vladimir m. braun - ukr

KITP, QCD and String Theory Integrability in QCD and beyond - p. 1/35

Integrability in QCD and beyond

Vladimir M. BraunUniversity of Regensburg

thanks to

Sergey Derkachov, Gregory Korchemsky and Alexander Manashov


Regensburg


Outline

In this talk:

■ Renormalization in conformally-covariant form■ Integrability on the example of three-quark operators in QCD■ Violation of integrability: Mass gaps■ SUSY extensions: From N = 0 to N = 4

■ Further QCD example: Polarized deep-inelastic scattering

not included:

■ High energy scattering in QCD

■ Full N = 4 SUSY and gauge/string correspondence


Heisenberg spin chain

1926: Heisenberg

H

s=1=2

= �

L

X

n=1

�

^

S

n

�

^

S

n+1

�

14

�

1981-83: Generalizations for arbitrary spin

H

s

=

L

X

n=1

H(J

n;n+1

)

J

n;n+1

(J

n;n+1

+ 1) = (

^

S

n

+

^

S

n+1

)

2

H(x) = (2s+ 1)� (x+ 1)

Complete integrability

m

Number of degrees of freedom =number of conservation laws

� Conceptually interesting� Powerfull machinery:

Algebraic Bethe Ansatz (ABA)Method of separation of

Variables (SoV)


at the same time, in QCD . . .

1973-74: Anomalous dimensions of leading twist operators

N

= C

F

�

�3�

2

(N+1)(N+2)

+ 4 (N+2)� 4 (1)

�

1976-78: High-energy (Regge) asymptotics of scattering amplitides

E

BFKL

N;�

= 2 (1)�

�

n+1

2

+ i�

�

�

�

n+1

2

� i�

�

1991: Anomalous dimensions of quark-antiquark-gluon operators ,N

!1

5 10 15 20 25 3010

15

20

25

30

35

40 γN

N

Exact analytic solution

Hidden symmetry?


Example:

Twist-three operators

,

Leading-twistthree-particle parton

distributions

E.g. baryon distribution amplitudes B = N;�; : : :

h0jq(z

1

)q(z

2

)q(z

3

)jB(p; �)i = : : :

Z

1

0

dx

1

dx

2

dx

3

Æ(

P

x

i

� 1) e

�ip(x

1

z

1

+x

2

z

2

+x

3

z

3

)

'

B

(x

i

; �

2

)

q

"

q

"

q

"

) '

�=3=2

�

(x

i

; �

2

) ;

q

"

q

#

q

"

)

(

'

�=1=2

N

(x

i

; �

2

)

'

�=1=2

�

(x

i

; �

2

)

� quark fields “live” on a light-ray z2 = 0


Problem: Prolifiteration of degrees of freedom

Moments of distribution amplitudes , local operators:

'(x

i

)! '(k

i

) =

Z

Dx

i

x

k

1

1

x

k

2

2

x

k

3

3

'(x

i

; �

2

)

q(z

1

)q(z

2

)q(z

3

) ! (D

k

1

+

q) (D

k

2

+

q) (D

k

3

+

q)

Mixing matrix: N = k

1

+ k

2

+ k

3

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30

E3/

2(N

)

N

γ n=0,1,...N=14

Example:qqq

� = 3=2

Rich spectrum of anomalous dimensions reflectscomplexity of genuine degrees of freedom


Conformal Symmetry on the Light-Cone: SL(2; R)

z ! z

0

=

az + b

z + d

; ad� b = 1 �(z)! �

0

(z) = �

�

az + b

z + d

�

� ( z + d)

�2j

�

j

q

= 1, jg

= 3=2 is conformal spin of the field

Generators obey the SL(2) algebra

L

�

�(z) = �

d

dz

�(z)

L

+

�(z) =

�

z

2

d

dz

+ 2j

�

z

�

�(z)

L

0

�(z) =

�

z

d

dz

+ j

�

�

�(z)

Casimir operators

L

2

= L

20

� L

0

+ L

+

L

�

L

2

�(z) = j(j � 1)�(z)

Summation of spins

L

2ik

=

P

�=0;1;2

(L

i;�

+L

k;�

)

2

L

2123

=

P

�=0;1;2

(L

1;�

+L

2;�

+L

3;�

)

2

� second-order differential operators on the space of (z

1

; z

2

; z

3

)


RG equations in SL(2)-covariant form

Light-ray operators

�

�

�

��

+ �(g)

�

�g

�

B = H �B ; B(z

1

; z

2

; z

3

) ' q(z

1

)q(z

2

)q(z

3

)

Two-particle structure:

H

qqq

= H

12

+H

23

+H

13

Renormalization = Displacement: Balitsky, V.B. 88’

H

12

B(z

1

; z

2

; z

3

) =

Z

D� !(�

1

�

2

) B(��

1

z

1

+ �

1

z

2

; ��

2

z

2

+ �

2

z

1

; z

3

)

where

R

D� =

R

d�

1

d�

2

d�

3

Æ(1��

1

��

2

��

3

), �� = 1� �


RG equations in SL(2)-covariant form — continued

Conformal symmetry ! [L

�;0

;H

12

℄ = 0

!(�

1

�

2

) = ��

2j

1

�2

1

��

2j

2

�2

2

'

�

�

1

�

2

��

1

��

2

�

�� = 1� �

Explicit calculation

1 2 3

) '(x) = Æ(x);

1 2 3

) '(x) = 1


RG equations in SL(2)-covariant form — Hamiltonian approach

B’FK’L 85’

write H as a function of Casimir operators L

2ik

= �

�

�z

i

�

�z

k

(z

i

�z

k

)

2

�

b

J

ik

(

b

J

ik

� 1)

1 2 3

=)

H

v12

= 2[ (J

12

)� (2)℄

H

�=3=2

qqq

= H

v

12

+H

v

23

+H

v

13

1 2 3

=)

H

(e)

1=2

=

1

L

212

=

1

J

12

(J

12

�1)

H

�=1=2

qqq

= H

�=3=2

qqq

�H

e

12

�H

e

23

Have to solve H

N;n

= E

N;n

N;n

— A Schr odinger equation with Hamiltonian H


Hamiltonian approach — continued

going over to local operators:

In helicity basis (BFKL 85’) obtain a

HERMITIAN Hamiltonianwith respect to the conformal scalar product

h

1

j

2

i =

Z

1

0

dx

i

Æ(

X

x

i

� 1)x

2j

1

�1

1

x

2j

2

�1

2

x

2j

3

�1

3

�1

(x

i

)

2

(x

i

)

that is, conformal symmetry allows for the transformation o f the mixing matrix:

0�

forward

non-hermitian

1A

| {z }

N + 1

)

0�

non-forward

hermitian

1A

| {z }

N(N + 1)=2

)

0�

hermitian

in onformal basis

1A

| {z }

N + 1


Hamiltonian approach — continued (2)

~ Eigenvalues (anomalous dimensions) are real numbers

~ “Conservation of probability” – possibility of meaningful

approximations to exact evolution

~ Systematic 1=N

expansion

E

N;n

= N

E

N;n

+N

�1

ÆE

N;n

+ : : :

N;n

=

(0)

N;n

+N

�2

Æ

N;n

+ : : :

with the usual quantum-mechanical expressions

ÆE

N;n

= jj

(0)

N;n

jj

�2

h

(0)

N;n

jH

(1)

j

(0)

N;n

i

etc.


Complete Integrability

� Conformal symmetry implies existence

of two conserved quantities:

[H; L

2

℄ = [H; L

0

℄ = 0

� For qqq and GGG states with maximum helicity and for qGq states at N

!1 there exists an

additional conserved charge

qqq

GGG

�=max

)

Q = [L

212

; L

223

℄

[H; Q℄ = 0

qGq

N

!1

)

Q = fL

2qG

; L

2Gq

g+

1

L

2qG

+

2

L

2Gq

[H; Q℄ = 0

BDM ’98

� Anomalous dimensions can be clas-

sified by values of the charge Q:

H = E

Q = q

=)

E = E(q)

� A new quantum number


WKB expansion of the Baxter equation ) 1/N expansion

G. Korchemsky ’95-’97

~ equation Q = q is much simpler as H = E

~ non-integrable corrections are suppressed at large N

~ can use some techniques of integrable models

E(N; `) = 6 ln � � 3 ln 3� 6 + 6

E

�

3

�

(2`+ 1)

�

1

�

2

�

5`

2

+ 5`� 7=6

�

�

1

72�

3

�

464`

3

+ 696`

2

� 802`� 517

�

+ : : :

� =

p

(N+3)(N+2)

An integer ` numerates the trajectories:(semiclassically quantized solitons: Korchemsky, Krichever, ’97)

� Integrability imposes a nontrivial analytic structure


WKB expansion of the Baxter equation ) 1/N expansion — continued

The ‘dispersion curve’ E(q)

-0.20 -0.10 0.00 0.10 0.2010.0

11.0

12.0

13.0

14.0

15.0

E(q

)

η3q/

Ν = 30

WKB

exactExample:qqq

� = 3=2

E(q) = 2 ln 2� 6 + 6

E

+

+ 2Re

3

Xk=1

(1 + i�

3

Æ

k

) +O(�

�6

)

Æ

k

are defined as roots of the cubic equation:2Æ

3

k

� Æ

k

� q=�

3

= 0

� =

p

(N+3)(N+2)


WKB expansion of the Baxter equation ) 1/N expansion — continued (2)

Double degeneracy:

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30

q/3

N

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30

E3/

2(N

)

η

N

l = 0

l = 2

l = 7

Example:qqq

� = 3=2

E(N; q) = E(N;�q)

q(N; `) = �q(N;N � `)

BDKM ’98


The ground state solution ( q = 0) for the wave function of �

�=3=2

For evenN q = 0 is the solution with lowest energy

x

1

x

2

x

3

N;q=0

(x

i

) = x

1

(1�x

1

)C

3=2

N+1

(1�2x

1

)

+ x

2

(1�x

2

)C

3=2

N+1

(1�2x

2

)

+ x

3

(1�x

3

)C

3=2

N+1

(1�2x

3

)

The lowest anomalous dimension

E(N; q = 0) = 4(N + 3) + 4

E

� 6

Here x1

; x

2

; x

3

are fractions of the �3=2 momentum carried by the three quarks


Breakdown of integrability: Mass gaps and bound states — ”experiment”

Simplest case:

Difference between �

�=3=2 and N(�)

�=1=2

H(") = H

3=2

� "

�

1

L

212

+

1

L

223

�

H(" = 1) = H

1=2

Flow of energy levels

(anomalous dimensions):


Breakdown of integrability: Mass gaps and bound states — ”theory”

In lower part of the qqq spectrum

‘Perturbation’ hq

0

j

1

L

212

+

1

L

223

jqi �

1

lnN

Level splitting �

1

ln

2

N

=)

Of order � lnN lower levels

have to be rediagonalized

hq

0

j

1

L

212

+

1

L

223

jqi �

1

lnN

os(�

q

� �

q

0

)

Phases of the cyclic permutations: (123)

P

�! (312)

Pjqi = �

q

jqi

θq

Matrix ( lnN � lnN ):

1

lnN

0BBBBBB�

1 �1=2 �1=2 1 : : :

�1=2 1 �1=2 �1=2 : : :

�1=2 �1=2 1 �1=2 : : :

......

......

. . .

1CCCCCCA

diag

�!

1

lnN

0BBBBBB�

lnN 0 0 0 : : :

0 lnN 0 0 : : :

0 0 0 0 : : :

......

......

. . .

1CCCCCCA

! A mass gap !


Breakdown of integrability: Mass gaps and bound states — ”interpretation”

�

�=3=2 wave function Nucleon and �

1=2 wave functions

'

�

3=2

(x

i

)

�

=

1

X

N=0

'

�

0

N;n=0

�

�

s

(�)

�

s

(�

0

)

�

N;n=0

�

x

1

(1�x

1

)C

3=2

N+1

(1�2x

1

) + x

2

(1�x

2

)C

3=2

N+1

(1�2x

2

)

+x

3

(1�x

3

)C

3=2

N+1

(1�2x

3

)

�

'

�

1=2

(x

i

)

�

= x

1

x

2

x

3

1

X

N=0

'

�

0

N;n=0

�

�

s

(�)

�

s

(�

0

)

�

N;n=0

�

P

(1;3)

N

(1� 2x

3

)� P

(1;3)

N

(1� 2x

1

)

�

1 2 3

free motion

scalar diquarks?

1 2 3

1] -+ ( )3[


Supersymmetry: from N = 0 to N = 4

A. Belitsky, S. Derkachov G. Korchemsky and A. Manashov [hep-th/0403085]

� One expects that SUSY enhances integrability as one goes fro m N = 0! N = 4

? QCD-like integrability is extended and new features may appear

� Need a universal approach to treat simultaneously the opera tor mixing in various

N�extended SYM

The light-cone superspace formalism

■ Quantize SYM in the light-cone gaugeA+

(x) = 0

■ Decompose fermions and gauge fields into “good” (+

; A

?

) and “bad” (�

; A

�

) components

■ Integrate out nonpropagating (“bad”) fields and rewrite the action in terms of propagating (“good”) fields

S

N

= S

N

(

+

; A

?

)

■ Combine “good” fields into light-cone superfields �(x; �A)

Kogut, Soper’70


N = 4 light-cone SYM

Light-cone superfield formulation of N = 4 SYM Mandelstam, Brink et al. ’83

S

N=4

=

Z

d

4

x d

4

� d

4

�

�

8<

:

12

�

�

a

�

�

2

+

�

a

�

23

gf

ab

�

1

�

+

�

�

a

�

b

�

��

+

1

�

+

�

a

�

�

b

�

�

�

�

�

12

g

2

f

ab

f

ade

�

1

�

+

(�

b

�

+

�

)

1

�

+

(

�

�

d

�

+

�

�

e

) +

12

�

b

�

�

�

d

�

�

e

�

9=

;

Complex scalar N = 4 chiral superfield

� = �

�1

+

A+ �

A

�

�1

+

�

�

A

+

i

2!

�

A

�

B

�

�

AB

�

1

3!

"

ABCD

�

A

�

B

�

C

�

D

�

1

4!

"

ABCD

�

A

�

B

�

C

�

D

�

+

�

A

� Chiral superfield spans all helicities of the particles in th e supermultiplet

� The anti-chiral superfield is not independent

� All single-trace light-cone operators are built from a sing le chiral superfield

O(Z

1

; : : : ; Z

L

) = trf�(z

1

; �

1

) : : :�(z

L

; �

L

)g


. . . and back from N = 4 to N = 0

“Method of truncation”: SN=4

! S

N=2

! S

N=1

! S

N=0

Brink et al. ’83

) N = 2 light-cone chiral superfield

�(x; �) = �

�1

+

A(x) + �

A

�

�1

+

�

�

A

(x) +

i

2!

"

AB

�

A

�

B

�

�(x)

) N = 1 light-cone chiral superfield

�(x; �) = �

�1

+

A(x) + � �

�1

+

�

�

A

(x)

) N = 0 light-cone chiral (super)field

�(x) = �

�1

+

A(x)

� chiral superfield � and its conjugate �

� are independent

� Single-trace light-cone operators can be built from two sup erfields

Integrable sector O

hiral

(Z

1

; : : : ; Z

L

) = Trf�(Z

1

) : : :�(Z

L

)g

Non-integrable sector O

mixed

(Z

1

; : : : ; Z

L

) = Trf�(Z

1

) : : :

�

�(Z

L

)g


One-loop dilatation operator in chiral sector

A two particle kernel

H

k;k+1

=

k

k+1

= V

k;k+1

(1� �

k;k+1

)

acts as a displacement in the light-cone superspace

Z = (z; �

A

) (A = 1; : : : ;N )

V

12

O (Z

1

; Z

2

) =

R

1

0

d�

(1��)�

2

h

2�

2

O (Z

1

; Z

2

)�O (�Z

1

+(1��)Z

2

; Z

2

)�O (Z

1

; �Z

2

+(1��)Z

1

)

i

�(x) = �

�1

+

A(x)+: : :

� Projection operator, (�

12

)

2

= �

12

�

12

O (Z

1

; Z

2

) =

12

(1 + Z

12

� �

Z

) O (Z; Z

2

)

��

Z=Z

2

+

12

(1 + Z

21

� �

Z

) O (Z

1

; Z)

��

Z=Z

1

Compare this with QCD expression for B(z

1

; z

2

; z

3

) = q

"

(z

1

)q

"

(z

2

)q

"

(z

3

):

H

12

B(z

1

; z

2

; : : :) =

R

1

0

�d�

1��

h

2B(z

1

; z

2

)�B(�z

1

+ (1� �)z

2

; z

2

)�B(z

1

; �z

2

+ (1� �)z

1

)

i

[For quarks jq

= 1, for chiral superfield j�

= �1=2, hence different power of �]


Dilatation operator on the light-cone superspace Z = (z; �)

■ The one-loop dilatation operator has the same, universal form for N = 0, N = 1,

N = 2 and N = 4 SYM

◆ Dilatation operator displaces superfields in the superspaceZ = (z; �

A

)

◆ The number of superchargesN defines the dimension of the superspace

◆ Natural generalization of the QCD formulae Z = (z; �

A

= 0)

■ Two reductions of the N = 4 dilatation operator

◆ Expand in even light-cone coordinates, project on � = 0

! Maximal helicity gluon operators H = SL(2;R) magnet[Braun,Derkachov,Manashov; Belitsky; G.K ’98]

◆ Expand in odd �-coordinates, project on zi

= 0

! BMN-like operators: O

s al

= Trf�

A

1

B

1

(0) : : : �

A

L

B

L

(0)g H = SO(6) magnet[Minahan, Zarembo ’02]

■ In general: collinear subgroup of the superconformal symme try group:Heisenberg (super) spin magnet with the SL(2jN ) symmetry


N = 4 beyond the light-cone

The case of N = 4 SYM is still special:

■ All light cone operators belong to the SL(2jN ) sector since �

�(Z) is not dynamicaly

independent

■ Integrability extends to all operators (including those wi th “bad” components) and the collinear

subgroup SL(2jN ) extends to the full superconformal symmetry group PSU(2; 2;N ).

[Beisert, Staudacher, ’03]

Why?◆ Crucial: Superconformal symmetry relates operators of different twist

◆ Superconformal transformations applied to light-cone operators �iDN

+

�

i yield combinations of

operators of different twist but having the same anomalous dimension

◆ This is not the case in QCD where operators of different twists are believed to be dynamically

independent

■ Probably persists to all loops


Outlook

■ One-loop dilatation operator (evolution equations) in QCD is/are integrable in somesectors◆ Classical bremsstrahlung: A �

d!

!

d�

�

◆ Cusp anomalous dimension: for N !1

N

� �

usp

(�

s

) logN

◆ Integrability appears as a consequence of the existence of massless vector particles

■ Offers powerful machinery

■ Open problems in QCD context:◆ Analytic structure of the spectrum — Parton interpretation in higher twists

◆ FromN = 4 toN = 0: Rethinking of the role of non-quasipartonic operators; properties of

anomalous dimensions in all twists

◆ Beyond one loop: Formal conformal limit

A = A

onformal

+

�(g)

g

�A

Integrability? Particular models?

◆ Breaking of integrability vs. breaking of conformal symmetry: physics issues?


Addendum: Polarized deep-inelastic scattering

Cross section: � = pq=M = E

l

�E

l

0

d

2

�

ddE

l

0

(#"+"#) =

8�

2

E

2

l

0

Q

4

�

�

F

2

(x;Q

2

) os

2

�2

+

2�

M

F

1

(x;Q

2

) sin

2

�2

�

;

d

2

�

ddE

l

0

(#"�"#) =

8�

2

E

l

0

Q

4

x

�

g

1

(x;Q

2

)

�

1+

E

l

0

E

l

os �

�

�

2Mx

E

l

g

2

(x;Q

2

)

�

At tree level:

g

1

(x) =

Z

1

�1

d�

2�

e

i�x

hp; s

z

j�q(0)6n

5

q(�n)jp; s

z

i

g

T

(x) =

1

2M

Z

1

�1

d�

2�

e

i�x

hp; s

?

j�q(0)

?

5

q(�n)jp; s

?

i

— quark distributions in longitudinally and transversely polarized nucleon, respectively


Polarized deep-inelastic scattering — continued

Beyond the tree level

hN(p; s)j�q(z

1

)G(z

2

)q(z

3

)jN(p; �)i =

= : : :

Z

1

�1

dx

1

dx

2

dx

3

Æ(x

1

+x

2

+x

3

) e

�ip(x

1

z

1

+x

2

z

2

+x

3

z

3

)

D

q

(x

i

; �

2

)

�q G q ) g

p�n

2

(x

B

; �

2

)

�q G q

GGG

)

) g

p+n

2

(x

B

; �

2

)

quark-gluon correlations in the nucleon


Renormalization of quark-gluon operators, flavor non-sing let

H

qGq

= N

H

(0)

�

2

N

H

(1)

;

H

(0)

= V

(0)

qg

(J

12

) + U

(0)

qg

(J

23

) ;

H

(1)

= V

(1)

qg

(J

12

) + U

(1)

qg

(J

23

) + U

(1)

qq

(J

13

) :

where

V

(0)

qg

(J) = (J + 3=2) + (J � 3=2)� 2 (1)� 3=4 ;

U

(0)

qg

(J) = (J + 1=2) + (J � 1=2)� 2 (1)� 3=4 ;

V

(1)

qg

(J) =

(�1)

J�5=2

(J � 3=2)(J � 1=2)(J + 1=2)

;

U

(1)

qg

(J) = �

(�1)

J�5=2

2(J � 1=2)

;

V

(1)

qq

(J) = (J)� (1)� 3=4 ;

U

(1)

qq

(J) =

12

[ (J � 1) + (J + 1)℄� (1)� 3=4 :


Open inhomogeneous spin chain

5 10 15 20 25 3010

15

20

25

30

35

40 γN

N

Exact analytic solution

� The lowest level is special

and is separated from the rest of

the spectrum by a “mass gap”

� This special level was found by ABH 91’ and it determines the e volution of g

2

(x;Q

2

) to the

leading logarithmic accuracy

� It corresponds to the particular combination of quark-anti quark-gluon operators that can be

reduced to the quark-antiquark twist-three operator using equations of motion

Detailed study:

Belitsky; Derkachov, Korchemsky, Manashov ’00


Flavor singlet qGq and GGG operators

BDM ’01

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16 18 20

EN

,α

N

qGq levels: crosses GGG levels: open circles

The highest qGq level is separated

from the rest by a finite gap and is

almost degenerate with the lowest

GGG state

� Interacting open and closed spin chains


Flavor singlet qGq and GGG operators: WKB expansion

Energies: j = N + 3

E

low

q

= N

�

2 [j℄ +

1

j

�

12

+ 2

E

�

�

2

N

�

ln j +

E

+

34

�

�

2

6

+ : : :

�

E

high

q

= N

�

4 ln j + 4

E

�

19

6

�

19

3j

2

�

+

23

n

f

+O(1=j

3

)

E

low

G

= N

�

4 ln j + 4

E

+

13

�

�

2

3

+

1

j

�

2�

2

�18

3

(ln j +

E

)�

�

2

3

��

+O(ln

2

j=j

2

)

To the leading logarithmic accuracy, the structure functio n g2

(x;Q

2

) is expressed through the

quark distribution

g

2

(x;Q

2

) = g

WW

2

(x;Q

2

) +

12

X

q

e

2q

Z

1

x

dy

y

�q

+

T

(y;Q

2

)


Approximate evolution equation for the structure function g

2

(x;Q

2

)

Introduce flavor-singlet quark and gluon transverse spin di stributions

Q

2

d

dQ

2

�q

+; S

T

(x;Q

2

) =

�

s

4�

Z

1

x

dy

y

h

P

T

qq

(x=y)�q

+; S

T

(y;Q

2

) + P

T

qg

(x=y)�g

T

(y;Q

2

)

i

Q

2

d

dQ

2

�g

T

(x;Q

2

) =

�

s

4�

Z

1

x

dy

y

P

T

gg

(x=y)�g

T

(y;Q

2

)

with the splitting functions

P

T

qq

(x) =

�

4C

F

1� x

�

+

+ Æ(1� x)

�

C

F

+

1

N

�

2�

�

2

3

��

� 2C

F

;

P

T

gg

(x) =

�

4N

1� x

�

+

+ Æ(1� x)

�

N

�

�

2

3

�

13

�

�

23

n

f

�

+N

�

�

2

3

� 2

�

+N

ln

1� x

x

�

2�

2

3

� 6

�

;

P

T

qg

(x) = �4n

f

�

x� 2(1� x)

2

ln(1� x)

�

:

vladimir m. braun - ukr

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