BFKL pomeron in the external field of the nucleus in (2 + 1)-dimensional QCD. Mihail Braun, Andrey Tarasov

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1 BFKL pomeron in the external field of the nucleus in (2 + 1)-dimensional QCD Mihail Braun, Andrey Tarasov

2 QCD at high energies Strong interaction at high energies is mediated by the exchange of BFKL pomerons BFKL pomerons In the limit N c!1 by triple pomeron vertex they split and fuse For hadron-nucleus scattering the relevant tree (fan) diagrams are summed by the Balitsky- Kovchegov (BK) evolution equation BFKL pomerons

3 Fan diagrams rapidity BK approximation can be justified if = exp P y is small triple pomeron coupling pomeron intercept For a large nuclear target, such that A 1/3 1, the tree diagrams indeed give the dominant contribution and loops can be dropped For supercritical BFKL pomeron with rapidity P > 0. Parameter grows 1) loop contribution becomes not small 2) one can not apply the perturbative approach

4 Loop diagrams With the growth of energy the role of pomeron loops becomes important On has to search for methods to take them into accounts Calculations of loops may become easier if one starts with the perturbative approach inside the nucleus from the start The nuclear field transforms the supercritical pomeron into a subcritical one with the intercept smaller than unity. Parameter vanishes with rapidity growth. Perturbative treatment becomes possible

5 Local reggeon field theory L = S + ( + )+g To go beyond the classical approximation we make a shift in the quantum field (y, b) = 1 (y, b)+ (y, b) L = 1 (S +2 ) ( + 1 )

6 Local reggeon field theory Propagator in the external field: the main equation of the (y, b y 0 b 0 ) =( 0 r 2 b)p (y, b y 0,b 0 )+2 (y, b)p (y, b y 0,b 0 ) with boundary conditions P (y, b y 0,b 0 )=0, y y 0 < 0, P(y 0,b y 0,b 0 )= 2 (b b 0 ) At large y the Green function behaves as the free Green function with sign opposite to and so vanishes at y!1

7 BFKL pomeron We use effective non-local field theory with L 0 = d 2 r 1 d 2 r 2 r 2 1r2 2 + H BFKL L = L 0 + L I + L E BFKL pomeron propagator interaction with the nuclear target Z L E = d 2 r 1 d 2 r 2 J Triple pomeron vertex L I = 2 2 sn c Z d 2 r 1 d 2 r 2 d 2 r 3 r 2 12 r2 23 r2 13 (y, r 1,r 2 ) (y, r 2,r 3 )K 31 (y, r 3,r 1 )+( $ )

8 Propagator in the external (y, r 1,r 2 ) = 2 with boundary conditions Z BFKL pomeron the main equation of the approach d 2 r 2 12 r 3 r13 2 P (y, r 1,r 3 )+P(y, r 2,r 3 ) P (y, r 1,r 2 ) r2 23 (y, r 1,r 3 )P (y, r 2,r 3 ) (y, r 2,r 3 )P (y, r 1,r 3 ) P (y = y 0,r 1,r 2 ; y 0,r 0 1,r 0 2)=r 2 1 r (r 1 r 0 1) 2 (r 2 r 0 2) Since the study is only possible numerically, to avoid using the singular initial condition, we shall consider a convolution with an arbitrary function Z P (y, r 1,r 2 )= d 2 r1d 0 2 r2p 0 (y, r 1,r 2 ; y 0,r1,r 0 2)r 0 2 1r 2 2 (r1,r 0 2) 0

9 BFKL pomeron With the chosen set of initial conditions, the convoluted BFKL pomeron propagator vanishes at large rapidity distances P (y =0,r 1,r 2 )=1 e c 1r 2 12 e b2 /c 2 initial condition for evolution of the convoluted propagator c1 = 10; c2 = φ=π/2 5 Calculation is possible only numerically p e-08 1e-07 1e-06 1e r 1 =r 2 [1/(GeV/c)]

10 BFKL pomeron in (2+1) dimensional QCD Propagator in the external (y, r 1,r 2 ) = 2 Z d 2 r 2 12 r 3 r13 2 P (y, r 1,r 3 )+P(y, r 2,r 3 ) P (y, r 1,r 2 ) r2 23 (y, r 1,r 3 )P (y, r 2,r 3 ) (y, r 2,r 3 )P (y, r 1,r 3 ) 2 s N c (r max 2,1 r 3 ) (r 3 r min 2,1 ) the kernel in (2+1)-dimensional QCD We expect to find analytical solution in (2+1) dimensional QCD J. Bartels, V.S. Fadin, L.N. Lipatov, Nucl. Phys. B 698 (2004) 255

11 BFKL pomeron in (2+1) dimensional QCD In terms of S-matrix S r2 r 1 (y) =1 r 2 r 1 (y) BK equation Pomeron r2 r 1 = Z r2 r 1 dr 0 S r2 r 0 S r0 r 1 S r2 r r2 r 1 = Z r2 r 1 dr 0 S r2 r 0 P r0 r 1 + P r2 r 0 S r0 r 1 P r2 r 1 r 2 r 1 (y) =e r 21y S r2 r 1 Q r2 r 1 (y) =e r 21y P r2 r 1 (y) = 2 = n o Q(y), (y) (y) = (0)[1 y (0)] 1 Q(y, y 0 )= 1 (y 0 ) (y)q(y 0,y 0 ) (y) 1 (y 0 ) Solution was found in matrix notation T (y, y 0 )Q(y 0 y 0 )T (y, y 0 )

12 BFKL pomeron in (2+1) dimensional QCD The initial condition for the BFKL pomeron propagator in the nuclear field may be chosen as: g r2 r 1 r 0 2 r0 1 (y0,y 0 )=(r 2 r 0 2) (r 2 r 0 2)(r 0 1 r 1 ) (r 0 1 r 1 ) g(y, y 0 )=T (y, y 0 )g(y 0 y 0 )T (y, y 0 ) Solution in matrix notation The solution factorizes: g r2 r 1 r 0 2 r0 1 (y, y0 )=e r 21(y y 0) U r22 0 (y, y 0 )U r1 0 1 (y, y 0 ) Z r U r (y, y 0 )= dr 0 (r r 0 )T r 0(y, y 0 )e y0 r 0 r 1 r2 0 g r2 r 1 r 0 2 r0 1 (y, y0 ) 6= 0 only if r 1 <r 0 1 <r 0 2 <r 2 r 0 1 r 0 2

13 BFKL pomeron in (2+1) dimensional QCD The pomeron self-mass: (y, r 2,r 1 y 0,r 0 2,r 0 1) = (8 2 sn c ) 2 Z max{r2,r 1 } min{r 2,r 1 } Z max{r 0 2,r0 1 } dr 3 min{r 0 2,r0 1 } dr 0 3g(y, r 2,r 3 y 0,r 0 2,r 0 3)g(y, r 3,r 1 y 0,r 0 3,r 0 1) r 3 It is trivial to see that this expression is zero due to the properties of the pomeron propagator r 0 3 r 0 3 Pomerons cannot form loops in the nuclear field. This implies that the BK equation gives the complete solution to the quantum field theory of interacting pomerons in the nuclear field in dimensions