Vertices of Three Reggeized Gluons and the Unitarity Corrections to the Propagator of Reggeized Gluons

Vertices of three reggeized gluons and the unitarity corrections to the propagator of reggeized gluons

S. Pozdnyakov(1,2), S. Bondarenko(2),

1) Saint Petersburg State University, Russia 2) Ariel University, Israel

1 / 25 Plan

1 The Lipatov’s eﬀective action 2 How correlators and evolution equations are derived from it? 3 One loop QCD Regge Field Theory 4 Vertices of three reggeized gluons Eﬀective action

The Lipatov’s eﬀective action is gauge invariant and written in the covariant form in terms of gluon ﬁeld v as

Z 4 1 a µν Seﬀ = − d x G G 4 µν a

1 h 2 + 2 − i + tr ( A+(v+) − A+ ) ∂ A + ( A−(v−) − A− ) ∂ A , N i a i a

where 1 R x± 0± ± ± g −∞ dx v± A±(v±) = ∂± O(x , v±); O(x , v±) = P e . g There are additional kinematical constraints for the reggeon ﬁelds

∂− A+ = ∂+ A− = 0 ,

[1] L. N. Lipatov, Nucl. Phys. B 452, 369 (1995); Phys. Rept. 286, (1997) 131;

3 / 25 In the framework with an external source of the color charge introduced, keeping only gluon ﬁeld depending terms, we rewrite action as

Z 1 S = − d 4 x G a G µν + v J−(v ) + v J+(v ) , eﬀ 4 µν a − − + +

Under variation on the gluon ﬁelds these currents reproduce the Lipatov’s induced currents

± ind ± δ v± J (v±) = (δ v±) j∓ (v±) = (δ v±) j (v±) ,

The classical equations of motion for the gluon ﬁeld vµ which arose from the action are the following:

µν µν b c µν + ν+ − ν− ( Dµ G )a = ∂µ Ga + g fabc vµ G = ja δ + ja δ

4 / 25 One-loop correction

a a We have found a solution to these equations of motion vi cl , v+ cl with NNLO precision and determined in the original Lagrangian:

a a a a a a a vi → vi cl + εi , v+ → v+ cl + ε+ , v− = 0 ,

+∞ +∞ ia X k ia a X k a vcl = g vk (A+, A−), v+cl = g v+k (A+, A−) k=0 k=0

and after integration over fluctuation εi,+ we have an expression for an effective one-loop action: Z 4 cl cl a + cl a 2 a Γ = d x LYM (vi , v+ ) − v+ cl Ja (v+ ) − A+ ∂i A− ı + ln 1 + G(v cl ) M(v cl ) , 2 which is the functional of only reggeized gluon fields.

5 / 25 The Lipatov’s eﬀective action formulated as RFT

In general case, consider the Lipatov’s effective action for reggeized gluons A±, formulated as RFT (Regge Field Theory) which can be obtained by an integration out the gluon fields v in the generating functional for the Seff [v, A]: Z eı Γ[A] = Dv eı Seff [v, A ]

6 / 25 The Lipatov’s eﬀective action formulated as RFT

The result of this action is that the eﬀective action Γ can be expanded in terms of reggeon ﬁelds A− and A+ as

X a1··· an Γ = A a1 ···A an K + ··· + A b1 ···A bm + + − ··· − b1··· bm − − n,m = 1 + = − A a ∂2 A a + A a K a b A b + ··· , + x i − x + x xy − − y that determines this expression as functional of reggeon ﬁelds and provides eﬀective vertices of the interactions of the reggeized gluons in the RFT calculus.

[2] S. Bondarenko, L. Lipatov, S. Pozdnyakov, A. Prygarin, Eur. Phys. J. C 77 (2017) no.9, 630.

7 / 25 A hierarchy of correlators in the formalism of Lipatov’s eﬀective action

Generating functional for calculation the correlators of the reggeon ﬁelds

Z Z Z 4 a − a + 4 a + a − Z[J] = Dexp ı Γ[A]−ı d x J−(x , x⊥) A+(x , x⊥) − ı d x J+ (x , x⊥) A−(x , x⊥)

a a1 ··· an a X ˜ a1 an < A±(x, η) >= K(η; x, x1 ··· , xn) < A± (x1) ··· A± (xn) > n = 1

Taking the derivative of this equation with respect to η, we can also obtain a BFKL-like evolution equation for reggeized gluon ﬁelds:

! ∂ ∂ X a a1 ··· aa < A a > = K˜(η) < A a1 ··· A an > ∂ η ± ∂ η ± ± n = 1

8 / 25 Results of variation on currents J

± 2 a1 a2 a1 a2 a1 b1 a2 ∂⊥ 1 < A± A∓ > = − ı δ δ±1 ±2 δ(x⊥ 1 − x⊥ 2) + K b < A± A∓ > + ··· . 1 ∓

a b ···bn a a1 am X ˆ 1 b1 bn a1 am < A± A± ··· A± > = K(η) < A± ··· A± A± ··· A± > . n = 1

Equations analogous to the hierarchy of Wilson line correlators in the Balitsky-JIMWLK formalism

∂ a a a δ X a b1···bn b b a a < A A 1 ··· A m > = Kˆ(η) < A 1 ··· A n A 1 ··· A m > . ∂ η ± ± ± δ η ± ± ± ± n = 1

9 / 25 Propagator of reggeized gluons

1) Eﬀective vertex of the interaction with NLO

2 ! a b + − a b δ Γ Kx y = Kx y = a b . δA+ x δA− y A+, A−, vf ⊥ = 0

2) Propagator of reggeized gluons in the form of the perturbative series Z Z ! ac ac 4 4 ab X bd dc Dxy = Dxy 0 − d z d w Dxz 0 Kzw k Dwy . k = 1

3) Final expression for the propagator with one-loop precision

2 Z d p i i Dac = δac δ(y − − x −) δ(x + − y +) D˜(p , η) e−ı pi (x −y ) , xy (2π)2 ⊥

ab Z 2 ab δ η (p2 ) 2 αs N 2 p⊥ D˜ (p⊥, η) = e ⊥ , (p ) = − d k⊥ p2 ⊥ 4 π2 2 2 ⊥ k⊥ ( p⊥ − k⊥ ) 10 / 25 One-loop correction from a hierarchy of correlators

Here in a diagrammatic form: the ovals are the correlators, the blocks are vertices (interaction kernel), and the bold dot is the bare propagator of the Reggeon ﬁeld D0:

a b 2 a + bc + a c δ ∂⊥ − (K b )− D0 − = δ .

11 / 25 Violation of kinematical conditions

2 a + a1 − aa1 + − 2 ∂⊥ x < A+(x , x⊥)A− (y , y⊥) > = − ı δ δ(x ) δ(y ) δ (x⊥ − y⊥) + Z 2 + 2 b1b2a + + b1 + b2 + a1 − + d z⊥dz1 d z1⊥K++− (x , z⊥; z1 , z1⊥; x⊥) < A+ (x , z⊥)A+ (z1 , z1⊥)A− (y , y⊥) > + Z 2 − 2 b1b2a − − b1 + b2 − a1 − + d z⊥dz1 d z1⊥K+−− (z⊥; z1 , z1⊥; x , x⊥) < A+ (x , z⊥)A− (z1 , z1⊥)A− (y , y⊥) > .

The third term in the r.h.s. of the equation depends on x− variable, whereas the correlator in the l.h.s. does not. This discrepancy means a violation of kinematical conditions, which in fact were known as not-absolute and therefore, generalizing the approach, we have to consider the reggeon ﬁelds as four-dimensional ones:

+ + − + + − + − A+(x , x⊥) → B+(x , x , x⊥) = A+(x , x⊥) + D+(x , x , x⊥) , D+(x , x = 0, x⊥) = 0

and

− + − − + − + − A−(x , x⊥) → B−(x , x , x⊥) = A−(x , x⊥) + D−(x , x , x⊥) , D−(x = 0, x , x⊥) = 0 .

12 / 25 Violation of kinematical conditions for correlator

a + a1 − a + − a1 − < A+(x , x⊥)A− (y , y⊥) > → < B+(x , x x⊥)A− (y , y⊥) > a + a1 − a + − a1 − = < A+(x , x⊥)A− (y , y⊥) > + < D+(x , x , x⊥)A− (y , y⊥) >

Treating the expression perturbatively and writing the correlators as propagators

a + b − + − ab < A+(x , x⊥)A−(y , y⊥) > = ı δ(x ) δ(y ) δ D(x⊥, y⊥) and

a + − a1 − aa1 + − − < D+(x , x , x⊥)A− (y , y⊥) >= ı G+ (x , x , x⊥; y , y⊥) ,

13 / 25 Violation of the reggeized form of the initial propagator

ab + − + − ab aa1 + − − D (x , x , x⊥; y , y , y⊥; Y ) = D (x⊥, y⊥; Y ) + G+ (x , x , x⊥; y , y⊥; Y ) aa1 + + − + G− (x , x⊥; y , y , y⊥; Y ) .

aa1 + Fourier transform of G+ with respect to x and x⊥ − y⊥ is

˜aa1 − G+ (p+, p⊥; y−, x ; Y ) = aa 2 ! 2 δ 1 2 2 Z − 2 (p⊥) Y dy = e (p⊥) Y − e2 (p⊥) Y − e−y − 1 G˜− 0 − G˜− 0 2 x− 0 0 x− p − (p2 ) Y y ⊥ ⊥

aa1 Correction G− can be directly obtained by change of the + sign in the expression on the − sign everywhere and corresponding change of the coordinates

This correction is large and does not suppressed perturbatively

14 / 25 ˜aa1 − − G+ (p+, p⊥; x , y ; Y ) =

aa 2 ! 2δ 1 2 Z −(p⊥)Y dy 1 − e(p⊥)Y − e−y − 1 G˜−0 − G˜−0 δ(p )δ(y −) . 2 x−0 0x− + p⊥ 0 y The last step in the Fourier transform is the transforms of the theta functions x − coordinat. We note that expression is zero at x − = 0, as it must be. Therefore, in order to preserve the property of the expression, we have to regularize the Fourier transform of the diﬀerence of the theta functions in this limit, we do so with the help of Sokhotski expressions: 1 1 1 + = 2 P( ) (1) p− + ı ε p− − ı ε p− where Z Z 1 f (p−) f (p−) − f (0) P( ), f (p−) = PV dp− = dp− . p− p− p− (2)

15 / 25 On reggeization of vertex of three reggeized gluons

Any vertex of the three Reggeon ﬁelds interactions is deﬁned as following

3 a b c µνρ δ Γ(vi cl (A+, A−), v+ cl (A+, A−)) Kx y z = a b c . δ Aµ δ Aν δ Aρ A+, A−, vf ⊥ = 0

η ++ ++ 2α N Z Z ++ K abc = K abc − s d s d 2ω K abc xzy;η xzy 0 4 ⊥ xzy;s (2π) 0 2 Z Z k i i 2 2 1⊥ −ı(y −ω )k1i ∗ d k⊥ d k1⊥ e . 2 2 k⊥ (k⊥ − k1⊥) where

abc ++ 1 abc 2 2 2 K = gf (θx+−z+ − θz+−x+ )δ δ ∂ . xzy 0 2 y⊥ x⊥ z⊥ y⊥ iy

16 / 25 The solution of this equation after Fourier transform leads to the reggeization of the vertex with the trajectory twice larger than the trajectory of the propagator of reggeized gluons.

¯abc Z 2 ∂K abc αs N 2 k1⊥ = 2ε(p2⊥)K¯ , ε(p2⊥) = − − d k⊥ ∂η (2π)2 2 2 p⊥ (k⊥ − p⊥)

¯abc + + ¯abc + + 2ηε(p2⊥) K x , z ; p⊥, p1⊥, p2⊥; η = K0 x , z ; p⊥, p1⊥, p2⊥; η e .

17 / 25 [3] S. Bondarenko, M.A. Zubkov, Eur.Phys.J. C 78 (2018) no.8, 617 The relation of reggeon correlators to the correlators of Wilson lines operators built from the longitudinal gluons is also clariﬁed and results of this relation, we hope, will help to understand general landscape of the high energy QCD approaches.

Correlators of Lipatov’s operators

1 R ∞ ± ± ∓ R ∞ ± ± ∓ g −∞ dx v±(x , x , x⊥) − g −∞ dx v±(x , x , x⊥) Oˆ(v±) = P e − P¯ e 2

a1 a2 an < T Oˆ1(v±) ⊗ T Oˆ2(v±) ⊗ · · · ⊗ T Oˆn(v±) > = ! δ n = C(R)n ( 2 g)n log Z[J] = C(R)n (2g)n δJa1 ··· δJan ∓ ∓ J = 0 Z Z J j ± ± X a ···a ı −2 n−2j b b dx ··· dx C 1 n ∂ (−ı) < A 1 ··· A n−2j > 1 n b1··· bn−2j 2 ⊥ ± ± j=0

18 / 25 Conclusion

1 This effective action formalism, based on the reggeized gluons as main degrees of freedom, can be considered as reformulation of the RFT calculus for the case of high energy QCD. 2 We calculated the one loop RFT contribution to the propagator of reggeized gluons in the framework of Lipatov’s effective action and Dyson-Schwinger hierarchy of the equations for the correlators of reggeized gluon fields. 3 We find additional part of the corrections that breaks the propagator’s reggeization. 4 The one loop QCD triple vertices will change the rapidity structure of the two reggeized gluons correlator as well and will allow understand better the quantum structure of the theory. 5 The continuation of high energy QCD Lipatov’s effective action to Euclidean space is performed.

19 / 25 Solution to these equations of motion

[2] S. Bondarenko, L. Lipatov and A. Prygarin, Eur. Phys. J. C 77 (2017) no.8, 527.

v+0 a = A+ a, (3) b − vi0 a = ρi (x⊥, x )Uab, (4) 2 v = − −1 (∂ ∂i U )ρb , (5) +1 a g + ab i h 1 i v = − −1 ∂j F + ∂ (∂j U )ρb − ∂ −1j + . (6) i1 a ji a g i ab j i a1

+ h R x 0+ 0+ − c −g R +∞ dx0+v (x0+,x−,x )T d i ab a g −∞ dx v+c (x ,x ,x⊥)T b + +d ⊥ U (v+) = −tr T Pe T Pe x . (7)

20 / 25 NNLO solution to these equations of motion

h 2 v = −1∂−1 ∂ j + +2 a − g + a1 b ic i c b b i c i + fabc 2∂−vi0∂+v1 + (∂i v0 )∂−v1+ + 2∂i (A+∂−v1 ) , (8)

h 1 v i = −1 ∂−1∂i L+ − (∂j ρb)( ∂ U ) 2a − a2 j g 2 − ab 1 − f v b ∂j v i c − ∂i v j c + v b (∂j v i c − ∂i v j c ) abc g j0 0 0 0j 1 1 i c b b i c − v0 v1+ + 2A+v1 + j b i c i b j c cde b j i i + ∂j (v0 v1 − v0 v1 ) + f vj0v0d v0e . (9)

21 / 25 Extending by including the quarks

We have added a part of the Lagrangian Lquark . We ﬁnd only one correction of the order g 2 and ε2. This addition will allow us to take into account the contribution with a quark loop in the gluon. Integrating out the ﬂuctuation we obtain the following expression for the contribution to the action: g 2 − tr G 0(y, x)(γν v (x))G 0(x, y)(γµv a(y)) , (10) 4 q aν q µ where Z d 4p eip(x−y) G 0(x, y) = (iγ ∂ν + m)∆ , ∆ = q ν x xy xy (2π)4 p2 − m2 + i0

Then after varying by vaρ(z) we obtain the following contributions to the equation of motion

g 2 j ρ (z) = − tr γρG 0(z, y)γµv (y)G 0(y, z)) (11) quark a 2 q a µ q

22 / 25 ρ We veriﬁed, that condition ∂ρjquark a = 0 is true. Then the classical i solutions have additional contributions vq2 and vq+2, respectively:

i −1h i i −1 + i vq2a = jquark 0a − ∂ ∂− jquark 0a , (12)

−1h − −1 + i vq+2a = jquark 0a − ∂+∂− jquark 0a , (13) where g 2 j ρ (z) = − tr γρG 0(z, y)γµv (y)G 0(y, z)) (14) quark a 2 q a µ q

23 / 25 We want to estimate the contribution of quarks to the reggeon propagator

Inserting obtained classical gluon fields solutions in the Eq. (??) action, we will obtain a action which will depend only on the reggeon fields. In order to calculate this action we need to know the components of the field strength tensor, with LO precision we have:

a a a a a G+ − 0 = 0 , Gi + 0 = ∂i A+ , Gi − 0 = − ∂− vi0 , Gi j 0 = 0 , (15)

and components of the ﬁeld strength tensor with included quarks

a a 2 a a 2 a G+ − q2 = 0 , Gi + q2 = g ∂i vq+2 , Gi − q2 = − g ∂− vqi2 , Gi j q2 = 0 , (16) that gives 1 G G µ ν = g 2 ( ∂ va ) ∂ v a + g 2 ∂ v a ∂ Aa . (17) 2 µ ν 0 q2 − i0 i q+2 − qi2 i +

24 / 25 Therefore, for the additional contributions to the eﬀective action from the inclusion of quarks we obtain to LO:

g 2 Z h S = − d 4x ∂ v a ∂ Ab + eﬀ q2 N i q+2 i − b a b a i N ∂− vqi2∂i A+ − vqi2 ∂i ∂−A+ = 0, (18)

In addition, it is a direct contribution to the eﬀective action:

 2 0 ν 0 µ a  g 2 δ tr Gq (y, x)(γ vaν (x))Gq (x, y)(γ vµ(y)) −  a b  . (19) 4 δA+ x δA− y A+, A− = 0 After the calculations, we showed that this contribution is also zero.

25 / 25