Second Order Twist Contributions to the Balitsky-Kovchegov Equation at Small-X: Deterministic and Stochastic Pictures

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 Weizsäcker-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 ⊥

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 |F+α(0, ξ−, x1⊥)F+α(0)F+β (0, ξ−, x2⊥)F+β (0)|P (4) 1 1 −iP xξ− Td(x, 2 ) = + dξ−e x⊥ 2πP D a + b E × P |F+α(0, ξ−, x⊥)γ WabF+α(0)|P (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]. W x⊥ One notes that by starting with the dipoles indepen- is the ”gauge link” or Wilson operator, which by choice of dent of the method of preparation of the double dipole 3 system, one suppresses the anomalous dimension arising First Order Twist Deterministic Equation in the Scaling Region (γ=1) in the Operator Product Expansion. However, this is 1 irrelevant for this discussion, as the discussion focuses on simply calculating the interaction of a double dipole interaction to analyze stochastic effects, which are inde- 0.8 pendent of the quadratic terms.

0.6

III. DETERMINISTIC EQUATIONS IN THE SCALING REGION 0.4

The twist expansion becomes important in the scal- 0.2 ing region, i.e. where the dipole sign is smaller that the inverse saturation momentum. In this region, the scat- 4 5 6 7 8 9 10 x Q tering amplitudes corresponds to the leading twist, with s the higher twist terms simply acting as a correction. Then, in [3] the equation for T in the extended geo- Figure 3. First Order Twist Deterministic Equation. Equa- metric scaling region (x . 1/QS0) becomes tion had been normalized to 1 and only relevant geometric ⊥ scaling region is shown. 1+2iνsp α¯S yχ(0,νsp) Td = (x Qs0) C(αs)e (7) ⊥ Where y is the rapidity and is equal to ln(x2E2), where Second Order Twist Deterministic Equation in the Scaling Region (γ=1) −3 αS NC ×10 E is the center of mass energy for the dipoles,α ¯S = π and χ(0, νsp) is an eigenvalue of the BK Kernel. Using [2] to further reﬁne this equation as is done in equation 0.45 (118), and noting that the logarithmic term comes from the integral arising in the BK equation as in equation 0.4 1 (120), I then write, with γ0 = 2 + iνsp, and C(αs) being some constant of the form αS × K 0.35

2 2 2γ Td(x , y) = C(αs) ln(x QS(y) )(xQ) 0.3 ⊥ ⊥ 2 2 2 ln (x⊥QS (y) ) 00 × e− 2α ¯S χ (γ0)y (8) 0.25 χ(γ0) α¯S y 4 5 6 7 8 9 10 Which, substituting for QS(y) = Qs0e 2γ0 yields

1 2γ Td(x , y) = C(αs) ln( )(x Qs0) Figure 4. Second Order Twist Deterministic Equation. Nor- ⊥ x2 Q2 ⊥ s0 malization is consistent with that applied to figure 3 above ⊥ !2 1 γ0 ln( )+α ¯ χ(γ0)y x2 Q2 S ⊥ S0 00 × e− 2γ0α¯S χ (γ0)y (9) second order twist are a factor of 103 smaller compared One notes that, for x QS . 1 the final term is very close to 1, thus I shall absorb⊥ it into the C term from this point to the first order twist. This gives a relative correction, forward. Figure 3 shows a representation of equation 9 with taking the final term of equation 9 to be ≈ 1 and taking x1 ≈ x2 ≈ x , of with normalization set such that the max value is 1 with ⊥ ⊥ ⊥ γ set to 1 for simplicity of presentation. One notes that this representation additionally simplified the final term 2γ Tdd/Td ∝ (x Qs0) ln (x Qs0) (11) of equation 9, the leading term of which gives the char- ⊥ ⊥ acteristic shape in the scaling region. Thus, by applying the above to equation 6, the higher twist term, following the earlier approach reads off as (with the C2 absorbing all leading terms) IV. STOCHASTIC EFFECTS IN THE SCALING REGION 2 2 2 2 2 2γ Tdd = C ln (x Qs0(y) )[x1 x2 Qs0] (10) ⊥ ⊥ ⊥ Figure 4 is another simplified representation of the de- From Munier [2], the equation for scattering ampli- S terministic equations, this time the second order twist in tude accounting for front fluctuations (Td stands for the D equation 10, which is scaled using the same normaliza- Stochastic version of Td which will be noted as Td for γδ tion as figure 3. One can see that the corrections from the clarity henceforth), with P (δ) = e− is the probability 4

Ratio of Stochastic To Deteministic First Order Corrections (γ=1) Ratio of Stochastic To Deteministic Second Order Corrections (γ=1)

1 1

0.9 0.9 0.8

0.7 0.8

0.6

0.7 0.5

0.4 0.6 0.3

0.2 4 5 6 7 8 9 10 4 5 6 7 8 9 10 x Qs

Figure 5. Ratio of Stochastic to Deterministic Equations in Figure 6. Ratio of Stochastic to Deterministic Equations in the First Order Twist the Second Order Twist

Ratio of Stochastic Corrections (Second Order to First) (γ=1) of having a front delayed by δ [2]. 1 ln 1 Z x2 Q2 S ⊥ S0 2 2 0.9 Td ∝ dδP (δ) ln(xperpQS(y) ) − δ 0 0.8 γ[ln( 1 δ] × − x2Q2 − e (12) 0.7 ln 1 Z x2 Q (y)2 2 2γ ⊥ S 1 0.6 ∝ [x QS0] dδ ln( ) − δ(13) ⊥ 2 2 0 x QS(y) 0.5 ⊥ 1 2 1 2γ ∝ ln ( 2 2 )[x Qs0] (14) 0.4 2 x QS(y) ⊥ ⊥ 0.3 Thus 4 5 6 7 8 9 10 x Qs S D 1 Td /Td ∝ ln( 2 2 ) (15) x QS0 ⊥ Figure 7. Ratio of Stochastic Equations in the Scaling Region 1 Then, in the second order twist, writing ln( 2 2 ) = x⊥Qs0 1 1 φ ≈ ln( 2 2 ) ≈ ln( 2 2 ) x1⊥Qs0 x2⊥Qs0 Figures 5-7 represent ratios of stochastic equations, with the max of each ratio in the relevant ranges–keeping 2 1 φ h 1 i 2γ[ln( 2 ) δ] S R γδ 2 − x1⊥x2⊥Qs0 − with those set in figure 4–set to 1. Figure 5 is the ratios Tdd = 0 dδe− C ln( x2 Q (y)2 ) − δ e ⊥ S of the first order twist equations, and figure 6 is the same (16) for the second order twist. One can see that the ratios Which taking δ ≈ φ + Constant, which is to say of terms grows substantially more quickly in the second large fluctuations which dominate other contributions, order twist. As expected, the stochastic terms contribute thus giving an approximate equation of larger corrections as xQ grows. Figure 7 represents the

S 2 2 2 2 2 2 2γ ratios of stochastic corrections (Second order divided by Tdd ≈ C e− ( ) (x QS0) (17) first order). γ ⊥ The most interesting result of this analysis is that the Thus second order stochastic term is larger than the factorization approach correction given by (T S)2 by a factor 1 d S D of Tdd/Tdd ∝ 2γ 2 2 (18) γ(x QS) ln(x QS) ⊥ ⊥ !2 1 S 2 S 2 2 (Td ) = Tdd (20) S S 1 γ ln (x Qs0) Tdd/Td ∝ (19) ⊥ 1 ln x2 Q2 ⊥ S This then gives that there is a correction about equivalent to the square of the gluon distribution function, 5 implying that the dynamics of the scattering may need component is inversely proportional to the gluon distri- corrections corresponding to ladder diagrams, thus con- bution of the target dipole. necting to approaches in BFKL and GGM [3]. In this work, I have presented a proof-of-concept of an approach that may be applied to later studies to solve the problem of a Stochastic version of the BK evolution V. SUMMARY AND CONCLUSIONS equations. Additionally this work may be implemented in the study of proper Deep-Inelastic scattering rather By implementing the stochastic corrections to the than in the toy model presented in this paper. second-order twist of a dipole-dipole interaction, it is ap- Further work in this vein would include a mechanism parent that, in the stochastic region, the factorization by which to measure this effect as including effects of approach does not properly capture the dynamics, lead- nuclear fluctuations on small nuclei. Such work would ing to a large correction on the second order term. Fur- require similar analysis of the JIMWLK equation. ther studies in this area, including fixing leading terms in a precise DIS context would allow for a better un- derstanding of this effect. This work has translated the work previously done on the fluctuations of higher twists VI. ACKNOWLEDGMENTS to the operator representation of the gluon distribution, as is seen in [33] [34] [34]. Further, this work shows, I would like to thank Alfred Mueller of Columbia Uni- through this connection, that the corrections to the scat- versity for his invaluable help with this paper. Without tering amplitude from the second order twist stochastic his support this paper could not have been written.

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