Low $x$ physics as an infinite twist (G)TMD framework: unravelling the origins of saturation

TL;DR

This paper demonstrates how low x physics can be reformulated within an infinite twist TMD/GTMD framework, unifying it with moderate x QCD descriptions and providing new insights into the origins of saturation effects.

Contribution

It introduces a comprehensive rewriting of low x physics using infinite twist TMD/GTMD frameworks, linking it with moderate x QCD and highlighting the role of higher-body PDFs in saturation.

Findings

Reformulation of low x physics in TMD/GTMD frameworks

Derivation of the BFKL limit as a kinematic boundary

Identification of three distinct saturation effects

Abstract

We show how the formulations of low $x$ physics involving Wilson line operators can be fully rewritten into an infinite twist TMD or GTMD framework, respectively for inclusive and exclusive observables. This leads to a perfect match between low $x$ physics and moderate $x$ formulations of QCD in terms of GTMDs, TMDs, GPDs or PDFs. We derive the BFKL limit as a kinematic limit and argue that beyond the Wandzura-Wilczek approximation, 3-body and 4-body unintegrated PDFs should be taken into account even in this regime. Finally, we analyze how saturation should be understood as three distinct effects: saturation through non-linearities in the evolution equations at small $x$, saturation through multiple interactions with slow gluons as TMD gauge links, and saturation as the enhancement of genuine twist corrections.

Equations

$$J(z)J(0)\to \sum _n{C}_n\left(z,\mu \right){\mathcal{O}}_n\left(\mu \right),$$

$$J(z)J(0)\to C_0\left(z,Y_c\right){\mathcal{O}}_0\left(Y_c\right)+{\alpha }_s{C}_1\left(z,Y_c\right){\mathcal{O}}_1\left(Y_c\right)+...,$$

$$x\cdot y=x^{+}{y}^{-}+x^{-}{y}^{+}+x_{\perp }\cdot y_{\perp }=x^{+}{y}^{-}+x^{-}{y}^{+}-\mathbf{x}\cdot \mathbf{y}.$$

$$A_{Y_c}^{\mu }(x)=\delta (x^{+})A_{Y_c}^{-}(x_{\perp }){\delta }^{\mu -}+O\left(m_T/\sqrt{s}\right),$$

$${\left[a^{+},b^{+}\right]}_{\mathbf{x},Y_c}^R\equiv \mathcal{P}{e}^{ig{\int }_{a^{+}}^{b^{+}}d{z}^{+}{T}_R^a{A}_{Y_c}^{a-}\left(z^{+},\mathbf{x}\right)},$$

$$U_{\mathbf{x},Y_c}^R={\left[-\infty ,+\infty \right]}_{\mathbf{x},Y_c}^R.$$

$$\frac{d}{d{Y}_c}\left(U_{{\mathbf{x}}_1,Y_c}^{R_1}...U_{{\mathbf{x}}_n,Y_c}^{R_n}\right)=-H\cdot \left(U_{{\mathbf{x}}_1,Y_c}^{R_1}...U_{{\mathbf{x}}_n,Y_c}^{R_n}\right).$$

$$\mathcal{A}=\int d^2{\mathbf{x}}_1...d^2{\mathbf{x}}_n{\mathcal{H}}_{Y_c}\left({\mathbf{x}}_1,...,{\mathbf{x}}_n\right)\frac{⟨P^{\left(\prime \right)}\left|U_{{\mathbf{x}}_1,Y_c}^{R_1}...U_{{\mathbf{x}}_n,Y_c}^{R_n}\right|P⟩}{⟨P|P⟩},$$

$$⟨P^{\prime }|P⟩=2P^{-}{\left(2\pi \right)}^3\delta \left(P^{\prime -}-P^{-}\right){\delta }^2\left({\mathbf{P}}^{\prime }-\mathbf{P}\right),$$

$$\int d{x}_1^{+}d{x}_2^{+}{d}^2{\mathbf{x}}_1{d}^2{\mathbf{x}}_2{\frac{⟨P\left|{\left[x_2^{+},x_1^{+}\right]}_{{\mathbf{x}}_2}{F}^{i-}\left(x_1\right){\left[x_1^{+},x_2^{+}\right]}_{{\mathbf{x}}_1}{F}^{j-}\left(x_2\right)\right|P⟩}{⟨P|P⟩}|}_{x_{1,2}^{-}=0}$$

$$=\frac{1}{2P^{-}}\int d{r}^{+}{d}^2\mathbf{r}{⟨P\left|{\left[0^{+},r^{+}\right]}_0{F}^{i-}\left(r\right){\left[r^{+},0^{+}\right]}_{\mathbf{r}}{F}^{j-}\left(0\right)\right|P⟩|}_{r^{-}=0},$$

$$F^{i-}(x)={\partial }^i{A}^{-}(x).$$

$$\int d{r}^{+}{d}^2\mathbf{r}{e}^{ix{P}^{-}{r}^{+}}{⟨P^{\prime }\left|F^{i-}\left(r\right){\mathcal{U}}_{\left[r,0\right]}^{\pm }{F}^{j-}\left(0\right){\mathcal{U}}_{\left[0,r\right]}^{\pm }\right|P⟩|}_{r^{-}=0}$$

$$\to \int d{r}^{+}{d}^2\mathbf{r}{⟨P^{\prime }\left|F^{i-}\left(r\right){\left[r^{+},\pm \infty \right]}_{\mathbf{r}}{\left[\pm \infty ,0^{+}\right]}_0{F}^{j-}\left(0\right){\left[0^{+},\pm \infty \right]}_0{\left[\pm \infty ,r^{+}\right]}_{\mathbf{r}}\right|P⟩|}_{r^{-}=0},$$

$${\mathcal{U}}_{\left[x,y\right]}^{\pm }=\left[\left(x^{+},x^{-},\mathbf{x}\right),\left(\pm \infty ,x^{-},\mathbf{x}\right)\right]\left[\left(\pm \infty ,x^{-},\mathbf{x}\right),\left(\pm \infty ,x^{-},\mathbf{y}\right)\right]\left[\left(\pm \infty ,x^{-},\mathbf{y}\right),\left(y^{+},x^{-},\mathbf{y}\right)\right],$$

$$\mathcal{A}$$

$$\times \left[\left(U_{\mathbf{b}+\bar{z}\mathbf{r}}^{R_1}{T}^{R_0}{U}_{\mathbf{b}-z\mathbf{r}}^{R_2}\right)-\left(U_{\mathbf{b}}^{R_1}{T}^{R_0}{U}_{\mathbf{b}}^{R_2}\right)\right],$$

$$\mathbf{k}\equiv {\mathbf{p}}_1+{\mathbf{p}}_2,$$

$$\mathbf{q}\equiv \frac{p_1^{+}{\mathbf{p}}_1-p_2^{+}{\mathbf{p}}_2}{p_1^{+}+p_2^{+}},$$

$$z\equiv \frac{p_1+}{p_0^{+}},\bar{z}\equiv \frac{p_2^{+}}{p_0^{+}}=1-z.$$

$$U_{\mathbf{b}+\bar{z}\mathbf{r}}^{R_1}$$

$$U_{\mathbf{b}-z\mathbf{r}}^{R_2}$$

$$\left({\partial }_i{U}_{{\mathbf{b}}_1}^R\right)=i{g}_s{T}_R^a{\int }_{-\infty }^{+\infty }d{b}_1^{+}{\left[-\infty ,b_1^{+}\right]}_{{\mathbf{b}}_1}^R{F}_a^{i-}\left(b_1^{+},0^{-},{\mathbf{b}}_1\right){\left[b_1^{+},+\infty \right]}_{{\mathbf{b}}_1}^R,$$

$$\mathcal{A}$$

$${\mathcal{A}}_g$$

$$\times \left(U_{\mathbf{b}+\bar{z}\mathbf{r}}^{R_1}-U_{\mathbf{b}}^{R_1}\right)T^{R_0}\left(U_{\mathbf{b}-z\mathbf{r}}^{R_2}-U_{\mathbf{b}}^{R_2}\right),$$

$${\mathcal{A}}_k^{\left(1\right)}$$

$$\times \left(U_{\mathbf{b}+\bar{z}\mathbf{r}}^{R_1}-U_{\mathbf{b}}^{R_1}\right)T^{R_0}{U}_{\mathbf{b}}^{R_2},$$

$${\mathcal{A}}_k^{\left(2\right)}$$

$$\times U_{\mathbf{b}}^{R_1}{T}^{R_0}\left(U_{\mathbf{b}-z\mathbf{r}}^{R_2}-U_{\mathbf{b}}^{R_2}\right).$$

$${\mathcal{A}}_g$$

$$\times \int d^2{\mathbf{b}}_1{d}^2{\mathbf{b}}_2{e}^{-i\left({\mathbf{k}}_1\cdot {\mathbf{b}}_1\right)-i\left({\mathbf{k}}_2\cdot {\mathbf{b}}_2\right)}\left({\partial }_i{U}_{{\mathbf{b}}_1}^{R_1}\right)T^{R_0}\left({\partial }_j{U}_{{\mathbf{b}}_2}^{R_2}\right)$$

$$\times \int d^2\mathbf{r}{e}^{-i\left(\mathbf{q}\cdot \mathbf{r}\right)}\left[-{\mathbf{r}}^i{\mathbf{r}}^j\mathcal{H}\left(\mathbf{r}\right)\frac{\left(e^{i\bar{z}\left({\mathbf{k}}_1\cdot \mathbf{r}\right)}-1\right)\left(e^{-iz\left({\mathbf{k}}_2\cdot \mathbf{r}\right)}-1\right)}{\left({\mathbf{k}}_1\cdot \mathbf{r}\right)\left({\mathbf{k}}_2\cdot \mathbf{r}\right)}\right].$$

$$\left(U_{\mathbf{b}+\bar{z}\mathbf{r}}^{R_1}-U_{\mathbf{b}}^{R_1}\right)T^{R_0}{U}_{\mathbf{b}}^{R_2}=\sum _{n=1}^{\infty }\frac{{\bar{z}}^n}{n!}\left[{\left(r_{\perp }\cdot {\partial }_{\perp }\right)}^n{U}_{\mathbf{b}}^{R_1}\right]T^{R_0}{U}_{\mathbf{b}}^{R_2}\equiv \sum _{n=1}^{\infty }{\mathcal{U}}_n.$$

Full text

1 Introduction

The continuity between moderate $x$ and low $x$ observables has been the subject of many studies in perturbative QCD. The factorization schemes involved in both cases seem indeed very different at first sight. For processes where the center-of-mass energy $s$ is of the same order as the hard partonic scale $Q^{2}$, i.e. at moderate $x=Q^{2}/s$, and for inclusive enough observables, colinear factorization applies. The simplicity of having a single hard scale carried by a single hard momentum allows for a standard Operator Product Expansion (OPE) to be performed. Such an OPE consists in the expansion of a bilocal operator into a discrete set of local operators $\mathcal{O}_{n}$, usually ordered according to their twist (dimension $-$ spin). For example for quark currents $J$ the OPE has the form:

[TABLE]

where $\mu$ is a renormalization scale and where divergences from the loop corrections to the Wilson coefficient $C_{n}\left(z,\mu\right)$ are cancelled via the renormalization of the local operator $\mathcal{O}_{n}\left(\mu\right)$. This renormalization allows to resum large logarithms of the hard scale $Q$. In the low $x$ limit, the hardest scale of the process is given by $s$, and the previous OPE is not convenient to address logarithms of this scale. The low $x$ OPE developed in Balitsky:1995ub ; Balitsky:1998kc ; Balitsky:1998ya , which we refer to as the shockwave framework, has a very different form:

[TABLE]

where $Y_{c}$ is a rapidity separation scale and where the spurious rapidity divergence from the 1-loop correction to the $C_{n+1}$ coefficient is cancelled by the Leading Logarithmic (LL) $Y_{c}$ evolution of the $n$-th operator $\mathcal{O}_{n}$, and so on and so forth. Such an evolution in $Y_{c}$ allows to resum large logarithms of $s$.

It is possible to take the low $x$ limit in Eq. (1) and to match it to the first powers of the partonic hard scale in Eq. (2). However, Eq. (1) is usually valid for the first few powers at best while Eq. (2) is valid for all powers of $Q$. On the other side, so far only the first few subleading $s^{-1/2}$ corrections to Eq. (2) have been computed Altinoluk:2014oxa ; Altinoluk:2015gia ; Altinoluk:2015xuy ; Agostini:2019avp ; Balitsky:2015qba ; Balitsky:2016dgz ; Balitsky:2017flc ; Balitsky:2017gis ; Chirilli:2018kkw ; Kovchegov:2015pbl ; Kovchegov:2016zex while Eq. (1) is valid for all powers of $s$.

Finding a general continuity between the moderate and low $x$ factorization schemes is very challenging due to the fact that the operators involved in these two schemes have different nature. While the moderate $x$ factorization schemes involve operators which consist of parton fields with the appropriate gauge links, the low $x$ schemes involve full Wilson line operators. The main focus of this paper is to address this challenge.

For several processes and in several kinematic regimes, a matching between the low $x$ Wilson line operators and some standard moderate $x$ distributions have been found. Transverse Momentum Dependent (TMD) distributions were first recovered in Dominguez:2010xd ; Dominguez:2011wm via a so-called correlation expansion, which was extended in Altinoluk:2018uax ; Altinoluk:2018byz for 3-particle final states and to infinite kinematic twist accuracy in Altinoluk:2019fui . The correspondance between low $x$ and TMD observables is the subject of many recent studies Metz:2011wb ; Akcakaya:2012si ; Dumitru:2016jku ; Marquet:2016cgx ; Boer:2017xpy ; Marquet:2017xwy . For a review on TMD gluon distributions at small $x$, the reader is referred to Petreska:2018cbf . The off-forward generalization of TMD distributions (Generalized TMD distributions, GTMD) and their Fourier transforms, the Wigner distributions, were also found in Hatta:2016dxp ; Boussarie:2018zwg . Generalized Parton Distributions (GPD), the off-forward extension of PDFs, were extracted via a twist expansion in Hatta:2017cte . In this paper, we develop a method to rewrite low $x$ physics in terms of TMD and Wigner distributions, which are known to span PDFs and GPDs as well Belitsky:2003nz ; Lorce:2011kd , allowing for a completely systematic rewriting of low $x$ observables in terms the distributions involved in the moderate $x$ regime as well. Our formulation of low $x$ physics is that of an infinite twist (G)TMD framework.

The question of gluon saturation is one of the most exciting topics in low $x$ physics. While the original Balitsky-Fadin-Kuraev-Lipatov (BFKL) description Kuraev:1977fs ; Balitsky:1978ic did not involve saturation effects, the more recent dipole Mueller:1989st ; Mueller:1993rr ; Mueller:1994gb and shockwave Balitsky:1995ub ; Balitsky:1998kc ; Balitsky:1998ya frameworks contain non-linear effects embedded in their evolution equation. More strikingly, the effective Feynman rules and Wilson line operators involved in the shockwave framework were shown to be perfectly compatible with earlier results McLerran:1993ni ; McLerran:1993ka ; McLerran:1994vd describing scattering off a heavy ion with large gluon occupancy taken into account as the very starting point. The low $x$ description of scattering off nuclear targets, known as the Color Glass Condensate (CGC) Gelis:2010nm , also has the exact same hierarchy of evolution equations as in Balitsky:1995ub ; Balitsky:1998kc ; Balitsky:1998ya . This evolution equation is called the Balitsky-Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (B-JIMWLK) evolution equation JalilianMarian:1997jx ; JalilianMarian:1997gr ; JalilianMarian:1997dw ; Kovner:1999bj ; Kovner:2000pt ; Weigert:2000gi ; Iancu:2000hn ; Ferreiro:2001qy and in the mean field approximation it reduces to the Balitsky-Kovchegov (BK) equation Kovchegov:1999yj which is known as the evolution equation in the dipole framework.

Saturation is usually understood as a combination of two distinct effects: gluon recombinations via non-linearities in the evolution equation due to $x$ being small Gribov:1984tu , and the importance of multiple scatterings due to the large gluon occupancy number for dense targets McLerran:1993ka ; McLerran:1993ni ; McLerran:1994vd . In this paper, we give a new point of view on saturation in terms of TMD physics. In particular, we distinguish 3 origins of saturation, whose effects can be studied separately. Moreover, we discuss the linear BFKL limit as a kinematic limit rather than a dilute limit.

We restrict ourselves to dilute-dense collisions where either the projectile is a photon (with or without virtuality), or where in a hadron-hadron collision the observed particles are forward enough for the so-called hybrid factorization ansatz Dumitru:2005gt ; Altinoluk:2011qy to apply. This ansatz relies on the assumption that the incoming projectile parton is produced close enough to the projectile hadron beam to be reliably described via colinear factorization. Then the observables are described as the convolution of a colinear parton distribution with a low $x$ amplitude where the incoming parton is treated as the projectile. The hybrid ansatz has been successfully applied to one-loop order Chirilli:2011km ; Chirilli:2012jd ; Stasto:2013cha ; Stasto:2014sea ; Altinoluk:2014eka ; Watanabe:2015tja ; Ducloue:2016shw ; Iancu:2016vyg ; Ducloue:2017dit ; Ducloue:2017mpb For central production, a more involved formalism similar to the one developed in Balitsky:2017flc ; Balitsky:2017gis should be used, but this is beyond the scope of this paper and we leave this case for future studies.

The paper is organized as follows. In Section 2, we give the computation steps to rewrite low $x$ amplitudes in a form that is compatible with an infinite twist TMD amplitude, involving all kinematic twist corrections to the 1-body and 2-body (half-)operators. In Section 3, we derive the cross section in the inclusive case, involving 2-body, 3-body and 4-body TMD distributions. We then show how PDFs appear in more inclusive observables and briefly discuss inclusive diffraction. In Section 4, we derive the cross section in the exclusive case, which involves a GTMD and show how GPDs appear for less exclusive observables. In Section 5, we show how the BFKL limit can be understood as a kinematic limit in the Wandzura-Wilczek approximation, and we give predictions beyond the WW approximation in terms of 2-body, 3-body and 4-body unintegrated PDFs. Finally, in Section 6, we discuss how saturation can be understood in terms of TMD physics and how one kind of saturation could also appear in the kinematic BFKL limit.

Notations and conventions

We consider the most generic small $x$ limit: $s$ is assumed to be much larger than any other scale, and our processes are assumed to have at least one partonic hard scale $Q$. Any number of hard or semi-hard scales can be involved.

We define lightcone directions $+$ and $-$ such that the projectile (resp. target) has a large momentum along the $+$ direction $p_{0}^{+}\sim\sqrt{s}$ (resp. along the $-$ direction $P^{-}\sim\sqrt{s}$). We denote transverse components as with a $\perp$ subscript in Minkowski space and by bold characters in Euclidean space. Thus, the scalar product is written as

[TABLE]

Our treatement of low $x$ physics relies on the covariant shockwave effective approach which is very similar to the CGC approach. Both of these frameworks are based on the separation of gluon fields in rapidity space: the QCD Lagrangian is separated111For central production in hadron-hadron collisions, additional separations would be introduced. into fast fields ($|k^{+}|>e^{-Y_{c}}p_{0}^{+}$) and slow fields ($|k^{+}|<e^{-Y_{c}}p_{0}^{+}$). We use the lightcone gauge $A^{+}=0,$ in which slow fields have the form

[TABLE]

where $m_{T}$ is a typical mass in the target. Note that the $\delta(x^{+})$ function accounts for the fact that the field is peaked around $x^{+}=0$, and should be treated as such rather than an actual $\delta$-distribution. The slow field is then treated as an external field, which allows for multiple interactions to be resummed into path-ordered Wilson lines. For a color representation $R$, the finite Wilson line operators are defined as

[TABLE]

and the more standard infinite lines are defined as

[TABLE]

Large logarithms of $s$ are resummed via the $Y_{c}$ evolution of the Wilson line operators, given by the B-JIMWLK hierarchy of evolution equations Balitsky:1995ub ; Balitsky:1998kc ; Balitsky:1998ya ; JalilianMarian:1997jx ; JalilianMarian:1997gr ; JalilianMarian:1997dw ; Kovner:1999bj ; Kovner:2000pt ; Weigert:2000gi ; Iancu:2000hn ; Ferreiro:2001qy , which can be written in a compact form as the action of the JIMWLK Hamiltonian $H$ on a functional of Wilson line operators

[TABLE]

Then the observables are given as the convolution of the hard part $\mathcal{H}_{Y_{c}}\left(\boldsymbol{x}_{1},...,\boldsymbol{x}_{n}\right)$, obtained through effective Feynman rules in the external slow (classical) field with a lower rapidity cutoff $Y_{c}$ for the fast (quantum) gluon fields, and the action of Wilson line operators at rapidity $Y_{c}$ on target states $\left\langle P^{\left(\prime\right)}|P\right\rangle$:

[TABLE]

where for exclusive observables $\mathcal{A}$ is the amplitude and the matrix element is off-diagonal, and for inclusive observables $\mathcal{A}$ is the cross section and the matrix element is diagonal. Hereafter, we drop the subscript $Y_{c}$ subscripts for convenience. We normalize our target states such that

[TABLE]

and use extensively relations similar to

[TABLE]

as a result of normalization (9) and translation invariance. We also use the fact that in lightcone gauge $A^{+}=0$ and in the eikonal approximation given in Eq. (4), the slow gluon field $F^{i-}$ simply reads

[TABLE]

Finally, the connection to standard parton distributions is always obtained by using small $x$ limits of these distributions as given by relations of type:

[TABLE]

where $\mathcal{U}_{\left[x,y\right]}^{\pm}$ are staple gauge links:

[TABLE]

whose transverse parts are subeikonal in $A^{+}=0$ gauge.

2 Low $x$ amplitudes as (G)TMD amplitudes

All the computation steps which we perform here would apply for generic shockwave amplitudes. We restrict ourselves to processes with 1 incoming particle of momentum $p_{0}$ in color representation $R_{0}$ and 2 outgoing particles of respective momenta $p_{1}$ and $p_{2}$ and in respective color representations $R_{1}$ and $R_{2}$, in the external field of a hadronic target. In the shockwave and CGC formulations of low $x$ physics, the amplitude for such a process has the form Altinoluk:2019fui

[TABLE]

where $\mathcal{H}\left(\boldsymbol{r}\right)$ is the hard part, and we defined

[TABLE]

and

[TABLE]

and where

[TABLE]

The study we perform here is independent of $\mathcal{H}$ and thus valid for any process with 2 outgoing particles, with or without masses or virtualities. Note that final state partons can hadronize for example via fragmentation functions, distribution amplitudes or NRQCD, without changing the validity of present study either. Also note that the true amplitude is given by the action of the Wilson line operators on target states, which we will introduce later in Section 3 in the inclusive case and in Section 4 in the exclusive case. For the moment, it is enough to keep the amplitude as an operator.

The equivalence we want to prove here relies on the following rewriting of Wilson lines in terms of their derivatives:

[TABLE]

and

[TABLE]

Indeed, derivatives of Wilson line operators are the main quantities to consider when trying to match the TMD formalism, as shown in Dominguez:2011wm . The derivative of a fundamental line is given by

[TABLE]

which allows to identify the $F_{a}^{i-}\left(b_{1}\right)$ field as the actual gluon field in a TMD operator. At moderate $x$ this gluon would be isolated from the slow gluons in the gauge links. The amplitude in Eq. (14) can be rewritten as a sum of three pieces:

[TABLE]

where the three pieces are defined respectively as

[TABLE]

[TABLE]

and

[TABLE]

The first piece is easy to rewrite thanks to Eqs. (18) and (19):

[TABLE]

This contribution is perfectly compatible with an all-kinematic-twists-resummed TMD amplitude for the first subleading-twist TMD half-operator.

The second and third pieces $\mathcal{A}_{k}^{\left(1\right)}$ and $\mathcal{A}_{k}^{\left(2\right)}$ contain both 1-gluon and 2-gluon contributions. The 1-gluon contributions were extracted and resummed in Altinoluk:2019fui , then compared to predictions from the kinematic-twist-resummed TMD framework developped in Kotko:2015ura ; vanHameren:2016ftb for several concrete examples. Let us recall the method which was used, and resum the 2-gluon contributions as well. The Taylor expanded form of the Wilson line operator involved in $\mathcal{A}_{k}^{\left(1\right)}$ can be written as

[TABLE]

Keeping in mind that $\mathcal{U}_{n}$ will be integrated as $\int\!d^{2}\boldsymbol{b}\,e^{-i\left(\boldsymbol{k}\cdot\boldsymbol{b}\right)}\,\mathcal{U}_{n}$, we can use integrations by parts to write

[TABLE]

Then an easy recursion shows

[TABLE]

Using

[TABLE]

and

[TABLE]

one can obtain

[TABLE]

A final resummation, using the relations

[TABLE]

and

[TABLE]

leads to

[TABLE]

Plugging Eq.(34) into Eq.(22) finally yields

[TABLE]

where a 1-gluon contribution and a 2-gluon contribution were explicitely extracted and power-resummed. Applying exactly the same method to the remaining piece, we obtain

[TABLE]

We can finally gather the 1-gluon and 2-gluon amplitudes. The 1-gluon amplitude reads:

[TABLE]

Single-scattering contributions like those in Eq. (37) were extracted for explicit processes in (Altinoluk:2019fui, ) and the consistency of the results was checked by comparing single-scattering cross sections derived with our method to those obtained in the so-called improved TMD formalism, which is a method to incorporate kinematic twists in TMD factorization. A perfect match was found for all processes considered.

The 2-gluon amplitude is given by:

[TABLE]

The crucial point to note is that Eqs. (37, 38) sum up exactly to the low $x$ amplitude in Eq. (14). As a result, this shows that any low $x$ amplitude of the form of Eq. (14) can be rewritten as the sum of all kinematic twist corrections to the single-scattering TMD amplitude and to the double-scattering (G)TMD amplitude (i.e. the first genuine twist correction). The notable absence of triple or higher scattering amplitudes is due to the eikonal approximation: a contribution with two derivatives hitting the same line i.e. with 2 low $x$ TMD gluons hitting the same parton, constitutes a gauge invariance fixing contribution: it was already taken into account either as part of a gauge link or as a kinematic twist correction. We thus expect a low $x$ amplitude with $n$ final state particles to have at most an $n$-scattering operator in its amplitude in the eikonal approximation. With subeikonal corrections, one could have higher genuine twist contributions.

In principle, Eqs. (37, 38) answer the long-sought equivalence between low $x$ and moderate $x$ formulations of factorization: modern formulations of low $x$ amplitudes can be rewritten as the sum over all twists of (G)TMD amplitudes in their small $x$ limit.

For the sake of the compactness of the notations let us introduce

[TABLE]

and

[TABLE]

so that the 1-gluon amplitude given in Eq. (37) can be written as

[TABLE]

and the 2-gluon amplitude given in Eq. (38) can be written as

[TABLE]

3 Inclusive cross sections

It is easy to obtain inclusive cross sections from our amplitudes given in Eq. (37) and in Eq.. (38). In order to account for the possible use of our results in the hybrid factorization ansatz, we average over incoming projectile color states, with an averaging factor $C_{0}=N_{c}$ for a quark, $C_{0}=N_{c}^{2}-1$ for a gluon, and $C_{0}=1$ for a photon. We also use the rapidities $y_{1}$ and $y_{2}$ of the outgoing particles. We can distinguish four contributions, depending on the number of gluons in the TMD half-operator in the amplitude and in the complex conjugate amplitude.

3.1 TMD cross sections

The 2-body contribution, given by the four diagrams in Figure 1, reads:

[TABLE]

the 3-body contributions are given by the diagrams with one gluon in the amplitude and two gluons in the complex conjugate amplitude, as in Figure 2, which add up to:

[TABLE]

and by those with two gluons in the amplitude and one in the complex conjugate amplitude as in Figure 3, which yield:

[TABLE]

Finally the 4-body contribution from the diagram in Figure 4, reads

[TABLE]

The inclusive (or incoherent) diffractive case is very similar to the fully inclusive case. The difference lies in the TMD operators. While the fully inclusive cross section involves $\langle P|\mathrm{tr}(\mathcal{O}_{\boldsymbol{x}}\mathcal{O}_{\boldsymbol{y}}^{\dagger})|P\rangle,$ the inclusive diffractive cross section involves $\langle P|\mathrm{tr}(\mathcal{O}_{\boldsymbol{x}}^{\left(1\right)}\mathcal{O}_{\boldsymbol{y}}^{\left(1\right)\dagger})|P\rangle,$ where $\mathcal{O}_{\boldsymbol{x}}^{\left(1\right)}$ and $\mathcal{O}_{\boldsymbol{y}}^{\left(1\right)\dagger}$ are the color singlet projections of the operators. In the CGC and dipole descriptions of low $x$ physics, this matrix element is often described as the $b$-dependent dipole scattering amplitude $\mathcal{N}(\boldsymbol{b},\boldsymbol{r})$. It is important to note that the $\boldsymbol{b}$ variable which appears in these matrix elements is the Fourier conjugate to the partonic transverse momentum in a TMD. As a result, it should not be interpreted as the physical impact parameter, which is the Fourier conjugate to the transverse momentum imbalance in incoming and outgoing target states in a GTMD or in a GPD. Instead, the $\boldsymbol{b}$ variable that appear in inclusive diffraction is actually the transverse coordinate variable involved in the Collins-Soper equation. This remark does not invalidate the description of the b-dependent dipole scattering amplitude, but it is important to keep in mind the nature of $\boldsymbol{b}$ when interpreting this quantity for inclusive observables.

3.2 Cross sections with a PDF

A parton distribution function (PDF) is the integral of a TMD w.r.t. its partonic transverse momenta. In order to obtain a cross section with a PDF instead of a TMD, one should consider an inclusive process where momentum $\boldsymbol{k}$ is not measured, and perform an expansion Eqs. (43, 44, 45, 46) in twists, by taking the leading power in the hard scale. This hard scale $Q$ can be given by a virtuality or an invariant mass and thus the expansion is process-dependent. However, we can easily see how the leading kinematic twist part of the leading genuine twist cross section, Eq. (43), can be rewritten with a PDF: one considers $\left|\boldsymbol{k}\right|\ll Q$ in the hard factors $\mathcal{I}_{\mathcal{H}}^{i}$ then integrates the cross section w.r.t. $\left|\boldsymbol{k}\right|$. For example for a photon-induced process the leading twist cross section becomes

[TABLE]

where one can easily identify a PDF in the $x\sim 0$ approximation in the second line.

4 Exclusive cross sections

4.1 GTMD cross sections

For exclusive cross sections, only the 2-body amplitudes contribute since the target matrix elements in this case is a color singlet operator. The off-diagonal matrix elements of the 2-body operators can easily be identified as GTMDs. Here, we denote the momentum imbalance as $\Delta_{\perp}$ rather than $k_{\perp}$ to match more standard notations for the GTMD. The exclusive cross section reads

[TABLE]

where $\mathrm{tr}_{c}$ is the trace over all remaining open color indices in the product of distributions. For example in a $g\rightarrow q\bar{q}$ cross section the last line in Eq. (48) would read

[TABLE]

The non-perturbative matrix elements involved in Eq. (48) are GTMDs. We would like to emphasize the fact that Eq. (48) is exact. Here, it shows a perfect match between exclusive low $x$ cross sections and twist-resummed GTMD cross sections in the small $x$ limit.

4.2 Cross sections with a GPD

The GPD limit is obtained from a GTMD cross section the same way the PDF limit is obtained from a TMD cross section, noting that a GPD is the integral of a GTMD with respect to partonic transverse momenta. One performs a twist expansion by taking $\boldsymbol{k}_{1,2}^{\left(\prime\right)}/Q\rightarrow 0$** **in the hard parts, then integrates over partonic transverse momenta. For example for photon-induced processes at leading twist we get

[TABLE]

At leading twist the gauge links do not contribute, and one can easily recognize leading twist GPDs in the last two lines.

5 The BFKL limit as a kinematic limit

The BFKL limit is usually understood as a weak field limit $gF\sim 0$, known as the dilute limit. In a recent study (Altinoluk:2019fui, ), it was shown how this limit could also be recovered by using the Wandzura-Wilczek approximation in the CGC and identifying all gluon distributions as the unintegrated PDF, which is justified at large $\left|\boldsymbol{k}\right|$. In this section, we aim at describing the BFKL limit as a kinematic limit rather than a weak field limit. BFKL is valid when all transverse momenta are of the order of the hard scale, and we are interested in studying BFKL beyond the WW approximation, so let us consider the limit of large partonic transverse momenta. By Fourier conjugation, this limit leads to the shrinking of transverse gauge links:

[TABLE]

This makes all gauge links unidimensional and in the same direction, which means all 2-body distributions can be rewritten as

[TABLE]

since the modification of gauge links between $x$ and [math] in the transverse plane is free up to small corrections. This unique distribution is the 2-body unintegrated PDF. In $A^{+}=0$ gauge, it can be rewritten as

[TABLE]

where one can explicitely identify the so-called nonsense polarization vector in lightcone gauge $\frac{\boldsymbol{k}^{i}}{\left|\boldsymbol{k}\right|}$.

The importance of gauge links at small $k_{\perp}$ and the shrinking of all TMD distributions into the unique unintegrated PDF was observed and confirmed numerically in Marquet:2016cgx ; Marquet:2017xwy .

Similarly to Eq.( 52), all 3-body distributions become

[TABLE]

where it is important to keep $g_{s}$ in the operator rather than the hard part. Indeed genuine twist corrections do not come with a perturbative $g_{s}$ suppression: the $g_{s}$ factor is in the non-perturbative matrix element, which means the 3-body contributions are of the same order of perturbation theory as the 2-body contributions. In most of the studies which appear in the BFKL literature, the 3- and 4-body unintegrated PDFs are not usually taken into account, with the ill-advised assumption that their contributions are $\alpha_{s}$- suppressed. Here, we observe that they are actually dropped as a Wandzura-Wilczek approximation, which does not have a perturbative origin. Understanding BFKL as a kinematic limit means that all genuine twist corrections should be taken into account. For example for photon-induced processes in the BFKL kinematic limit and in $A^{+}=0$ gauge, Eqs. (43, 44, 45, 46) become respectively the 2-body contribution

[TABLE]

the 3-body contributions

[TABLE]

and

[TABLE]

and finally the 4-body contribution

[TABLE]

We emphasize that all four contributions are of the same order in perturbation theory, and neglecting the contributions from genuine higher twist unintegrated PDFs is justified as a Wandzura-Wilczek approximation rather than a perturbative suppression. The validity of this approximation should be evaluated for each process.

6 The origins of saturation

In our formulation of low $x$ physics, saturation can be understood as three distinct effects.

First of all, a well known form of saturation is evolutional, and arises from the non-linearity of the B-JIMWLK hierarchy of evolution equations and its truncated and approximated daughter equations (see Figure 5). This non-linearity is expected to slow down the growth in $s$ of low $x$ cross sections Gribov:1984tu , thus contributing to restoring the unitarity of the S matrix.

The other effects are due to multiple scattering via interactions with slow gluons, but we distinguish two types of such effects. We refer to the first type, described in Figure 6, as the kinematic saturation. It is linked to the gauge link structures of the gluon distributions. Indeed, the gauge links account for multiple scatterings from slow gluons, and the importance of such gauge links is that they can be used as a probe for multiple scattering effects. These effects have been investigated recently in Marquet:2016cgx ; Marquet:2017xwy . They are expected to appear at small $\left|\boldsymbol{k}\right|$, since all TMD distributions reduce to the unintegrated PDF in the large $\left|\boldsymbol{k}\right|$ limit regardless of their gauge link structure, as discussed earlier. By studying the behavior of different distributions all along the $\left|\boldsymbol{k}\right|$ range, it was shown that indeed distributions with distinct gauge link structures have to be distinguished at low $\left|\boldsymbol{k}\right|$ while at large $\left|\boldsymbol{k}\right|$ all distributions are the same. This kind of multiple scattering is thus due to the presence of a large transverse coordinate region, conjugate to $\left|\boldsymbol{k}\right|$, to fill with the soft gluons in that kinematic regime.

Finally the last type of saturation, described in Figure 7, to which we refer as genuine saturation, is due to genuine twist corrections. In addition to the gluons forming the gauge links, the extra gluons from higher twist operators can contribute to multiple scattering effects. Given that the genuine twist corrections in an operator are obtained in physical gauges by the insertion of a gluon field along with the coupling constant $g_{s}$ and the appropriate gauge links, the genuine twist corrections are not perturbatively suppressed as assumed implicitely in most studies involving unintegrated PDFs222Obviously this remark only concerns non-perturbative targets. BFKL resummation is valid in full generality for soft gluon exchanges between perturbative objects, for example Mueller-Navelet jets Mueller:1986ey .: the $g_{s}$ factor is part of the non-perturbative matrix elements and neglecting them is tantamount to using the Wandzura-Wilczek approximation. This kind of saturation effects would appear even in the high $\left|\boldsymbol{k}\right|$ BFKL regime if one does not restrict oneself to this unquantified approximation, whose validity should be tested in a process-dependent way. In the CGC picture, the large gluon occupancy number in a dense target leads to the scaling $g_{s}F\sim 1$, which leads to an expected enhancement of genuine twist corrections. In that sense, genuine saturation can be understood as the invalidation of the Wandzura-Wilczek approximation.

7 Discussions

We have found that any low $x$ cross section for a process of type $p_{0}H\rightarrow p_{1}p_{2}X$, where $H$ is a hadron and $X$ remnants are not measured, can be rewritten into an infinite twist TMD cross section. Similarly, any low $x$ exclusive cross section for a process of type $p_{0}H\rightarrow p_{1}p_{2}H^{\prime}$, where $H$ (resp. $H^{\prime}$) is an incoming (resp. outgoing) hadron, can be rewritten into an infinite twist GTMD cross section. Even though we have restricted ourselves to the case where we have one particle in the initial state and two particles in the final state, all the steps of our study can be applied for processes with more than two particles in the final state, or more than one particle in the initial state. Thus, we bridged one of the main gaps between low $x$ and moderate $x$ formulations of perturbative QCD: the apparent difference between the involved non-perturbative matrix elements.

We have also given a new interpretation of saturation in the low $x$ regime and distinguished three types of saturation: evolutional, kinematic and genuine. In principle, each type of saturation can be studied separately from the others and there are easy ways to distinguish them. For example, studying high $\left|\boldsymbol{k}\right|$ processes on dense targets and on dilute targets would probe genuine saturation alone. On the other hand, small $\left|\boldsymbol{k}\right|$ observables would be probes of both genuine and kinematic saturation on dense targets, and of kinematic saturation alone on dilute targets.

A study of angular correlations can be performed as in (Dumitru:2016jku, ), in the whole kinematic range in $\left|\boldsymbol{k}\right|$ using our results and a tensorial decomposition of the involved TMD distributions. It would be very insightful in future studies to focus on subeikonal corrections to low $x$ physics and try to match a TMD framework, similarly to what was performed in this article.

Finally, the most powerful feature of our formulation in terms of standard parton distributions is the possibility to resum easily logarithms of $Q$ and $|\boldsymbol{k}|$ using the known evolution equations for TMD distributions and Sudakov resummations. This could help solve the observed negativity issues for low $x$ cross sections (see for example Stasto:2013cha ; Stasto:2014sea ; Watanabe:2015tja ; Ducloue:2016shw ). Indeed the resummation of colinear logarithms via a similarly colinearly-improved low $x$ evolution equation Iancu:2015vea ; Iancu:2015joa ; Hatta:2016ujq is one of the most efficient tools to deal with the issue. A complementary approach to the improved-JIMWLK evolution would be to first apply the $Y_{c}$ evolution using the regular JIMWLK equation, then rewrite the evolved observables in terms of TMD distributions as was done in this article, and finally resum logarithms of the hard scale and Sudakov logarithms using standard TMD methods. For the leading twist TMD operators, one could also use the evolution equations derived in Balitsky:2015qba ; Balitsky:2016dgz .

Acknowledgments

RB thanks P. Taels and R. Venugopalan for stimulating discussions. The work of TA is supported by Grant No. 2017/26/M/ST2/01074 of the National Science Centre, Poland. The work of RB is supported by the U.S. Department of Energy, Office of Nuclear Physics, under Contracts No. DE-SC0012704 and by an LDRD grant from Brookhaven Science Associates.