Spatial image of reaction area from scattering.II: On connection between the differential cross-sections in transverse momentum and in nearest approach parameter

0.3 MB·English

by  N. Bobrovskaya

Spatial image of rea
tion area from s
attering. II :
On
onne
tion between the di(cid:27)erential
ross-se
tions in transverse momentum and in nearest approa
h
parameter.
∗
N.Bobrovskaya and A.N.Vall
Department of Theoreti
al Physi
s, Irkutsk State University, Irkutsk, 664003 Russia
†
M.V.Polyakov
Institut fur Theoretis
he Physik II Ruhr-Universitaet Bo
hum, NB6 D-44780 Bo
hum, Germany
‡
A.A.Vladimirov
Bogoliubov Laboratory of Theoreti
al Physi
s, JINR, 141980, Mos
ow Region, Dubna, Russia
8
0 C
0 The
onne
tionbetweendi(cid:27)erentia~bl
rossse
tionsofparti
le
reationontransSveOr(se2,m1)omentum
2 and on nearest app~broa
h parameter is investigated in the
ontext of formalism algebra.
Where parameter
hara
terizes parti
le
reation area. This distribution tightly
on
erned with
n
spatial stru
tureof parti
les intera
tion andallows intuitive physi
interpretation. It is shown that
a b
J area of large transverse momentum1
/a2rriesbiqn the1main
oqntribution to distribution fun
tiCon on
in ba
kward semisphere in interval ≤ ∼ , where is momentum of the parti
le . The
1
left border of inequation de(cid:28)nes by Geizenberg un
ertainty relation on parameters "momentum -
1
radius of lo
alization area" . The spatial stru
ture of
reation area in transverse momentumplane
C
] is a set of dis
rete axially~q-symm~ebtri
al (for spinless parti
les ) zones. The re
eived
onne
tion
h between
rossse
tionson ⊥ and isexa
tanddon't
onne
twithanymodel. Soabilityappearsto
C
-p awnaaslyrez
eetivheedspanateiaxlas
ttrrue
ltautrioenofbettawrgeeetnu<sinb2±g e>xpaenrdim<enctoasl2dθa±ta>offorpaarntyi
lAe+Ban→gleCd+istDribpurtoio
nes.seIst
p A B
e in
enter of parti
les and mass frame, where average is gobi2ng on
orresbp2oqn2d>ing~2d/i4(cid:27)erential
h
ross-se
tions<. cIotsi2sθs±ho>w>n1t/h8at quantum-me
hani

onstrain on spe
truγm+p( π0+p ) brings
[ etone
rognystEraγin=5Gev. . Asappli
ationitis
onsideredthepro
ess → atphoton
2
v PACSnumbers:
8
9
3 C
DIFFERENTIAL CROSS-SECTION ON THE TRANSVERSE MOMENTUM OF PARTICLE
3
.
9
0 In thiCs paper we will reSprodu
e in detail the derivation of
onne
tion between the di(cid:27)erential
ross-se
tion in
7 parti
le momentum and -matrix element [1℄. As in the framework of the same s
heme the similar
al
ulation for
b
0 the
ross-se
tionon nearest approa
h parameter [2℄ will be made.
A+B C +D C
v: Let us examine~qthe pro
ess → C for two physi
a~µl,
qa,sǫe>s. In the (cid:28)rst
ase parti
le is
reated with
i (cid:28)xed momentum , the se
ond, parti
le is
reated in state | . The di(cid:27)erential
ross-se
tion on transverse
X C
momentum of the parti
le is
r
a dσ± 1 dN±
= .
dΩ n n TV ~u dΩ (1)
q~ 1 2 q~
| |
~u n ,n A B ( ) C
1 2
Here is relative speed of initial parti
les, are densities of parti
les and , ± means that the parti
le
z A N
s
atters to the forward and ba
kward half-sphere ( -axis is dire
ted along the momentum of parti
le ), and is
C D V T
the total number of parti
les and
reation events in all spa
e (volume ) during in(cid:28)nite time ( ). This is
N = d~q d~q <~q;~q Fˆ in> 2 = N(ǫ) =
1 1
| | | |
Z ǫ=±1
X
= d~q q2dq dΩ <~q ,ǫ q2 q2;~q Fˆ in> 2 , (2)
1 q~ | ⊥ − ⊥ 1| | |
ǫX=±1Z q
where
in>=(2π)3n1/2n1/2a+(p~ ) a+(p~ ) 0> (2π)3n1/2n1/2 p~ ;p~ > .
| 1 2 1 2 | ≡ 1 2 | 1 2
2
A B Fˆ S S =I+iFˆ
The
reationoperatorsarerelatedtoparti
les and ,operator is
onne
tedwith -matrixbyrelation .
Translational invarian
e allows present the matrix element in the form:
<f Fˆ in>=δ(4)(q q )<f Aˆin> ,
in f
| | − | | (3)
therefore:
TV
N = d~q q2dq dΩ δ(4)(q+q P)
(2π)4 1 q~ 1− ×
ǫ=±1Z
X
<~q ,ǫ q2 q2;~q Aˆ in> 2 =
×| ⊥ − ⊥ 1| | |
= TV q q2dq dΩ δ(E + (~q P~)2+m2 P0) (4)
(2π)4 q~ q − D− ×
ǫ=±1Z q
X
<~q ,ǫ q2 q2;~q =P~ ~q Aˆ in> 2 .
×| ⊥ − ⊥ 1 − | | |
q
P =P +P δ C
1 2
Here, the four-momentum . In (4) the - fun
tion determines the energy shell for parti
le rea
tion :
E + (~q P~)2+m2 P0) =0 .
q − D− (5)
(cid:26) q (cid:27)q=q˜
Let us de(cid:28)ne:
E = (~q P~)2+m2
D − D
q
qE E
λ(~q )= q D . (6)
⊥ (q2P0−(~q·P~)Eq)q=q˜
Then the following relation is valid:
δ(E + (~q P~)2+m2 P0)=λ(~q ) δ(q q˜) .
q − D− ⊥ − (7)
q
q
Integrating (4) over we get:
TV
N = dΩ q2 λ(~q )
(2π)4 q~ ⊥ ×
ǫ=±1Z
X
(8)
<~q ,ǫ q2 q2;~q =P~ ~q A in> 2 , q =q˜ , q = ~q .
× | ⊥ − ⊥ 1 − | | | | |
q
C
So, we obtain a relation for parti
le transverse momentum distribution:
dN± TV
= q2λ(~q ) <~q , q2 q2;~q =P~ ~q A in> 2 , q =q˜.
dΩ (2π)4 ⊥ | ⊥ ± − ⊥ 1 − | | | (9)
q~
q
Let us de(cid:28)ne:
A(ǫ)(~q)=<~q ,ǫ q2 q2;~q =P~ ~q A p~ ;p~ > , q =q˜.
⊥ − ⊥ 1 − | | 1 2 (10)
q
Then (9) takes a view:
dN±
=(2π)2TVn n q2λ(~q ) A(±)(~q)2 , q =q˜.
1 2 ⊥
dΩ | | (11)
q~
The di(cid:27)erential
ross-se
tionis resulted by substitution (11) to relation (1). dσ±
Here we mark an important noti
e. An angle dependents in
ross se
tion dΩq~ appears not only through variable
~q q˜ A B C
⊥
, but ingeneral
asealsothrough . Inthe frameofparti
les and
entreofmass(
.m.f.) theparti
le energy
s=(p +p )2
A B
de(cid:28)ned only through square of invariant mass . And it is
1
E⋆ = (s+m2 m2 ) .
C 2√s C − D (12)
3
So:
q˜=p⋆ = (E⋆)2 m2 .
C C − C
q
E⋆
And C isn't depend for s
attering angle (the star means
.m.f.). The situation
hanges in the labor frame (l.f.). In
κ
this
ase there are
riti
al parameter whi
h is [3℄
√s p⋆
κ= · C
m p
C A
·
κ>1 C
. In the
ase the
onne
tion betweenparti
le momentum and s
atteringanglehasa single meaningand has
a view:
√sE⋆p cos(θ)+(E +m )[s(p⋆)2 m2p2 sin2(θ)]1/2
q˜=p = C A A A C − C A ,
C (E +m )2 p2 cos2(θ) (13)
A B − A
0 < θ < π
where is s
attering angle in l.f. (there isn't the maximum angle limitation). We will ba
k to dis
uss this
expression below.
~b
THE DIFFERENTIAL CROSS-SECTION ON NEAREST APPROACH PARAMETER IN C.M.F.
C ~µ,q,ǫ> out>
Nowwedis
ussthe
ase,whentheparti
le
reatesinstate| . Choosingforstate| the
orresponding
C D
basis, we obtain following relation for the total number of and parti
les
reation events [2℄:
N =(2π)2 d~q q2dq dΩ < ~µ,q,ǫ;~q Fˆ in> 2 .
1 µ~ 1
| | | | (14)
ǫ=±1Z
X
< ~µ,q,ǫ;~q
1
CAs distin
t from the previous
ase, the state | is not a state wi<th ~µa,dqe,(cid:28)ǫ;n~qe moFˆmeinntu>m. So the pdaσ±rti
le

ross-se
tion in this state
an't be expressed through the matrix element 1 | | . To (cid:28)nd dΩµ~ we
C
will use a state expansion on states with the (cid:28)xed parti
le transversemomentum [2℄:
1
< ~µ,q,ǫ;~q = ξ¯(~q ,~µ)<~q ,ǫ q2 q2;~q dΩ .
1 | (2π)2 ⊥ ⊥ − ⊥ 1| q~ (15)
Z q
Substituting this expansion to (14) we obtain:
1
N = d~q q2dq dΩ dΩ dΩ ξ¯(~q ,~µ) ξ(~k ,~µ)
(2π)2 1 µ~ q~ ~k ⊥ ⊥ ×
ǫ=±1Z
X (16)
F(~q ,~q ) F¯(~k ,~q ) , ~q = ~k =q ,
⊥ 1 ⊥ 1
× | | | |
where we use a notation:
<~q ,ǫ q2 q2;~q Fˆ in>=F(~q ,~q ) .
⊥ − ⊥ 1| | ⊥ 1 (17)
q
Let us make an identi
al transformationin (16):
F(~q ,~q )=[F(~q ,~q ) F(~k ,~q )]+F(~k ,~q ) ,
⊥ 1 ⊥ 1 ⊥ 1 ⊥ 1
−
F¯(~k ,~q )=[F¯(~k ,~q ) F¯(~q ,~q )]+F¯(~q ,~q ) .
⊥ 1 ⊥ 1 ⊥ 1 ⊥ 1
−
Then:
1
N = Re d~q q2dq dΩ dΩ dΩ ξ¯(~q ,~µ) ξ(~k ,~µ)F(~q ,~q )2
(2π)2 1 µ~ q~ ~k ⊥ ⊥ | ⊥ 1 | −
ǫ=±1Z
X
1 1
d~q q2dq dΩ dΩ dΩ ξ¯(~q ,~µ) ξ(~k ,~µ)
− (2π)22 1 µ~ q~ ~k ⊥ ⊥ × (18)
ǫ=±1Z
X
F(~q ,~q ) F(~k ,~q )2 , ~q = ~k =q .
⊥ 1 ⊥ 1
×| − | | | | |
4
ξ(~k ,~µ)
⊥
The se
ond term in (18) is turned to zero, owing to the
ompleteness of the system of basi
fun
tions [2℄ :
dΩ ξ¯(~q ,~µ) ξ(~k ,~µ) δ(~k ~q ) .
µ~ ⊥ ⊥ ⊥ ⊥
∼ −
Z
Finally we obtain:
1
N = Re d~q q2dq dΩ dΩ dΩ ξ¯(~q ,~µ) ξ(~k ,µ~)F(~q ,~q )2 .
(2π)2 1 µ~ q~ ~k ⊥ ⊥ | ⊥ 1 | (19)
ǫ=±1Z
X
N ~q
⊥
Su
h representation for is universal in the meaning that it follows both di(cid:27)erential
ross-se
tionsin terms of
µ~ N
and alsoin terms of . ThisaFˆllowsAˆmarkoutsingularfa
torsin ,
on~qne
tedwithin(cid:28)nite time andvolume. Turning
1
to the matrix elements from to and integrating over momentum , we get:
VT
N = Re q2dq dΩ dΩ dΩ ξ¯(~q ,~µ) ξ(~k ,~µ)
(2π)6 µ~ q~ ~k ⊥ ⊥ ×
ǫ=±1Z
X
(20)
δ(E + (~q P~)2+m2 P0) A(ǫ)(~q)2 .
× q − D− | |
q
~µ q
If we integrate this
orrelation over the parameter and momentum then we automati
ally obtain relation (11).
~k
⊥
But we will integrate over momentum . So we have:
VT
N = Re q2dq dΩ dΩ κ(µ) ξ¯(~q ,~µ)
(2π)6 µ~ q~ ⊥ ×
ǫ=±1Z
X
(21)
δ(E + (~q P~)2+m2 P0) A(ǫ)(~q)2 ,
× q − D− | |
q
where [4℄
2π2 √π iµ 1 2
κ(µ)= ξ(~k ,~µ) dΩ = = √π Γ( + ) .
⊥ ~k ch(πµ) Γ(iµ + 3)2 2 4 (22)
Z | 2 4 | (cid:12) (cid:12)
(cid:12) (cid:12)
q (cid:12) (cid:12)
Further integrating over in relation (21) we obtain:
VT
N = Re dΩ dΩ κ(µ) ξ¯(~q ,~µ) q2λ(~q ) A(ǫ)(~q)2 ,q =q˜.
(2π)6 µ~ q~ ⊥ ⊥ | | (23)
ǫ=±1 Z
X
Taking into a

ount (11) we
an rewrite this expression in the following form:
1 dN(ǫ)
N = Re dΩ dΩ κ(µ) ξ¯(~q ,~µ) , q =q˜.
(2π)2 µ~ q~ ⊥ dΩ (24)
ǫ=±1 Z q~
X
N A B N
Thisrelationforthetotalnumberofevents isrightinanyframeofparti
les and . Crossingfrom todi(cid:27)erential
q˜
distribution demands (cid:28)xation of frame. If the initial state set in
.m.f, then don't depend of the s
attering angle.
~µ
And in this
ase we have for the di(cid:27)erential distribution on :
dN± 1 dN±
= κ(µ) Re dΩ ξ¯(~q ,~µ) , q =q˜,
dΩ (2π)2 q~ ⊥ dΩ (25)
µ~ Z q~
where
(s+m2 m2 )2
q˜2 = C − D m2 .
4s − C
q =q˜
There are everywhere below (rea
tion surfa
e).
µ b
Using relations between and
1
µ=(b2q2 )1/2 , dΩ =q2tanh(πµ)d~b , d~µ=µ dµ dϕ, d~b=b db dϕ ,
µ~
− 4 (26)
5
~b
we get a distribution on . So we have for di(cid:27)erential
ross-se
tions:
dσ± 1 dσ±
= κ(µ) Re dΩ ξ¯(~q ,~µ) ,
dΩ (2π)2 q~ ⊥ dΩ
µ~ Z q~
dσ± q2λ(~q )
=(2π)2 ⊥ A(ǫ)(~q)2 ,
dΩ ~u | |
q~
| |
qE E (27)
q D
λ(~q )= ,
⊥ q2P0 (~q P~)E
q
− ·
E = (~q P~)2+m2 .
D − D
q
Aˆ Sˆ <f S in>=<f in>+iδ(4)(q q )<f Aˆin>
f in
The matrix element isdσ
±onne
ted with -matrix by relation~µ | | | − | | .
Now, let us integrate dΩµ~ (27) over the dire
tion of ve
tor :
dσ± 1 q q dσ±
dµ = 2π µth(πµ)κ(µ)Z dΩq~( q2−q⊥2 P−12+iµ( q2−q⊥2 )dΩq~) . (28)
p p
Here we used an integral representation of
one fun
tion [5℄:
2π
dϕ (u u2 1cos(ϕ θ))−1/2+iµ =2πP (u ) .
0− 0− − −1/2+iµ 0 (29)
Z0 q
~µ
Subsequent transformationof di(cid:27)erential
ross-se
tionon
onne
t with turning to hyperboli
variables:
q ~q
u=(u ,u ,u ) , u = , ~u= ⊥ , u2 =u2 u2 u2 =1
0 1 2 0 q2 q2 q2 q2 0− 1− 2 (30)
− ⊥ − ⊥
p p
In this variables the di(cid:27)erential volume is:
d~q d~u du dϕ
⊥ 0
dΩ = = =
q~ q q2 q2 u3 u2 (31)
− ⊥ 0 0
ϕ ~q p
⊥
where is the azimuth angle of ve
tor . dσ± ϕ
Taking into a

ount that the di(cid:27)erential
ross-se
tion dΩq~ does not depend on we obtain:
∞
dσ± du dσ±
0
dµ =µtanh(πµ)κ(µ)Z u0 P−12+iµ(u0)(cid:18)dΩq~(cid:19) , (32)
1
u
0
where the angular part of the
ross-se
tionin the right part of integral is expressed through the variable . Finally
µ θ ϕ
we represent the di(cid:27)erential
ross-se
tionon through the di(cid:27)erential
ross-se
tionon s
atteringangle. Let and
~q
be axial and azimuth angles of the momentum . Let us turn to integrating over this angles in the expression (28).
f(~q)
For this we noti
e that for arbitrary fun
tion the following integral relation is right:
f(~q) d~q = q2dq dΩ f(~q)= q2dq dΩ f(~q ,ǫ q2 q2) ,
q~ ⊥ − ⊥ (33)
Z Z ǫX=±1Z q
1
dΩ=sinθdθdϕ , dΩ = dq~ .
q~ ⊥
q q2 q2
− ⊥
p
From this follows that:
dΩ f(~q)= dΩ f(~q ,ǫ q2 q2) ,
q~ ⊥ − ⊥ (34)
Z ǫX=±1Z q
6
θ ϕ
or in terms of and angles
1 2π
dΩ f(~q ,q =+ q2 q2)= dz dϕf(~q ,q =qz) ,
q~ ⊥ 3 − ⊥ ⊥ 3
Z q Z0 Z0
0 2π (35)
dΩ f(~q ,q = q2 q2)= dz dϕf(~q ,q =qz) ,
q~ ⊥ 3 − − ⊥ ⊥ 3
Z q −Z1 Z0
z =cosθ , f(~q ,q =qz) f(~q) .
⊥ 3
≡
So relations between variables are
+ q2 q2 ,
qz = − ⊥ forward half-sphere , q⊥ =q 1 z2 .
(−pq2−q⊥2 ba
kward half-sphere. p −
p
We turn to integrating over angles in the integral (28). We obtain:
1
dσ+ 1 dz 1 dσ
= µtanh(πµ)κ(µ) P ( ) ,
dµ 2π z −1/2+iµ z dz
Z
0
0 (36)
dσ− 1 dz 1 dσ
= µtanh(πµ)κ(µ) P ( ) .
dµ 2π z −1/2+iµ z dz
−Z1 | | | |
dσ C A+B C+D
where dz isthedi(cid:27)erential
ross-se
tiononthe
osineofs
atteringangleofparti
le inthe → pro
ess.
dσ± ϕ
Here is taken into a

ount that dΩq~ do not depend of variable , therefore
dσ± 1 dσ
= .
dΩ 2π dz
q~
Unifying relations (22) , ( 27) , ( 36) and taking into a

ount that:
1
κ(µ)ξ¯(~q ,~µ)dΩ =1 ,
(2π)2 ⊥ µ~
Z
σ(AB CD)
we obtain the norm of the di(cid:27)erential
ross-se
tionon the total
ross-se
tion →
∞ 1
dσ(ǫ) dσ N
dµ= dz = =σ(AB CD) .
dµ dz TVn n ~u → (37)
ǫX=±1Z0 (cid:18) (cid:19) −Z1 (cid:18) (cid:19) 1 2| |
dσ± dσ±
As follows from (27), in
ontrast to dΩq~ the di(cid:27)erential
ross-se
tion dΩµ~ is not positively sign determined on all
µ dσ±
interval. Contribution of negative value area of dΩµ~ redu
es e(cid:27)e
tively to de
reasing of the total event number of
C C
parti
le
reation. So, this spatial area we
an interpret as area where taking pla
e an absorption of parti
les .
µ
But the total number of asymptoti
states with de(cid:28)ne regulates by the relation (37).
dσ
APPLICATION TO SIMPLE MODELS dΩ IN C.M.F.
t A+B A+B
Asanexamplewedis
ussanone-parti
leex
hangein -
hannel,forelasti
→ s
attering. Corresponding
u
0

ross se
tion as fun
tion of has a polar view:
dσ α2 α2N 1 α2N u2
= N = 0 = 0 0 ,
dΩ 0(t M2)2 (2q2)2(z z)2 (2z q2)2(u ε )2 (38)
− 0− 0 0− z0
7
25 €€€€€€€€€€e€€x€€€p€€€@€€Π€€€€€b€€€€q€€€€€€D€€€€€€€€€€€€€€d€€€Σ€€€€€-€€€€
qΣ- HAB®CDL db
20
15
II
10
R
III 0 I

5
bq R1
0.5 1 1.5 2 2.5
-5

a. b.
b bq<1/2
F1/I2G<. 1b:qa.)D√is2tribution fun
tionon in theCmobdqe&l o√f2one parti
le ex
hange. C(cid:21) area forbidden byun
ertaintyr~belation,
(cid:21) area of
reation parti
les , (cid:21) area of absorption parti
les . b)The zone stru
ture of plain in the
C
model ofone parti
leex
hange. ZoneI(cid:21) area forbidden byun
ertaintyrelation, zone II (cid:21)area of
reation parti
les ,zone III
C
(cid:21) area of absorption parti
les .
z0 =1.001
here
2M2s M2
z = 1+ = 1+ ,
0 λ(s,m2,m2) 2q2
A B
M λ(x,y,z) = (x2+y2+z2 2xy 2xy 2yz α
(cid:21) massof ex
hange parti
le, − − − )(cid:21) wellknown the triangle fun
tion, (cid:21)
the
oupling
onstant,
q2λ˜(~q ) (s2 (m2 m2)2)2
N = (2π)2 ⊥ = (2π)2 − 1− 2 .
0 ~u 16s3
| |
σ±
In this
ase for the
ross-se
tion we obtain:
α2N 2π
σ± = 0 ,
(2q2)2z (z 1) (39)
0 0
∓
where
1
dσ dz , ε = 1
∞ dz
dσ(ǫ)  0
σ(ε) = dµ= R (cid:0) (cid:1) .
Z0 (cid:18) dµ (cid:19)  0 dσ dz , ε = 1 (40)
dz −
−1
 R (cid:0) (cid:1)
The integral(32) with su
h
rossse
tion is
omputed analyti
ally. The normalizedistribution on nearestapproa
h
parameter takes a view:
1 dσε q2btanh(πµ)κ(µ)z ε ε ε ε
= 0− P − + P1 − ,
σε db 2 cosh(µπ) z0 " iµ−1/2(cid:16)z0 (cid:17) z02−1 iµ−1/2(cid:16)z0 (cid:17)# (41)
p
ε = 1 P1 (x)
where ± , and iµ−1/2 (cid:21) asso
iated
ones fun
tion. The plot of (41) is shown on Fig.1a.
z >1 [ 1,1]
0
Sin
e sothe argumentof
onesfun
tions belong to segment − . The
onesfun
tionispositivewith su
h
m > 0 ε = 1
argument (and asso
iated
ones fun
tions with too). So the
ross-se
tion is positive de(cid:28)ned on . On
ε= 1 ~b R R R
0 1 2
− Rth2e=-~p2la/n4eq2divides into zones with radiuses , and Fig.1b.
Here 0 de(cid:28)nesaborderofforbidden area,where Geizenbergun
ertainty relationis broken(phasespa
e
C R < b < R C b > R
0 1 1
of parti
le is less then allowable). Area de(cid:28)nes spatial area of parti
le
reation. And area
is area where abIs(bo)rptio1n of partRi
les<isbt<akRen pla
e relating to equalibty>(R37). At that, if 1w0e−s5ymboli
ally take density
0 1 1
of distribution ∼ in area , when density in area would be ∼ . It is made
onditional
8
I(b) R 1/q I(b) exp( πbq)
upon that exponential de
rease of in the ba
kward sphere determines by radius ∼ i.e. ∼ −
bq 1
at ≫ . R M2
The left border of 2-zone is de(cid:28)ned by zero of expression (41). In the area of small values of parameter s the

ross-se
tion(41) turn into zero on
√7 2 2 2M2s M2
µ = µ = +2 ǫ+ ǫ2+O(ǫ3) , ǫ = .
0 2 √7 21√7 λ(s,m2,m2) ∼ s
1 2
M2 1
In the area s ≪ the
ross-se
tiontakes a view:
1 dσ− tanh(πµ)κ(µ) 7 4µ2 M2
= q2b − +O( ) ,
σ− db cosh(µπ) " 8 s # (42)
1 dσ+ tanh(πµ)κ(µ) cosh(µπ) M2
= q2b +O( ) .
σ+ db cosh(µπ) " π s # (43)
<b2>
CONNECTION BETWEEN AND THE CROSS-SECTION ON TRANSVERSE MOMENTUM OF
C A B
PARTICLE IN C.M.F OF AND PARTICLES
<b2 >
Let us
al
ulate the an averagevalue of nearest approa
hparameter square ± . We have by de(cid:28)nition:
1 ∞ dσ±
<µ2 > = µ2 dµ ,
± σ± dµ (44)
Z0
σ±
where was de(cid:28)ned in (40). So:
1 1
<b2 > = <µ2 >+ , q = q˜.
± q2 ± 4 (45)
(cid:16) (cid:17)
dσ±
Substituting the representation for dµ from (32), into (44) and taking into a

ount that (
ompare with (22) and
(29)):
∞ dx
κ(µ) = 2π P (x) ,
iµ−1/2 x
Z1
we obtain
2π 1 dσ±dxdu
<b2 > = (µ2+ )P (x)P (u ) 0dΩ .
± q˜2σ± 4 iµ−1/2 iµ−1/2 0 dΩ x u µ (46)
Z 0
P (u ) u
iµ−1/2 0 0
Thedi(cid:27)erentialequationon
onesfun
tions withargument
anberepresentedinfollowingform[5℄:
1 d2 d
(µ2+ )P (u ) = (u2 1) 2u P (u ) .
4 iµ−1/2 0 − 0− du2 − 0du iµ−1/2 0
0 0
(cid:16) (cid:17)
The left part of this expressionwent asunderintegral fa
torinto relation (46) and its substitution allowsus integrate
µ
over . From the
ompleteness relation of
ones fun
tions follow [2℄:
∞
P x P u dΩ = δ(x u ) .
iµ−1/2 iµ−1/2 0 µ 0
− (47)
Z0
(cid:0) (cid:1) (cid:0) (cid:1)
x
Next, we
an integrate overvariable . After some transformations we obtain:
2π 2 dσ±du
<b2 > = 0 .
± q˜2σ± Z u30 dΩ u0 (48)
9
u θ
0
Crossing in this relation from the variable to the s
attering angle , we obtain:
2 1 dσ+
cos2θ dcosθ , ε = 1
q2σ+ dcosθ
<b2ε > =  2 Z00cos2θ dσ− dcosθ , ε = 1 . (49)
q2σ− dcosθ −
Z−1

So, (cid:28)nally we have:
8 s
<b2 > = <cos2θ > .
± λ(s,m2,m2 ) ± (50)
C D
where
1 1 dσ
z2 dz , ε = 1
σ+ dz
<cos2θε > =  1 Z00 dσ , z = cosθ ,
 z2 dz , ε = 1
σ− dz −
Z−1
<b2 > 
and ± is de(cid:28)ned by relation (44).
It follows:
1
<µ2 > = 2<cos2θ > .
± ± −4 (51)
µ µ2 >0
The parameter is real number, so . From it and (51) follows an important physi
al inequality:
1
<cos2θ > > .
±
8 (52)
µ = b2q2/~2 1/4
~ Let us dis
uss the naturbe2of this inequality. It follows from the realitSyOo(f2p,a1r)ameter − , where
(cid:21) Plank
onstant. Here is an eigen valueb2oqf2K>az~im2/i4r's operator on -group[2℄. Sppe
trumµof this operator
in Hilbert spa
e of states satis(cid:28)es
ondition , and that is provide a reality of parameter . By itself this
C
inequality has an quantum nature and show us the fa
t that parti
le
an not
reates in phase spa
e less when it
allows by Geizenberg's un
ertainty relation.
<b2 > <cosθ2 > A+B C+D
We noti
e on the fa
t that relation (50) between ± and ± right for any pro
ess → in
A B

enter of mass frame and parti
les.
In the model with one-parti
le ex
hange (38) it is easy to obtain that
z 1
<cos2θ > = 2z2(z 1)ln 0∓ z +2z2 .
± ± 0 0∓ z ∓ 0 0
0
(cid:16) (cid:17)
So it follows:
1 z 1
<b2 > = 4z2(z 1)ln 0∓ 2z +4z2 .
± q2"± 0 0∓ z0 ∓ 0 0#
(cid:16) (cid:17)
z =1+M2/2q2
0
Let us analyzethis expressionsasfun
tions of parameter (relation (38)). The analyzeshows that
<cos2θ >
±

hanges in borders:
1
< <cos2θ > 6 1 ,
+
3
1
0.23 . <cos2θ > < ,
−
3
z 1
0
when
hanges in interval from to ∞.
10
~b
DIFFERENTIAL CROSS SECTION ON NEAREST APPROACH PARAMETER IN L.F.
B C
As itwasnotein the previousse
tion, in thelaborframeof parti
leenergyandmomentumof parti
le depend
q = q˜ µ
of s
attering angle (13). Consequen
e of this fa
t is appearan
e on the rea
tion plane as the parameter
b SO (2,1)
µ
dependen
e from the s
atteringangle, if we use relation (26) for the transitionto the parameter . algebra
µ
implements on basis fun
tions, for whi
h the parameter is natural variable. Completeness and orthogonality of
µ
basis aredσappeared in terms of it. If we take in relation (24) as an independent parameter, then di(cid:27)erential
ross
se
tion dµ is de(cid:28)ned unambiguously and doesn't depend of frame. But at the same time the physi
al meaning of
(b,q ) µ b
⊥
this parameter isn't
lear, only as phase spa
e measure . In
.m.f. transition in the integral (24) from to
µ b
does not
hange anything
onsiderably, and di(cid:27)erential distributions on and on are~bequivalent. But in l.f. the
di(cid:27)erentialdistribution
hangesradi
ally. Letus
rossin theintegral(24)tothevariable usingrelations(26). Than
we have:
1 dN(ǫ)
N = Re d~b dΩ ϑ[b R (q˜)]q2tanh(πµ) κ(µ) ξ¯(~q ,~µ) ,
(2π)2 q~ − 0 ⊥ dΩ (53)
ǫ=±1 Z Z q~
X
here
1 ~
µ=(b2q2 )1/2 , R (q˜)= , ϑ(x) , q =q˜,
0
− 4 2q˜ −dis
ontinuous fun
tion
q˜=p
C
and de(cid:28)nes in (13)
~b
From this it follows for the di(cid:27)erential
ross se
tion on :
dσ(ǫ) 1 dσ(ǫ)
= Re dΩ ϑ[b R (q˜)]q2th(πµ) κ(µ) ξ¯(~q ,~µ) .
d~b (2π)2 ǫ=±1Z q~ − 0 ⊥ dΩq~ (54)
X
~b θ ϕ ~q
⊥
Integrating left and right part over the dire
tion of ve
tor and
rossing to angle variables and of ve
tor we
obtain the result, whi
h is analogous to relations (36):
1
dσ+ b dz 1 dσ
= ϑ[b R (q˜)]q2tanh(πµ) κ(µ) P ( ) ,
db 2π z − 0 −1/2+iµ z dz
Z
0
0 (55)
dσ− b dz 1 dσ
= ϑ[b R (q˜)]q2tanh(πµ) κ(µ) P ( ) .
db 2π z − 0 −1/2+iµ z dz
−Z1 | | | |
b
From
omparison of (55) and (36) it follows that di(cid:27)eren
es between distributions on the parameter in
.m.f. and
C s t
in l.f. generates by di(cid:27)eren
e of the dependen
e of parti
le momentum from kinemati
variables and for this
q˜= p
C
two
ases. From the expli
it expression (36) it follows that it is a (cid:29)uent fun
tion of angle and with
ertain
q˜
a

ura
yit ispossible
rossfrom to somemidvalue in integrals(55). Forexample, we re
eivethat on the ba
kward
semisphere:
q˜(z) q˜(z¯) , z =cos(θ) , z¯= 0.5 ,
≃ −
q˜(z) q˜(z¯)
and it isn't depend on angle. Then the distribution (55)
oin
ides with the distribution (36) at ≃ . More
C
detailed stru
ture of parti
le
reation area re(cid:29)e
ts in a generalized distribution fun
tion:
d2σ b 1 1 dσ
= ϑ[b R (q˜)]q2tanh(πµ) κ(µ) P ( ) , 1<z <1 ,
db dz 2π · − 0 z −1/2+iµ z dz − (56)
| | | |
1
q =q˜=p (z) , µ=(b2q2 )1/2 .
C
− 4
z dσ b
It gives us a tomographi
pi
ture of partial integrals over to the spatial distribution db in the whole interval of .
q
⊥
It is similar to Wigner fun
tion [6℄ From it we
an get expressions for the di(cid:27)erential
ross se
tion on (11), for
b
di(cid:27)erential
ross se
tion on (36, 55) and norm relations (37).