1. Introduction
Figure 1. Examples of ladder diagrams with Glauber gluons exchanged (a) between active particles and spectators; (b) between active particles. |
Figure 2. Elastic scattering diagrams between spectators, where the dot lines represent Glauber gluons. |
2. Glauber gluons in SCET
Table 1. Relevant modes in the propagation of particles collinear to ${n}^{\mu }=\tfrac{1}{\sqrt{2}}(1,\vec{n})$ and the power counting for them in the Feynman gauge, where ${\bar{n}}^{\mu }=\tfrac{1}{\sqrt{2}}(1,-\vec{n})$ and $n\cdot {p}_{n\perp }=\bar{n}\cdot {p}_{n\perp }=0$. |
Modes | Fields | Momenta scales $(n\cdot p,\bar{n}\cdot p,{p}_{n\perp })$ | Infrared power counting |
---|---|---|---|
Collinear quarks | ξn | Q(λ2, 1, λ) | λ |
Collinear gluons | ${A}_{n}^{\mu }$ | Q(λ2, 1, λ) | λ |
Soft quarks | ψs | Q(λ, λ, λ) | λ3/2 |
Soft gluons | ${A}_{s}^{\mu }$ | Q(λ, λ, λ) | λ |
Ultrasoft quarks | ψus | Q(λ2, λ2, λ2) | λ3 |
Ultrasoft gluons | ${A}_{{us}}^{\mu }$ | Q(λ2, λ2, λ2) | λ2 |
Glauber gluons | ${A}_{{nG}}^{\mu }$ | Q(λ2, λb, λ)( b = 1,2) | ${\lambda }^{1+\tfrac{b}{2}}$ |
2.1. Power counting for couplings between Glauber gluons and other particles
Table 2. The infrared power counting for couplings involving Glauber gluons, where ${\bar{n}}^{\mu }=\tfrac{1}{\sqrt{2}}(1,-\vec{n})$. |
Couplings | Fields | Power counting |
---|---|---|
Glauber gluons and ultrsoft gluons | (AnG,Aus) | λ or higher |
Glauber gluons and ultrsoft fermions | (AnG,Aus) | ${\lambda }^{\tfrac{3}{2}}$ or higher |
Glauber gluons | (AnG) | ${\lambda }^{\tfrac{b}{2}}$ or higher |
Glauber gluons and soft gluons | (AnG,${A}_{\bar{n}G}$,As) | λ or higher |
Glauber gluons and soft gluons | (AnG,As) | ${\lambda }^{\tfrac{b}{2}}$ or higher |
Glauber gluons and soft fermions | (AnG,ψs) | ${\lambda }^{\tfrac{b}{2}}$ or higher |
Glauber gluons and collinear fermions | (AnG,ξn) | ${\lambda }^{\tfrac{b}{2}-1}$ or higher |
Glauber gluons and collinear gluons | (AnG,An) | ${\lambda }^{\tfrac{b}{2}-1}$ or higher |
Glauber gluons and collinear fermions | (AnG,${\xi }_{\bar{n}}$) | ${\lambda }^{1-\tfrac{b}{2}}$ or higher |
Glauber gluons and collinear gluons | (AnG,${A}_{\bar{n}}$) | ${\lambda }^{1-\tfrac{b}{2}}$ or higher |
2.2. Leading power action including Glauber gluons
3. Elastic scattering effects in hadron collisions
3.1. Elastic scattering effects without interactions between spectators and active particles
Figure 3. An example of diagrams without the active-spectator coherence. |
3.2. Cancellation of final state interactions
Figure 4. An example of diagrams with the active-spectator coherence. |
4. Glauber gluons coupling to active particles
4.1. Eikonal approximation in couplings between collinear particles and soft (ultrasoft) gluons
4.2. Eikonalization of couplings between active particles and Glauber gluons in ${ \mathcal H }(P,\bar{P},{q}^{+},{q}^{-})$
Figure 5. Couplings between Glauber gluons and an active particle, where H represents the hard vertex and dot lines represent Glauber gluons. Other parts of the whole diagram are not displayed explicitly. |
4.3. Glauber gluons exchanged between active and soft particles
4.4. Glauber gluons exchanged between spectators and active particles
4.5. Glauber gluons exchanged between active particles
5. Cancellation of spectator–spectator and spectator-soft Glauber exchanges in ${ \mathcal H }(P,\bar{P},{q}^{+},{q}^{-})$
Figure 6. An example of diagrams with the active-spectator coherence which obstructs the summation over all possible Glauber interactions between spectators, where the dot lines represent the Glauber gluons. |
Figure 7. (a) An example of diagrams with the Glauber gluons A+G couple to spectators after the non-Glauber couplings. (b)An example of diagrams with the Glauber gluons A+G couple to plus-collinear spectators before non-Glauber couplings. (c)An example of diagrams with the Glauber gluons A+G couple to minus-collinear spectators before non-Glauber couplings. The Glauber gluons are represented by dot lines in these diagrams. |
Figure 8. Some examples of spectator–spectator ladder diagrams with soft gluons connecting different ladders. |
5.1. $\widetilde{{n}_{+}}\cdot x$-evolution in ${ \mathcal H }(P,\bar{P},{q}^{+},{q}^{-})$
5.2. Couplings involving Glauber gluons A+G in ${ \mathcal H }(P,\bar{P},{q}^{+},{q}^{-})$
5.3. Cancellation of Glauber gluon A+G
5.4. Cancellation of Glauber gluons A−G
6. Graphic cancellation of Glauber gluons
6.1. $\widetilde{{n}_{+}}\cdot x$ evolution of plus-collinear particles
6.2. $\widetilde{{n}_{+}}\cdot x$ evolution of minus-collinear and soft particles
6.3. Glauber graphs in ${ \mathcal H }(P,\bar{P},{q}^{+},{q}^{-})$
Figure 9. An example of diagrams with Glauber couplings between non-Glauber couplings and the final cut, where the dot lines represent Glauber gluons and the circle lines represent collinear gluons. |
6.4. Cancellation of spectator–spectator and spectator-soft Glauber subgraphs
Figure 10. An example of cancellation of spectator-spectator and spectator-soft Glauber subgraphs. |
Figure 11. An example of the summation over spectator-spectator elastic scatterings. |
7. Conclusions and discussions
Acknowledgments
Appendix A. Time evolution operator of the effective theory
Figure A1. Examples of tree level elastics scattering processes induced by a Glauber exchange. Figure (a) shows the scattering between n-collinear and $\bar{n}$-collinear quarks. Figure (b) shows the scattering between n-collinear quarks and transverse polarized $\bar{n}$-collinear gluons. Figure (c) shows the scattering between n-collinear and soft quarks. Figure (d) shows the scattering between n-collinear quarks and soft gluons. |
Figure A2. Examples of tree level elastics scattering processes induced by two sequential Glauber gluons, where qs and ${q}_{s}^{{\prime} }$ represent soft gluons. |
Appendix B. Active particles and spectators in hard collisions
Figure B1. An example of parents of active particles. k1 is a parent of active particles. |
Figure B2. An example of how +-momenta flow affects the definition of plus-collinear active particles and spectators, where the dot line l represents a Glauber gluon. |
Appendix C. Reparameterization invariance of the effective theory
1. | 1. The subscript p of the collinear fields ξn,p and ${A}_{n,p}^{\mu }$ is arbitrary by order Qλ2 |
2. | 2. Transformations on nμ and ${\bar{n}}_{\mu }$ reads $\begin{eqnarray}\begin{array}{l}(a)\left\{\begin{array}{l}{n}_{\mu }\to {n}_{\mu }+{{\rm{\Delta }}}_{\mu }^{\perp }\\ {\bar{n}}_{\mu }\to {\bar{n}}_{\mu }\end{array}\right.\quad (b)\left\{\begin{array}{l}{n}_{\mu }\to {n}_{\mu }\\ {\bar{n}}_{\mu }\to {\bar{n}}_{\mu }+{\varepsilon }_{\mu }^{\perp }\end{array}\right.\\ \quad \times (c)\left\{\begin{array}{l}{n}_{\mu }\to (1+\alpha ){n}_{\mu }\\ {\bar{n}}_{\mu }\to (1-\alpha ){\bar{n}}_{\mu }\end{array}\right.\end{array}\end{eqnarray}$ should not change physical results, where the infinitesimal parameters satisfy the condition $\bar{n}\cdot {{\rm{\Delta }}}^{\perp }=n\cdot {{\rm{\Delta }}}^{\perp }=\bar{n}\cdot {\varepsilon }^{\perp }=n\cdot {\varepsilon }^{\perp }=0$. |
Table C1. Transformations of relevant operators under the type-2 reparameterization. |
Type 2a | Type 2b | Type 2c |
---|---|---|
${n}_{\mu }\to {n}_{\mu }+{{\rm{\Delta }}}_{\mu }^{\perp }$ | nμ → nμ | nμ → (1 + α)nμ |
${\bar{n}}_{\mu }\to {\bar{n}}_{\mu }$ | ${\bar{n}}_{\mu }\to {\bar{n}}_{\mu }+{\varepsilon }_{\mu }^{\perp }$ | ${\bar{n}}_{\mu }\to (1-\alpha ){\bar{n}}_{\mu }$ |
$n\cdot {{ \mathcal D }}_{{nG}}\to n\cdot {{ \mathcal D }}_{{nG}}+{{\rm{\Delta }}}^{\perp }\cdot {{ \mathcal D }}_{{nG}}^{\perp }$ | $n\cdot {{ \mathcal D }}_{{nG}}\to n\cdot {{ \mathcal D }}_{{nG}}$ | $n\cdot {{ \mathcal D }}_{{nG}}\to (1+\alpha )n\cdot {{ \mathcal D }}_{{nG}}$ |
${{ \mathcal D }}_{{nG}\mu }^{\perp }\to {{ \mathcal D }}_{{nG}\mu }^{\perp }-{{\rm{\Delta }}}_{\mu }^{\perp }\bar{n}\cdot {{ \mathcal D }}_{{nG}}-{\bar{n}}_{\mu }{{\rm{\Delta }}}^{\perp }\cdot {{ \mathcal D }}_{{nG}}^{\perp }$ | ${{ \mathcal D }}_{{nG}\mu }^{\perp }\to {{ \mathcal D }}_{{nG}\mu }^{\perp }-\varepsilon \cdot {{ \mathcal D }}_{{nG}}-{n}_{\mu }{\varepsilon }^{\perp }\cdot {{ \mathcal D }}_{{nG}}^{\perp }$ | ${{ \mathcal D }}_{{nG}\mu }^{\perp }\to {{ \mathcal D }}_{{nG}\mu }^{\perp }$ |
$\bar{n}\cdot {{ \mathcal D }}_{{nG}}\to \bar{n}\cdot {{ \mathcal D }}_{{nG}}$ | $\bar{n}\cdot {{ \mathcal D }}_{{nG}}\to \bar{n}\cdot {{ \mathcal D }}_{{nG}}+{\varepsilon }^{\perp }\cdot {{ \mathcal D }}_{{nG}}^{\perp }$ | $\bar{n}\cdot {{ \mathcal D }}_{{nG}}\to (1-\alpha )\bar{n}\cdot {{ \mathcal D }}_{{nG}}$ |
$n\cdot {{ \mathcal D }}_{\bar{n}G}\to n\cdot {{ \mathcal D }}_{\bar{n}G}+{{\rm{\Delta }}}^{\perp }\cdot {{ \mathcal D }}_{\bar{n}G}^{\perp }$ | $n\cdot {{ \mathcal D }}_{\bar{n}G}\to n\cdot {{ \mathcal D }}_{\bar{n}G}$ | $n\cdot {{ \mathcal D }}_{\bar{n}G}\to (1+\alpha )n\cdot {{ \mathcal D }}_{\bar{n}G}$ |
${{ \mathcal D }}_{\bar{n}G\mu }^{\perp }\to {{ \mathcal D }}_{\bar{n}G\mu }^{\perp }-{{\rm{\Delta }}}_{\mu }^{\perp }\bar{n}\cdot {{ \mathcal D }}_{\bar{n}G}-{\bar{n}}_{\mu }{{\rm{\Delta }}}^{\perp }\cdot {{ \mathcal D }}_{\bar{n}G}^{\perp }$ | ${{ \mathcal D }}_{\bar{n}G\mu }^{\perp }\to {{ \mathcal D }}_{\bar{n}G\mu }^{\perp }-\varepsilon \cdot {{ \mathcal D }}_{\bar{n}G}-{n}_{\mu }{\varepsilon }^{\perp }\cdot {{ \mathcal D }}_{\bar{n}G}^{\perp }$ | ${{ \mathcal D }}_{\bar{n}G\mu }^{\perp }\to {{ \mathcal D }}_{\bar{n}G\mu }^{\perp }$ |
$\bar{n}\cdot {{ \mathcal D }}_{\bar{n}G}\to \bar{n}\cdot {{ \mathcal D }}_{\bar{n}G}$ | $\bar{n}\cdot {{ \mathcal D }}_{\bar{n}G}\to \bar{n}\cdot {{ \mathcal D }}_{\bar{n}G}+{\varepsilon }^{\perp }\cdot {{ \mathcal D }}_{\bar{n}G}^{\perp }$ | $\bar{n}\cdot {{ \mathcal D }}_{\bar{n}G}\to (1-\alpha )\bar{n}\cdot {{ \mathcal D }}_{\bar{n}G}$ |
${\xi }_{n}\to (1+{{/}\!\!\!\!{{\rm{\Delta }}}}^{\perp }{/}\!\!\!\!{\bar{n}}){\xi }_{n}$ | ${\xi }_{n}\to (1+{{/}\!\!\!\!{\varepsilon }}^{\perp }\tfrac{1}{\bar{n}\cdot {{ \mathcal D }}_{{nG}}}{{/}\!\!\!\!{{ \mathcal D }}}_{{nG}}^{\perp }){\xi }_{n}$ | ξn → ξn |
${\xi }_{\bar{n}}\to (1+{{/}\!\!\!\!{{\rm{\Delta }}}}^{\perp }\tfrac{1}{n\cdot {{ \mathcal D }}_{\bar{n}G}}{{/}\!\!\!\!{{ \mathcal D }}}_{\bar{n}G}^{\perp }){\xi }_{\bar{n}}$ | ${\xi }_{\bar{n}}\to (1+{{/}\!\!\!\!{\varepsilon }}^{\perp }{/}\!\!\!\!{n}){\xi }_{\bar{n}}$ | ${\xi }_{\bar{n}}\to {\xi }_{\bar{n}}$ |
WnG → WnG | ${W}_{{nG}}\to \left(1-\tfrac{1}{\bar{n}\cdot {{ \mathcal D }}_{{nG}}}{\varepsilon }^{\perp }\cdot {{ \mathcal D }}_{{nG}}^{\perp }\right){W}_{{nG}}$ | WnG → WnG |
${W}_{\bar{n}G}\to \left(1-\tfrac{1}{n\cdot {{ \mathcal D }}_{\bar{n}G}}{{\rm{\Delta }}}^{\perp }\cdot {{ \mathcal D }}_{\bar{n}G}^{\perp }\right){W}_{\bar{n}G}$ | ${W}_{\bar{n}G}\to {W}_{\bar{n}G}$ | ${W}_{\bar{n}G}\to {W}_{\bar{n}G}$ |
ψs → ψs | ψs → ψs | ψs → ψs |
${A}_{s}^{\mu }\to {A}_{s}^{\mu }$ | ${A}_{s}^{\mu }\to {A}_{s}^{\mu }$ | ${A}_{s}^{\mu }\to {A}_{s}^{\mu }$ |