Spatial image of reaction area from scattering I: SO_mu(2.1) algebra formalism

0.25 MB·English

by  M. V. Polyakov

Spatial image of rea
tion area from s
attering. I:
SO (2.1)
µ
algebra formalism.
∗
M. V. Polyakov
Institut fu(cid:4)r Theoretis
he Physik II Ruhr-Universitaet Bo
hum, NB6 D-44780 Bo
hum, Germany
†
O.N. Soldatenko, A.N.Vall
Department of Theoreti
al Physi
s, Irkutsk State University, Irkutsk, 664003 Russia
‡
A.A.Vladimirov
Bogoliubov Laboratory of Theoreti
al Physi
s, JINR, 141980, Mos
ow Region, Dubna, Russia
8 We develop general formalism of how to relate s
attering amplitudes for ex
lusive pro
esses to
0
spatialimageoftargethadron. Morepre
iselyweshowhowtodeterminethespatialdistributionof
0
outgoingparti
lesinthespa
eofso-
allednearestapproa
hparameter. Thisparameter
hara
terizes
2
theposition in the
oordinate spa
e thepla
e where the(cid:28)nal parti
le is produ
ed.

n
a PACSnumbers:
J

2
1 INTRODUCTION

]
h Famous experiment by Hofstadter et al. [1℄ demonstrated that hadrons are not point-like parti
les
p
- and they have non-trivial spatial stru
ture. Re
ently interest to the spatial images of hadrons was
p
revived with development of the theory of hard ex
lusive pro
esses, whi
h
an be systemati
ally
e
h studied using formalism of Generalized Parton Distributions (GPDs) (for reviews see [2℄). It was
[
demonstrated [3℄ that GPDs provide an information about spatial distribution of partons in the

3 transverse plane.
v
7 In the present paper we develop general formalism of how to relate s
attering amplitudes for
5 ex
lusive pro
esses to spatial image of target hadron. More pre
isely we show how to determine
8
the spatial distribution of outgoing parti
les in the spa
e of so-
alled nearest approa
h parameter.
2
. This parameter
hara
terizes the position in the
oordinate spa
e the pla
e where the (cid:28)nal parti
le
8
0 is produ
ed.
7
One-parti
le states of a free relativisti
parti
le are de(cid:28)ned by the 10-parametri
group of motion
0 M P
µν ν
: of Minkowski spa
e, whi
h is given by generators of the Poin
are group: and . States
an be
v
i spe
i(cid:28)ed by
hoosing one or another subgroup. It was found that for des
ription of deep inelasti
X
pro
esses it is useful to
onsider the "in(cid:28)nite momentum frame" [4, 5℄. We remind brie(cid:29)y ne
essary
r
a for this paper ingredients.
The three following generators of the Poin
are group:
1 1
B = (K +J ) , B = (K J ) , J ,
1 1 2 2 2 1 3
√2 √2 − (1)
with
K = M , ǫ J = M ,
i i0 ijk k ij
2
satisfy the system of
ommutation relations:
[B ,B ] = 0 ,
1 2
[J ,B ] = iB ,
3 1 2
[J ,B ] = iB ,
3 2 1
−
[B2 +B2,J ] = 0 , (2)
1 2 3
[p ,B ] = iδ p+ ,
i j ij
[p+,B ] = [p+,J ] = 0 .
i 3
p p+ = 1 (p + p )
Here i is the operator of 3-momentum, √2 0 3 . The above
ommutation relations
B B J E(2)
1 2 3
imply that generators , and form the algebra. Casimir operator of this algebra is
B2 = B2+B2
1 2
. Be
ause of the last
ommutation relation, this algebra
an be realized in the three-
p+ = const ~λ,p+
dimension momentum spa
e restri
ted by the
ondition on states | i. These states
satisfy equations:
B2 ~λ,p+ = λ2 ~λ,p+ ,
| i | i
B ~λ,p+ = λ ~λ,p+ ,
1 1
| i | i
B ~λ,p+ = λ ~λ,p+ , (3)
2 2
| i | i
λ2 = λ2 +λ2 .
1 2
B B B2 p+ = const
1 2
Operators , , on the surfa
e have
ompli
ated expression. However in the
p /p 1
limiting
ase of ⊥ ≪ the
orresponding operators are given by simple expression:
∂ ∂
B = ip+ , B = ip+ .
1 2
− ∂p − ∂p (4)
1 2
Substituting these operators into equations (3), we obtain the following expression for the
orre-
sponding wave fun
tion in momentum representation:
~
p~ λ
p~ ,p+ ~λ,p+ = exp i ⊥ · .
h ⊥ | i p+ ! (5)
Now if we introdu
e dimensional parameter
~
λ
~
b = ,
p+
~
b
than the wave fun
tion (5) has the form of the eikonal kernel in the plane of impa
t parameter .
~λ,p+
It allows us to interpret in the in(cid:28)nite momentum frame the eigenstates of the algebra (2) | i,
as the states with (cid:28)xed spatial position in the transverse plane [6, 7℄. In the in(cid:28)nite momentum
frame the transformation from the transverse momentum to the transverse position representation is
redu
ed to simple two-dimensional Fourier transformation (see eq. (5)). That is the transformation
whi
h relates the GPDs to the parton distributions in the spatial transverse plane [3℄. Note that
su
h interpretation is possible only in the in(cid:28)nite momentum spa
e, in order to ensure the
ondition
p /p 1
t/⊥Q2 ≪1 . Fotr the hard ex
lusive pro
esses the latterQ
2ondition
orresponds to the
ondition
≪ , here is the 4-momentum transfer squared and is the hard s
ale for the pro
ess. First,
t/Q2
in experiments the ratio
an be not be not very small and
orre
tions to the eikonal pi
ture
3

an be sizeable. Se
ondly, the simple two-dimensional Fourier transform
an not be used to obtain
spatial distributions of partons for hard pro
esses with large transverse momentum transfer (like
hadron form fa
tors, wide angle Compton s
attering, et
. [8℄) Therefore our aim here is to develop
general formalism whi
h would allow "spatial imaging" of hadrons in arbitrary referen
e frame.
We
onsider another subalgebra of Poin
are group. This is subalgebra of operators:
1
d = M p .
i p2 ij j (6)
p2 = const
It is realized on the surfa
e . Below we analyze in detail this algebra and use it for
des
ription and building one parti
le states with de(cid:28)ned quantum numbers.
OPERATOR OF THE NEAREST APPROACH
r
q

q
r r
x(t) n
A

r
r
r d
p d ^
A
r
O n Z
A

X~(t) ~q
FIG. 1: A
lassi
al traje
tory of an asymd~ptoti
ally free parti
le with momentum is pro
eeded into rea
tion area, is

hara
terized bya ve
torof maximalapproa
h tothesele
t point"O". Inthelabor framethis pointisa
enter oftarget, in
the
enter mass frame it is a meetingpointof beams.
Let us
onsider a traje
tory of a free spinless parti
le, whi
h moves witharbitrary initial
onditions
(Fig.1) [9℄:
q
i
X (t) = (t t )+ξ , i = 1,2,3 ,
i 0 i
~q2 +m2 − (7)
~q pm
with - momentum of the parti
le , - mass of the parti
le. A distan
e between the parti
le and
the
oordinate origin "O" is
D(t) = (X~(t)X~(t))12 .
t = τ D(τ) τ
At some moment of time the distan
e has a minimal value. The time is de(cid:28)ned by
the extremum
ondition:
d 1 d
D(t) = X (t) X (t) = 0 .
i i
dt t=τ D(t) dt t=τ (8)
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
Using (7) , we obtain:
~
(~uξ)
τ = t ,
0 − u2
q
i (9)
u = .
i
~q2 +m2
t = τ D(t) p
Substituting into , we get the following expression for
omponents of the ve
tor of the
d
i
nearest approa
h :
u
i ~
d = X (τ) = (~uξ)+ξ .
i i −u2 i (10)
L , k = 1,2,3
k
Let us introdu
e the operator of orbital momentum , in usual way:
L = ε ξ q .
k klm l m
(11)
Then the expression (10)
an be rewritten as:
1 1
d = ε q L = M q .
i q2 ijk j k q2 ij j (12)
X~(t)
We see that for ea
h
lassi
al traje
tory of an asymptoti
ally free parti
le, whi
h moves out
~
~q d
from the rea
tion area with momentum , we
an (cid:28)nd the ve
tor of the nearest approa
h of the
traje
tory to the point "O". We
an interpret this ve
tor as an e(cid:27)e
tive
oordinate of the parti
le

reation region. ~
d
Note that for the pro
ess reversed in time, the ve
tor
orresponds to the impa
t parameter.
The impa
t parameter is useful for geometri
al interpretation of s
attering whi
h is based on the
eikonal approximation for the s
attering amplitude. In the eikonal approximation the states with
de(cid:28)nite impa
t parameter of the initial parti
le are de(cid:28)ned as semi-
lassi
al approximation of states
with de(cid:28)nite orbital momentum. The impa
t parameter representation is a good toll for studying
di(cid:27)ra
tive pro
esses, see e.g [10℄. However for the small value of impa
t parameter semi-
lassi
al
approa
h is not valid. It is espe
ially
learly seen from the un
ertainty prin
iple: the states with
(cid:28)nitemomentumandin(cid:28)nitesimalimpa
tparameterareforbidden. So,therearesigni(cid:28)
antquantum
S
e(cid:27)e
ts in the region of small impa
t parameters. Also a
al
ulation of matrix elements of -matrix
needs
onsist~ent quantum-me
hani
al de(cid:28)nition of states. In this
ase it is a state with a de(cid:28)ne
d
value of the .
QUANTUM DESCRIPTION OF THE NEAREST APPROACH PARAMETER.
Using a standard quantization pro
edure based on the
anoni
al
ommutation relations:
[ξ q ] = ıδ ,
i j ij
5
we get a system of
ommutation relations:
1
d = (ε q L iq ) ,
i q2 ijk j k − i
d = (d )+ , [d q2] = 0 ,
i i i
[d L ] = iε d ,
i j ijk k
−
[L L ] = iε L ,
i j ijk k
i
[d d ] = − ε L ,
i j q2 ijk k (13)
i
[d q ] = iδ q q ,
i j ij − q2 i j
2i
[d ,q q ] = iδ q +iδ q q q q ,
i j k ij k ik j − q2 i j k
....................
et
.
d L SO(3,1) q2 = const
i j
We see that operators and form algebra of group on the sphere . Corre-
sponding Casimir operator of this algebra be
omes a number:
1 1
Cˆ = d2 L2 .
− q2 ≡ q2
d , d , L
1 2 3
Operators form a nontrivial subalgebra:
i
[d d ] = L ,
1 2 −q2 3
[d1L3] = id2 , (14)
−
[d L ] = id .
2 3 1
SO(2,1)
This is algebra of group. Its properties are investigated and presented in details in mono-
graph [11℄. This group is non
ompa
t and it has three series of unitary representations in spa
e
of quadrati
ally integrated fun
tions: general, non
ompa
t and dis
reet. Casimir operator of this
algebra is:
1
Kˆ = d2 L2 , d2 = d2 +d2 , [d Kˆ] = 0 , [d Kˆ] = 0 .
⊥ − q2 3 ⊥ 1 2 1,2 3 6 (15)
Kˆ
We interpret the eigenvalue of this operator in the
ontinuous spe
trum as a quantum-me
hani
generalization of a squared e(cid:27)e
tive distan
e of the
reated parti
le, as it was de(cid:28)ned above. Here
z
we note, that relations (14) suppose a de(cid:28)nite
hoi
e of the dire
tion. We link this dire
tion with
p~
A
dire
tion of the momentum of proje
tile parti
le .
A realizationof the algebra (14) is
onne
ted
losely to the geometri
alproperties of the transverse
momentum spa
e - the spa
e on whi
h this algebra is realized. Let us
onsider unitary transforma-
d d
1 2
tions of the momentum spa
e generated by the generators and [12℄:
q = e ip~d~q eip~d~ , i = 1,2 ,
i′ − · i ·
~p = p~n , ~n = (cosϕ , sinϕ) , ~n d~= n d +n d , < p < + . (16)
1 1 2 2
· −∞ ∞
6
p q
If we di(cid:27)erentiate this relations in respe
t to , we get the following Lee equation for i′:
dq′ ~n ~q′
i = n · q′ , i = 1,2 ,
dp i − q2⊥ i
(17)
′
q (p = 0) = q .
i i
This equation
an be easily integrated:
q + qsinh(p)+~n ~q cosh(p) ~n ~q n
q′ = i q · ⊥ q − · ⊥ i , i = 1,2 .
i (cid:16) cosh(p)+ (~n·q~⊥) sinh(p) (cid:17) (18)
q q q
~κ ~p
It is
onvenient to introdu
e new parameter related to the parameter by:
p
~κ = q~n tanh ,
q (19)
(cid:18) (cid:19)
then we have:
(~κ~q)(1 B) (~κ~q) 1
q = 1+ − κ +Bq 1+ − ~q ~κ ,
i′ κ2 i i q2 ≡ ⊕ (20)
(cid:16) (cid:17)(cid:16) (cid:17)
(cid:2) (cid:3)
with
κ2
B = 1 .
s − q2
SO(2,1)
We
an see from (20) , that the motion of the transverse momentum spa
e is the nonlinear
transformation. The operation ⊕ is not a group operation. Be
ause of a produ
t of two su
h
operations
ontains not only the resulting operation ⊕, but also a rotation. But the operation ⊕
ds
has a set of group properties. In parti
ular, transformations (20) posses the invariant interval :
ds2 = G dq dq ,
ik i k
q2 q q
i k
G = δ + , i,k = 1,2 ,
ik q2 ik q2 (21)
3 (cid:18) 3 (cid:19)
q2 = q2 q2 .
3 −
⊥
G
ik
Ri

i tensor
orresponding to the metri
tensor has the form:
1 q q
i k
R = δ + .
ik −q2 ik q2 (22)
3 (cid:18) 3 (cid:19)
The
orresponding s
alar
urvature is:
2
R = .
−q2 (23)
~q
We see that the transver~κse momentum p~κla2ne wqh2i
h is obtaine by transformations (20) from (cid:28)xed ⊥
by
hanging parameter in the region | | ≤ , is the spa
e with
onstant negative
urvature.
q q > 0
3 3
Let us now show that transformations (20) does not
hange the sign of , i.e. forward ( )
q < 0
3
and ba
kward ( ) semi-spheres are invariant subspa
es. Applying the transformation (16) to
q
3
we get another Lee equation:
dq′ ~n ~q′
3 = · q′ , q′(p = 0) = q .
dp − q2⊥ 3 3 3 (24)
7
Integrating it we obtain:
~n ~q′
′
q = q exp · dp .
3 3 − q2⊥ (25)
(cid:26) Z (cid:27)
On the other hand, from (18) we
an
on
lude that:
qsinh(p)+~n ~q cosh(p)
~n ~q′ = q · ⊥ q .
· ⊥ cosh(p)+ ~n·q~⊥ sinh(p) (26)
q q q
Substituting this expression to (25) and integrating it we arrive at:
1
′ ~κ ~q −1 κ2 2
q3 = q3 1+ q·2⊥ 1− q2 . (27)
(cid:18) (cid:19) (cid:18) (cid:19)
Combining this expression and relation (20), (cid:28)nally we get:
q′ q q
3 = 3 = 3 = .
q2 q′2 q2 q2 q3 inv (28)
− − | |
⊥ ⊥
p p
The
onsequen
e of this invarian
e is appearan
e of the signature in amplitudes (
ross-se
tions)
C
whi
h
orresponds to
reating of the parti
le to the forward or ba
kward semi-sphere.
At the end of this se
tion we note that nonlinear transformations (20)
an be linearized in the
u ,u ,u
0 1 2
three-dimension spa
e on hyperboloid. Indeed, if we introdu
e new variables :
q ~q
u = (u0,u1,u2) , u0 = q , ~u = q⊥ , u2 = u20 −u21 −u22 = 1 , (29)
3 3
than the transformations (20)
an be written in these variables as:
′
u = Λ(~κ)u ,
(30)
Λ(~κ)
with the transformation matrix :
κ κ
1 2
1
q q
 
1 κ 1 B 1 B
Λ(~κ) =  1 B + − κ2 − κ κ 
B  q κ2 1 κ2 1 2  (31)
 
 
 κ 1 B 1 B 
 2 − κ κ B + − κ2 
 q κ2 1 2 κ2 2 
 
 
1
κ2 2
Det(Λ(~κ)) = 1 , B = 1 .
− q2
(cid:18) (cid:19)
SO (2,1)
µ
REPRESENTATION OF ALGEBRA IN THE TRANSVERSE MOMENTUM SPACE
Letus obtaingroup basi
fun
tions, whi
h realizestates witha de(cid:28)ne valueof theCasimiroperator
L
3
(15) and de(cid:28)nite value of the third proje
tion of orbital momentum . For that we introdu
e
spheri
al
oordinates in the momentum spa
e:
q = qsinθcosϕ ,
1
8
q = qsinθsinϕ ,
2
q = qcosθ .
3
d , L
i 3
Using an expli
it expression for operators , we obtain:
i ∂ sinϕ ∂
d = cosϕcosθ sinθcosϕ ,
1
q ∂θ − sinθ ∂ϕ − (32)
(cid:16) (cid:17)
i ∂ cosϕ ∂
d = sinϕcosθ + sinθsinϕ ,
2
q ∂θ sinθ ∂ϕ −
(cid:16) (cid:17)
∂
L = i .
3
− ∂ϕ
The Casimir operator is:
1 ∂2 cosθ ∂ cos2θ ∂2
Kˆ = − cos2θ + 3cosθsinθ + 2cos2θ .
q2 ∂θ2 sinθ − ∂θ sin2θ ∂ϕ2 − (33)
!
(cid:16) (cid:17)
Let us
onsider a system of equations:
KˆΨ = b2Ψ , L Ψ = mΨ .
3
(34)
m = 0
To make analysis simpler we
onsider (cid:28)rst spe
ial
ase of . For this
ase we have only one
ordinary di(cid:27)erential equation:
d2f (1 x2)2 df
(1 x2)2 + − +4q2b2f = 0 ,
− dx2 x dx (35)
with
1 θ
Ψ(θ) = f(θ), x = tan .
cosθ 2 (36)
θ = π/2
This equation
an be redu
ed to the Riemann equation. The point is the singular point of
0 θ < π/2
equation (35). Regular and
ontinuous solution for the region ≤ is the so
alled
ones
fun
tion:
1 1
Ψ(θ) = P , 0 θ < π/2
1/2+iµ
cosθ − cosθ ≤
(cid:18) (cid:19)
(37)
µ = q2b2 1/4 1/2 , q2b2 1/4 ,
− ≥
P (z) (cid:0) (cid:1)
ν
with (cid:21) the Legendre fun
tions.
π/2 θ < π
To (cid:28)nd the solution in the ba
kward semi-sphere ≤ , we note that equation (35) is
x 1/x cosθ cosθ
invariant under the transformation → , whi
h
orresponds to transformation → − .
π < θ π
So, regular and
ontinuous solution in the region 2 ≤ has the following form:
1 1 π
Ψ(θ) = P , < θ π .
1/2+iµ
−cosθ − −cosθ 2 ≤ (38)
(cid:18) (cid:19)
Finally,
ombining (37) and (38) we obtain:
q q
Ψ(~q ) = P , q = qsinθ ,0 θ π .
1/2+iµ
⊥ q2 q2 − q2 q2 ! ⊥ ≤ ≤ (39)
− −
⊥ ⊥
p p
9
Kˆ
There is an important physi
al
onstraint on the spe
trum of the operator :
1
b2 .
≥ 4q2 (40)
This
onstraint follows from the Heisenberg un
ertainty prin
iple (cid:21) a parti
le with (cid:28)xed (cid:28)xed energy

an not be
reated in the region of the
oordinate transverse plane of arbitrary small radius.
m = 0
The generalization for
ase 6 is simple, the solution of Eqs. (34) has the form:
1 1
Ψ(θ,ϕ) = P|m| eimϕ , 0 θ π , 0 ϕ 2π ,
cosθ 1/2+iµ cosθ ≤ ≤ ≤ ≤
−
| | (cid:18)| |(cid:19) (41)
µ = q2b2 1/4 1/2 , q2b2 1/4 , m = 0, 1, 2,... .
− ≥ ± ±
(cid:0) (cid:1)
The set of fun
tions (41) forms
omplete and normalizable basis. The de
omposition of fun
tions
m = 0
in this basis is known as Fo
k-Meller expansion and for
ase it was investigated by V.A.Fo
k
[13℄.
SO (2,1)
µ
For the s
attering problem, we are interested in, only the plane wave on group are
relevant. These plane waves
orrespond to the main series of irredu
ible representations on the
group fun
tion spa
e, this series is generated by ⊕ operation (20). The situation is similar to a
SO(3,1)
problem of building of a relativisti

on(cid:28)gurational spa
e on Lorentz group . In our
ase the
role of
oordinate operators isplayed by motiongenerators inthe momentumspa
e. The momentum
E2 q2 = m2
spa
e is represented on positive part of the two-sheet hyperboloid − . This problem was
solved in Refs. [14, 15℄. It was shown, that the relativisti

on(cid:28)gurational spa
e
onjugate to the
momentum spa
e in Fourie-transformation mean. In our
ase the role of the unitary transformation
x q
kernel from to -spa
e is played by the plain waves on hyperboloid, so
alled Shapiro's fun
tions
SO(3,1)
[16℄. These fun
tions implements the main series of unitary representation group.
ψ(~q )
We do not gi~ve detailed
al
ulations, we just dis
uss main step. Let the state ⊥ diagonalizes
(~nd ) ~n ~p
A
an operator ⊥ , with is a ve
tor perpendi
ular to the momentum :
~
(~nd ) ψ(~q ) = ψ(~q ) .
⊥ ⊥
onst ⊥ (42)
Now, using the transformation (20) we obtain that fun
tion
q2
f(~q ) = 1 ψ(~q )
⊥
⊥ s − q2 ⊥ (43)
satis(cid:28)es the following fun
tional equation:
f(~q ~κ) = f(~q ) f(~κ) .
⊥ ⊕ ⊥ · (44)
~
exp i~q b
Usual plain waves ⊥ ·
orrespond to translations of the Eu
lidean plain for the
ase when
(cid:16) (cid:17)
⊕ operation is just a sum operation.
~κ
Obviously, the relation (44) is not satis(cid:28)ed for arbitrary , but only for lines of a
oordinate
κ = κ (t)
i i
grid in the momentum spa
e. Let be a parametri
representation of those lines. We
an
assume without loss of generality that:
dκ
κ (t = 0) = 0 , i = n , n2 = 1 , i = 1,2 .
i i
dt (45)
(cid:18) (cid:19)t=0
10
κ = 0
i
Expanding the equation (44) in the vi
inity of the point and using that:
dq′ q q
i = δ i k ,
ik
dκ − q2 (46)
(cid:18) k(cid:19)~κ=0
f(~q )
we obtain the following equation for ⊥ :
∂f
(δ x x )n = 2iβf ,
ik i k k
− ∂x (47)
i
with
q ∂f 1
i
x = , = iα , β = (α~ ~n) , i,k = 1,2 .
i k
q x − 2 ·
(cid:18) k (cid:19)~x=0
It is a di(cid:27)erential equation in partial derivative of the (cid:28)rst order. Its solution is determined up to
ζ
arbitrary fun
tion of the s
alar de(cid:28)ned as:
1 x2 ~n ~q
ζ = − , z = · ⊥ .
(1 z)2 q
−
β
Tψ(h~qe
)onstant is arbitrary as well. This arbitrariness isKˆ(cid:28)xed by the
ondition that the fun
tion
⊥ must be the eigenfun
tion of the Casimir operator :
Kˆψ(~q ) = b2ψ(~q ) .
⊥ ⊥ (48)
f
Eventually this gives the
orresponding equation for the fun
tion :
∂2f ∂f b2q2
(x x δ ) +2~x f = 0 .
i k ik
− ∂x ∂x · ∂~x − 1 x2 (49)
i k
−
The system of equations (47) and (49) is equivalent to the system (42) and (48). Its solution gives
us the basis fun
tions of plane waves type. Let us introdu
e the notation:
ψ(~q ) ξ(q~ ,~µ) ,
⊥ ≡ ⊥
~n ~µ = ~nµ µ
with - the two-dimension elementary ve
tor, introdu
ed in equation (42), . was de(cid:28)ned
in (37). The (cid:28)nal result is the following:
1
+iµ
q q ~n q~ −2
ξ(q~ ,~µ) = − · ⊥
⊥ q2 q2 q2 q2 !
− −
⊥ ⊥
1
= up0(u n)−2+ipµ ; (u n) = u0 ~u ~n
· · − ·
1
+iµ
−2 (50)
= u u u2 1cos(ϕ θ) ,
0 0 − 0 − −
(cid:20) q (cid:21)
q~ = (q cosϕ , q sinϕ) , ~n = (cosθ,sinθ) ,
⊥ ⊥ ⊥
n = (1,~n) , n2 = n2 ~n 2 = 0 ,
0 −
n
with - a three dimensional light-like ve
tor on hyperboloid.
f ψ
From the relation (43) between and it follows that:
1
f f(q~ ,~µ) = (u n)−2+iµ .
≡ ⊥ · (51)