+iv , ¿T-5-r ; ¿L"p>r) (D-29)
where V —
p Furthermore (Slater. 1960) a-i
-1
- 170 -
Taking into account that x* = -x and 2Rea+l=b we can prove F
(o-, b ; * ) =
£
• F
(íX-t-i , b j x )
Henee
(^ v- _ u vi F ty •+• i •*l*v i ^T*' / ^L' P v '
4- £.
(D-31) _ CCÍ-I + — (2-p^V .
*
.
?
Since C can be taken real it is obvious that rg(r) andvff(r) are real. Let us now determine the valué of C.The normalization- condition is
(E'Í T£
CE)
= é ce -e»)
JMn
Henee
Since the integrand does not goes :• to z e r o w h e n M « « , we can substitute g and f by their as^mptotic expansions. As it is
- 171 -
well
known
(SlaterB
1960)
e
x
+ e,
.
-LÍ2L x
rW Since Re a = (>f +.1) and b = .-'2.Y+1 then the first term decreases as x > and the second as x 1
and for large
x the first term is dominant in our case since X"= -x 0 Henee r t 2-T+ ^
ü
;
v
- -2TV ^
- £ )r Taking into account-
(D-31)
ur>
2.
4
2C
' TIZ" - ^
1+
- \rc T ,-)\
(D-33)
Using (D-32) and (D-33) we get immediatly (D-34).
c= -I
If we want to compute N(E S 9) taking into account the final síate interaction in the app^oximation outlined in (1-8) we Bust multiply (III.2-10) by 172 -
axi
JHI
JNl
Ftz,
(D-35)
where the superscript zero means that Z=0 o r isasmall nonvanishing number to be identified as the nuclear radius, Henee
F(_Z,E)-
We introduce (D-36)
= / l_
Froip (D-29) i t i s c l e a r therefore
t h a t f o r s m a l l Z , F ( a 9 b ; z )-?> 1 and
P
fflr.
ftr\
Henee nu F(ZlE1
-
which can be vrritten as
- 173
J
Since S is a number cióse to 1 in most casess than the last factor can be approximated by 1 and we get eq(III02« 12) if r is assumed to be the nuclear radiuso A study of F(Z 9 E) is given in the article (Weidenmüller9 1961) and in (Blatt9 1952),
- 174 -
APPENDIX
E
SOME INTEGRALS OFFREQUENT USE
In the calculation of a transition probability when some of final partióles are not observed there fre~ = quently arise invariant integráis some of xíhich are inclu ded below, Let q be a time-like (four) vector such that q°¿M.+M.£0 9 where M. 9 M. are the masses of two paríicles with momenta p i 9 p 9 » If
(see 1.6-9)
then
(E-l) _pc _ p j 0 Cp,?j.) =
n
>^
c i
=: n
Hc - H ^ Z -»- \L-j•] -
175
O=
c
dr¿ & (=m 15 p « = m 9 ) s it follows ,1
dL
(l ;
o
(-1 ; k y J>fe.,fea-)
A
But
dL
TTT^j
The second térra of the r.h.s. of the above expression gives a vanishing cofitribution 9 when X —^o« 5 to any integral of the form
- 181 -
for
e v e r y c o n t i r a o u s and bounded f u n c t i o n
square 0 ¿ x ¿ 1,
f(xsy)
in
the
0 ¿ y ¿ 1,
On t h e oüiher h a ú d 5 when A —> 0 s
A
zp-
^
± _ _
-_ _L
m
'o
o
Px
r
du
+
-f. o
We have also
du
4.
0
STP**"-»-
t.'-a')
and since, whenA —^. °o 9
O
0
d*
X2"'U
d^ - | i x i a ^ An
"^
+
ü (VA)
for any continous bounded function f(x 9 y) in the integratioñ square9 it will follow
.
•-••
-
182
-
'
.1
2-
?Í
f or
Consequently «, t
(I; ¿j;
C a l c u l a t i o n of J^ Jc=
I
Ü
member that
L Px1
(Behrends9 '
9 with
1956)
Pj4
= xPl
+
ít_»
Pfl
sand
re-
P^ = mi,1 >
- 183 -
Let
-
E¿,
; and w É i t e
h
©
tu.. 2.
=
Having
thüs
computed
explicity
C J,. o „ N 5 the (1 ; 2j ;3jcr-)5 - 187 -
calculation of Vr(p ,p.) has been accomplished up to terms of the form o(A ) 9
O(1/Á) 5
which we shall neglect0 Having
in mind that VÍ?" is to b:e sandwished between ü- and u. 9 the following relations can be used:
1
JCT"
where 9 is either Y or r" ys 9 the upper (lowep) signs res> refemng to the case ^ » y}* Cvhfs-) * ¿--i CY?^) pectively 9 and
A •= Z Fz
B> •=
&' =
(pi-po +
Pj ivif -i- CFj.- F-J )
F 2 rm,
^ F,
m,
1
T-,
fe- =.
- 188 -
Consequently
-i-
where
S S _
_
C© - F,") tcí-V, © •+• C 1 - © t^t-K ís) Cu.-- 2 0 ¿í/vih 6 UJ
R 5
Uüj,
4.
—
ZUJ
-l 3 .-Yellow Report . KOFOED-HAUSEN 9 0 . \\. (1 9 6 2 ) . -Handbuch áer Physik Ul/2. LEE, T.D., YANG, C.H.(1956 ) . - Phys.Rev, l^S, 254(1956). LEE 5 Y.K, et al. (1963).» Phys, Rev.Lett. l_£s 253 (1963). LUFDY 9 A . (1 9 5 0 ) . - PropersS iri Elei.'.e nt ary Partióles and Cosr.ic Ray Phys ics". VolV.- North-Holland Pub » Co . Amsterdam . MANDL , F. (1960) .-CERi-i 60-L:-0 . -Yellov; Reporta NEWTOK, T.D, 5
WIG«£R C.F, (1 349 ) . -Re v . Mod . Phys .2_1_, 400,
PAULI s V/ „ (1 9 3 5 ) . -Zeeman Verhandelingen . iíaríinus Nijhoff. Haag, ! PAULI 5 i-:.(1936).-Ann. de 1' Institut Henri M n c a r é
6_B137O
PLANO, R.J. (1960),- Phys.Rev. 119, 1400, PRESIÓN,, M.A.(1952).- "Physics of the nucleus" . -AddisonVJesley Pub . Co . London . ROSE 5 H . E . (1 9 5 2 ) "Elementary Theory of angular moinentum" „John Wiley and Son s „ London . ROSENFELD 9 A.H. et al. (i 965) .-Rev. of Modera Phys. 37 B 633 SAKURAI 3 J.J.(1964),-"Invariance principies and elementary Darticles",-Princeton University Press. SCHWEBER 9 S.S,(1961),-"An introduction to relativistic Quantum field theory".-Row . Peterson and Co.Evanston,, ülllinois
- 19!
SHERR et al. (1949 ) . -Phys . Rev „ 7JL 9 2 8 2 « SIRLINj A.(1962),-"Weak Interactions".-Benjamín p,1730 SLATER9 L.Jo(1960).-"Confluent Hypergeometric functions",Cambridge Univ. Press» SOÜRIAU, J.M.(1961).-"Geometrie et relativité".-Hermann,París . TOLH0EK5 H.A,(1963)0-"Selected topics in nuclear theory".International Summer School. Low-Tatra Mountains.IAEA. WAPSTRA, A,Ha (1955) .-Physica _21_9 367. WEINDENMULLER, H.A.(1961).-Rev.Mod.Phys. 33 ,571(1961 ) , WU 9 C 0 S 0 et al.(1957) .-Phys.Rev9 l_0_5_s 1413, WU 9 C .S . ( 1960 ) .-"Theo.retical physics on the XXth , Century" . • Po249,,
ZYRYANOVA9 L.N.(19 63).-"Once-Forbidden beta-transitions",Pergamon Press.
J.E.N. 186-DF/I 57 Junta de Energía Nuclear, División de Física, Madrid " B é t á and m u o n decay"
J.E.N. 186-DF/l 57 Junta de Energía Nuclear, División de Física, Madrid.
"Beta and muon decay" GMJNDO, A., PASCUAL, P . (1967) 196 pp.
GALINDO, A., PASCUAL/ P. (1967) 196 pp. These notes represent a serie of lectures delivered by the authors in the Junta de Energía Nuclear, during the Spring térra of 1965, They were devoted to gpaduate students interested In the Theory of Elementary Partióles.. Special emphasis was focussed Into the coraputational problema. Chapter I 1s a review of basic principies (Dirac equatiqn, transitioñ probabilities,- final state Interactions..) which will be neéded later. In Chapter II the four-ferm1on punctual Interaction 1s discussed. Chapter I I I is devoted to the study of beta-decay; theraalnemphasis Is given to the dedüction of the formulae corresponding to electron-antinéutrino correlation, electrón T , energy spectrum, lifetimes, asymmetry of electrons emitted from polarized nu-
These notes represent a serie of lectures delivered by the authors in the Junta de Energía Nuclear, during the Spring term of 1965. They were devoted to gradúate students interested in the Theory of Elementary Particles. Special emphasis was focussed into the computational problems. Chapter I Is a review of basic principies (Dirac equation, transitioñ probabilities, final state interactions.¿)which will be needed later. In Chapter II the four-fermion punction Interaction is discussed, Chapter I I I i s devoted to the study of beta-decay; theraainemphasis is given to the deduction of the formulae corresponding to electron-antineutrino correlation, electrón energy spectrum, lifetimes, asymmetry of electrons emitted from polarized nu-
J. E.N. 186-DF/I 57
J. E.N. 186-DF/l 57
Junta de Energía Nuclear, División de Física, Madrid. " B e t a and m u o n decay"
Junta de Energía Nuclear, División de Física, Madrid
"Beta and muon decay" . GALINDO, A , ( PASCUAL, P . (1967) 196pp.
GAL INDO, A., PASCUAL, P. (1967) 196 pp. These notes represent a serie of lectures delivered by the authors 1n the Junta de Energía Nuclear, during the Spring term of 1965. They were devoted to gradúate students interested in the Theory of Elementary Partióles. Special emphasis was focussed into the computational problems. Chapter I i s a review of basic principies (Dirac equation, transitioñ probabilities, final state interactions..) which will be needed later» In Chapter II the four-fermion punctual interaction 1s discussed. Chapter I I I Is devoted to the study of beta-decay; the main emphasis is given to the deduct'ion
'These note? represent a serie of lectures delivered by the authors in the Junta de Energía Nuclear, during the Spring term of 1965. They were devoted to gradúate students interested In the Theory of Elementary Particles. Special emphasis was focussed 1nto the computational problems. Chapter I Is a review of basic principies (Dirac equation, transítion probabilities, final state interactions..) which will be needed later. In Chapter II the four-fermion punction interaction is discussed. Chapter I I I is devoted to the study of betardecay; the main emphasis i s given to the deduction
of the formulae corresponding to electron-antineutrino correlation, electrón energy spectrum, lifetimes, asymmetry of electrons emitted from polarized nu-
of the formulae corresponding to electron-antineutrino correlation, electrón enerav soectrum. lifetimes. asvmmetrv of elBr.+rnns ñmittfiri frnm nnlari7fiH nu-
clei, electrón and neutrino polarization and time reversal invariance in beta decayw In Chapter IV we deal with the decay of polarized muons with radiative corrections. Chapter V is devoted to an introduction to C.V.C theory,
clei, electrón and neutrino. polarization and time reversal invariance in beta declay. In Chapter IV we deal with the decay of polarized muons with radiative corrections. Chapter V is devoted to an introduction to C.V.C. theory.
clei, electrón and neutrino polarization and time reversal invariance in beta decay. In Chapter IV we dea! with the decay of polarized muons with radiative corrections. Chapter V is devoted to an introduction to C.V.C. theory.
clei, electrón and neutrino polarization and time reversal invariance in beta decay. In Chapter IV we deal with the decay of polarizedrauonswith radiative corrections. Chapter V is devoted to an introduction to C.V.C. theory.
,J.E.N. 186-DF/T57 Junta de Energía Nuclear, División de Física, Madrid "Desintegración beta y d e s i n t e g r a c i ó n d e l muon" GAL INDO, A., PASCUAL, P. (1967) 196 pp.
J. E.N. 186-DF/l 57 Junta de Energía Nuclear, Diyisión de Física, Madrid " D e s i n t e g r a c i ó n beta y d e s i n t e g r a c i ó n d e l muon" GAL1NDO, A., PASCUAL, P. (1967) 196 pp.
Estas notas corresponden a una serie de seminarios dados por los autores en l a Junta de Energía Nuclear, durante l a primavera de 1965. Estos fueron dedicados a licenciados interesados en l a teoría de las Partículas Elementales. Se cuidaron, especialmente, los problemas de cálculo.
Estas notas corresponden a una serie de seminarios dados por los autores en l a Junta de Energía Nuclear, durante l a primavera de 1965. Estos fueron dedicados a licenciados interesados en l a teoría de las Partículas Elementales. Se cuidaron, especialmente, los problemas de cálculo.
El Capítulo I está dedicado a los principios básicos (ecuación de Dirac, pro habilidades de transición, interacción entre estados f i n a l e s . . . ) que se necesitaran después. En el Capítulo II se discute l a interacción puntual de cuatro f e r uñones. El Capítulo I I I está dedicado al estudio de l a desintegración beta; se hace especial hincapié en l a deducción de las fórmulas correspondientes a l a
El Capítulo I está dedicado a los principios básicos (ecuación de Dirac, pro habilidades de transición, Interacción entre estados finales...)que se necesitaran después* En el Capítulo II se discute la interacción'puntual de cuatro f e r niones. El Capítulo I I I está dedicado al estudio de l a desintegración beta; se hace especial hincapié en l a deducción de las fórmulas correspondientes a l a
J. E.N. 186-DF/l 57 Junta de Energía Nuclear, División de Física, Madrid.
"Desintegración beta y desintegración del muon"
J.E.N. 186-DF/l 57 Junta de Energía Nuclear, División de Física, Madrid
"Desintegración beta y desintegración del muon"
GALINDO.A., PASCUAL, P . (1967) 196 pp.
GAL INDO, A., PASCUAL, P . (1967) 196 pp.
Estas notas corresponden a una serie de seminarios dados por los autores en 1.a Junta de Energía Nuclear, durante l a primavera de 1965. Estos fueron dedicados a licenciados Interesados en l a teoría de las Partículas Elementales. Se cuidaron, especialmente, los problemas de cálculo. El Capítulo I está dedicado a los principios básicos (ecuación de Dirac, pro habilidades de transición, interacción entre estados f i n a l e s . - . ) que se necesitaran después. En el Capítulo I I se discute l a interacción puntual de cuatro f e r miones. El Capítulo I I I está dedicado al estudio de la desintegración beta; se hace especial hincapié en l a deducción de las fórmulas correspondientes a l a
Estas notas corresponden a una serie de seminarios dados por los autores en l a Junta de Energía Nuclear, durante l a primavera de 1965. Estos fueron dedicados a licenciados interesados en l a teoría de las Partículas Elementales. Se cuidaron, especialmente, los problemas de cálculo El Capítulo I está dedicado a los principios básicos (ecuación de Dirac, probabilidades de transición, interacción entre estados f i n a l e s . . . ) que se necesitaran después. En el Capítulo I I se discute la interacción puntual de cuatro f e r miones. El Capítulo I I I está dedicado al estudio de l a desintegración beta; se hace especial hincapié en la deducción de las fórmulas correspondientes a l a
correlacio'n electrón-antineutrino, espectro energético de los electrones, vidas medias, asimetría de los electrones emitidos por núcleos polarizados, polarización del electrón y del neutrino e invariancia bajo la inversión temporal. Eh el Capítulo IV se trata la desintegración de muones polarizados con correcciones radiactivas. El Capítulo V está dedicado a una introducción a la teoría C.V.C. ' '
correlación electrón-antineutrino, espectro energético de los electrones, /vidas;medias, asimetría de los electrones emitidos por núcleos polarizados, polarización del electrón y del neutrino e invariancia bajo la inversión temporal. En el Capítulo IV se trata la desintegración de muones polarizados con correcciones radiactivas. El Capítulo V está dedicado a una introducción a la teoría C.V.C. ' -
correlación electrón-antineutrino, espectro energético de los electrones, vidas medias, asimetría de los electrones emitidos por núcleos polarizados, polarización del electrón y del neutrino e invariancia bajo la inversión -tera-a. poral. En el Capítulo IV se trata la desintegración.de muones polarizados con correcciones radiactivas. El Capítulo V está dedicado a una introducción, a la teoría C.V.C.
correlación electrón-antineutrino, espectro energético de los electrones, vidas medias, asimetría de los electrones emitidos por núcleos polarizados, polarización del electrón y del neutrino e invariancia bajo la inversión tem poral. En el Capítulo IV se trata la desintegración de muones polarizados con correcciones radiactivas. El Capítulo V está dedicado a una introducción a la teoría C.V.C.
