- Email: [email protected]
Special relativity, phase space and cross sections
i7
SPECIAL
Notation.
RELATIVITY,
4-vector in c . m . p = (w,~); in lab P = (W,~),
Solid-angle element d~ = 2~d cos 0; pZ = w2 . ~2 = m 2 i s ~n i n v a r x a n t . C r o s s
PHASE
SPACE,
AND
CROSS
SECTIONS
Rn, Invariant Volume in n-Body Momentum Space
T = W-re.
Ausefuli.... iaoti4d'p
cl~= Z~d cos O.
section 0 is invariant,
Rz
:'lPtl/W-~7", %
=
S g = fp d do ½Dldwdo
* [dwidw z
= (~7Z/4S) f d m f z dzm z 3 . z
Lorentz Transformation w Px
-ff
V
If 6 and e are m e a s u r e d
0 0
0
Px
to t h e t r a n s f o r m a t i o n
i 0
0 t
PV P
Pl = tan 6 =
(/0 °/0 ,~
py
0
:
Pz
0
if p a r t i c l e i i s b e a m ,
v:
(w~+~z)/~,
with respect
-~W+Vi~icosO
as
2 i s target, then ( W z , ~ ~) : (mz,~) and
= (2Tr)464(p . ~ q i )
(i=i,Z),
(4)
fuse (6), below].
(5)
~ + m~2+ m ~ + rn~2.
(6) (3,1aS)
~f = ~
= +ZlTil I;'t d cos 0.
F o r e l a s t i c s c a t t e r i n g ( m i = m ' t , m 2 = rn*z) , (4) and (5) i n c . m . t = _~z
(~,cm) become
li - cos~l = -C~ z s i n Z O / Z ,
.
,
(tZ)
,
ci~)
where Tif is an invariant matrix element. F is M~ller's invariant flux Z 2 Z 2 f a c t o r . F = (Pi" PZ ) - PlP2" In .... y s y s t e m w h e r e ~ i a n d ~2 . . . . . Ui . . . . . r : w i w z l ~ i - ~'21 ~ : P~/~I. U f is b . . . . Z. target ~ 2 ~ 01, then
I~t I~-;"
d~
ITI z
d Ldl~ IPS
d~
= ~ f
{~t [ ' and(12) yields --~--s
ITI z
4~2cos20/2.
(5,el)
T h e n o r m a l i z a t i o n i s s u c h t h a t the o p t i c a l t h e o r e m r e a d s
using (4,1ab), (4,el), and {Z),
l m T It=0 = Z [ ~ i l q ~ - ~ t o t •
T~ = 2~tZm2s sin2 (~)(useInl Two-Body States. s+mt
2
-mZ
wt
Energies Z
~2 . pf
=p;
~,
in 2 = ~:m~ + ij m iZ2 3
Z
Z
lc~iati.
Z
'
~ i r a ~ ....
lie , ) I
=~
T h e c h o i c e of E q . ( i t ) i m p l i e s a p a r t i c u l a r n o r m a l i z a t i o n of a n y s p i n o r s t h a t m a y o c c u r in T. t T h e a d v a n t a g e of t h i s normmalization i s that it g r e a t l y i i s i m p l i f i e s the s t r u c t u r e of T by p u t t i n g f a c t o r s s u c h a s ~ ~-~ into the p h a s e s p a c e w h e r e t h e y r e a l l y b e l o n g . In a d d i t i o n , the l a b e l s , i, f, r e f e r to
[s-(mi+m2)Zl[s-(mi-m2)Z].
(8)
specific spin (helicity) states, implicit.
s o that the u s u a l " a v e r a g e and s u m " r u l e i s
then
= const. (i,j = i,Z, 3) [ f o l l o w s f r o m (6)]
m i . k = ~."a i + 2 m t 2 3 4 = c o n s t . l
(i5)
(7)
a n d m o m e n t a in c . m .
= 2 ~ m i + m i 2 3 4 = const. i
f ....
Z L e t m i j = (Pi + Pj) ' e t c . ;
3- and 4 - B o d y S t a t e s . i
(it)
(4,ell
u = (mi2 - n~Z)Z/s- z~Z(l+cos 0) = ( m t 2 - m ; ) Z / s -
F o r elastic scattering,
Ze I
in g ..... 1 if} ~ If> .
JIT,,IZdL~Slm~;,f,'..,~.)
F o r e l a s t i c s c a t t e r i n g i n c. in. ,
dt
~ i=i
or
r = I ~ Im 2 = s : m ~ 2 + m ~ + 2 W i r e Z = ( m I + mmz)Z + 2 T ~ r n z ,
ystem
~P(KL)
Note that R n = (Zw) 3n-4 f dLIPS.
~ = ~-~ f [rifJ z dLmSIs;q t, ..-,%)
'~)
In lab s y s t e m P 2 = (mz' ~)' and writing W = m + T,
In . . . . .
i (-~) 3n
i
2
t ....
Z
qn [ q i = ( e l ' q~i)]' d e f i n e L . . . . t z I n v a r l a n t P h a s e S p a c e
dLIPS(S;ql ' ...,qn)
f + Z ~ I' + 2'.
u = (p~ - pz) z = (p~ - p~)Z
ta qi' "'''
F o r I + Z -- n particles or f ~ n particles,
= m~i+m*i2- 2(wiw'i-Ti.7?,
_Z
d3~
~nd 7 , = ~ - f ~ ,
s : (p~ + p2) 2 : m~ + ~2 + ~(w~2 - ~i" ~ ) '
G . . . . . 1 relati . . . . .
Rn=Id(~)~Rn-k+t'
F o r a s y s t e m of n p a r t i c l e s w i t h o v e r a l l Inur-111omenturn p a n d f i n a l m ....
four-vectors that correspond to a resonant state.
t = (p*i-pi)2
<
/
C r o s s S e c t i o n s and D e c a y R a t e s t
w h e r e Q2 = M2 and f = (e + e ' ) / ( E + M). T h e s e e q u a t i o n s f o l l o w f r o m e x a m p l e (b), p. 34 of H a g e d o r n . ~ T h e y a r e p a r t i c u l a r l y u s e f u l w h e n ~ i s a s u m of
Notation:
I then
(Rk7
(
A Useful T r a n s f o r m a t i o n : Consider two 4-vectors G = (E, Q) and q = (e, ~). In the r e s t f r a m e of ~ [QI = (M,~)], q b e c o m e s (q ~ q')
Invariants.
(an) ,
(l~n k+l)l
(z)
G e n e r a l L .... tz Transformation [characterized by ~, with V = ( t ' ~ ) " i / 2 a n d . 7 = ~Yf: w:~W~.7; 7=~-7 W+w ?+I "
e,:o.~/~
k + i, • - • , n
k+l,...,n
or as N ~ K, L
IF, I / ~ , I ~ 1 : Ipzl : ~mZ = I~fl~z/~"
~" ~:
N~K,
Lt, ~,..., k
-2 "i = f + T f / Z m f .
F o r m i = m Z,
R e l a t i o n f o r F a c t o r i n g R n (see. e. g . , H a g e d o r n , p. 93*):
W r i t e N ~ i , Z, • • . , k,
•
i~lsin e
Px
Recurrence
a x i s x,
( i , j , k = i, Z, 3, 4.)
(9)
(lO)
* R . H a g e d o r n , R e l a t i v i s t i c K i n e r n a t i c s ~ W. A . B e n j a m i n , N e w Y o r k , 1964. # S e e , f o r e x a m p l e , C h a p s . f a n d Z of H. P i l k u h n , T h e I n t e r a c t i o n s of Hadrons., John Wiley & Sons, New York, f967.
SPECIAL
Notation.
RELATIVITY,
4-vector in c . m . p = (w,~); in lab P = (W,~),
Solid-angle element d~ = 2~d cos 0; pZ = w2 . ~2 = m 2 i s ~n i n v a r x a n t . C r o s s
PHASE
SPACE,
AND
CROSS
SECTIONS
Rn, Invariant Volume in n-Body Momentum Space
T = W-re.
Ausefuli.... iaoti4d'p
cl~= Z~d cos O.
section 0 is invariant,
Rz
:'lPtl/W-~7", %
=
S g = fp d do ½Dldwdo
* [dwidw z
= (~7Z/4S) f d m f z dzm z 3 . z
Lorentz Transformation w Px
-ff
V
If 6 and e are m e a s u r e d
0 0
0
Px
to t h e t r a n s f o r m a t i o n
i 0
0 t
PV P
Pl = tan 6 =
(/0 °/0 ,~
py
0
:
Pz
0
if p a r t i c l e i i s b e a m ,
v:
(w~+~z)/~,
with respect
-~W+Vi~icosO
as
2 i s target, then ( W z , ~ ~) : (mz,~) and
= (2Tr)464(p . ~ q i )
(i=i,Z),
(4)
fuse (6), below].
(5)
~ + m~2+ m ~ + rn~2.
(6) (3,1aS)
~f = ~
= +ZlTil I;'t d cos 0.
F o r e l a s t i c s c a t t e r i n g ( m i = m ' t , m 2 = rn*z) , (4) and (5) i n c . m . t = _~z
(~,cm) become
li - cos~l = -C~ z s i n Z O / Z ,
.
,
(tZ)
,
ci~)
where Tif is an invariant matrix element. F is M~ller's invariant flux Z 2 Z 2 f a c t o r . F = (Pi" PZ ) - PlP2" In .... y s y s t e m w h e r e ~ i a n d ~2 . . . . . Ui . . . . . r : w i w z l ~ i - ~'21 ~ : P~/~I. U f is b . . . . Z. target ~ 2 ~ 01, then
I~t I~-;"
d~
ITI z
d Ldl~ IPS
d~
= ~ f
{~t [ ' and(12) yields --~--s
ITI z
4~2cos20/2.
(5,el)
T h e n o r m a l i z a t i o n i s s u c h t h a t the o p t i c a l t h e o r e m r e a d s
using (4,1ab), (4,el), and {Z),
l m T It=0 = Z [ ~ i l q ~ - ~ t o t •
T~ = 2~tZm2s sin2 (~)(useInl Two-Body States. s+mt
2
-mZ
wt
Energies Z
~2 . pf
=p;
~,
in 2 = ~:m~ + ij m iZ2 3
Z
Z
lc~iati.
Z
'
~ i r a ~ ....
lie , ) I
=~
T h e c h o i c e of E q . ( i t ) i m p l i e s a p a r t i c u l a r n o r m a l i z a t i o n of a n y s p i n o r s t h a t m a y o c c u r in T. t T h e a d v a n t a g e of t h i s normmalization i s that it g r e a t l y i i s i m p l i f i e s the s t r u c t u r e of T by p u t t i n g f a c t o r s s u c h a s ~ ~-~ into the p h a s e s p a c e w h e r e t h e y r e a l l y b e l o n g . In a d d i t i o n , the l a b e l s , i, f, r e f e r to
[s-(mi+m2)Zl[s-(mi-m2)Z].
(8)
specific spin (helicity) states, implicit.
s o that the u s u a l " a v e r a g e and s u m " r u l e i s
then
= const. (i,j = i,Z, 3) [ f o l l o w s f r o m (6)]
m i . k = ~."a i + 2 m t 2 3 4 = c o n s t . l
(i5)
(7)
a n d m o m e n t a in c . m .
= 2 ~ m i + m i 2 3 4 = const. i
f ....
Z L e t m i j = (Pi + Pj) ' e t c . ;
3- and 4 - B o d y S t a t e s . i
(it)
(4,ell
u = (mi2 - n~Z)Z/s- z~Z(l+cos 0) = ( m t 2 - m ; ) Z / s -
F o r elastic scattering,
Ze I
in g ..... 1 if} ~ If> .
JIT,,IZdL~Slm~;,f,'..,~.)
F o r e l a s t i c s c a t t e r i n g i n c. in. ,
dt
~ i=i
or
r = I ~ Im 2 = s : m ~ 2 + m ~ + 2 W i r e Z = ( m I + mmz)Z + 2 T ~ r n z ,
ystem
~P(KL)
Note that R n = (Zw) 3n-4 f dLIPS.
~ = ~-~ f [rifJ z dLmSIs;q t, ..-,%)
'~)
In lab s y s t e m P 2 = (mz' ~)' and writing W = m + T,
In . . . . .
i (-~) 3n
i
2
t ....
Z
qn [ q i = ( e l ' q~i)]' d e f i n e L . . . . t z I n v a r l a n t P h a s e S p a c e
dLIPS(S;ql ' ...,qn)
f + Z ~ I' + 2'.
u = (p~ - pz) z = (p~ - p~)Z
ta qi' "'''
F o r I + Z -- n particles or f ~ n particles,
= m~i+m*i2- 2(wiw'i-Ti.7?,
_Z
d3~
~nd 7 , = ~ - f ~ ,
s : (p~ + p2) 2 : m~ + ~2 + ~(w~2 - ~i" ~ ) '
G . . . . . 1 relati . . . . .
Rn=Id(~)~Rn-k+t'
F o r a s y s t e m of n p a r t i c l e s w i t h o v e r a l l Inur-111omenturn p a n d f i n a l m ....
four-vectors that correspond to a resonant state.
t = (p*i-pi)2
<
/
C r o s s S e c t i o n s and D e c a y R a t e s t
w h e r e Q2 = M2 and f = (e + e ' ) / ( E + M). T h e s e e q u a t i o n s f o l l o w f r o m e x a m p l e (b), p. 34 of H a g e d o r n . ~ T h e y a r e p a r t i c u l a r l y u s e f u l w h e n ~ i s a s u m of
Notation:
I then
(Rk7
(
A Useful T r a n s f o r m a t i o n : Consider two 4-vectors G = (E, Q) and q = (e, ~). In the r e s t f r a m e of ~ [QI = (M,~)], q b e c o m e s (q ~ q')
Invariants.
(an) ,
(l~n k+l)l
(z)
G e n e r a l L .... tz Transformation [characterized by ~, with V = ( t ' ~ ) " i / 2 a n d . 7 = ~Yf: w:~W~.7; 7=~-7 W+w ?+I "
e,:o.~/~
k + i, • - • , n
k+l,...,n
or as N ~ K, L
IF, I / ~ , I ~ 1 : Ipzl : ~mZ = I~fl~z/~"
~" ~:
N~K,
Lt, ~,..., k
-2 "i = f + T f / Z m f .
F o r m i = m Z,
R e l a t i o n f o r F a c t o r i n g R n (see. e. g . , H a g e d o r n , p. 93*):
W r i t e N ~ i , Z, • • . , k,
•
i~lsin e
Px
Recurrence
a x i s x,
( i , j , k = i, Z, 3, 4.)
(9)
(lO)
* R . H a g e d o r n , R e l a t i v i s t i c K i n e r n a t i c s ~ W. A . B e n j a m i n , N e w Y o r k , 1964. # S e e , f o r e x a m p l e , C h a p s . f a n d Z of H. P i l k u h n , T h e I n t e r a c t i o n s of Hadrons., John Wiley & Sons, New York, f967.