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Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

Infrared-finite Obser vables in N=4 Super Yang-Mills Theor y
L.Bork2 , D.Kazakov
1

1,2

, G.Var tanov1 and A.Zhiboedov3,

1

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 2 Institute for Theoretical and Experimental Physics 3 Physics Depar tment, Moscow State University

21st August 2009

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

Outline
1

Introduction N=4 Syper Yang Mills Theory Gluon scattering amplitudes Infrared Divergences Weak Coupling Case Strong Coupling Case Cancellation of IR Divergences Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theory Summary and Outlook

2

3

4

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

Outline
1

Introduction N=4 Syper Yang Mills Theory Gluon scattering amplitudes Infrared Divergences Weak Coupling Case Strong Coupling Case Cancellation of IR Divergences Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theory Summary and Outlook

2

3

4

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

N=4 Super Yang-Mills Theor y

N = 4 Syper Yang-Mills theory is the most supersymmetric theory possible without gravity Field content: 1 massless gauge boson, 4 massless (Majorana) spin 1/2 fermions, 6 real (or 3 complex) massless spin 0 bosons All fields are in adjoint representation of the gauge group (Take SU (Nc )) The theory is exactly scale invariant, conformal field theory at quantum level, i.e. the function identically vanishes at all orders of PT

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

N=4 Super Yang-Mills Theor y

N = 4 Syper Yang-Mills theory is the most supersymmetric theory possible without gravity Field content: 1 massless gauge boson, 4 massless (Majorana) spin 1/2 fermions, 6 real (or 3 complex) massless spin 0 bosons All fields are in adjoint representation of the gauge group (Take SU (Nc )) The theory is exactly scale invariant, conformal field theory at quantum level, i.e. the function identically vanishes at all orders of PT

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

N=4 Super Yang-Mills Theor y

N = 4 Syper Yang-Mills theory is the most supersymmetric theory possible without gravity Field content: 1 massless gauge boson, 4 massless (Majorana) spin 1/2 fermions, 6 real (or 3 complex) massless spin 0 bosons All fields are in adjoint representation of the gauge group (Take SU (Nc )) The theory is exactly scale invariant, conformal field theory at quantum level, i.e. the function identically vanishes at all orders of PT

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

AdS/CFT Correspondence
Nc (planar limit) is expected to be integrable and solvable Maldacena's conjecture: Planar Limit of N=4 SYM at strong coupling is dual to weakly coupled type II b supergravity in 10 dimensional AdS5 S5 space. How might PT series be organized to produce simple strong coupling result? The amplitudes on shell possess IR singularities which should cancel in observables. What are the observables in the strong coupling limit?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

AdS/CFT Correspondence
Nc (planar limit) is expected to be integrable and solvable Maldacena's conjecture: Planar Limit of N=4 SYM at strong coupling is dual to weakly coupled type II b supergravity in 10 dimensional AdS5 S5 space. How might PT series be organized to produce simple strong coupling result? The amplitudes on shell possess IR singularities which should cancel in observables. What are the observables in the strong coupling limit?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

AdS/CFT Correspondence
Nc (planar limit) is expected to be integrable and solvable Maldacena's conjecture: Planar Limit of N=4 SYM at strong coupling is dual to weakly coupled type II b supergravity in 10 dimensional AdS5 S5 space. How might PT series be organized to produce simple strong coupling result? The amplitudes on shell possess IR singularities which should cancel in observables. What are the observables in the strong coupling limit?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

AdS/CFT Correspondence
Nc (planar limit) is expected to be integrable and solvable Maldacena's conjecture: Planar Limit of N=4 SYM at strong coupling is dual to weakly coupled type II b supergravity in 10 dimensional AdS5 S5 space. How might PT series be organized to produce simple strong coupling result? The amplitudes on shell possess IR singularities which should cancel in observables. What are the observables in the strong coupling limit?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

Outline
1

Introduction N=4 Syper Yang Mills Theory Gluon scattering amplitudes Infrared Divergences Weak Coupling Case Strong Coupling Case Cancellation of IR Divergences Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theory Summary and Outlook

2

3

4

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

Gluon scattering amplitudes

Colour decomposition of amplitudes in N=4 SYM theory for Nc A
(l ) n

=g

n-2

(

g 2 Nc l ) 16 2

Tr (T
perm

a ( 1 )

, ..., T

a (n)

)An (a

(l )

( 1)

, ..., a

(n )

),

where An - physical amplitude, An - par tial amplitude, ai - is color index of i - th external "gluon" Maximal helicity violating (MHV) amplitudes (two negative helicities and the rest positive) have observed a simple structure on tree level (and even in loops) and one can speculate that this is the consequence of SUSY
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

N=4 Super Yang Mills Theor y Gluon scattering amplitudes

Gluon scattering amplitudes

Colour decomposition of amplitudes in N=4 SYM theory for Nc A
(l ) n

=g

n-2

(

g 2 Nc l ) 16 2

Tr (T
perm

a ( 1 )

, ..., T

a (n)

)An (a

(l )

( 1)

, ..., a

(n )

),

where An - physical amplitude, An - par tial amplitude, ai - is color index of i - th external "gluon" Maximal helicity violating (MHV) amplitudes (two negative helicities and the rest positive) have observed a simple structure on tree level (and even in loops) and one can speculate that this is the consequence of SUSY
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Outline
1

Introduction N=4 Syper Yang Mills Theory Gluon scattering amplitudes Infrared Divergences Weak Coupling Case Strong Coupling Case Cancellation of IR Divergences Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theory Summary and Outlook

2

3

4

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Per turbation theory
Bern, Dixon & Smirnov's conjecture:


M

( L) n l

() An /A f (l ) ()M
( 1) n

(L)

(0) n

Mn 1 +
L= 1

g 2 Nc M 16 2

L

(L) n

() = exp
l =1 (l ) (l )

g 2 Nc 16 2

(l ) + C (l ) + En ()

(l )

f (l ) () = f0 () + f1 () + 2 f2 () 1 8


(l )

Mn ()

=

exp - 1 4


l =1

g 2 Nc 16 2
l (l )

l

(l ) 2G0 K + (l )2 l (1) n

(l )

n

i =1

µ2 -si ,i

l +1

+

l =1

g Nc 16 2
( 1) 4

2

K F

(0)

F

(0) =

1 log 2

2

-t -s

+ 42

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Per turbation theory
Bern, Dixon & Smirnov's conjecture:


M

( L) n l

() An /A f (l ) ()M
( 1) n

(L)

(0) n

Mn 1 +
L= 1

g 2 Nc M 16 2

L

(L) n

() = exp
l =1 (l ) (l )

g 2 Nc 16 2

(l ) + C (l ) + En ()

(l )

f (l ) () = f0 () + f1 () + 2 f2 () 1 8


(l )

Mn ()

=

exp - 1 4


l =1

g 2 Nc 16 2
l (l )

l

(l ) 2G0 K + (l )2 l (1) n

(l )

n

i =1

µ2 -si ,i

l +1

+

l =1

g Nc 16 2
( 1) 4

2

K F

(0)

F

(0) =

1 log 2

2

-t -s

+ 42

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Per turbation theory
Bern, Dixon & Smirnov's conjecture:


M

( L) n l

() An /A f (l ) ()M
( 1) n

(L)

(0) n

Mn 1 +
L= 1

g 2 Nc M 16 2

L

(L) n

() = exp
l =1 (l ) (l )

g 2 Nc 16 2

(l ) + C (l ) + En ()

(l )

f (l ) () = f0 () + f1 () + 2 f2 () 1 8


(l )

Mn ()

=

exp - 1 4


l =1

g 2 Nc 16 2
l (l )

l

(l ) 2G0 K + (l )2 l (1) n

(l )

n

i =1

µ2 -si ,i

l +1

+

l =1

g Nc 16 2
( 1) 4

2

K F

(0)

F

(0) =

1 log 2

2

-t -s

+ 42

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Cusp anomalous dimension
Cusp anomalous dimension appears in RG eq. for the expectation value of a Wilson line with a cusp


Loop expansion
(1)

K =
l =1 ( 2)

g 2 Nc 16 2

l

K

(l )

K = 8, K = -162 , K = 1764 , ... It also controls the large spin limit of anomalous dimension of leading-twist operators ¯ Oj q (+ D+ )j q j = 1 K () log j + O(j 0 ), 2 j

( 3)

and large x limit of the DGLAP kernel for p.d.f. P
gg

=

1 K () + ..., 2 (1 - x )+

1

x 1,

(j ) = -
0

dx x

j -1

P (x )

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Cusp anomalous dimension
Cusp anomalous dimension appears in RG eq. for the expectation value of a Wilson line with a cusp


Loop expansion
(1)

K =
l =1 ( 2)

g 2 Nc 16 2

l

K

(l )

K = 8, K = -162 , K = 1764 , ... It also controls the large spin limit of anomalous dimension of leading-twist operators ¯ Oj q (+ D+ )j q j = 1 K () log j + O(j 0 ), 2 j

( 3)

and large x limit of the DGLAP kernel for p.d.f. P
gg

=

1 K () + ..., 2 (1 - x )+

1

x 1,

(j ) = -
0

dx x

j -1

P (x )

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Cusp anomalous dimension
Cusp anomalous dimension appears in RG eq. for the expectation value of a Wilson line with a cusp


Loop expansion
(1)

K =
l =1 ( 2)

g 2 Nc 16 2

l

K

(l )

K = 8, K = -162 , K = 1764 , ... It also controls the large spin limit of anomalous dimension of leading-twist operators ¯ Oj q (+ D+ )j q j = 1 K () log j + O(j 0 ), 2 j

( 3)

and large x limit of the DGLAP kernel for p.d.f. P
gg

=

1 K () + ..., 2 (1 - x )+

1

x 1,

(j ) = -
0

dx x

j -1

P (x )

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Outline
1

Introduction N=4 Syper Yang Mills Theory Gluon scattering amplitudes Infrared Divergences Weak Coupling Case Strong Coupling Case Cancellation of IR Divergences Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theory Summary and Outlook

2

3

4

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Strong coupling expansion/AdS
Classical solution (Alday & Maldacena) for the scattering amplitude
E M4 () = exp[-Scl ]


E Scl =

+
1

1 2

g 2 Nc µ2 IR -s /2

µ2 IR -s

/2

+
µ2 IR -t /2

µ2 IR -t

/2

g 2 Nc 2

-

(1-log 2)

+
,

g 2 Nc 8

log2 ( s ) + c + O() t

K (g 2 )

g 2 Nc

G0 (g 2 )

g2N

c

1-log 2 2

,

for g 2 Nc

All orders proposal for K by Beiser t, Eden & Staudacher from integrability requirement consistent with weak and strong coupling expansion

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Strong coupling expansion/AdS
Classical solution (Alday & Maldacena) for the scattering amplitude
E M4 () = exp[-Scl ]


E Scl =

+
1

1 2

g 2 Nc µ2 IR -s /2

µ2 IR -s

/2

+
µ2 IR -t /2

µ2 IR -t

/2

g 2 Nc 2

-

(1-log 2)

+
,

g 2 Nc 8

log2 ( s ) + c + O() t

K (g 2 )

g 2 Nc

G0 (g 2 )

g2N

c

1-log 2 2

,

for g 2 Nc

All orders proposal for K by Beiser t, Eden & Staudacher from integrability requirement consistent with weak and strong coupling expansion

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Strong coupling expansion/AdS
Classical solution (Alday & Maldacena) for the scattering amplitude
E M4 () = exp[-Scl ]


E Scl =

+
1

1 2

g 2 Nc µ2 IR -s /2

µ2 IR -s

/2

+
µ2 IR -t /2

µ2 IR -t

/2

g 2 Nc 2

-

(1-log 2)

+
,

g 2 Nc 8

log2 ( s ) + c + O() t

K (g 2 )

g 2 Nc

G0 (g 2 )

g2N

c

1-log 2 2

,

for g 2 Nc

All orders proposal for K by Beiser t, Eden & Staudacher from integrability requirement consistent with weak and strong coupling expansion

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summar y and Outlook

N=4 SYM Theory: Weak Coupling Case AdS/CFT Correspondence: Strong Coupling Case

Strong coupling expansion/AdS
Classical solution (Alday & Maldacena) for the scattering amplitude
E M4 () = exp[-Scl ]


E Scl =

+
1

1 2

g 2 Nc µ2 IR -s /2

µ2 IR -s

/2

+
µ2 IR -t /2

µ2 IR -t

/2

g 2 Nc 2

-

(1-log 2)

+
,

g 2 Nc 8

log2 ( s ) + c + O() t

K (g 2 )

g 2 Nc

G0 (g 2 )

g2N

c

1-log 2 2

,

for g 2 Nc

All orders proposal for K by Beiser t, Eden & Staudacher from integrability requirement consistent with weak and strong coupling expansion

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Cancellation of IR divergences

How and where the IR divergences cancel?

What is left after cancellation of IR divergences?

Which quantities (S-matrix elements, x-sections, etc) might have a simple (exact) solution?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Cancellation of IR divergences

How and where the IR divergences cancel?

What is left after cancellation of IR divergences?

Which quantities (S-matrix elements, x-sections, etc) might have a simple (exact) solution?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Cancellation of IR divergences

How and where the IR divergences cancel?

What is left after cancellation of IR divergences?

Which quantities (S-matrix elements, x-sections, etc) might have a simple (exact) solution?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Cancellation of IR divergences

How and where the IR divergences cancel?

What is left after cancellation of IR divergences?

Which quantities (S-matrix elements, x-sections, etc) might have a simple (exact) solution?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Outline
1

Introduction N=4 Syper Yang Mills Theory Gluon scattering amplitudes Infrared Divergences Weak Coupling Case Strong Coupling Case Cancellation of IR Divergences Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theory Summary and Outlook

2

3

4

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Electron-quark scattering

Born Vir tual Correction d d d d =
0

Vir tual

Real Emission

2 2E 2

s 2 + u 2 - t t2 s 4

2

µ2 s




=
vir t

d d

1 - 2C
0

F

µ2 -t

(

2 3 + + 8) 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Electron-quark scattering

Born Vir tual Correction d d d d =
0

Vir tual

Real Emission

2 2E 2

s 2 + u 2 - t t2 s 4

2

µ2 s




=
vir t

d d

1 - 2C
0

F

µ2 -t

(

2 3 + + 8) 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Electron-quark scattering

Born Vir tual Correction d d d d =
0

Vir tual

Real Emission

2 2E 2

s 2 + u 2 - t t2 s 4

2

µ2 s




=
vir t

d d

1 - 2C
0

F

µ2 -t

(

2 3 + + 8) 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission µ2 -t µ2 -t


d d

=
real

d d

2C
0

F

s 4


( (

23 + + 8) 2

+

2 s CF 2 E 4

µ2 s

f1 + f2 ),

where the functions f1 and f2 in the c.m. frame are (c = cos ) (c 3 + 5c 2 - 3c + 5) log( 1-c ) + (1 - 2 (1 - c )(1 + c )2 1 f2 = - (1 - c )(c 3 + 5c 2 - 3c (1 - c 2 )2 1 1-c + (1 - c )(3c 3 + 15c 2 + 77c - 31) log( )+ 2 2 f
1

= -2

c 2 )(c - 11)/4 + 5) log2 ( 1-c ) 2

1 (1 - c 2 )(5c 2 - 42c - 23) 2 1+c +(1 + c )2 (c 2 + 5c + 3) 2 - 12(9c 2 + 2c + 5)Li2 ( ). 2
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission µ2 -t µ2 -t


d d

=
real

d d

2C
0

F

s 4


( (

23 + + 8) 2

+

2 s CF 2 E 4

µ2 s

f1 + f2 ),

where the functions f1 and f2 in the c.m. frame are (c = cos ) (c 3 + 5c 2 - 3c + 5) log( 1-c ) + (1 - 2 (1 - c )(1 + c )2 1 f2 = - (1 - c )(c 3 + 5c 2 - 3c (1 - c 2 )2 1 1-c + (1 - c )(3c 3 + 15c 2 + 77c - 31) log( )+ 2 2 f
1

= -2

c 2 )(c - 11)/4 + 5) log2 ( 1-c ) 2

1 (1 - c 2 )(5c 2 - 42c - 23) 2 1+c +(1 + c )2 (c 2 + 5c + 3) 2 - 12(9c 2 + 2c + 5)Li2 ( ). 2
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission µ2 -t µ2 -t


d d

=
real

d d

2C
0

F

s 4


( (

23 + + 8) 2

+

2 s CF 2 E 4

µ2 s

f1 + f2 ),

where the functions f1 and f2 in the c.m. frame are (c = cos ) (c 3 + 5c 2 - 3c + 5) log( 1-c ) + (1 - 2 (1 - c )(1 + c )2 1 f2 = - (1 - c )(c 3 + 5c 2 - 3c (1 - c 2 )2 1 1-c + (1 - c )(3c 3 + 15c 2 + 77c - 31) log( )+ 2 2 f
1

= -2

c 2 )(c - 11)/4 + 5) log2 ( 1-c ) 2

1 (1 - c 2 )(5c 2 - 42c - 23) 2 1+c +(1 + c )2 (c 2 + 5c + 3) 2 - 12(9c 2 + 2c + 5)Li2 ( ). 2
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial state splitting d d d d where for f3 = Qf2 - =^ t =
split

1 s 2

1

dz
0

µ2 Qf2 µ2 s




Pqq (z ) µ2 -t


d 0 (pz ) d f1 + f3 ),

=C
split

F

2 s 2E 2 2

(-

1 1-c 2(1 - c )(c 3 + c 2 - 33c + 7) log( ) (1 - c )2 (1 + c )2 2 1+c +12(9c 2 + 2c + 5)Li2 ( ) - (1 + c )2 (c 2 + 5c + 3) 2 2 1 - (1 - c )(1 + c )(11c 2 - 19) . 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial state splitting d d d d where for f3 = Qf2 - =^ t =
split

1 s 2

1

dz
0

µ2 Qf2 µ2 s




Pqq (z ) µ2 -t


d 0 (pz ) d f1 + f3 ),

=C
split

F

2 s 2E 2 2

(-

1 1-c 2(1 - c )(c 3 + c 2 - 33c + 7) log( ) (1 - c )2 (1 + c )2 2 1+c +12(9c 2 + 2c + 5)Li2 ( ) - (1 + c )2 (c 2 + 5c + 3) 2 2 1 - (1 - c )(1 + c )(11c 2 - 19) . 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial state splitting d d d d where for f3 = Qf2 - =^ t =
split

1 s 2

1

dz
0

µ2 Qf2 µ2 s




Pqq (z ) µ2 -t


d 0 (pz ) d f1 + f3 ),

=C
split

F

2 s 2E 2 2

(-

1 1-c 2(1 - c )(c 3 + c 2 - 33c + 7) log( ) (1 - c )2 (1 + c )2 2 1+c +12(9c 2 + 2c + 5)Li2 ( ) - (1 + c )2 (c 2 + 5c + 3) 2 2 1 - (1 - c )(1 + c )(11c 2 - 19) . 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial state splitting d d d d where for f3 = Qf2 - =^ t =
split

1 s 2

1

dz
0

µ2 Qf2 µ2 s




Pqq (z ) µ2 -t


d 0 (pz ) d f1 + f3 ),

=C
split

F

2 s 2E 2 2

(-

1 1-c 2(1 - c )(c 3 + c 2 - 33c + 7) log( ) (1 - c )2 (1 + c )2 2 1+c +12(9c 2 + 2c + 5)Li2 ( ) - (1 + c )2 (c 2 + 5c + 3) 2 2 1 - (1 - c )(1 + c )(11c 2 - 19) . 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free obser vable = inclusive cross-section

d d = - 2E
2 2

=
observ

d d

22

+
vir t

d d

2 3

+
real

d d

2 2 split

c + 2c + 5 (1 - c )2

2

s CF 1-c (c 3 + 5c 2 - 3c + 5) log2 2 (1 - c )(1 + c )2 2

1 1-c + (7c 3 + 19c 2 - 55c - 3) log - (1 + c )(3c 2 + 21c + 2) 2 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Outline
1

Introduction N=4 Syper Yang Mills Theory Gluon scattering amplitudes Infrared Divergences Weak Coupling Case Strong Coupling Case Cancellation of IR Divergences Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theory Summary and Outlook

2

3

4

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

From par tial amplitudes to cross-sections
To obtain the cross sections from par tial amplitudes one have to compute the square of them. In the the planar limit it is just:
± ± n (p1 , ..., pn ) = g 2n-4

(

g 2 Nc 2 ) 16 2

l colors

An A |An (a
(l )

(l )

(l ) n

= , an )|2

2g

2n-4

N

n-2 c

2 (Nc - 1)(

g 2 Nc 2l ) 16 2

(1)

, ..., a

(n-1)

perm

Then the cross-section is
± ± d n (pin ) = n (p1 , ..., pn )d k ,

where d k is the phase space of the outgoing par ticles: d k D (pin - pfin )S
n k 2 + (pk )d D pk ,

where Sn - is so called measurement function and integration goes over D = 4 - 2 dimensions.
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Vir tual Correction (MHV) d d d d
--++ 2 2 Nc s4 (s2 + t 2 + u 2 ) 2 8E s2 t 2 u 2

=
0

µ2 s



=

2 N E2

2 c

µ2 s



3 + c2 (1 - c 2 )2

--++

=
vir t

2 2 Nc 2 8E

µ2 s



Nc s 4 2 s 2 t 2 u

2

-
2

8 2

((

µ2 µ2 ) +( ) )s -t -u

2

µ2 µ2 µ2 µ2 ) )t +(( ) + ( ) )u 2 + (( ) + ( s -t s -u +

-s -s t 16 2 2 2 (s + t + u 2 ) + 4(u 2 log2 ( ) + t 2 log2 ( ) + s2 log2 ( )) 3 t u u 2 N E2
2 c

=

µ2 s

2

Nc 16 3 + c 2 4 -2 + 2 (1 - c 2 )2

5 + 2c + c 2 1-c log( ) (1 - c 2 )2 2

+(c -c )

+

16(3 + c 2 ) 2 16 1-c 1+c - log( ) log( ) 3(1 - c 2 )2 (1 - c 2 )2 2 2
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Vir tual Correction (MHV) d d d d
--++ 2 2 Nc s4 (s2 + t 2 + u 2 ) 2 8E s2 t 2 u 2

=
0

µ2 s



=

2 N E2

2 c

µ2 s



3 + c2 (1 - c 2 )2

--++

=
vir t

2 2 Nc 2 8E

µ2 s



Nc s 4 2 s 2 t 2 u

2

-
2

8 2

((

µ2 µ2 ) +( ) )s -t -u

2

µ2 µ2 µ2 µ2 ) )t +(( ) + ( ) )u 2 + (( ) + ( s -t s -u +

-s -s t 16 2 2 2 (s + t + u 2 ) + 4(u 2 log2 ( ) + t 2 log2 ( ) + s2 log2 ( )) 3 t u u 2 N E2
2 c

=

µ2 s

2

Nc 16 3 + c 2 4 -2 + 2 (1 - c 2 )2

5 + 2c + c 2 1-c log( ) (1 - c 2 )2 2

+(c -c )

+

16(3 + c 2 ) 2 16 1-c 1+c - log( ) log( ) 3(1 - c 2 )2 (1 - c 2 )2 2 2
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Vir tual Correction (MHV) d d d d
--++ 2 2 Nc s4 (s2 + t 2 + u 2 ) 2 8E s2 t 2 u 2

=
0

µ2 s



=

2 N E2

2 c

µ2 s



3 + c2 (1 - c 2 )2

--++

=
vir t

2 2 Nc 2 8E

µ2 s



Nc s 4 2 s 2 t 2 u

2

-
2

8 2

((

µ2 µ2 ) +( ) )s -t -u

2

µ2 µ2 µ2 µ2 ) )t +(( ) + ( ) )u 2 + (( ) + ( s -t s -u +

-s -s t 16 2 2 2 (s + t + u 2 ) + 4(u 2 log2 ( ) + t 2 log2 ( ) + s2 log2 ( )) 3 t u u 2 N E2
2 c

=

µ2 s

2

Nc 16 3 + c 2 4 -2 + 2 (1 - c 2 )2

5 + 2c + c 2 1-c log( ) (1 - c 2 )2 2

+(c -c )

+

16(3 + c 2 ) 2 16 1-c 1+c - log( ) log( ) 3(1 - c 2 )2 (1 - c 2 )2 2 2
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Vir tual Correction (MHV) d d d d
--++ 2 2 Nc s4 (s2 + t 2 + u 2 ) 2 8E s2 t 2 u 2

=
0

µ2 s



=

2 N E2

2 c

µ2 s



3 + c2 (1 - c 2 )2

--++

=
vir t

2 2 Nc 2 8E

µ2 s



Nc s 4 2 s 2 t 2 u

2

-
2

8 2

((

µ2 µ2 ) +( ) )s -t -u

2

µ2 µ2 µ2 µ2 ) )t +(( ) + ( ) )u 2 + (( ) + ( s -t s -u +

-s -s t 16 2 2 2 (s + t + u 2 ) + 4(u 2 log2 ( ) + t 2 log2 ( ) + s2 log2 ( )) 3 t u u 2 N E2
2 c

=

µ2 s

2

Nc 16 3 + c 2 4 -2 + 2 (1 - c 2 )2

5 + 2c + c 2 1-c log( ) (1 - c 2 )2 2

+(c -c )

+

16(3 + c 2 ) 2 16 1-c 1+c - log( ) log( ) 3(1 - c 2 )2 (1 - c 2 )2 2 2
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (MHV)
2

d d 14

(--+++)

=
Born

2 N E2

2 c

µ2 s

Nc 2

1 8(3 + c 2 ) 2 (1 - c 2 )2

16 (2 - 3) 1 1-c 1+c 2 2 + log( log( )+ )+ (1 + c )2 2 (1 - c )2 2 (1 - c 2 )2 (1 - )2 + 12(3 + c 2 )(log(1 - ) - log( )) + Finite par t (1 - c 2 )2

What is ? One has a singularity as 1. This is the cut-off in external momenta of the scattered gluon: |p| E (1 - ). 2 This allows one to distinguish identical final gluons, so that the gluon scattered at angle has non-zero momentum

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (MHV)
2

d d 14

(--+++)

=
Born

2 N E2

2 c

µ2 s

Nc 2

1 8(3 + c 2 ) 2 (1 - c 2 )2

16 (2 - 3) 1 1-c 1+c 2 2 + log( log( )+ )+ (1 + c )2 2 (1 - c )2 2 (1 - c 2 )2 (1 - )2 + 12(3 + c 2 )(log(1 - ) - log( )) + Finite par t (1 - c 2 )2

What is ? One has a singularity as 1. This is the cut-off in external momenta of the scattered gluon: |p| E (1 - ). 2 This allows one to distinguish identical final gluons, so that the gluon scattered at angle has non-zero momentum

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (MHV)
2

d d 14

(--+++)

=
Born

2 N E2

2 c

µ2 s

Nc 2

1 8(3 + c 2 ) 2 (1 - c 2 )2

16 (2 - 3) 1 1-c 1+c 2 2 + log( log( )+ )+ (1 + c )2 2 (1 - c )2 2 (1 - c 2 )2 (1 - )2 + 12(3 + c 2 )(log(1 - ) - log( )) + Finite par t (1 - c 2 )2

What is ? One has a singularity as 1. This is the cut-off in external momenta of the scattered gluon: |p| E (1 - ). 2 This allows one to distinguish identical final gluons, so that the gluon scattered at angle has non-zero momentum

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (MHV)
2

d d 14

(--+++)

=
Born

2 N E2

2 c

µ2 s

Nc 2

1 8(3 + c 2 ) 2 (1 - c 2 )2

16 (2 - 3) 1 1-c 1+c 2 2 + log( log( )+ )+ (1 + c )2 2 (1 - c )2 2 (1 - c 2 )2 (1 - )2 + 12(3 + c 2 )(log(1 - ) - log( )) + Finite par t (1 - c 2 )2

What is ? One has a singularity as 1. This is the cut-off in external momenta of the scattered gluon: |p| E (1 - ). 2 This allows one to distinguish identical final gluons, so that the gluon scattered at angle has non-zero momentum

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (MHV)
2

d d 14

(--+++)

=
Born

2 N E2

2 c

µ2 s

Nc 2

1 8(3 + c 2 ) 2 (1 - c 2 )2

16 (2 - 3) 1 1-c 1+c 2 2 + log( log( )+ )+ (1 + c )2 2 (1 - c )2 2 (1 - c 2 )2 (1 - )2 + 12(3 + c 2 )(log(1 - ) - log( )) + Finite par t (1 - c 2 )2

What is ? One has a singularity as 1. This is the cut-off in external momenta of the scattered gluon: |p| E (1 - ). 2 This allows one to distinguish identical final gluons, so that the gluon scattered at angle has non-zero momentum

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (Anti MHV)
2

d d 14

(--++-)

=
Born

2 N E2

2 c

µ2 s

Nc 2

1 8(3 + c 2 ) 2 (1 - c 2 )2

(6 2 - 3 + 30)c 2 + (10 2 - 57 + 66) 6(c 2 + 3) log 2 64 + + - + 2 )3 2 )2 3(1 - c 3(1 - c (1 - c 2 )2 4(c 2 - 6c + 21) 1 + - (1 - )c 3c 2 - 24c + 85 1-c log - log (1 - c )(1 + c )3 2 (1 - c )(1 + c )3 2 + 32(1 - c ) 16 (5 - c ) - + (c -c ) (1 - c 2 )2 (1 + - (1 - )c ) 3(1 + c )3 (1 + - (1 - )c )3

+Finite par t

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (Anti MHV)
2

d d 14

(--++-)

=
Born

2 N E2

2 c

µ2 s

Nc 2

1 8(3 + c 2 ) 2 (1 - c 2 )2

(6 2 - 3 + 30)c 2 + (10 2 - 57 + 66) 6(c 2 + 3) log 2 64 + + - + 2 )3 2 )2 3(1 - c 3(1 - c (1 - c 2 )2 4(c 2 - 6c + 21) 1 + - (1 - )c 3c 2 - 24c + 85 1-c log - log (1 - c )(1 + c )3 2 (1 - c )(1 + c )3 2 + 32(1 - c ) 16 (5 - c ) - + (c -c ) (1 - c 2 )2 (1 + - (1 - )c ) 3(1 + c )3 (1 + - (1 - )c )3

+Finite par t

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (Matter)( = 1) Fermions d d 14
¯ (--+q q ) 2

=
Real

2 N E2

2 c

µ2 s

Nc 2

4 32(79 - 25c 2 ) 3(1 - c 2 )2 )

64(3 - c )2 64(3 + c )2 1-c 1+c ) log( + )+ log( ) + Finite par t 3 (1 - c )(1 + c ) 2 (1 - c )3 (1 + c ) 2 Sfermions d d 14
~~ ¯ (--+q q )

=
Real

2 N E2

2 c

µ2 s

2

Nc 2

128(10 + 7c 2 ) 1 - (1 - c 2 )2

192(5 - c ) 192(5 + c ) 1-c 1+c - log( )- log( ) + Finite par t (1 + c )3 2 (1 - c )3 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (Matter)( = 1) Fermions d d 14
¯ (--+q q ) 2

=
Real

2 N E2

2 c

µ2 s

Nc 2

4 32(79 - 25c 2 ) 3(1 - c 2 )2 )

64(3 - c )2 64(3 + c )2 1-c 1+c ) log( + )+ log( ) + Finite par t 3 (1 - c )(1 + c ) 2 (1 - c )3 (1 + c ) 2 Sfermions d d 14
~~ ¯ (--+q q )

=
Real

2 N E2

2 c

µ2 s

2

Nc 2

128(10 + 7c 2 ) 1 - (1 - c 2 )2

192(5 - c ) 192(5 + c ) 1-c 1+c - log( )- log( ) + Finite par t (1 + c )3 2 (1 - c )3 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Real Emission (Matter)( = 1) Fermions d d 14
¯ (--+q q ) 2

=
Real

2 N E2

2 c

µ2 s

Nc 2

4 32(79 - 25c 2 ) 3(1 - c 2 )2 )

64(3 - c )2 64(3 + c )2 1-c 1+c ) log( + )+ log( ) + Finite par t 3 (1 - c )(1 + c ) 2 (1 - c )3 (1 + c ) 2 Sfermions d d 14
~~ ¯ (--+q q )

=
Real

2 N E2

2 c

µ2 s

2

Nc 2

128(10 + 7c 2 ) 1 - (1 - c 2 )2

192(5 - c ) 192(5 + c ) 1-c 1+c - log( )- log( ) + Finite par t (1 + c )3 2 (1 - c )3 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial and final state splitting (MHV) Initial d d 14 Nc 2 - Final d d 14 Nc 2
(--+++) (--+++)

=
InSplit

2 N E2

2 c

µ2 s

µ2 Qf2

4(c 2 + 3) 1 - (1 - c 2 )2

log

1-c 1+c + log 2 2

-

8(c 2 + 3) 1- log (1 - c 2 )2

16 (2 - 3) + Finite par t (1 - 2 )(1 - c 2 )2

=
FnSplit

2 N E2

2 c

µ2 s



µ2 Qf2



1 4(c 2 + 3) 1- - log + Finite par t (1 - c 2 )2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial and final state splitting (MHV) Initial d d 14 Nc 2 - Final d d 14 Nc 2
(--+++) (--+++)

=
InSplit

2 N E2

2 c

µ2 s

µ2 Qf2

4(c 2 + 3) 1 - (1 - c 2 )2

log

1-c 1+c + log 2 2

-

8(c 2 + 3) 1- log (1 - c 2 )2

16 (2 - 3) + Finite par t (1 - 2 )(1 - c 2 )2

=
FnSplit

2 N E2

2 c

µ2 s



µ2 Qf2



1 4(c 2 + 3) 1- - log + Finite par t (1 - c 2 )2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial and final state splitting (MHV) Initial d d 14 Nc 2 - Final d d 14 Nc 2
(--+++) (--+++)

=
InSplit

2 N E2

2 c

µ2 s

µ2 Qf2

4(c 2 + 3) 1 - (1 - c 2 )2

log

1-c 1+c + log 2 2

-

8(c 2 + 3) 1- log (1 - c 2 )2

16 (2 - 3) + Finite par t (1 - 2 )(1 - c 2 )2

=
FnSplit

2 N E2

2 c

µ2 s



µ2 Qf2



1 4(c 2 + 3) 1- - log + Finite par t (1 - c 2 )2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial and final state splitting (Anti MHV) Initial d d 14
(--++-) 2 2 Nc E2

=
InSplit

µ2 s



µ2 Qf2



Nc 2

1

4(c 3-15c 2 + 51c - 45) 1-c log (1 - c )2 (1 + c )3 2

1 + - c (1 - ) 8(c 2 + 3) 16(c 2 - 3c + 3) log log + (c -c ) + - (1 - c )2 (1 + c )3 2 (1 - c 2 )2 4 - c 8 (1 - )6 (2 2 + 3 + 6) - 4c 6 (1 - )4 ( 4 + 10 3(1 - c 2 )2 ((1 + )2 - c 2 (1 - )2 )3 -23 2 + 114 - 33) + 2c 4 (1 - )2 (39 5 - 102 4 + 86 3 + 658 2 + 183 - 312) +4c 2 ( 8 - 12 7 + 39 6 + 216 5 - 42 4 - 720 3 - 421 2 + 300 + 208) -(1 + )3 (2 5 - 9 4 + 63 3 + 455 2 + 579 + 198) Final d d 14
(--++-) 2

3

+ Finite par t

=
FnSplit

N E2

2 c

µ2 s



µ2 Qf2



Nc 1 4(c 2 + 3) log - (2 2 - 9 + 18) + F.p. 2 (1 - c 2 )2 3
Infrared-finite Observables in N=4 SYM

D.Kazakov


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial and final state splitting (Anti MHV) Initial d d 14
(--++-) 2 2 Nc E2

=
InSplit

µ2 s



µ2 Qf2



Nc 2

1

4(c 3-15c 2 + 51c - 45) 1-c log (1 - c )2 (1 + c )3 2

1 + - c (1 - ) 8(c 2 + 3) 16(c 2 - 3c + 3) log log + (c -c ) + - (1 - c )2 (1 + c )3 2 (1 - c 2 )2 4 - c 8 (1 - )6 (2 2 + 3 + 6) - 4c 6 (1 - )4 ( 4 + 10 3(1 - c 2 )2 ((1 + )2 - c 2 (1 - )2 )3 -23 2 + 114 - 33) + 2c 4 (1 - )2 (39 5 - 102 4 + 86 3 + 658 2 + 183 - 312) +4c 2 ( 8 - 12 7 + 39 6 + 216 5 - 42 4 - 720 3 - 421 2 + 300 + 208) -(1 + )3 (2 5 - 9 4 + 63 3 + 455 2 + 579 + 198) Final d d 14
(--++-) 2

3

+ Finite par t

=
FnSplit

N E2

2 c

µ2 s



µ2 Qf2



Nc 1 4(c 2 + 3) log - (2 2 - 9 + 18) + F.p. 2 (1 - c 2 )2 3
Infrared-finite Observables in N=4 SYM

D.Kazakov


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial and final state splitting (Anti MHV) Initial d d 14
(--++-) 2 2 Nc E2

=
InSplit

µ2 s



µ2 Qf2



Nc 2

1

4(c 3-15c 2 + 51c - 45) 1-c log (1 - c )2 (1 + c )3 2

1 + - c (1 - ) 8(c 2 + 3) 16(c 2 - 3c + 3) log log + (c -c ) + - (1 - c )2 (1 + c )3 2 (1 - c 2 )2 4 - c 8 (1 - )6 (2 2 + 3 + 6) - 4c 6 (1 - )4 ( 4 + 10 3(1 - c 2 )2 ((1 + )2 - c 2 (1 - )2 )3 -23 2 + 114 - 33) + 2c 4 (1 - )2 (39 5 - 102 4 + 86 3 + 658 2 + 183 - 312) +4c 2 ( 8 - 12 7 + 39 6 + 216 5 - 42 4 - 720 3 - 421 2 + 300 + 208) -(1 + )3 (2 5 - 9 4 + 63 3 + 455 2 + 579 + 198) Final d d 14
(--++-) 2

3

+ Finite par t

=
FnSplit

N E2

2 c

µ2 s



µ2 Qf2



Nc 1 4(c 2 + 3) log - (2 2 - 9 + 18) + F.p. 2 (1 - c 2 )2 3
Infrared-finite Observables in N=4 SYM

D.Kazakov


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial state splitting (Matter) ( = 1) Fermions d d 14 +
¯ (--+q q )

=
InSplit

2 N E2

2 c

µ2 s

µ2 Qf2

Nc 2

4 32(79 - 25c 2 ) 3(1 - c 2 )2 )

64(3 + c )2 64(3 - c )2 1-c 1+c ) log( )+ log( ) + Finite par t 3 (1 - c )(1 + c ) 2 (1 - c )3 (1 + c ) 2

Sfermions d d 14
~~ ¯ (--+q q )

=
InSplit

2 N E2

2 c

µ2 s



µ2 Qf2



Nc 2

128(10 + 7c 2 ) 1 - (1 - c 2 )2

192(5 - c ) 192(5 + c ) 1-c 1+c log( )- log( ) + Finite par t - (1 + c )3 2 (1 - c )3 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial state splitting (Matter) ( = 1) Fermions d d 14 +
¯ (--+q q )

=
InSplit

2 N E2

2 c

µ2 s

µ2 Qf2

Nc 2

4 32(79 - 25c 2 ) 3(1 - c 2 )2 )

64(3 + c )2 64(3 - c )2 1-c 1+c ) log( )+ log( ) + Finite par t 3 (1 - c )(1 + c ) 2 (1 - c )3 (1 + c ) 2

Sfermions d d 14
~~ ¯ (--+q q )

=
InSplit

2 N E2

2 c

µ2 s



µ2 Qf2



Nc 2

128(10 + 7c 2 ) 1 - (1 - c 2 )2

192(5 - c ) 192(5 + c ) 1-c 1+c log( )- log( ) + Finite par t - (1 + c )3 2 (1 - c )3 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Initial state splitting (Matter) ( = 1) Fermions d d 14 +
¯ (--+q q )

=
InSplit

2 N E2

2 c

µ2 s

µ2 Qf2

Nc 2

4 32(79 - 25c 2 ) 3(1 - c 2 )2 )

64(3 + c )2 64(3 - c )2 1-c 1+c ) log( )+ log( ) + Finite par t 3 (1 - c )(1 + c ) 2 (1 - c )3 (1 + c ) 2

Sfermions d d 14
~~ ¯ (--+q q )

=
InSplit

2 N E2

2 c

µ2 s



µ2 Qf2



Nc 2

128(10 + 7c 2 ) 1 - (1 - c 2 )2

192(5 - c ) 192(5 + c ) 1-c 1+c log( )- log( ) + Finite par t - (1 + c )3 2 (1 - c )3 2

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free sets (for any arbitrary )
(--++) (--+++) (--+++) (--+++) FnSplit

A

MHV

=

1 2

d d 14

+
Vir t

d d 14

+
Real

d d 14

+
InSplit

d d 14

B

AntiMHV

=

1 2

d d 14

(--++)

+
Vir t

d d 14

(--++-)

+
Real

d d 14

(--++-)

+
InSplit

d d 14

(--++-) FnSplit

C

Matter

=

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
Real

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
InSplit

d d 14

¯ ~~ ¯ (--+q q ,q q ) FnSplit

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free sets (for any arbitrary )
(--++) (--+++) (--+++) (--+++) FnSplit

A

MHV

=

1 2

d d 14

+
Vir t

d d 14

+
Real

d d 14

+
InSplit

d d 14

B

AntiMHV

=

1 2

d d 14

(--++)

+
Vir t

d d 14

(--++-)

+
Real

d d 14

(--++-)

+
InSplit

d d 14

(--++-) FnSplit

C

Matter

=

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
Real

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
InSplit

d d 14

¯ ~~ ¯ (--+q q ,q q ) FnSplit

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free sets (for any arbitrary )
(--++) (--+++) (--+++) (--+++) FnSplit

A

MHV

=

1 2

d d 14

+
Vir t

d d 14

+
Real

d d 14

+
InSplit

d d 14

B

AntiMHV

=

1 2

d d 14

(--++)

+
Vir t

d d 14

(--++-)

+
Real

d d 14

(--++-)

+
InSplit

d d 14

(--++-) FnSplit

C

Matter

=

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
Real

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
InSplit

d d 14

¯ ~~ ¯ (--+q q ,q q ) FnSplit

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free sets (for any arbitrary )
(--++) (--+++) (--+++) (--+++) FnSplit

A

MHV

=

1 2

d d 14

+
Vir t

d d 14

+
Real

d d 14

+
InSplit

d d 14

B

AntiMHV

=

1 2

d d 14

(--++)

+
Vir t

d d 14

(--++-)

+
Real

d d 14

(--++-)

+
InSplit

d d 14

(--++-) FnSplit

C

Matter

=

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
Real

d d 14

¯ ~~ ¯ (--+q q ,q q )

+
InSplit

d d 14

¯ ~~ ¯ (--+q q ,q q ) FnSplit

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free observables Registration of two fastest gluons of positive chirality A
MHV =1/3

+B

AntiMHV =1

Registration of one fastest gluon of positive chirality A
MHV =1/3

+B

AntiMHV =1/3

+C

Matter =1

Anti MHV cross-section B
AntiMHV =1

+C

Matter =1

Finite Par t

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free observables Registration of two fastest gluons of positive chirality A
MHV =1/3

+B

AntiMHV =1

Registration of one fastest gluon of positive chirality A
MHV =1/3

+B

AntiMHV =1/3

+C

Matter =1

Anti MHV cross-section B
AntiMHV =1

+C

Matter =1

Finite Par t

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free observables Registration of two fastest gluons of positive chirality A
MHV =1/3

+B

AntiMHV =1

Registration of one fastest gluon of positive chirality A
MHV =1/3

+B

AntiMHV =1/3

+C

Matter =1

Anti MHV cross-section B
AntiMHV =1

+C

Matter =1

Finite Par t

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free observables Registration of two fastest gluons of positive chirality A
MHV =1/3

+B

AntiMHV =1

Registration of one fastest gluon of positive chirality A
MHV =1/3

+B

AntiMHV =1/3

+C

Matter =1

Anti MHV cross-section B
AntiMHV =1

+C

Matter =1

Finite Par t

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

Infrared-free observables Registration of two fastest gluons of positive chirality A
MHV =1/3

+B

AntiMHV =1

Registration of one fastest gluon of positive chirality A
MHV =1/3

+B

AntiMHV =1/3

+C

Matter =1

Anti MHV cross-section B
AntiMHV =1

+C

Matter =1

Finite Par t

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Toy model: electron-quark scattering Gluon scattering in N=4 Super Yang-Mills Theor y

The simplest IR finite answer so far (Qf = E ): N=4 SYM Anti MHV

d d 14 -

=
AntiMHV

2 2 Nc E2

3 + c2 (1 - c 2 )2
1c - 2

(c 4+ 2c 3+ 4c 2+ 6c+19) log2 ( Nc 2 2 (1 - c )2 (1 + c)4 (c 2 + 1) log( 1+c ) log( 2 (1 - c 2 )2
1-c 2

)

+2

(c 4- 2c 3+ 4c 2- 6c+19) log2 ( (1 - c )4 (1 + c )2

1c + 2

)

-8 +2

)

-

6 2 (c 2 - 1) + 5(61c 2 + 99) 9(1 - c 2 )2 ) -2 (11c 3-31c 2- 47c-133) log( 3(1 + c )3 (1 - c )2
1c - 2

(11c 3+31c 2- 47c+133) log( 3(1 - c )3 (1 + c )2

1c + 2

)

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Summary
In observable cross-sections the IR divergences do cancel in accordance with Kinoshita-Lee-Nauenberg theorem
1

d

incl obs

=
n =2 0

Qf2 dz1 q1 (z1 , 2 ) µ
0

1

Qf2 dz2 q2 (z2 , 2 ) µ
4 L=0

n

1

dxi qi (xi ,
i =1 0 L 2

Qf2 )â µ2

â d

2 n

(z1 p1 , z2 p2 , ...)Sn ({z }, {x }) = g

g2 16

Finite d L (s, t , u , Qf2 )

The simple structure of the MHV amplitude DOES NOT reveal at the level of IR finite cross-sections; In some cases the cancellation of complicated functions occurs, though not always; The dependence on the scale Qf which characterizes the asymptotic states is left.
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Summary
In observable cross-sections the IR divergences do cancel in accordance with Kinoshita-Lee-Nauenberg theorem
1

d

incl obs

=
n =2 0

Qf2 dz1 q1 (z1 , 2 ) µ
0

1

Qf2 dz2 q2 (z2 , 2 ) µ
4 L=0

n

1

dxi qi (xi ,
i =1 0 L 2

Qf2 )â µ2

â d

2 n

(z1 p1 , z2 p2 , ...)Sn ({z }, {x }) = g

g2 16

Finite d L (s, t , u , Qf2 )

The simple structure of the MHV amplitude DOES NOT reveal at the level of IR finite cross-sections; In some cases the cancellation of complicated functions occurs, though not always; The dependence on the scale Qf which characterizes the asymptotic states is left.
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Summary
In observable cross-sections the IR divergences do cancel in accordance with Kinoshita-Lee-Nauenberg theorem
1

d

incl obs

=
n =2 0

Qf2 dz1 q1 (z1 , 2 ) µ
0

1

Qf2 dz2 q2 (z2 , 2 ) µ
4 L=0

n

1

dxi qi (xi ,
i =1 0 L 2

Qf2 )â µ2

â d

2 n

(z1 p1 , z2 p2 , ...)Sn ({z }, {x }) = g

g2 16

Finite d L (s, t , u , Qf2 )

The simple structure of the MHV amplitude DOES NOT reveal at the level of IR finite cross-sections; In some cases the cancellation of complicated functions occurs, though not always; The dependence on the scale Qf which characterizes the asymptotic states is left.
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Summary
In observable cross-sections the IR divergences do cancel in accordance with Kinoshita-Lee-Nauenberg theorem
1

d

incl obs

=
n =2 0

Qf2 dz1 q1 (z1 , 2 ) µ
0

1

Qf2 dz2 q2 (z2 , 2 ) µ
4 L=0

n

1

dxi qi (xi ,
i =1 0 L 2

Qf2 )â µ2

â d

2 n

(z1 p1 , z2 p2 , ...)Sn ({z }, {x }) = g

g2 16

Finite d L (s, t , u , Qf2 )

The simple structure of the MHV amplitude DOES NOT reveal at the level of IR finite cross-sections; In some cases the cancellation of complicated functions occurs, though not always; The dependence on the scale Qf which characterizes the asymptotic states is left.
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Summary
In observable cross-sections the IR divergences do cancel in accordance with Kinoshita-Lee-Nauenberg theorem
1

d

incl obs

=
n =2 0

Qf2 dz1 q1 (z1 , 2 ) µ
0

1

Qf2 dz2 q2 (z2 , 2 ) µ
4 L=0

n

1

dxi qi (xi ,
i =1 0 L 2

Qf2 )â µ2

â d

2 n

(z1 p1 , z2 p2 , ...)Sn ({z }, {x }) = g

g2 16

Finite d L (s, t , u , Qf2 )

The simple structure of the MHV amplitude DOES NOT reveal at the level of IR finite cross-sections; In some cases the cancellation of complicated functions occurs, though not always; The dependence on the scale Qf which characterizes the asymptotic states is left.
D.Kazakov Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Outlook

What are the IR safe observables in the strong coupling limit? Which IR finite quantities have a simple (integrable) structure? What are the true scale invariant quantities in conformal theories?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Outlook

What are the IR safe observables in the strong coupling limit? Which IR finite quantities have a simple (integrable) structure? What are the true scale invariant quantities in conformal theories?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Outlook

What are the IR safe observables in the strong coupling limit? Which IR finite quantities have a simple (integrable) structure? What are the true scale invariant quantities in conformal theories?

D.Kazakov

Infrared-finite Observables in N=4 SYM


Introduction Infrared Divergences Cancellation of IR Divergences Summary and Outlook

Outlook

What are the IR safe observables in the strong coupling limit? Which IR finite quantities have a simple (integrable) structure? What are the true scale invariant quantities in conformal theories?

D.Kazakov

Infrared-finite Observables in N=4 SYM