Next-to-leading corrections to BFKL and BKP equations

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Nuclear Physics B (Proc. Suppl.) 245 (2013) 188–194 www.elsevier.com/locate/npbps

Next-to-leading corrections to BFKL and BKP equations L. N. Lipatov St. Petersburg State University, Theoretical Physics Department Petersburg Nuclear Physics Institute Gatchina, 188300, St. Petersburg, Russia

Abstract We consider the BFKL and BKP equations in the leading and next-to-leading approximations in QCD and in N=4 SUSY. In QCD the spectrum of the Pomeron singularities of the t-channel partial waves is discreet. The BKP equation in LLA is integrable. The effective action for the reggeized gluons gives a possibility to calculate the next-to-leading corrections to the gluon Regge trajectory and Reggeon couplings. We discuss the M¨obius invariance of the integral kernels of the BFKL and BKP equations in the next-to-leading order for the color singlet and adjoint representations. In the N = 4 SUSY the BFKL intercept can be calculated in the strong coupling expansion due to the AdS/CFT correspondence. We discuss also the generalization of the effective action approach to the gravity and supergravity. In these models the gravito-graviton scattering amplitude is calculated in the double-logarithmic approximation.

1. Pomeron and Odderon At high energy collisions in QCD the final particles are produced essentially in the multi-Regge kinematics corresponding to the large pair energies in comparison with their transverse momenta and momentum transfers. The total cross section for hadron scattering should satisfy the Froissart bound. A(s, 0) < c ln2 s . s→∞ 2s Further, for the difference of total cross sections for the particle-particle and particle-anti-particle interactions the Pomeranchuck theorem lim σt (s) = 

σtpp (s) − σtp p¯ (s) =0 s→∞ σtpp (s) lim

should be fullfilled. In the framework of the Regge model [1] the special Regge poles - Pomeron and Odderon are introduced to provide the corresponding contributions to the cross sections σt (s) ∼ sΔP , σtpp − σtp p¯ ∼ sΔO , 0920-5632/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.nuclphysbps.2013.10.038

Their Regge trajectories are assumed to be linear ωr (t) ≈ Δr + αr t , ΔO < ΔP  1 , where Δr are the intercepts. 2. Gribov Pomeron calculus It turns out, that there could be more complicated contributions to the scattering amplitudes - Mandelstam cuts, corresponding to the exchange of several Pomerons in the crossing channel [2]. The corresponding singularities at j = j(t) appear in the t-channel partial waves φ j (t) according to the fact, that these functions satisfy multi-particle unitarity conditions [3]. The effective quantum field theory describing all Mandelstam contributions was constructed by V. Gribov [4]. In this theory the clusters of particles are ordered in their rapidities  k2 + m2r + k 1 yr = ln  , yr − yr−1 1. 2 k2 + m2r − k

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and the interaction among these clusters is performed by the exchange of Pomerons. Assuming, that the Regge trajectory for Pomeron is linear, its Green function has a non-relativistic form G0 =

1 E+Δ−

k2

, E = −ω , α =

2m

The BFKL equation for the Pomeron wave function can be written as follows [6] E Ψ( ρ1 , ρ2 ) = H12 Ψ( ρ1 , ρ2 ).

1 . 2m

The total cross-section grows in LLA

In this case the Gribov effective action for Pomerons can be written as follows  S = dy d2 ρ , L = φ∗ (∂y − Δ)φ +

4. Balitsky-Fadin-Kuraev-Lipatov equation

σt ∼ sΔ , Δ = −

The BFKL Hamiltonian is given below H12 = h12 + h∗12 , h12 = ln (p1 p2 )+ 1 1 (ln ρ12 )p1 + (ln ρ12 )p2 + 2γE , p1 p2

1 |∂μ φ|2 + iλφ∗ φ2 + ... 2m

One of its possible solutions obtained by V. Gribov corresponds to the weak coupling with universal Pomeron residues and vanishing diffractive processes at small momentum transfers q Δ = 0 , γii (0) = const , γir (0) = 0 ,

We used here the complex notations ρ12 = ρ1 − ρ2 , ρr = xr + iyr . Hamiltonian H is M¨obius invariant [7] ρk →

which is similar to the scattering with the t-channel exchange of a photon or a graviton.

aρk + b . cρk + d

Conformal weights are given below m=γ+

3. Gluon reggeization in QCD In QCD the Born amplitude at high energies responds to the helicity conservation  

AB MAB = 2s g T Ac  A δλA λA

√

s cor-

1 g T Bc  B δλB λB . t

In the leading logarithmic approximation (LLA) the gluon is reggeized M(s, t) = M|Born sω(t) , α s ln s ∼ 1 . Its Regge trajectory is given below ω(−|q|2 ) ≈ −

α s Nc g2 |q2 | ln 2 , α s = . 2π 4π λ

The amplitudes for produced gluons with the definite helicity in multi-Regge kinematics has the factorized form [5] BFKL ∼ M2→2+n

...Cn+1,n

sω1 1 sω2 2 d1 gT C ... 2,1 c c 2 1 |q1 |2 |q2 |2

n+1 sωn+1 q2 q∗1 , C = . 2,1 q∗2 − q∗1 |qn+1 |2

α s Nc E0 . 2π

n n 1  = γ − , γ = + iν , m 2 2 2

in terms of anomalous dimensions γ and conformal spin n. The energy is given by the expression E = m + m , m = ψ(m) + ψ(1 − m) + 2γE . The Pomeron intercept Δ = g2 Nc /π2 is positive in LLA and one should take into account mult-reggeon exchanges. 5. BKP equation in multi-color QCD The wave function satisfying theBartels-KwiecinskiPraszalowicz equation for the composite state of several gluons EΨ = HΨ , H =

 T k T l k
−Nc

Hkl

has the property of holomorphic factorization at large Nc  Ψ( ρ1 , ... ρn ) = ar,s Ψr (ρ1 ...ρn )Ψ s (ρ∗1 ...ρ∗n ). r,s

If we introduce the monodromy matrix by the definition   A(u) B(u) t(u) = L1 (u)...Ln (u) = , C(u) D(u)

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 Lk (u) =

u + ρk pk −ρ2k pk

pk u − ρk pk

 ,

its trace commutes with the hamiltonian T (u) = A(u) + D(u) , [T (u), h] = 0 and one obtains an integrable XXX spin model [8]. Moreover, H has the dual symmetry [9]. In the case of the BFKL equation for the composite state in the adjoint representation we have an integrable open Heisenberg spin chain [10].

and construct the effective action for the reggeon interactions [12]  S = T r d4 x(LY M + V+ ∂2μ A− + V− ∂2μ A+ ), ⎞ ⎛  x± ⎟ 1 ⎜⎜⎜⎜   ±⎟ v± (x )d(x ) ⎟⎟⎠ . V± = − ∂± P exp ⎝−g g −∞ The Feynman rules in the momentum space for this action are known [13]. 8. Next-to-leading octet BFKL kernel

6. Spectrum of Pomerons in QCD In the next-to-leading order for the BFKL equation one should take into account the effects of the asymptotic freedom

The infrared stable BFKL kernel in two loops for the octet states in N=4 SUSY K( q1 , q1 ; q) = Kr + δ2 ( q1 − q1 ) q12 q22 Ω . was constructed recently [14]. Here Kr is the contribution from the production of real particles and Ω is the Regge trajectories in two loops

1 α s (k2 ) = . 2 2π β0 ln Λ2k QCD

In this case its solution has the form  ⎞ β 12ω ⎛ ⎞ ⎛   ∞ ⎜⎜⎜ k2 ⎟⎟⎟i ν ⎜⎜⎜ Γ 12 + iν ⎟⎟⎟ 0 n  ⎟⎟⎠⎟ fn (k) = d ν ⎜⎜⎝ 2 ⎟⎟⎠ ⎜⎜⎜⎝  . 1 ΛQCD −∞ Γ 2 − iν and we have a discreet spectrum of Pomerons, which allows us to describe the gluon distributions g(x, k2 ) measured in HERA [11] ωn ≈

0.5 , g(x, k2 ) = 1 + 0.95n

∞ 

cn x−ωn fn (k).

n=1

It turns out, that the essential transverse momenta in the Pomeron wave functions rapidly grow with the quantum number n k¯ n ≈ ΛQCD e4n , k¯ 3 ≈ 10 T ev and the new physics BSM is essential for the theoretical description of the HERA data [11]

Ω = ωg (− q12 ) + ωg (− q22 ) − 2ωg (− q 2 ) ⎞ ⎛ α s Nc  α Nc  ⎜⎜⎜ q12 q22 ⎟⎟⎟ =− 1 − ζ(2) ln ⎜⎝ 4 ⎟⎠ . 2π 2π q The total kernel can be reduced to the M¨obius invariant form in the momentum space with the use of the infinitesimal similarity transformation [15] (4) (4) (4) ˆ = Kˆ non , Kˆ r(4) = Kˆ inv + Kˆ non , [Kˆ B , O]

where

1   ˆ 2 ˆ 2 ˆ B Oˆ = ln q1 q2 , K . 4

9. Amplitude A2→4 in N = 4 SUSY With the use of the dispersion relations for the production amplitude in the multi-Regge kinematics [16] and the solution of the BFKL equation for the octet state [14] we can calculate the remainder factor R = BDS for the BDS amplitude A2→4 /A2→4

7. High energy effective action in QCD Using the Gribov idea about the locality of reggeon interactions in the rapidity space y=

1 k + |k| ln , |y − y0 | < η , η  ln s , 2 k − |k|

one can introduce the gluon and reggeized gluon fields vμ (x) = −iT a vaμ (x) , A± (x) = −iT a Aa± (x)

R eiπδ = cos πωab + i  f =

∞

−∞

|w|2iν dν ν2 +

n2 4

∞ a  (−1)n eiφn f , 2 n=−∞



−1 Φ(ν, n) √ u2 u3

ω(ν,n) .

The unharmonic ratios are defined as follows ss2 s1 t3 s3 t1 , u2 = , u3 = u1 = s012 s123 s012 t2 s123 t2

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and we introduced the variables  1 − u1 − u2 − u3 u2 iφ e , cos φ = . w= √ u3 2 u2 u3 The parameters δ and ωab are known in all orders in terms of the cusp anomalous dimension [16] δ=

γK γK |w|2 , ωab = ln ln |w|2 . 8 8 |1 + w|4

The product of impact factors was found in [17] ⎞ ⎛ 2 2 ⎟⎟ ⎜⎜⎜ Eνn /8 3 n + ζ(2)⎟⎟⎟⎠ . + Φ = 1 − a ⎜⎜⎝ 2 n 2 2 2 (ν + 4 )

11. BFKL equation in N = 4 SUSY The eigenvalue of the BFKL kernel in two loops for the Pomeron wave function can be written as follows [19] ω = 4 aˆ χ(n, γ) + 4 aˆ 2 Δ(n, γ) , aˆ =

In N=4 SUSY it has the hermitian separability [20] Δ(n, γ) = φ(M) + φ(M ∗ ) − where M = γ +

ρ(M) + ρ(M ∗ ) , 2ˆa/ω

|n| 2,

Eigenvalues of the BFKL kernel ω(ν, n) = −aEν,n −

FL a2 νn

ζ(2) , β (M) = ρ(M) = β (M) + 2

,



are known in leading Eνn = −

|n|/2 ν2 +

n2 4

|n| + 2 ψ(1 + iν + ) + 2γE , 2

and next-to-leading orders [14] FL = 3ζ(3) − ζ(2) Eνn − νn

⎛  ⎜⎜ ψ (1 + iν + − ⎜⎜⎜⎝ 2



1 |n| ν 4 ν2 + 2

2 − n4  n2 3



4

⎞ |n| iνψ (1 + iν + |n|2 ) ⎟⎟⎟ 2) − ⎟⎟⎠ . 2 ν2 + n4

10. Integral kernel for Odderon state NLO Odderon Hamiltonian is written below [18] Kˆ =

3 

Kˆ kr + Kˆ 123 , K123 = g4 R123 + perm.

R123 = − −

 1 q2 + k1 1 |q2 + k1 |2 q∗ 1 q3 ln |k|1 |k|3 q∗1 q3 k1 k3 q∗2 + k1∗ 4|q2 |2

1 |q2 + k3 |2 q1 q3 1 q2 + k3 ln + c.c. |k|1 |k|3 q1 q∗3 k1 k3 q∗2 + k3∗ 4|q2 |2

One can introduce color symmetric, anti-symmetric and abelian kernels d , K d = K f + K af + K sa . Kˆ 12 = Kˆ 12

In particular, for the Odderon equation we should use the color symmetric kernel. The expected value of the Odderon intercept in NLO is ΔBLV = 0.

Ψ



M+1 2



4

−

Ψ

  M 2

4

and the maximal transcendentality [21] because     φ = 3ζ(3) + Ψ (M) − 2Φ + 2β (M) Ψ(1) − Ψ(M) ,   ∞  (−1)k Ψ(k + 1) − Ψ(1)  Ψ (k + 1) − . Φ= k+M k+M k=0 The predicted singularities of the anomalous dimension for the twist-2 operators at ω = 0 are in an agreement with the direct calculations in N = 4 SUSY up to 5 loops [22, 23, 24]. 12. Pomeron and graviton in N=4 SUSY The BFKL kernel in a diffusion approximation j = 2 − Δ − D ν2 , γ = 1 +

k
Triple gluon interactions in complex notations are

g2 Nc , 16π2

j−2 + iν 2

at large coupling constants satisfies the energymomentum conservation requirement providing that D = Λ. Due to the AdS/CFT correspondence [25, 26, 27] this approximation corresponds to the linear Regge trajectry j=2+

α R2 t , t = E 2 /R2 , α = Δ. 2 2

With the use of the known values of the anomalous dimensions at large coupling constants one can calculate several terms of the strong coupling expansion for the Pomeron intercept at large λ = g2 Nc [22, 28, 29] 1 2 + 6ζ3 2 1 + ... j = 2 − √ − + −3/2 + λ λ2 4λ λ

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The slope of the anomalous dimension is known exactly 1 λ2 λ 2 λ3 + − 2 24 2 24 5 242 √ √ λ I3 ( λ) 7 λ4 11 λ5 + − + ... = − √ . 20 244 35 245 4 I2 ( λ) The perturbation series for γ at large Nc ∞  γ(ω) = λk ck (ω) , ω = j − 1 γ (2) = −

k=1

has a finite radius of convergency, because λ−k 1 cr lim ck (ω) = 3/2 √ a, k→∞ k 2 π where λcr (ω) and a can be found in terms of parameters of the diffusion approximation for the BFKL equation ω = ω0 (λ) − D(λ) ν2 + ...

It will be interesting to calculate the intercept of the Pomeron ω0 (λ) and the diffusion coefficient D(λ) in this model for arbitrary ω by comparing above formulas for ck (ω) with the explicit results for γ obtained in 5 loops. 13. High energy action in gravity

gμν = ημν + hμν , ∂+ A++ = ∂− A−− = 0 . The action for the high energy gravity has the form  √  1 d4 x −g R + ΔL , S =− 2κ  1  − 2 ++ ∂+ j ∂μ A + ∂− j+ ∂2μ A−− . ΔL = 2 The introduced currents j± satisfy the Hamilton-Jacobi equations ±

±

±

A2→n =

ω(q2 ) ω(q2 ) s 1 s2 2   2 μ ν  1 −s Γμν Γρ1 σ1 2 ...Γρρσσ 2 q1 q2

,

where the graviton-graviton-reggeized graviton vertex    κ  Γμμ Γνν + Γμν Γνμ Γμμνν = 4 can be expressed in terms of the gluon-gluon-reggeized gluon vertex 

Γμμ = −δμμ +

pμA pμB + pμA pμB pA p B

+

q2 pμB pμB 2(pA pB )2

.

where Cσ is the corresponding gluon vertex and  A   pB 2 2 p N = q1 q2 − kpA kpB is proportional to the bremstrahlubng factor in QED. 15. Graviton trajectory in super-gravity

One can construct also the effective action for the reggeized gravitons in gravity [30]. For this purpose we introduce the fields h and A describing the gravitons and reggeized gravitons

±

Production amplitudes in gravity at LLA have the factorized form [31, 32]

Similarly the reggeon–reggeon-graviton vertex can be written as follows  κ Γρσ = CρCσ − Nρ Nσ , 4

through the solution of the relations  λcr ω0 (λcr ) . ω = ω0 (λcr ) , a = D(λcr )

μν

14. High energy amplitudes in gravity

±

g ∂μ ω ∂ν ω = 0 , j = 2x − ω . One can find their perturbative solution  2 1 ∂ρ h±± + ... ∂± j∓ = h±± − hρ± − 2 ∂± On the other hand one can construct this solution as an extremal value of the Hamilton-Jacobi functional  τ e(τ) μν ω = min dτ (pμ xμ − g pμ pν ). xν ,pμ −∞ 2

Graviton Regge trajectory can be written as follows [31, 32, 33]  q2 d2 k α 2 f (k, q) , ω(q ) = π k2 (q − k)2 where



f (k, q) = (k, q − k)2 −q2 +

1 1 + k2 (q − k)2



κ2 N (k, q − k) , α = 2 . 2 8π

We inluded here the gravitino contribution from the Nextended supergravity. The gravitino action is given below  N  1 μνρσ 4 d x S 3/2 = −  ψ¯ rμ γ5 γν ∂ρ ψrσ . 2 r=1 The graviton Regge trajectory contains the infrared and ultraviolet divergencies   |q|2 N − 4 |Λ|2 2 2 ω(q ) = −α|q| ln 2 + ln 2 . 2 λ |q|

L.N. Lipatov / Nuclear Physics B (Proc. Suppl.) 245 (2013) 188–194

The ultraviolet divergencies is fictive because in one loop the gravity is renormalizable. The ultraviolet cutoff Λ should be substituted by a quantity proportional to the invariant s, which would lead to the doublelogarithmic terms. These terms in each orders of the perturbation theory were calculated in Ref. [35] basing on the approach developed for the gauge theories in Ref. [36].

16. Double-logarithms in gravity It is convenient to write the Mellin representation for scattering amplitudes in the double-logarithmic (DL) approximation A(s, t) = ABorn s−α|q|

2

ln

|q|2 λ2

Φ(ξ) ,

where s |q|2

ξ = α |q|2 ln2

193

17. Eikonal DL approximation One can use in gravity the eikonal resummation ADL (s, t) = −α|q|2 ln

−2is s

|q|2 μ2



d2 ρ ei q ρ (eiδDL ( ρ, ln s) − 1) ,

The eikonal phase here can be calculated in the DL approximation  2 d q −i q ρ s κ2 δDL ( ρ, ln s) == e Φ(ξ) , 2 (2π)2 |q|2 where ξ = α |q|2 ln2

s . |q|2

The expansion of the eikonal expression up two two loops A4N=8 =

ln3 |q|s2 κ 2 s2 2 2 (−iπs)α |q| 3 |q|2

is in an agreement with explicit results [37, 38].

and  Φ(ξ) =

a+i∞

a−i∞

 ω s dω fω . 2πi ω |q|2

The infrared evolution equation suggested earlier for the gauge theories [36] can be written for super-gravity in the form [35]  fω = 1 + α|q|

2

 d fω N − 6 fω2 − . dω ω 2 ω2

Its solution in terms of parabolic cylinder function can be presented as follows  x2  fω 2 1 d = ln e 4 D 6−N (x) , √ 2 ω 6−N b dx where x = below

ω √ . b

The perturbative expansion of Φ is given

Φ(ξ) = 1 −

−

N − 4 ξ (N − 4)(N − 3) ξ2 + 2 2 2 4!

N−4 ξ3 (5N 2 − 26N + 36) + ... 8 6!

It turns out, that for N > 5 the scattering amplitude divided by the Born term tends to zero at large energies.

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