Communications in Theoretical Physics ›› 2022, Vol. 74 ›› Issue (11): 115202. doi: 10.1088/1572-9494/ac841c
• Particle Physics and Quantum Field Theory • Previous Articles Next Articles
Gao-Liang Zhou1,(
), Zheng-Xin Yan1, Xin Zhang1, Feng Li2
Received: 2022-05-10
Revised: 2022-06-30
Accepted: 2022-07-26
Published: 2022-10-28
Contact:
Gao-Liang Zhou
E-mail:zhougl@alumni.itp.ac.cn
Gao-Liang Zhou, Zheng-Xin Yan, Xin Zhang, Feng Li, Commun. Theor. Phys. 74 (2022) 115202.
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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."
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) | ${\lambda }^{1+\tfrac{b}{2}}$ |
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 |
Figure 3.
An example of diagrams without the active-spectator coherence."
Figure 4.
An example of diagrams with the active-spectator coherence."
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."
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."
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."
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."
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."
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."
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 |
| ${A}_{s}^{\mu }\to {A}_{s}^{\mu }$ | ${A}_{s}^{\mu }\to {A}_{s}^{\mu }$ | ${A}_{s}^{\mu }\to {A}_{s}^{\mu }$ |
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