Second Order Twist Contributions to the Balitsky-Kovchegov Equation at Small-X: Deterministic and Stochastic Pictures
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Second Order Twist Contributions to the Balitsky-Kovchegov Equation at small-x: Deterministic and Stochastic Pictures S. K. Grossberndt Department of Physics, Columbia University (Dated: January 5, 2021) I study the second order twist corrections to a toy model of a dipole-dipole interaction in the context of a both deterministic and stochastic effects. This work is done in the high NC limit in the Bjorken picture. I show that the correction to the second twist terms of the stochastic picture suggest additional importance of the second twist correction in the stochastic model as compared to the deterministic model. I. INTRODUCTION is performed on this cascade, one reaches the Balitsky- Kovchegov (BK) evolution equation which is unitary and does not diffuse into the IR, distinguishing it from ear- There has been a large body of work in the field lier equations and generates a saturation scale that grows of high energy QCD related to the scattering for vir- with energy: allowing for the suppression of the non- tual photons on bound states [1] [2] [3] [4] [5]. Im- perturbative QCD physics that may enter through the portant results in this field include the parton model initial conditions [19][20] [21] [22]. As described by Bal- of Deep Inelastic scattering, first given by Feynman [6] itsky, the relevant logarithms in this equation may arise and later modified by Bjorken and Paschos [7], and the from the expansion of non-local operators which gives rise DGLAP evolution equations [8][9][10], both cornerstones to a twist interpretation of the equation, where higher of perturbative QCD. This work was originally performed twists represent the original dipole branching and further in Lorentz-covariant Feynman diagrams techniques [11] scattering off of the nucleon [20] [22]. Balitsky developed [12], however, one may also proceed with these calcula- the structure functions as a sum over integer moments tions using light cone perturbation theory as set forth in the factorization theorem as a twist expansion, with by Lepage and Brodsky [13], in their paper following the 1 the twists corresponding to damping by powers of 2 , an work of Bjorken, Kogut and Soper [14]. This method Q allows the analysis to proceed via the light cone wave approach refined and expanded upon in [23] [24] [25] [26] function, which allows for the description of the Fock . This twist expansion appears explicitly in the BFKL approach, the anomalous dimension of the twist-2 oper- state of a hadron as a function of gluon and quark num- 2 bers. Further, as outlined in [13], the use of the light ators sets the Q dependence in the deep inelastic mo- cone gauge, which simplifies the gluon field to two inde- ments [27]. In the derivation of equation 2 of [20], the pendent components{the transverse components{and a use of light cone perturbation theory allows for the fac- single dependent component. Such work often is done torization of the relevant diagrams [19] which often allows for the diagrams to be represented in the form of Muller using large NC perturbation theory and the small-x regime [15], where the amplitudes may be descried us- Vertices [11]. ing the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation However, this equation assumes a linear scaling in the [16][17]. In these pictures, one must consider the equa- bulk of the phase space, which allows for unrestrained tions via careful choice of approximations and frame. To growth of density of the dipoles in this equation. To this end, one may consider the following simplifications: correct for this, one argues for a modeling of the non- take a target that is ultra-relativistic, thus defining the linearity using recombination in analogy to branching- ”Infinite Momentum Frame" or Bjorken Frame [4] and diffusion models. This process is captured using stochas- setting the momentum of the target much larger than tic corrections of tip fluctuations, which may not be arXiv:2003.09303v3 [hep-ph] 31 Dec 2020 the mass and the center of mass energy is high (thus the modeled analytically, and front fluctuations, which will Bjorken-x, which is in exact analogy to the Feynman-x, be considered in this paper [2]. There has been re- is small); ignore the evolution of the quark distributions cent work in Monte Carlo analysis to correspond to an- of the hadron in this frame as the distribution functions alytical calculations at infinite-time limits [28]. Such of such run significantly slower than the gluon distribu- corrections are motivated by their applicability to the tion; assume that the coupling constant is fixed (or at Fischer-Kolmogorov-Petrivsky-Piscounouv equation and least αs << 1). These assumptions give rise to the dou- its mapping to the BK equation [29]. Thus, for the pur- ble logarithmic approximation of the DGLAP evolution poses of this paper, I shall consider the equations without equations [3] . such corrections to be the "Deterministic equations" and One may consider DIS in the rest-frame of the nu- those with such to be the "Stochastic equations". cleon. In this frame, the virtual photon fluctuates into a This paper aims to connect the work done on these quark-antiquark pair, that, in the large-Nc limit may de- fluctuations on the second order twist corrections with velop into a cascade of gluons which may be described the operator language of Balitsky [20]. This will pro- by the Mueller dipole model [18]. When resumation vide corrections to the DGLAP equation to account for 2 2 + lower values of Q , taking the BFKL approach rather the gauge A+ = 0, γ Wab may be set to 1. The choice of than via string operators, which are an equivalent ap- this gauge is the light cone gauge used in [3] and is chosen proach to higher twist correction to DIS [30]. This paper as to connect equation 2 to equation 3. This connection takes an approach similar to [31], further connecting the is motivated in [32], and uses the fact that the light cone OPE and factorization approaches, then extending into gauge gives a representation of hadronic states in terms of the stochastic world. the partonic basis that does set the Weizs¨acker-Williams distribution as the number density of gluons. Equation 2 is the transform of a Transverse Momentum Dependent II. CALCULATING THE SECOND ORDER distribution which is not gauge invariant when utilized TWIST to count the number of gluons; a correction for which involves choosing the gauge link to cancel in the partic- To investigate the dynamics of the second order ular gauge, thus, leading to the method outlined in [32], twist contribution in the stochastic picture, I must first and is seen to be equivalent to the Wilson line approach evaluate the deterministic picture. In order to do this, I in [33]. This process is illustrated in figure 1, with the utilize a simplified picture where the probe may be one or relevant integral to be calculated. In this gauge, more dipoles prepared in some manner that is irrelevant to this discussion and the scatting target is an individ- F+α = @+Aα ual dipole. This toy model will allow us to consider the Using this approach allows for the application of Opera- relevant evolution equations without needing to concern tor Product Expansion to derive equation 3 from equa- ourselves with the physical compilations of true Deep In- tion 2. For the next-order twist, I consider the case of elastic Scattering or the process by which an additional a state of two dipoles scattering off of the target proton. dipole is produced from the original. Then, define the For the purposes of this discussion, I shall consider the contribution from each twist to the forward scattering method of preparation of this system to be irrelevant, but amplitude, defined as I require that the impact parameter of both dipoles be equal to zero, and that the dipoles have transverse size T = 1 − ST = Td + Tdd + Tddd + :::: (1) x1 and x2 respectively. This results in a system where ? ? With Td being the contribution from a single dipole (lead- a set of two dipoles each scatter off of the target once as ing twist), Tdd from two dipoles (higher twist) and so illustrated in figure 2. This yields the factorized equa- forth. The first twist scattering amplitude of a dipole tion, // This yields in the mean field picture, following with transverse size x and Bjorken variable x is given as [32]. ? x1? Which gives x2? x ⊥ P 1 1 1 Z + + 1 T x; x ; ; = dξdξ1eip xξ− eip x1ξ− P dd 1 2 2 + x1? x2? p D a a b 1 b E Z + × P jF+α(0; ξ−; x1?)F+α(0)F+β (0; ξ−; x2?)F+β (0)jP (4) 1 1 −iP xξ− Td(x; 2 ) = + dξ−e x? 2πP D a + b E × P jF+α(0; ξ−; x?)γ WabF+α(0)jP (2) Figure 2. Second order Twist similar analysis to that for the quark distribution in [11], Figure 1. Lowest Twist Gluon Distribution. and noting a CF arises from the splitting of the state into two dipoles as can be seen in equation (74) of [2], 2 2 1 1 αsπ 1 2 Tdd = CF Td(x; x2 )Td(x1; x2 ) (5) Td = x xG(x; ) (3) 1? 2? 2NC ? x ? 1 Which, taking x1 ≈ x and x1 ≈ x2 ≈ 2 I require the gluons to form a color singlet, as the dipole ? ? Q must also be a color singlet. By setting the scattering off 2 4 αsCF π 2 2 2 2 of a dipole, rather than utilizing a Muller vertex, I escape Tdd = 2 x1 x2 xG (x; Q ) (6) 4N ? ? the divergences that necessitate renormalization at scale C 1 2 Q, but this scale is identified with 2 = Q as in [3].