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Effective actions for high energy scattering in QCD and in gravity
L. N. Lipatov Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg, Russia QFTHEP2011, So chi Content 1. Gribov Pomeron calculus 2. BFKL equation 3. Effective action for high energy QCD 4. Production amplitudes in N = 4 SUSY 5. Pomeron and graviton in N = 4 SUSY 6. High energy amplitudes in gravity and supergravity 7. Effective action for reggeized gravitons 8. Discussion
1

1

Scattering amplitudes at high energies
High energy kinematics s = 4E
2

-t = q 2 ,

|q | E

1

t-channel partial wave expansion A (s, t) = s
a-i p a+i

d p ((-s) - ps ) f (t) , p = ±1 2 i

Regge pole hypothesis and Pomeron tra jectory 2 (t) p , P (-q 2 ) P - P q 2 , p = 1 f (t) = - p (t) Asymptotics of elastic amplitudes and cross-sections A(s, t) is 2 (-q 2 )s
P -P q
2

, = 2 (0) s

P

2

2

Grib ov Pomeron calculus
Multi-particle t-channel unitarity
t f

(t)
n

( dn |fn) |

2

Mandelstam cut contribution AM (s, t) -i d k (k , q - k ) s
2 2 2-P k2 -P (q -k)2

Separation of particles in their rapidities 0 < y1 < ... < yn < ln s , yk - yk
-1

1, y =

1 E+p ln 2 E-p

Gribov's Pomeron action S= ( )2 dy d (y - ) + + ( 2 + 2 ) + ... 2m
2

3

3

Gluon reggeization in QCD
QCD Born amplitude 1 gT t

AB MAB (s, t)|

B or n

= 2s g T

c A A A

A

c B B

B

B

Leading Logarithmic Approximation M (s, t) = MB
or n

(s, t) s

(t)

,

Its region of applicability g2 s ln s 1 , s = 4 s Nc (-|q | ) = - 4 2
2

1

Gluon tra jectory in LLA s Nc |q 2 | |q |2 - ln 2 dk 2 2 |k | |q - k | 2
2 4

4

Amplitudes in multi-Regge kinematics

BF K M22+L n

s 1 1 2 gT |q1 |

d1 c2 c

1

s 2 C (q2 , q1 ) 2 2 ...g T |q2 |

dn cn+1 c

n

C (q

n+1

n+1 sn+1 , , qn ) |qn+1 |2

s Nc r = - 2

2 |qr | 1 ln 2 - µ

q2 q1 , C (q2 , q1 ) = , t = q2 - q1 5

dn |M2
n

2+n

|

2

5

BFKL equation (1975)

Balitsky-Fadin-Kuraev-Lipatov equation s Nc E0 12 2 Hamiltonian for the Pomeron wave function 1 1 2 = (ln |12 | )p1 p2 + (ln |12 |2 )p p2 + ln |p1 p2 |2 - 4 (1) , 1 p1 p p1 p2 2 E (1 , 2 ) = H (1 , 2 ) , t s , = -
12

H

12

= 1 - 2 , r = xr + iyr , = 4Nc ln 2 /

M¨ obius invariance and Pomeron intercept k E=
m

ak + b , m = + n/2 , m = - n/2 , = 1/2 + i , ck + d +
m

,

m

g 2 Nc = (m) + (1 - m) - 2 (1) , = ln 2 2
6

6

Effective action in QCD
1 ln 2 + |k | , |y - y0 | < , << ln s - |k |

Locality of the theory in the rapidity space y=
k k

Gluon and Reggeized gluon fields
a vµ (x) = -iT a vµ (x) , A± (x) = -iT a Aa (x) , A± (x) = 0 ±

Effective action for their interactions (L., 1995) S= d4 x L
QC D x 2 2 + T r(V+ µ A- + V- µ A+ ) ,

1 V+ = - + P exp -g g

+

v+ (x )d(x )
-

+

1 = v+ - g v+ v+ + ... +

7

7

Pro duction amplitudes in N = 4 SUSY
Relative correction R to the BDS amplitude A
2 4

(F.,L., 2011) 1 u1 - 1
(,n)

R ei

a = cos ab +i 2

(-1)n e
n=-

in

-

|w|2i d (, n) 2 + n2 4

,

ss2 s1 t3 s3 t 1 u2 2 u1 = , u2 = , u3 = , |w| = , s012 s123 s012 t2 s123 t2 u3 |w |2 1 - u1 - u2 - u3 K ln cos = , = , 4 2 u2 u3 8 |1 + w | (, n) = -aE
,n ab

=

K ln |w|2 , 8 | n| )-2 (1) , 2
n2 4 3

-a 2 (

n

+3 (3)) , E

n

=-

1 | n| 2 2 +
|n| 2

n2 4

+2 (1+i +

n

=-

2

(1 + i +

2i (1 + i + |n| )- 2 2 2 + n 4
8

| n| 2 - ) 1 - (2) E n - 2 4 2 + n 4

8

Pomeron and graviton in N = 4 SUSY
BFKL Pomeron in a diffusion approximation j = 2 - - D 2 , = 1 + j-2 + i 2

Constraint from the energy-momentum conservation D= AdS/CFT relation for the graviton Regge tra jectry R2 2 2 j =2+ t , t = E /R , = 2 2 Large coupling asymptotics for and (KLOV, BPST) = - j - 2 + , = 1 g 2 Nc

9

9

Perturbation theory in gravity
Einstein-Hilbert action S
EH

=-

1 2

2

d4 x -g R , R = Rµ g Riemann tensor

µ

Rµ = Rµ, , R

µ,

= - + µ µ µ

- µ

Christophel symbol and gravity field
µ

=

1 g 2

(µ g + gµ - gµ ) , g

µ

= µ + hµ

General coordinate transformation h
µ

= Dµ + D µ , Dµ = µ - µ

10

10

High energy amplitudes in gravity
Production amplitudes in LLA (L.L. (1982)) A
2 n
2 (q1 ) 1 2 q1 2 (q2 ) 2 2 q2

=-

s2 µ µ

s

s
1 1

2 2

...

Graviton-graviton-reggeon vertex µ µ = (µµ + µ µ ) 4 Gluon-gluon-reggeized gluon vertex µµ = -µµ pA pB + pA pB q 2 pB pB µ µµ µµ µ + + pA pB 2 (pA pB )2 pB pA -B k pA kp

Reggeonreggeon-graviton vertex

= (C C - N N ) , N = 4
11

22 q1 q2

11

Graviton tra jectory and amplitudes
Graviton Regge tra jectory (L. (1982)) a 2 (q ) = f (k , q ) = (k , q - k )2 q 2 d2 k 2 f (k , q ) , a = , k 2 (q - k )2 8 2 1 1 + k2 (q - k )2 Gravitino action S
3 /2 N µ

- q2 +

N (k , q - k ) 2

1 =- 2

d4 x
r =1

¯r µ 5

r

Amplitudes in the DL approximation A=A
B orn

s

-a ln

q2 2

2 d f N - 4 f ad s , f = 1 - a 2- f 2 i 2 2 d

12

12

Effective action for gravity
Locality in the rapidity space y= 1 ln 2
k k

+ |k | , |y - y0 | < , << ln s - |k |

Reggeized graviton fields A
++

(x) = A

--

(x) = 0 , + A

++

(x) = - A-- (x) = 0

Effective action for the high energy gravity (L. 2011) 1 S=- 2 dx
4

1 2 -g R + + j - µ A 2

++

2 + - j+ µ A

--

Hamilton-Jacobi equation for effective currents g
µ

µ j ± j

±

= 0 , ± j

=h

±±

-h

±

1 - h 2 ±

2 ±±

+ ...

13

13

Effective currents for sho ck waves
Aichelburg - Sexl metric x- 8 dxµ dx +a ln |- | (x- ) (dx- )2 , a = G µ , z = a 2 x |- | x 2 Effective current for the shock wave 1 , f (z ) = + 2 1 + 2z 2
2

(ds)2 =

µ

- | + ln f (z ) - 1 z j = -a µ ln | x 4 f 2 (z )
+ 2 | + a - + j = -a ln | x - 2

Perturbative expansion x 2|- | x a 3 x µ µ )- - 2|- | - x
+

x - 2| | x

+ ...

Variational principle for j j+ =
x-

g
-

++

- - (y - , (y - )) + (- )2 ,

j + =0 - (y + )

14

14
1. 2. 3. 4. 5. 6. 7. 8.

Discussion

Locality of reggeon interactions in the rapidity space. BFKL equation for Pomeron wave function High energy effective action for gluons in QCD. Pomeron-graviton duality in N = 4 SUSY. Multi-regge processes in gravity. The graviton tra jectory and double logarithms. Effective action for the high energy gravity. Hamilton-Jacobi equation for effective currents.

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