High energy QCD at NLO: from light-cone wave function to JIMWLK evolution

Journal of High Energy Physics, May 2017

Soft components of the light cone wave-function of a fast moving projectile hadron is computed in perturbation theory to the third order in QCD coupling constant. At this order, the Fock space of the soft modes consists of one-gluon, two-gluon, and a quark-antiquark states. The hard component of the wave-function acts as a non-Abelian background field for the soft modes and is represented by a valence charge distribution that accounts for non-linear density effects in the projectile. When scattered off a dense target, the diagonal element of the S-matrix reveals the Hamiltonian of high energy evolution, the JIMWLK Hamiltonian. This way we provide a new direct derivation of the JIMWLK Hamiltonian at the Next-to-Leading Order.

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High energy QCD at NLO: from light-cone wave function to JIMWLK evolution

Received: October High energy QCD at NLO: from light-cone wave function to JIMWLK evolution Michael Lublinsky 0 1 2 Yair Mulian 0 2 Open Access 0 c The Authors. 0 0 Beer-Sheva 84105 , Israel 1 Physics Department, University of Connecticut 2 Department of Physics, Ben-Gurion University of the Negev Soft components of the light cone wave-function of a fast moving projectile hadron is computed in perturbation theory to the third order in QCD coupling constant. At this order, the Fock space of the soft modes consists of one-gluon, two-gluon, and a quark-antiquark states. The hard component of the wave-function acts as a non-Abelian background eld for the soft modes and is represented by a valence charge distribution that accounts for non-linear density e ects in the projectile. When scattered o a dense target, the diagonal element of the S-matrix reveals the Hamiltonian of high energy evolution, the JIMWLK Hamiltonian. This way we provide a new direct derivation of the JIMWLK Hamiltonian at the Next-to-Leading Order. NLO Computations; QCD Phenomenology Contents 1 Introduction and summary Basics of JIMWLK Light cone QCD Hamiltonian Field quantisation Light cone wave-function of a fast hadron Eigenstates of free Hamiltonian Eikonal approximation for QCD Hamiltonian Eikonal scattering LO JIMWLK Hamiltonian NLO JIMWLK Hamiltonian The light cone wave function at NLO Third order perturbation theory Matrix elements Technical aspects of the calculation Computation of the NLO wave function Quark anti-quark state Two gluon state One gluon state The nal result Computation of qq Computation of Computation of Computation of Computation of Computation of JJSSJJ and JJJSJ Virtual contributions The NLO JIMWLK Hamiltonian assembled Reduction of the (LO)2 contribution C Integrals and Fourier transformations D Properties of the NLO JIMWLK kernels The phase of the wave function G Supplementary for section 3 G.1 Supplement for computation of G.2 Supplement for computation of G.3 Supplement for computation of G.4 Supplement for computation of G.5 Supplement for computation of G.6 Supplement for computation of H Supplement for section 4 H.1 Supplement for computation of qq H.2 Supplement for computation of H.3 Supplement for computation of H.4 Supplement for computation of JJSSJJ Colour Glass Condensate (CGC). functional equation of the form Introduction and summary dO = HJIMWLK O : Here the rapidity Y JIMWLK was argued to be incomplete). to the lowest order (LO) in recent progress in s corrections to the triple Pomeron vertex. framework. rst part, the light coordinate). Soft gluons with momenta smaller than the cuto do not participate in scattering. at NLO schematically has the form j i = (1 ) j no soft gluonsi + ) j one soft gluoni ) j two soft gluonsi + gs2 6 j quark antiquarki : rather than being sourced by any topology. in which s corrections enhanced by the density were resummed. the cuto space. In fact, is a rapidity evolution operator = e Y HJIMWLK Y HJIMWLK + Y 2 HJ2IMWLK : : : de ned as HJIMWLK d Y j Y =0 our direct calculation. check on our calculation. devoted to the calculation of Basics of JIMWLK tonian formalism. Light cone QCD Hamiltonian nates, four-vectors are x = (x+; x ; x), where x+ 0 + x3 and x x3 stand metric [86]. The light-cone gauge: Aa+ = Aa0 + Aa3 = 0: sketched in appendix A: HLC QCD dx d2x 1 a(x ; x) a(x ; x) + gf abcAibAjc: i 2 (1; 2) is a transverse component index; denotes Dirac's 4-component quark spinor,1 with free Hamiltonian H0 given by: (@iAja)2 + i +y @@i+@i The interaction Hamiltonian Hint reads gf abcAibAjc@iAja + f abcf adeAibAjcAidAje @+ (Aib@+Aic) @1+ (Ajd@+Ae) j @+ (Aib@+Aic) @1+ ig2 +ytatb iAia @1+ ( +yta +) + 2g2 1 gauge group. Field quantisation gauge elds: Aia(x) = Z 1 dk+ Z aia(k+; k)e ik x + aiay(k+; k)eik x : p+) (2)(k Transforming to coordinate space, aia(k+; k) = e ik z aia(k+; z) ; aiay(k+; k) = eik z aiay(k+; z); 1The avour index is suppressed in this section. p+) (2)(x b (k+; k)e ik x + d y(k+; k)eik x 1 = 2 = 2 1 1 2 : b 1 (k+; k); b y(p+; p)o = n 2 d 1 (k+; k); d y(p+; p)o 2 = (2 ) b 1 (k+; x); b y(p+; y)o = n 2 d 1 (k+; x); d y(p+; y)o 2 = 2 H0 = Z 1 dk+ Z (2 )2 2k+ aiay(k+; k) aia(k+; k) + X hb y(k+; k) b (k+; k) d (k+; k) d y(k+; k) the commutation relation becomes: For the quark elds: +(x) = = 12 The polarisation vectors are: Z 1 dk+ Z + 12 = 2 = The anti-commutation relations: Light cone wave-function of a fast hadron , which is implicitly related are not are also referred as valence modes. valence modes, and A for the soft gluon eld. of partons in the LCWF get shifted up ( gure 1): j iY0 = j i : j iY = j i get lifted above Ignoring the modes below momenta. be to Eigenstates of free Hamiltonian aia(k+; k) j0i = 0; bi (k+; k) j0i = 0; di (k+; k) j0i = 0; for any One gluon state. jgia(k)i (2 )3=2 j0i gia(k+; z) (2 )1=2 j0i ; These is a normalised state with the normalisation Two gluon state. Dgjb(p) jgia(k) E = ab ij (2)(k p) (...truncated)


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Michael Lublinsky, Yair Mulian. High energy QCD at NLO: from light-cone wave function to JIMWLK evolution, Journal of High Energy Physics, 2017, pp. 97, Volume 2017, Issue 5, DOI: 10.1007/JHEP05(2017)097