Finiteness of multi-loop superstring amplitudes

16 Pages·0.18 MB·English

by  G. S. Danilov

Finiteness of multi-loop superstring amplitudes
G.S. Danilov∗
8
Petersburg Nuclear Physics Institute
9
9
1
Gatchina, St.-Petersburg 188350, Russia

n
a
J

5

1 Abstract
v
3
1 Superstring amplitudes of an arbitrary genus are calculated through super-Schottky
0
1
parameters by a summation over thefermion strings. For acalculation of divergent multi-
0
8
9 loop fermion stringamplitudes asupermodularinvariant regularization procedureis used.
/
h
A cancellation of divergences in the superstring amplitudes is established. Grassmann
t
-
p
variables are integrated, the superstring amplitudes are obtained to be explicitly finite
e
h
: and modular invariant.
v
i
X
r
a
∗
E-mail address: [email protected]
0
1 Introduction
During years, a great deal of efforts in superstring theory [1] has been invested [2—10] in a
construction of a perturbation series for interaction amplitudes. Especially, difficulties arose in
acalculationofpartitionfunctionsandoffieldvacuumcorrelatorsforRamondstringswherethe
desired values can not be derived by an obvious extension of boson string results [11]. Finally,
by a method given in [7, 9] the partition functions and the superfield vacuum correlators were
calculated [9, 10] in terms of super-Schottky parameters for all the fermion strings. Multi-
loop superstring amplitudes could be obtained by a summation of the fermion strings, but
every fermion string amplitude is divergent. Though the divergences are expected [2, 4, 12]
to be canceled in the full superstring amplitude, up to now they kept the desired superstring
amplitudes from being calculated. In this paper a supermodular invariant calculation [13] of
divergent fermion string amplitudes is proposed. So, superstring amplitudes calculated by a
summation over the fermion string ones are surely invariant under supermodular group. We
establish a cancellation of divergences in the above superstring amplitudes. Moreover, on
integration over Grassmann moduli we obtain expressions for the superstring amplitudes that
are explicitly finite and supermodular invariant. Details of this construction are need yet to be
clarify, but the paper mainly completes a building of the superstring perturbation series that,
in turn, opens opportunities for a wide investigation of superstrings. A topic of an essential
interest that can be advanced in the next future is a summation of the superstring perturbation
series in an infrared energy region of interacted states. Amplitudes of genus going to infinity
dominate in this case, the discussed infrared asymptotics are expected to be quite different
from each of multi-loop amplitudes taken in the infrared limit. Perhaps, they can be applied
to particle interactions below the Plank mass. Another goal could be a summation of the
superstring perturbation series for massless string states provided that both a number thereof
and their energies tend to infinity. It might be applied to a creating of the Universe.
1
As it is usually, a genus-n closed superstring amplitude A with m legs is given by
n,m
m
′ (n) (n,m)
A = dq dq dt dt Z ({q ,q })E ({t ,t },{q ,q };{p }) (1)
n,m N N r r L,L′ N N L,L′ r r N N j
Z N r=1 L,L′
Y Y X
where {p } are particle momenta, the overline denotes the complex conjugation and L (L′)
j
labels superspin structures of right (left) superfields defined on the complex (1|1) superman-
ifolds [14]. Every superspin structure L = (l ,l ) presents a superconformal extension of the
1 2
(l ,l ) = n (l ,l ) ordinary spin one [15]. The genus-n theta function characteristics (l ,l )
1 2 s=1 1s 2s 1 2
S
can be restricted by l ∈ (0,1/2). The prime denotes a product over those (3n − 3|2n − 2)
is
super-Schottky parameters {q } that are chosen as moduli, (3|2) thereof being fixed com-
N
mon to all the genus-n supermanifolds by a super-M¨obius transformation. Partition functions
(n)
Z ({q ,q }) are calculated from equations [7, 9] expressing that the superstring amplitudes
L,L′ N N
are independent of a choice of two-dim. metrics and of a gravitino field. The vacuum expec-
(n,m)
tation E ({t ,t },{q ,q };{p }) of the vertex product is integrated over supermanifods
L,L′ r r N N j
t = (z |θ ) where z is a local complex coordinate and θ is its odd partner. Moduli are
r r r r r
integrated over the fundamental domain [16]. Among other things, this domain depends on
L through terms proportional to Grassmann super-Schottky parameters since, generally, the
supermodular changes of moduli and of supercoordinates depend on superspin structure [16].
To calculate A a regularization procedure is necessary because every term in (1) is diver-
n,m
gent [17]duetodegeneration ofRiemannsurfaces. Ifacutoff[18]ofmodularintegrals isused, it
is necessary yet to verify the supermodular invariance of the calculated amplitudes because the
cutoff [18] violates the supermodular group. So we use a supermodular invariant regularization
procedure given in this paper. Also, we regularize the integrals over z that are ill defined too.
j
A construction of supermodular covariant functions needed for a regularization of the modular
integrals is complicated by a dependence on the superspin structure of supermodular changes of
super-Schottky parameters [16]. As example, the sum over (L,L′) in (1) calculated with {q }
N
common to all the superspin structures is non-covariant under the supermodular group though
each of the (L,L′) terms is covariant [16] under the group considered. Hence we perform a
singular transformation [19] to a new parameterization P where transition groups are split,
split
2
the supermodular group being reduced to the ordinary modular one. While the superstring in
not invariant under the above transformation, it is useful to integrate over Grassmann variables
because in this case the integration region is independent of the above Grassmann ones.
The P parameterization is considered in Sec.2. In Sec.3 regularized expressions for
split
superstring amplitudes are given. It is argued that amplitudes of an emission of a longitudinal
polarized gauge boson vanish in our scheme as it is required by the gauge symmetry. In Sec.4
the cancellation of divergences in superstring amplitudes and non-renormalization theorems
[4] are verified. On integration over the P Grassmann variables the expressions for the
split
amplitudes are derived that are free from divergences and supermodular invariant as well.
2 Regularization of modular integrals
As it was mentioned above, we perform a singular t → tˆ = (zˆ|θˆ) superholomorphic trans-
formation [13, 19] to a P parameterization where transition groups contain no Grassmann
split
parameters:
1
z = f (zˆ)+f′(zˆ)θˆξ (zˆ), θ = f′(zˆ) 1+ ξ (zˆ)ξ′ (zˆ) θˆ+ξ (zˆ) , f (zˆ) = zˆ+y (zˆ). (2)
L L L L 2 L L L L L
q (cid:20)(cid:18) (cid:19) (cid:21)
Here the ”prime” symbolizes zˆ-derivative, ξ (zˆ) is a Grassmann function and y (zˆ) is propor-
L L
tional to Grassmann modular parameters. On the t supermanifold the rounds about (A ,B )-
s s
cycles are associated with super-Schottky transformations (Γ (l ),Γ (l )) where every A -
a,s 1s b,s 2s s
cycle is a Schottky circle. In this case [8, 9] Γ (l = 0) = I, Γ2 (l = 1/2) = I, but
a,s 1s a,s 1s
Γ (l = 1/2) 6= I. So a square root cut on the z-plane appears for every l 6= 0 with endcut
a,s 1s 1s
points to be inside corresponding Schottky circles. For a handle s, the super-Schottky trans-
formation is determined by a multiplier k and two unmoved points (u |µ ) and (v |ν ) where
s s s s s
µ and ν are Grassmann partners of u and, respectively, of v . In the P parameterization
s s s s split
the same (A ,B ) rounds are associated with transformations (Γˆ (l ),Γˆ (l )). Hence
s s a,s 1s b,s 2s
Γ (l )(t) = t Γˆ (l )(tˆ) , Γ (l )(t) = t(s) Γˆ (l )(tˆ) (3)
b,s 2s b,s 2s a,s 1s a,s 1s
(cid:16) (cid:17) (cid:16) (cid:17)
3
where t(s)(tˆ) is obtained by 2π-twist of t(tˆ) on the complex zˆ-plane about the Schottky circle
ˆ
assigned to a particular handle s. In this case Γ (l ) may only give a sign of fermion fields
a,s 1s
ˆ ˆ
and Γ (l ) is a Schottky transformation with a multiplier k and two unmoved local points
b,s 2s s
uˆ and vˆ . So both Γˆ (l ) and Γˆ (l ) do not contain Grassmann modular parameters. Since
s s a,s 1s b,s 2s
every super-Schottky group depends, among other things, on (2n−2) Grassmann moduli, the
(1) (2)
transitionfunctionsin(2)necessary dependon(2n−2) Grassmannparameters(λ ,λ )where
j j
j = 1...n−1. The equations similar to (3) were already used [16] in a calculation of the acting
of supermodular transformations on supercoordinates and on modular parameters. Unlike [16],
eqs.(3) are satisfied only if the transition functions in (2) have poles in a fundamental region of
zˆ-plane,singularpartsbeingproportionaltoGrassmannparameters. Wetake1 thempossessing
(n−1) poles zˆ of an order 2. For even superspin structures we choose the above poles among n
j
zeros of the fermion Green function Rf(zˆ,zˆ ) calculated for zero Grassmann moduli.2 For odd
L 0
superspin structures the poles can be chosen by a similar way [19]. We take zˆ common to all
0
(1) (2)
superspin structures. In this case supermodular changes of (λ ,λ ) are independent of the
j j
superspin structure and the supermodular group in the P representation is mainly reduced
split
totheordinarymodularone. Thesingular partsof(2)aredetermined byaconditionthatabove
modular group is isomorphic to the supermodular one in the super-Schottky parameterization.
From this condition, it is follows [19] that near every pole zˆ (zˆ ;L)
j 0
[1+ξ (zˆ)ξ′ (zˆ)] λ(2) ∂Rf(zˆ,zˆ ) λ(1)λ(2)ξ (zˆ)∂2ln[Rf(zˆ,zˆ )]
ξ (zˆ) ≈ L L j L 0 +λ(1) + j j L L 0 ,
L Rf(zˆ,zˆ ) [Rf(zˆ,zˆ )] ∂ j  2[Rf(zˆ,zˆ )]2 ∂ ∂
L 0 L 0 zˆ0 L 0 zˆ zˆ0
 λ(2)ξ (zˆ)f′(zˆ)∂Rf(zˆ,zˆ ) λ(1)ξ (zˆ)f′(zˆ)
f (zˆ) ≈ j L L L 0 + j L L (4)
L [Rf(zˆ,zˆ )]2 ∂ Rf(zˆ,zˆ )
L 0 zˆ0 L 0
where the ”prime” symbol denotes ∂ . The calculation [13, 19] of both y (zˆ) and ξ (zˆ) is
zˆ L L
quite similar to that in Sec. 3 of [16]. The set of (2) and of (3) determines both t and q
N
in terms of tˆ and of {qˆ } up to SL transformations of t where {qˆ } = {qˆ ,λ(1),λ(2)} and
N 2 N ev j j
1
An another choice of the poles is discussed in [19].
2See Sec. 4 of [9] where RLf(zˆ,zˆ0) is denoted as Rf(z,z′).
4
ˆ
{qˆ } = {k ,uˆ ,vˆ }. We consider the {r,j} set of the solutions fixed by
ev s s s
t(tˆ;{qˆ };L;r,j); q ({qˆ };L;r,j) : µ = ν = 0, u = uˆ , v = vˆ , u = uˆ . (5)
N N N r r r r r r j j
Every solution is obtained by a SL transformation M(r,j;r ,j ) of the (r = r ,j = j ) one as
2 0 0 0 0
t(tˆ;L;r,j) = M(r,j;r ,j )t(tˆ;L;r ,j ), {P(r,j)} = M(r,j;r ,j ){P(r ,j )}. (6)
0 0 0 0 0 0 0 0
where {P(r,j)} = {(u |µ ),(v |ν )}. The {k } multipliers are the same for all (r,j). The P
s s s s s split
partition functions Zˆ(n) ({qˆ ,qˆ }) can be derived by a going to the P variables in (1) as
L,L′ N N split
ZˆL(n,L)′({qˆN,qˆN}) = FL({qˆN};r,j)FL′({qˆN};r,j)Z˜L(n,L)′({qN,qN})|(uˆr −uˆj)(vˆr −uˆj)|2 (7)
where F ({qˆ };r,j) is the Jacobian of the transformation and q = q ({qˆ };L;r,j). Further-
L N N N N
˜(n)
more, Z ({q ,q }) being multiplied by the factor behind it, is just the partition function in
L,L′ N N
(1), if (u ,v ,u ,µ ,ν ) are fixed in (1) as in (5) to be common to all genus-n supermanifolds
r r j r r
(for details, see eq.(132) in [9]). Under the SL transformations (6) this factor is re-defined
2
by a factor that arises in the Jacobian due to parameters of these transformations depend on
{q }. As the result, (7) appears invariant under the transformations (6). In the same way the
N
partition functions in (1) being multiplied by the product of the moduli differentials, are invari-
ant under the discussed transformations. Supermodular invariant function Y({qˆ ,qˆ };zˆ ,zˆ )
N N 0 0
used in a regularization scheme is constructed as
[Y ({qˆ ,qˆ };zˆ ,zˆ )]2n−1(2n+1)
1 N N 0 0
Y({qˆ ,qˆ };zˆ ,zˆ ) = (8)
N N 0 0
Y ({qˆ ,qˆ };zˆ ,zˆ )
2 N N 0 0
with Y ({qˆ ,qˆ };zˆ ,zˆ ) ≡ Y and Y ({qˆ ,qˆ };zˆ ,zˆ ) ≡ Y defined to be
1 N N 0 0 1 2 N N 0 0 2
Y = Zˆ(n)({qˆ ,qˆ }) and Y = Zˆ(n)({qˆ ,qˆ }) (9)
1 L,L N N 2 L,L N N
L∈X{Lev} L∈Y{Lev}
where {L } is the set of 2n−1(2n+1) even spin structures, Zˆ(n)({qˆ ,qˆ }) is defined by (7) and
ev L,L N N
{qˆ }-set is common to all superspin structures. Since both Y ({qˆ ,qˆ }) and Y ({qˆ ,qˆ }) re-
N 1 N N 2 N N
ceive the same factor under modular transformation of qˆ -parameters, the right side of (8)
N
5
is invariant under supermodular transformations. In addition, it tends to infinity, if Rie-
mann surfaces are degenerated. Indeed, if a particular handle, say s, become degenerated,
the corresponding Schottky multiplier k tends to zero. In this case both the nominator
s
and the denominator in (8) tend to infinity [9], but terms associated with l = 0 have an
1s
additional factor |k |−1 → ∞ in a comparison with those associated with non-zero l . So
s 1s
Y({qˆ ,qˆ };zˆ ,zˆ ) → ∞. If a even spin structure of a genus-n > 1 is degenerated into odd spin
N N 0 0
structures, the partition functions tend to zero [10] while not vanishing, if it is degenerated into
even spin ones. So again Y({qˆ ,qˆ };zˆ ,zˆ ) → ∞. Hence to regularize the desired integrals we
N N 0 0
introduce in the integrand (1) a multiplier
(n)
B ({qˆ ,qˆ };zˆ ,zˆ ;δ ) = {exp[−δ Y({qˆ ,qˆ };zˆ ,zˆ )]} (10)
mod N N 0 0 0 0 N N 0 0 sym
where δ > 0 is a parameter and the right side of (10) is symmetrized over all the sets of
0
(n−1) zeros of the fermion Green function Rf(zˆ,zˆ ). By the above reasons, (10) vanishes, if
L 0
Riemann surfaces become degenerated that provides the finiteness of the modular integrals in
(1). Therightsideof(10)isinvariantundertheSL transformations(6)ofthe{(u |µ ),(v |ν )}
2 s s s s
set. In addition, it is invariant under those L transformation of {uˆ ,vˆ } accompanied by a
2 s s
(1) (2)
corresponding L -transformation of zˆ and of (λ ,λ ), which reduce three (uˆ ,vˆ ,uˆ ) values
2 0 j j r r j
for particular (r,j) to the fixed ones uˆ = uˆ(0), vˆ = vˆ(0) and uˆ = uˆ(0) common to all spin
r r r r j j
structures. For a given tˆ = (zˆ |θˆ = 0) one can calculate its image t˜= (z |θ˜(z )) under the
0 0 0 0
mapping (2). It is evidently that t˜is defined modulo L -transformations. In the considered
2
f
case the transition functions have no poles because zeros zˆ (zˆ ;L) of R (zˆ,zˆ ) are always
j 0 L 0
different from zˆ . Simultaneously, so far as zˆ (zˆ ;L) is changed under fundamental group
0 j 0
transformations, eqs.(3) are satisfied only if every this transformation is accompanied by an
(1) (2)
appropriate change of the (λ ,λ ) parameters that is calculated from (4). Because the above
j j
(1) (2)
change of (λ ,λ ) does not depend on the superspin structure, (10) is invariant under the
j j
super-Schottky transformations of t˜.
6
3 Superstring amplitudes
Theintegralsover z in(1)areilldefined whenallthevertices tendtocoincide, or, alternatively,
j
all they are moved away from each other. In addition, there is no a region in the {p p } space
j l
where all the nodal domain integrations giving raise to poles and to threshold singularities of
A would be finite together. As it is usual [20], each of the above integrals is calculated at
n,m
ReE2 < 0 where E is a center mass energy in the channel considered. Then it is extended to
j j
ReE2 > 0 by an analytical continuation in E2. To regularize the t integrals we need functions
j j j
depending on two more supermanifold points t = (t ,t ) in addition to {t }. One receives
a −1 0 j
in hands the above t points multiplying (1) by the unity arranged to be a square in the same
a
(n)
integrals, every integral I = 1 being
LL′
1 dtdt
I(n) = I(n)(t,t), I(n)(t,t) = D(t)[J (t;L)+J (t;L′)][2πiω(L)−2πiω(L′)]−1
LL′ n 2πi LL′ LL′ s s sr
Z
×D(t)[J (t;L)+J (t;L′)], D(t) = θ∂ +∂ . (11)
r r z θ
Here J (t;L) are the genus-n superholomorphic functions [9] having periods, D(t) is the spinor
s
derivative and ω (L) is a supermanifold period matrix dependent on the superspin structure
sr
[6, 9]. Due to D(t)J (t;L) = 0, both J (t;L′) and J (t;L) could be omitted, but they are
r s r
remained to provide the integrand to have no cuts on the supermanifold. Integrating (11)
(n)
by parts one obtains that I = 1 as it was announced. With (11), we define a regularized
LL′
superstring amplitude A ({δ}) with m > 3 as
n,m
m 0
′ (n) (n,m) (n)
A ({δ}) = dq dq dt dt Z E dt dt I (t ,t )
n,m Z N N N! r=1 r r!L,L′ L,L′ L,L′ a=−1 a a LL′ a a 
Y Y X Y
×B(n) ({q ,q };zˆ ,zˆ ;δ ) B(n)({t ,t};{q ,q };{δ };L,L′) (12)
mod N N 0 0 0 jl a a N N jl
(Yjl)
(n) (n,m)
where t = (z |θ). BothZ andE arethesameasin(1), thearguments beingomittedfor
0 0 L,L′ L,L′
brevity. The(jl)symbollabelspairsofthevertices, δ > 0areparametersand{δ} = (δ ,{δ }).
jl 0 jl
Further, zˆ = zˆ (z ) is calculated together with its Grassmann partner θ(z ) from (2) taken at
0 0 0 0
ˆ
θ = 0, z = z and θ = θ(z ). At {δ > 0} every factor in the (jl) product tends to zero at |z −
0 0 jl j
z | → 0 and at |z −z | → ∞. Explicitly they are given in [13]. The superstring amplitude A
l j l n,m
7
is defined as A ({δ → 0}) calculated in line with the usual analytical continuation procedure
n,m
[20] for the integrals over nodal domains giving rise to poles and threshold singularities of A .
n,m
The (3|2) super-Schottky parameters are no moduli, say, they are µ = ν = 0, u = u(0),
r0 r0 r0 r0
v = v(0) and u = u(0) common to all the supermanifolds. So, {(u |µ ),(v |ν )} = {P(r ,j )}.
r0 r0 j0 j0 s s s s 0 0
Due to the previous Section, {qˆ } for every superspin structure L can be calculated as qˆ =
N N
qˆ ({q (r,j)};L;r,j) through any {q (r,j)} = ({k },{P(r,j)} where {P(r,j)} is obtained by
N N N s
a transformation (6) of {P(r ,j )}. The result is independent of the choice of (r,j).
0 0
Theintegrationsbeingwelldefined, (12)canberearrangedbyasuitableSL -transformation
2
M˜ to the integral over all (3n|2n) super-Schottky parameters and over (m−3|m−2) values
(0) (0) (0)
among {(z |θ )}, the rest being fixed as {z } = (z = z ,z = z ,z = z ), θ = θ = 0 as
j j b 1 1 2 2 3 3 1 2
˜(n) (n,m)
A ({δ}) = dq dq dt dt Z ({q ,q })K ({z },{q ,q },zˆ ,zˆ ,{p })
n,m N N 0 0 L,L′ N N L,L′ b N N 0 0 j
L,L′Z N !
X Y
(n) (n)
×B ({qˆ ,qˆ };zˆ ,zˆ ;δ )I (t ,t ) (13)
mod N N 0 0 0 LL′ 0 0
where the Z˜(n) ({q ,q }) partition function is symmetrical in the super-Schottky parameters
L,L′ N N
and the factor just behind it is given by
m m
(n,m) (n,m)
K ({z },{q ,q },zˆ ,zˆ ,{p }) = dz dz dθ dθ dt ,dt E
L,L′ b N N 0 0 j r r r r −1 −1 L,L′
! !
Z r=4 r=3
Y Y
×|(z(0) −z(0))(z(0) −z(0))|2I(n)(t t ) B(n)({t ,t }{δ };L,L′) (14)
1 3 2 3 LL′ −1 −1 jl a a jl
(Yjl)
(n,m) (n,m) (n)
where E is the same as in (12) and the factor between E ({t ,t }) and I (t t )
L,L′ L,L′ j j LL′ −1 −1
is due to the fixing of the {z } set. The modular parameters in (13) are integrated over
b
the fundamental domain [16] that is invariant under SL transformations. In addition, they
2
(0) (0) (0)
are restricted by both z , z and z to be outside all the Schottky circles. In (13) the
1 2 3
k multipliers are the same as in (12) and the {(u |µ ),(v |ν )} = {P} set is related with
s s s s s
{P(r ,j )} in (12) by the SL -transformation M˜ as {P(r ,j )} = M˜{P}. Just as in (12), the
0 0 2 0 0
{qˆ } set is calculated in term of {P(r,j)} given through {P} by
N
{P(r,j)} = M(r,j;r ,j ){P(r ,j )} = M(r,j;r ,j )M˜{P} (15)
0 0 0 0 0 0
8
where M(r,j;r ,j ) is defined in (6). Parameters of the transition matrix in (15) depend on
0 0
the super-Schottky parameters assigned to the r handle and on (uˆ ,vˆ ) in (5), but (13) is
r r
independent of (uˆ ,vˆ ) due to L symmetry discussed just below eq.(10).
r r 2
In (13), after a suitable rewriting of the integrals over the nodal domains the regularization
factors B(n)({t ,t }{δ };L,L′) can be removed from the integral. Hence the gauge symmetry
jl a a jl
inherent to massless modes presents though in A ({δ}) it is violated due to these factors.
n,m
4 Finiteness of the superstring amplitudes
Divergences due to a degeneration of a handle are already known [5, 16] to be canceled in the
superstring amplitudes. Additional divergences could be when clusters Cl of handles arise,
the sizes being small compared with distances to vertexes (except may be to a solely dilaton-
vacuum transition vertex). In this case, however, leading divergences in A disappear due
n,m
to integrations in (13) over Grassmann modular parameters associated with the Cl cluster
above. Indeed, here a dependence on modular parameters {q } of the Cl cluster is factorized
N1
in the partition functions. Besides, when all the vertexes are separated from the Cl cluster
the integrand (14) ceases to depend on {q } except only on the limiting point u , which the
N1 0
Cl cluster is contracted to. This u dependence in (14) is removed by a boost of the vertex
0
co-ordinates and of the modular parameters of the remainder. If in u the dilaton-vacuum
0
transitionvertex issituated, anadditional{q }dependence arises in(14)solely astheI(n)(t,t¯)
N1 LL′
factor (11). Owing to the above structure of the integrand (13), two Grassmann parameters
associated with the Cl cluster, say, (µ ,ν ) ∈ {q }, are removed from the integrand (13) by
r r N1
an SL transformation M of {P } = {(u |µ ),(v |ν )} ∈ {q } as {P˜ } = M {P˜ } where
2 r N1 s s s s N1 N1 r N1
{P˜ }{(u˜ |µ˜ ),(v˜ |ν˜ )} ∈ {q }. The desired M has a form (2) with transition functions f (z)
N1 s s s s N1 r r
and ξ (z) instead of f and ξ where
r L L
(z −u˜ ) µ (z −v˜ )−ν (z −u˜ )
r r r r r
M : f (z) = z +µ ν , ξ (z) = . (16)
r r r r r
(u˜ −v˜ ) u˜ −v˜
r r r r
9