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a41ee0c132bf0c83b3331da25a62569882faf1a7 | subsection | 46 | 65 | Mathematical background | \frac{\partial ^{} }{\partial b^{} } F(x,y) \right|_{y=b} &= \left( \frac{\partial ^{} F}{\partial y^{} } \frac{\partial ^{} y}{\partial b^{} } \right)_{y=b} = f(x,b(x)).Combining the above relations results in thmleibnizrule.[Leibniz's rule for two independent variables]
If a function f = f(x,y,z) and \partial _x f a... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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7b9b2f592d23384b8cc89fcacaaa18f9d1d40f7d | subsection | 47 | 65 | Mathematical background | \frac{\partial ^{} }{\partial a^{} } F(x,y,z) \right|_{z=a} &= \left( \frac{\partial ^{} F}{\partial z^{} } \frac{\partial ^{} z}{\partial a^{} } \right)_{z=a} = f(x,y,a(x,y)) \\
\left. | {
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
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d0245c8315644bffd5d52c15c610b89464fe6132 | subsection | 48 | 65 | Mathematical background | \frac{\partial ^{} }{\partial b^{} } F(x,y,z) \right|_{z=b} &= \left( \frac{\partial ^{} F}{\partial z^{} } \frac{\partial ^{} z}{\partial b^{} } \right)_{z=b} = f(x,y,b(x,y)).Combining the above relations with the facts that \mathrm {d}_x y = 0, \mathrm {d}_x b = \partial _x b, \mathrm {d}_x a = \partial _x a, and \ma... | {
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d9ec5e8834ac35f913d04aa58f2c56f6f450fa07 | subsection | 49 | 65 | Mathematical background | Then\begin{matrix}
\text{the total} \\
\text{rate of change} \\
\text{of quantity $\phi $} \\
\text{in $^{\mathrm {e}}$}
\end{matrix} \hspace{7.11317pt} = \hspace{7.11317pt}
\begin{matrix}
\text{the inward flux} \\
\text{of $\phi $ across the} \\
\text{boundary $^{\mathrm {e}}$} \\
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
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e790a46fac72f9353f4751c03365e9f87f381a0c | subsection | 50 | 65 | Mathematical background | In addition, while the integral statement of conservation used in its derivation (cf. deffundamentalconservationequation) does not utilize the divergence theorem and thus does not require the material be continuous within , additional theory is required in order to state an analogous integral relation for discontinuous... | {
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
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5bda271937bdaee4826a1ec9b176e10d99aa28a6 | subsection | 51 | 65 | Mathematical background | \ \mathrm {d},where the jump operator \left.{\cdot } \right. is given by defjump.Applying the continuous divergence theorem (cf. thmdivergencetheorem) to both regions and taking the sum yields\int _{} \left( \nabla \cdot \underline{j}^- + \nabla \cdot \underline{j}^+ \right) \ \mathrm {d} =
\int _{} \left( \underline{j... | {
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
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8682ee49c29ad2240f8c85fd6651d7467359bb80 | subsection | 52 | 65 | Mathematical background | \ \mathrm {d},where \underline{w} = \underline{w}(\underline{x},t) is the propagation velocity of exterior surface and discontinuity surface ; vector \underline{\hat{n}} is the outward-facing-unit normal; and the jump operator \left.{\cdot } \right. is given by defjump.Applying the continuous form of Reynolds transport... | {
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
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ff54815559f68226460261dd4e04ada19670e882 | subsection | 53 | 65 | Mathematical background | \ \mathrm {d}.A time-invariant volume implies that \underline{w} = \underline{0} on and therefore \int _\phi \underline{w} \cdot \underline{\hat{n}}\ \mathrm {d} = 0, which substituted into thmgeneralizedreynoldstransporttheorem produces the result.A surface may have a surface velocity \underline{w} with non-zero magni... | {
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
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83ba173dd6e9f7dd3a0b542b4215d3d30bdae491 | subsection | 54 | 65 | Mathematical background | Then\begin{matrix}
\text{the total} \\
\text{rate of change} \\
\text{of quantity $\phi $} \\
\text{within $^{\mathrm {e}}$}
\end{matrix} \hspace{7.11317pt} = \hspace{7.11317pt}
\begin{matrix}
\text{the inward flux} \\
\text{of $\phi $ across the} \\
\text{boundary $^{\mathrm {e}}$} \\
\end{matrix} \hspace{7.11317pt} +... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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a5abdfd0f9323e6ed405e11a48112570f982616a | subsection | 55 | 65 | Mathematical background | However, if there is an additional fluctuation of mass unrelated to the fluid velocity \underline{u} along \cap ^{\mathrm {e}}, there will clearly be a vector with non-zero magnitude defined by the difference \underline{r} = \underline{w} - \underline{u} with \Vert \underline{r} \Vert \ne 0 on \cap ^{\mathrm {e}}.The a... | {
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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771e58141567839795b7ff464c8a79e0a65b374b | subsection | 56 | 65 | Mathematical background | Regardless, the following theorem is used to define the boundary conditions for any quantity from either interior or exterior domain at the material interface :[Discontinuity equation]
A field \phi = \phi (\underline{x},t) defined within an arbitrarily-fixed volume ^{\mathrm {e}}(\underline{x}) \subset (\underline{x},... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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854cdf856420cd6256f8448fff62d59a18927363 | subsection | 57 | 65 | Mathematical background | \ \mathrm {d}^{\mathrm {e}} = - \int _{^{\mathrm {e}}} \left( \phi \underline{u} + \underline{j} \right) \cdot \underline{\hat{n}}\ \mathrm {d}^{\mathrm {e}}
+ \int _{^{\mathrm {e}}} \mathring{f}_{} \ \mathrm {d}^{\mathrm {e}} + \int _{^{\mathrm {e}}} \mathring{f}_{} \ \mathrm {d}^{\mathrm {e}}.Applying discontinuous d... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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c8eb30bd22c6e915bd1d6fee9dafe9379f95cee2 | subsection | 58 | 65 | Analytic solution in | This section provides all relevant calculations associated with secr3analyticsolution. | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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afc678bb1302b9abd78f9d442a9fdb6c0e1ea0da | subsection | 59 | 65 | Linear hyperbolic equation analyticyvelocityproblem | The final y component of velocity u_y^{\mathrm {a}} must satisfy incompressibility-relation analyticincompressibleconservationofmass. | {
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} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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f9785ee86da7b5dcdb55304858625ae645746d47 | subsection | 60 | 65 | Linear hyperbolic equation analyticyvelocityproblem | This section derives Equation duzdz,duxdx,analyticyvelocityproblem in detail.To begin, the following derivatives will be required:\frac{\partial ^{} H}{\partial x^{} } &= \frac{\partial ^{} S}{\partial x^{} } - \frac{\partial ^{} B}{\partial x^{} } \\
\frac{\partial ^{} H}{\partial t^{} } &= \frac{\partial ^{} S}{\par... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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2a2b063a4649f42cbb65470efaf49b2a6c242afe | subsection | 61 | 65 | Invariant coordinate constant0 | The relationship between the two right-most differential terms from lagrangecharpit can be stated as\frac{\mathrm {d}^{} z}{\mathrm {d} y^{} } &= \frac{1}{H} \left( z \frac{\partial ^{} H}{\partial y^{} } + S \frac{\partial ^{} B}{\partial y^{} } - B \frac{\partial ^{} S}{\partial y^{} } \right) \\
\frac{\mathrm {d}^{}... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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73152c541b8051c3254cd403604b1d8720da5770 | subsection | 62 | 65 | Invariant coordinate constant1 | The relationship between the two left-most differential terms from lagrangecharpit can be stated as\frac{\mathrm {d}^{} u_y^{\mathrm {a}}}{\mathrm {d} y^{} } &= -A - G u_y^{\mathrm {a}} \\
\frac{\mathrm {d}^{} u_y^{\mathrm {a}}}{\mathrm {d} y^{} } + \frac{u_y^{\mathrm {a}}}{H} \frac{\partial ^{} H}{\partial y^{} } &= -... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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509d5066383d4a61f7bb50cf06eceb38ea51998b | subsection | 63 | 65 | Surface expressions analyticyvelocitysb | Evaluating expression analyticyvelocity at the upper surface,u_y^{\mathrm {a}}(x,y,z=S,t)
= &- \frac{1}{H} \int _y \left[ H \frac{\partial ^{} u_{x \mathrm {S}}^{\mathrm {a}}}{\partial x^{} } + \frac{\partial ^{} H}{\partial t^{} } - \Vert \hat{\underline{k}} - \nabla S \Vert \mathring{S}- \Vert \nabla B - \hat{\underl... | {
"cite_spans": []
} | 1805.11175 | On mass conservation for ice sheets | [
"Evan M. Cummings"
] | [
"physics.flu-dyn"
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95bb5b7982bef08543bed306c1f0add65d0545fb | subsection | 64 | 65 | Python source code | All source code used to generate the results of this work are provided in this section.language=python,
showtabs=true,
tab=,
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basicstyle=,stringstyle=,
showstringspaces=false,
alsoletter=1234567890,
otherkeywords= , }, {, %, &, ,
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"Evan M. Cummings"
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ad643de383f466ec5fcb33d7a9ed1424ba4cde4e | abstract | 0 | 84 | Abstract | We present an all-loop dispersion integral, well-defined to arbitrary
logarithmic accuracy, describing the multi-Regge limit of the 2->5 amplitude in
planar N=4 super Yang-Mills theory. It follows from factorization, dual
conformal symmetry and consistency with soft limits, and specifically holds in
the region where th... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
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e8f9306f5f860edb15bf175808423336d21a7fe8 | subsection | 1 | 84 | Introduction | From phenomenological studies to implications for quantum gravity, the description of scattering at high energies, or in the (multi-)Regge limit, is a subject with rich history and impact on many branches of theoretical physics. In the simplest case of 2\rightarrow 2 scattering, where the limit corresponds to a center-... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
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1f50e6887c4463bf2c8e3815b31e6f867bb582b0 | subsection | 2 | 84 | Introduction | These developments render the MRK as one of the best sources of `boundary data' , , , for determining the six-gluon amplitude in general kinematics through five loops, by exploiting its analytic structure with the help of the bootstrap method , , , , , .It would be of course very exciting if higher-point amplitudes in ... | {
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... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
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55b897ad6bc3e6012aed9a104875b6f7ea1294d5 | subsection | 3 | 84 | Introduction | The second significant result of this paper is the extraction of the NLO correction to the central emission vertex, from the known NLLA contribution to the 2-loop 7-particle MHV amplitude , see also refs. , for earlier work on the LLA contribution.More precisely, since the aforementioned NLLA contribution had been prev... | {
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logarithmic accuracy | [
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a8db9ccfa93891f108400c1e18998674d8efbfc2 | subsection | 4 | 84 | Multi-Regge kinematics | Let us start by recalling the precise definition of the limit and the kinematic region we will be considering, mostly following the conventions of . We will be keeping the number of particles N general, since in the next section we will first be reviewing the N=6 case, before focusing on N=7 for the rest of the paper.I... | {
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logarithmic accuracy | [
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bb6d94ae3a9970135385757eb077c36a4b5e5dc5 | subsection | 5 | 84 | Multi-Regge kinematics | Thus different helicity configurations are only distinguished by the helicities h_1,\ldots ,h_{N-4} of the produced gluons, for which we define the BDS-normalized ratio\begin{}R\end{}_{h_1\ldots h_{N-4}} \equiv \frac{A_N(-,+,h_1,\ldots ,h_{N-4},+,-)}{A_N^{\textrm {BDS}}(-,+,h_1,\ldots ,h_{N-4},+,-)}\,.Here A_N^{\textrm... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
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"James Drummond",
"Claude Duhr",
"Falko Dulat",
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6a329eb5f1f2ab215b1ec463e41336af92f8858b | subsection | 6 | 84 | Multi-Regge kinematics | In the right-hand side, the surviving transverse cross ratios describing the limit are chosen slightly differently compared to another widely used convention in the literature,z_i=-w_i=\frac{({\bf x}_1 -{\bf x}_{i+3})\,({\bf x}_{i+2} -{\bf x}_{i+1})}{({\bf x}_1 -{\bf x}_{i+1})\,({\bf x}_{i+2} -{\bf x}_{i+3})}\,,so as t... | {
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"sourc... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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5e15fb83b5ab6fd33912cd7c71e596a1081c478c | subsection | 7 | 84 | Multi-Regge kinematics | In this paper we will be focusing on the regions where all produced particles have their energy components flip sign under analytic continuation, which amounts , to transforming a single cross-ratio as follows,\tilde{u}\equiv U_{2,N-1}\rightarrow e^{-2\pi i}U_{2,N-1}\,.After analytic continuation, the ratio \begin{}R\e... | {
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"source_ref_id": "b73663136bdd738d18... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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17b7ef679de83a8729287fe2bc8fc3ab8173e529 | subsection | 8 | 84 | Symmetries and soft limits | Let us now discuss the relevant discrete symmetries of multi-Regge kinematics. Parity P corresponds to spatial reflection, or more correctly to the exchange of the two factors comprising the Lorentz group, SO(3,1)\simeq SL(2)\times SL(2). As such, it can be shown (for example by reflecting along the x-axis) that it rev... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
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d264a29d9a7f42d7133b74f5a0de7e6c8bb804c1 | subsection | 9 | 84 | Symmetries and soft limits | As a consequence of the fact that the BDS amplitude captures all soft/collinear divergences of the amplitudes, the BDS-normalized ratio (REF ) is finite in the limit where the momentum of any of the produced gluons becomes soft, and reduces to the same quantity with one leg less.More concretely, in terms of the transve... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
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"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
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"Bram Verbeek"
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1b6f34eb71f7fcc8f5b37df454b93f1a358778a3 | subsection | 10 | 84 | Single-valued multiple polylogarithms | N-particle amplitudes in MRK are expected to be described by single-valued iterated integrals on the space
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... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
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"Vittorio Del Duca",
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6abaa870304223c36d0056d9694a1d50de930289 | subsection | 11 | 84 | Single-valued multiple polylogarithms | As an example we give its action on weight 1 and 2 polylogarithms below,{\bf {s}}\left(G_{a}(z)\right)\equiv \begin{}G\end{}_{a}(z)&=G_{a}(z)+G_{\bar{a}}(\bar{z}),\\
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
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"James Drummond",
"Claude Duhr",
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"Robin Marzucca",
"Georgios Papathanasiou",
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d2ec4e05a8d5caf3d85426a98c828dcffbce4783 | subsection | 12 | 84 | The BFKL equation at finite coupling | In this section, we will obtain a dispersion integral describing the multi-Regge limit of the 2\rightarrow 5 amplitude that is well-defined at any logarithmic accuracy, based on the eikonal approach of . We review the basic ingredients of this approach for the 2\rightarrow 4 amplitude in subsection REF , before extendi... | {
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208a9d55761a6bfd85ea4dc33cb69b79a8ea0dc1 | subsection | 13 | 84 | 6-points | For 2\rightarrow 4 scattering in the multi-Regge limit, the six-point remainder function {R}_6 in the region where we analytically continue the energy components of all produced particles is given by the all-order dispersion relationNote that we have (z_i/\bar{z}_i)^{n/2}=(-1)^n(w_i/\bar{w}_i)^{n/2} when converting any... | {
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2c8e669c5927987af3bfdc0887f6b3a4018a0e66 | subsection | 14 | 84 | 6-points | The two formulations are in fact equivalent via a change of integration contour , as we explain below.
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a362b307f6c7b2e247b48b59400922377f101cc0 | subsection | 15 | 84 | 6-points | In fact, a more detailed analysis of the soft limit reveals that it separately restricts \omega (\nu ,n) and \tilde{\Phi }(\nu ,n) to obey the exact bootstrap conditions ,\omega (\pm \pi \Gamma ,0) = 0, \quad \textrm {and}\,\quad \mathrm {Res}_{\nu =\pm \pi \Gamma }\tilde{\Phi }(\nu ,0)=\pm \frac{1}{\pi a},where by vir... | {
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9ef3279f63443e8aeb52035c6bad3e1fee54b37e | subsection | 16 | 84 | 6-points | It is possible to preserve the symmetry of the integral by averaging between both choices, in order to finde^{{R}_6+i\delta _6} &= \frac{i\Gamma }{2}{\displaystyle \textrm {Res}\\\nu =-\pi \Gamma }\left(\frac{\Phi (\nu ,0)}{\nu ^2-\pi ^2\Gamma ^2}|z|^{2i\nu }e^{-L\omega (\nu ,0)}\right)-\frac{i\Gamma }{2}{\displaystyle... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
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d8093a5982f44c6cc2deef3a661c893d1e993de3 | subsection | 17 | 84 | 6-points | In the last equality we have combined the two contour integrals by introducing the Cauchy principal value \begin{}P\end{}In fact, we could have equally well arrived at the expression (REF ) by virtue of the Sokhotski–Plemelj theorem on the real line,
\frac{1}{x\pm i0}=\mp \pi \delta (x)+\mathcal {P}\left(\frac{1}{x}\ri... | {
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"raw": "V. S. Fadin and L. N. Lipatov, BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at N=4 SUSY, Phys. Lett. B706 (2012) 470–476, [1111.0782].",
... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
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eb5910344ff05c8982fc13081c80b775c1b3113e | subsection | 18 | 84 | 6-points | Focusing on the helicity configuration most commonly found in the literature, see e.g. , the analogue of (REF )-(REF ) is\begin{}R\end{}_{+-}e^{ i \delta _6(z)} =2\pi i f_{+-}\,,f_{+-}= \frac{a}{2} \sum _{n=-\infty }^{\infty }\left(\frac{z}{\bar{z}}\right)^{\frac{n}{2}}
\int _{-\infty }^{\infty }\frac{d\nu }{2\pi }\,\t... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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432d4f6a490ecc8707f4fdbe72b0cfc6664b660c | subsection | 19 | 84 | 7-points MHV | Armed with intuition from the six-point case, we now move on to propose an all-loop dispersion-type formula for the 2\rightarrow 5 amplitude in MRK, again in the region where we analytically continue the energy components of all produced particles. Our strategy will be as follows:A similar strategy for obtaining disper... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
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"James Drummond",
"Claude Duhr",
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67ded9b7d404830cc1bc070e85fc79adc2bb2312 | subsection | 20 | 84 | 7-points MHV | REFTo make contact with other notations used in the literature, f_{+++} is denoted as f_{\omega _2\omega _3} in , , and similarly the six-gluon analogue f_{++} of the previous section is denoted as f_{\omega _2}.,e^{{R}_7(z_1,z_2) + i \delta _7(z_1,z_2)} =2\pi i f_{+++}\,,\begin{aligned}
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
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c36a928ce1ec3daf10b52802a11cc981873a306e | subsection | 21 | 84 | 7-points MHV | Hence the two must be related by parity.\chi ^+(\nu ,n)\chi ^-(\nu ,n)=\tilde{\Phi }(\nu ,n)\,.Finally, C^+(\nu _1,\nu _2,n_1,n_2) is the central emission block, a new ingredient in the BFKL approach to the heptagon compared to the hexagon, first computed to leading order in .Next, we consider the three soft limits whe... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
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"Claude Duhr",
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"Robin Marzucca",
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39629d7b9df36c47d5a7c912133e236d2d4a2a26 | subsection | 22 | 84 | 7-points MHV | Furthermore, it is easy to check that the above relations hold if the residues on those poles are equal to\underset{{\nu _1=\pi \Gamma }}{\mathrm {Res}}\left(\chi ^+(\nu _1,0)C^+(\nu _1,0,\nu _2,n_2)
\chi ^-(\nu _2,n_2)\right)&=i \chi ^+(\nu _2,n_2)\chi ^-(\nu _2,n_2)\,,\\
\underset{{\nu _2=-\pi \Gamma }}{\mathrm {Res}... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
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af5e9dceb6a651f1b388a3f927bea69c80a5ea40 | subsection | 23 | 84 | 7-points MHV | There, we see that if after we close the contour at infinity, we receive a contribution from a residue on the real line in any of the integration variables, the integral left to do in the other integration variable will have the same pole structure as the hexagon integral of figure REF , whereby the poles pinch the con... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
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0be36cdd6cd60ac32cfbca8775710935d0e07f7c | subsection | 24 | 84 | 7-points MHV | \end{aligned}Then, we can rewrite our original contour as\int \tfrac{d\nu _1 d\nu _2 f(\nu _1,\nu _2)}{(\nu _1-\pi \Gamma -i0)(\nu _1-\nu _2+i0)(\nu _2+\pi \Gamma +i0)}&=\int \tfrac{d\nu _1 d\nu _2 f(\nu _1,\nu _2)}{(\nu _1-\pi \Gamma +i0)(\nu _1-\nu _2+i0)(\nu _2+\pi \Gamma +i0)}\\
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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44bccaea25bc866d216b5e9da4e6e01e9b7abf8d | subsection | 25 | 84 | 7-points MHV | The reason is that due to the bootstrap conditions (REF ) and (REF )-(), also the integrand of these simple integrals becomes identical to the hexagon integral in (REF ), up to factors independent of the integration variable,\begin{aligned}2\pi i\int \tfrac{d\nu _1 f(\nu _1,-\pi \Gamma )}{(\nu _1-\pi \Gamma +i0)(\nu _1... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
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"Georgios Papathanasiou",
"Bram Verbeek"
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4049f6681a5a14d3b83058da4a5d2206a8febc97 | subsection | 26 | 84 | 7-points MHV | We stress that although the above formula, which is the 7-point analogue of (REF ), holds independently of how we choose to close the integration contours, it is only valid in the region z_1\ll 1,\,z_2\gg 1 which is convenient for expanding at weak coupling. This is because the \nu _1=\nu _2 residue is a simple integra... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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c27b9d1dff8c9d93a896448daa169ad45015273b | subsection | 27 | 84 | Summary and extension to any helicity | In the previous section, for simplicity we focused on the MHV 2\rightarrow 5 amplitude. Here we will present the generalization of the all-loop dispersion integral (REF )-(REF ), as well as the exact bootstrap conditions (REF )-() that are obeyed by the building blocks, for arbitrary helicity configurations.Using defin... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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bca25e5584e0bc2256e44025d2813f017cfd742d | subsection | 28 | 84 | Summary and extension to any helicity | In addition, we have expressed the integrand in terms of the rescaled quantitiesIn more detail, the generalization of (REF ) to arbitrary helicity follows from \chi ^+ C^+\chi ^-\rightarrow \chi ^{h_1} C^{h_2}\chi ^{-h3}, which can then be recast in the form (REF ) after we plug in the solution of (REF ) and (REF ) for... | {
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dec6cbb5f11f22f7ce95e98500039a54e7688e1c | subsection | 29 | 84 | Summary and extension to any helicity | A crucial property that however follows immediately from the above representation, is thatH(\nu ,0)=1 \,\,\Rightarrow \,\,I^h(\nu ,0)=1\,.The most significant advantage of defining a rescaled central emission block as in (REF ), is that it allows us to formulate separate exact bootstrap conditions for the latter: Along... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
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"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
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"Bram Verbeek"
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1e2f4291487cc13db900626ebf053d548939aa52 | subsection | 30 | 84 | Summary and extension to any helicity | Imposing this on (REF )-(REF ) implies\tilde{C}^-(\nu _1,n_1,\nu _2,n_2)=\overline{H}(\nu _1,n_1) C^+(\nu _1,n_1,\nu _2,n_2)H(\nu _2,n_2)\,,allowing us to obtain (REF )-() for h=- from h=+, also with the help of (REF ). Note that the last formula implies that we only need consider \omega , \tilde{\Phi }, H and \tilde{C... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
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"Claude Duhr",
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3dce9beae55b8e399c955993c844295f846efe4a | subsection | 31 | 84 | From symbols to functions in MRK | In the previous section, we succeeded in obtaining an all-order dispersion integral describing the multi-Regge limit of the 2\rightarrow 5 amplitude of any helicity configuration, that is well defined at arbitrary logarithmic accuracy. In order to complete the description we also need to determine the building blocks o... | {
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logarithmic accuracy | [
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955c89e46075a8a5acbbcd60a53fc0906580c74d | subsection | 32 | 84 | Maximal degree of logarithmic divergence from the OPE | Let us start by stating the main result of this subsection: We will prove that the N-point L-loop remainder function R_N^{(L)} in general kinematics may always be written asR_N^{(L)}=\sum _{0\le j_1+\ldots +j_{N-5}\le L-1}\log ^{j_1} U_1\ldots \log ^{j_{N-5}} U_{N-5} f_{j_1,\ldots , j_{N-5}}where the functions f_{j_1,\... | {
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bd5d373537b1d8cb93c8deb98e5aa6a293f0650b | subsection | 33 | 84 | Maximal degree of logarithmic divergence from the OPE | We can then decompose the Wilson loop into excitations of this flux tube \psi _i, with energy E_i, momentum p_i and helicity m, corresponding to the three isometries of the square,\begin{}W\end{}_N=\sum _{\psi _1,\ldots , \psi _{N-5}}e^{\sum _{j}^{N-5}(-\tau _j E_j+i p_j \sigma _j+i m_j\phi _j)}\begin{}P\end{}(0|\psi _... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
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60aaa243a7f11099163f297c50c0e028ad4ae144 | subsection | 34 | 84 | Maximal degree of logarithmic divergence from the OPE | With the help of the last two formulas, we can show that indeed\begin{aligned}\frac{1}{1+e^{2\tau _{2j+1}}}&=\frac{{\langle } {-j-2}, {-j-1}, {j+2}, {j+3}{\rangle }{\langle } {-j-1}, {-j}, j+1, {j+2}{\rangle }}{{\langle } {-j-2}, {-j-1}, {j+1}, {j+2}{\rangle }{\langle } {-j-1}, {-j}, j+2, {j+3}{\rangle }}=U_{-j-1,j+2}\... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
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c12cf313471f7aca5efdbc8a30d316fa6e173dee | subsection | 35 | 84 | Maximal degree of logarithmic divergence from the OPE | (REF ). Then by virtue of (REF ) and (REF ), the same will be true for R_N^{(L)}, which thus completes the proof.A very similar statement also holds true beyond the MHV case, where it is more convenient to consider the entire superamplitude, rather than its gluonic component alone. This is then dual to a super-Wilson l... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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2939e1ebcb20b2813fb08d56e181c0522a7d6931 | subsection | 36 | 84 | The function-level 7-particle 2-loop MHV amplitude in MRK | In this subsection, we will promote the known 2-loop symbol of the heptagon remainder function in the multi-Regge limit of ref. , as defined in (REF ) for N=7, and in the region corresponding to the analytic continuation (REF ), to a function R_7^{(2)}.Let us start by reviewing the relevant information from the aforeme... | {
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"sou... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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61b7f4c3a634790ec5f418170613a864f4d688fe | subsection | 37 | 84 | The function-level 7-particle 2-loop MHV amplitude in MRK | Here we fix all remaining ambiguity, and show that\begin{aligned}4g(\rho _1,\rho _2)=&2 \begin{}G\end{}_{0,1,1/\rho _1}\left(1/\rho _2\right)-2 \begin{}G\end{}_{1,1,1/\rho _1}\left(1/\rho _2\right)-\begin{}G\end{}_{1/\rho _1}\left(1/\rho _2\right) \begin{}G\end{}_{0,1}\left(1/\rho _2\right)\\
&-\begin{}G\end{}_{0}\left... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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c30b59cc58db9db66d2d4c7a72de22ce6aa8da79 | subsection | 38 | 84 | The function-level 7-particle 2-loop MHV amplitude in MRK | (REF ). Specializing the discussion of subsection REF to the seven-particle amplitude, we infer that the relevant class of functions for describing it are single-valued A_2 polylogarithms. The construction of a basis of such functions at any weight is given in appendix REF . Then, if we know the symbol of any function ... | {
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"sou... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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eb74ddc95ae2e3c7e3bf606200b3deaccddbb0ba | subsection | 39 | 84 | The function-level 7-particle 2-loop MHV amplitude in MRK | The second limit is related to the first one by target-projectile symmetry, which leaves g invariant, g(\rho _1,\rho _2)=g(1/\rho _2,1/\rho _1). Therefore it will not provide any new information, since our ansatz already respects this symmetry. Finally, the third limit \rho _2\rightarrow \rho _1 also sets the coefficie... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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35300c917a8c213180b5320ccd756dda1c1c2894 | subsection | 40 | 84 | All function-level 2-loop MHV amplitudes in MRK | Quite interestingly, from the result of the previous section, we can also obtain all 2-loop MHV amplitudes, in any region in which the adjacent particles k+3,k+4,\ldots ,l+3 have their energy signs flipped. In particular, we will show that in the region in question, we have\frac{R_{N[k+3,l+3]}^{{(2)}}}{2\pi i}
=
\sum _... | {
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"so... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
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"Claude Duhr",
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adbba0a19910a3b428484c1ae60468916bdf9948 | subsection | 41 | 84 | All function-level 2-loop MHV amplitudes in MRK | Forgetting target-projectile symmetry momentarily, the weight-1 SVMPLs that can appear in N-point scattering in MRK will be\log |v_i|\,,\quad \log |1-v_i|\,,\quad \log |v_i-v_j|\,,\quad i<j=k,\ldots ,l-1\,.This is a consequence of the fact that in the regions we are considering, the multi-Regge limit is described by a ... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
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ed7344bfdf452042e7c8882ed7ecc9604b75ae55 | subsection | 42 | 84 | Extracting the NLO central emission block | Let us now combine the knowledge of seven-gluon amplitudes in MRK we have gathered so far, namely the function-level 2-loop MHV case of section , and the dispersion integral governing any helicity configuration to all loops (REF )-(REF ). By matching the perturbative two-loop result to the weak coupling expansion of th... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
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7c8c309a6bd8f1247121054410ec0373740b71a1 | subsection | 43 | 84 | Extracting the NLO central emission block | If we denote the perturbative expansion of the rescaled central emission block as\tilde{C}^+(\nu _1,n_1,\nu _2,n_2)=\tilde{C}^{(0)}(\nu _1,n_1,\nu _2,n_2)+a \tilde{C}^{(1)}(\nu _1,n_1,\nu _2,n_2) +\begin{}O\end{}(a^2)\,,we find the result for the \mathcal {O}(a) correction to the central emission block:\frac{\tilde{C}^... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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8790df7240347830c9f51fcb2968a4aa1531d82e | subsection | 44 | 84 | Extracting the NLO central emission block | In other words they only appear when we expand the arguments of the gamma functions in the inverse transformation from the u_i to the \nu _i.We may readily check that our expression (REF )-(REF ) indeed obeys the \mathcal {O}(a) expansion of the exact bootstrap conditions (REF )-(REF ). For completeness, let us also me... | {
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"raw": "J. Bartels, L. N. Lipatov, and A. Sabio Vera, N=4 supersymmetric Yang Mills scattering amplitudes at high energies: The Regge cut contribution, Eur. Phys. J. C65 (2010) 587–605, [0807.0894].",
... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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8ecb70dc99343c7bb628bbdc011f497ebc7e2609 | subsection | 45 | 84 | Extracting the NLO central emission block | Finally, we may invert (REF )-(REF ) in order to obtain equivalent perturbative expansions for the \chi ^{\pm } and C^+ building blocks of the BFKL approach,Note that here we have redefined \chi ^{\pm } and C^+, compared to e.g. , as follows: \chi ^{\pm }_{\text{here}}=i [\chi ^{\pm }_{\text{there}}+\begin{}O\end{}(a)]... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
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"Falko Dulat",
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99bf3c8a34c496e50449c46cb158c642c7e5a53e | subsection | 46 | 84 | Building the Fourier-Mellin representation | Here we would like to describe a procedure that can take us from the amplitude in multi-Regge kinematics to its corresponding Fourier-Mellin (FM) representation. As we have recalled in subsections REF and REF , in multi-Regge kinematics the amplitude exhibits divergent logarithms which take the form of powers of the \l... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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fb42080c76d552c9b952a547ad1ef228ac0e8079 | subsection | 47 | 84 | Building the Fourier-Mellin representation | Thus we obtain an expression for the holomorphic part of the amplitude in MRK of the form\mathcal {A}^{(L)}(z_1,\hat{z}_2) = \sum _{p,q,r,s} \log ^p \tau _1 \log ^q \tau _2 \log ^r z_1 \log ^s \hat{z}_2 f_{pqrs}(z_1,\hat{z}_2)\,,where f_{pqrs}(z_1,\hat{z}_2) are linear combinations of polylogarithms which are analytic ... | {
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"raw": "F. C. Brown, Multiple zeta values and periods of moduli spaces \\mathcal {M}_{0,n}(\\mathbb {R}), Annales Sci.Ecole Norm.Sup. 42 (2009) 371, [math/0606419].",
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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... |
11ed8430f9cc16af067e67ed1eeacd5f70015a69 | subsection | 48 | 84 | Building the Fourier-Mellin representation | While the formula (REF ) provides the explicit Taylor expansions, we empirically find that f_{pqrs} is always decomposable into sums of the following much simpler type involving only simple harmonic sums of depth one,H_{k_1,k_2,\lbrace r_i\rbrace ,\lbrace s_i\rbrace ,\lbrace t_i\rbrace }(z_1,z_2)=\sum \limits _{n_1,n_2... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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122196cd09135cb88611aa08b0210d9736d461d6 | subsection | 49 | 84 | Building the Fourier-Mellin representation | Secondly, the
reduced sum (REF ) is particularly well suited to expressing its single-valued completion through
a Fourier-Mellin integral.To find a Fourier-Mellin representation for the single-valued completion of each of the f_{pqrs} we begin by specifying the following prescription to be applied to the summations in ... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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7f35cb12e2b47671e31056e5f26a55b662940c50 | subsection | 50 | 84 | Building the Fourier-Mellin representation | Next we specify how to continue the harmonic sums,&Z_{r}(n_j-1)\rightarrow \\
&\quad \frac{(-1)^{r-1}}{(r-1)!}\Bigl [\psi ^{(r-1)}\bigl (\tfrac{n_j}{2}+i\nu _j\bigr )+(-1)^{r-1}\psi ^{(r-1)}\bigl (1+\tfrac{n_j}{2}-i\nu _j\bigr ) -2\delta _{r,{\rm odd}}\psi ^{(r-1)}(1)\Bigr ]\,.Finally we provide a prescription for the ... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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62467b28873fbb68f0b0bfa2357640e85cd07f61 | subsection | 51 | 84 | Building the Fourier-Mellin representation | It is easy to see that for the holomorphic pole the right hand side of (REF ) reduces to their initial quantities once we make use of (REF ) and similarly for (REF ).The prescription (REF ), (REF ), (REF ) is a conjectural method for producing the single-valued completion of a given holomorphic function. Once the presc... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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495e4a5500fde3b24ecc2b9281d0583cced34a5d | subsection | 52 | 84 | Building the Fourier-Mellin representation | Furthermore the derivative of the gamma functions with mixed arguments is already of the single-valued form and requires no rational terms. Altogether this makes it easy to express the integrand in terms of the D,N,V,E,M basis.It remains to note that to return to the variable z_2 instead of \hat{z}_2=1/z_2 we simply re... | {
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"raw": "J. M. Drummond, G. Papathanasiou, and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 07... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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ecced5c6d0899442d43b5ea87eac87ff3116219d | subsection | 53 | 84 | A worked example | We provide here a demonstration of the entire algorithm on a simple weight-three SVMPL whose holomorphic part admits a representation of the form (REF ). We begin with a function with only positive powers of z_1 and \hat{z}_2 in its Taylor expansion,G_{1-z_1,1-z_1,0}(\hat{z}_2) - G_{1,1,0}(\hat{z}_2) &= \log (\hat{z}_2... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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7984f129d895e99c2bfea7a970b912f77a5d9f0d | subsection | 54 | 84 | A worked example | Following the prescription in (REF ) we have&\mathcal {G}_{1-{z}_1,1-{z}_1,0}(\hat{z}_2) - \mathcal {G}_{1,1,0}(\hat{z}_2) \\
&\qquad =\sum _{-\infty <n_1,n_2<\infty } \int _{-\infty }^{\infty }\frac{d\nu _1}{2\pi }\int _{-\infty }^{\infty }\frac{d\nu _2}{2\pi }z_1^{i\nu _1+\frac{n_1}{2}} \bar{z}_1^{i\nu _1-\frac{n_1}{... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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6e9568c3983186db887d0026ac8b9f48fb3d66de | subsection | 55 | 84 | A worked example | In the above expression this means that \tilde{C}^{(0)} and M acquire arguments with canonical sign and the sign of N_2 and V_2 get flipped. | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
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0f53089ec2389f3a08be05419a50c992ea1c04d7 | subsection | 56 | 84 | Higher-loop NLLA predictions | In the previous section, we used the 2-loop MHV heptagon amplitude in the multi-Regge limit, that we
promoted from symbol to function in section , in order to extract the
NLO central emission block (REF )-(REF ). Here, we will use this result, together with the analogous weak coupling expansion of the BFKL eigenvalue (... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
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317c2d9bc6f76f29610797175fd547ebe8166cf4 | subsection | 57 | 84 | Higher-loop NLLA predictions | Separating the coefficients of this expansion into real and imaginary parts, we may define them as\begin{}R\end{}_{h_1,h_2,h_3}\left(\tau _{1},{z_{1}},\tau _{2},{z_{2}}\right)e^{i\delta _7(z_1,z_2)} &=1\,+ 2\pi i \sum _{\ell =1}^{\infty }\sum _{i_1,i_{2}=0}^{\ell -1}a^\ell \,\left(\prod _{k=1}^{2}\frac{1}{i_k!}\log ^{i... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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accf5d6f7c1e2868b78749a92b5c728534b1bc0d | subsection | 58 | 84 | Higher-loop NLLA predictions | The perturbative coefficients (REF ) will be a linear combination of the respective coefficients of all the terms in the right-hand side of (REF ). However for the hexagon amplitudes \begin{}R\end{}_{h_1h_2} they have already been obtained up to at least 7 loops , , , so we only need to focus on the last term that con... | {
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... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
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fd8901ee4c87b14bc3dac9e3824b1f6a1b348fc6 | subsection | 59 | 84 | A nested sum evaluation algorithm | After we expand the integrand in (REF ) at weak coupling, we close the integration contour below (above) the real axis for \nu _1 (\nu _2), and use Cauchy's theorem to express it as a sum over the enclosed residues, with the infinite semicircles giving a vanishing contribution due to |z_1|^{2|\text{Im}(\nu _1)|}, |z_2|... | {
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"source_ref... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
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6c6321d55d1feaab20b6b43bce759f68a0791034 | subsection | 60 | 84 | A nested sum evaluation algorithm | In this manner, and after we set k=|n_1|, l=|n_2|, (REF ) becomes a sum of terms of the general form\sum _{k,l=1}^\infty \frac{z_1^k}{k^{r_1}}\frac{z_2^{-l}}{l^{r_2}}\frac{\Gamma (k+l)}{\Gamma (1+k)\Gamma (1+l)}\prod _{m_i,m^{\prime }_i,m^{\prime \prime }_i}\psi ^{(m_i)}(k+1)\psi ^{(m^{\prime }_i)}(l+1)\psi ^{(m^{\prim... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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556d20232773bfa3f7e5b2ee955ec7e33481de91 | subsection | 61 | 84 | A nested sum evaluation algorithm | After soaking up the gamma function dependence of (REF ) into a rational factor times a binomial coefficient,{k+l k}=\frac{\Gamma (k+l+1)}{\Gamma (k+1)\Gamma (l+1)}\,,shifting the summation variable l\rightarrow j=k+l, and partial fractioning with respect to k, the latter formula splits into terms that look like\sum _{... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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7d342a10b061c59c316e22ca8ad2946f1c6d9975 | subsection | 62 | 84 | A nested sum evaluation algorithm | After shifting the summation variable j\rightarrow i=j-\max (1,a)+1 for each different a, and partial fractioning in i, we reduce all terms (REF ) in our expression for \tilde{f}^{h}_{h_1h_2h_3} into simple sums of the form\sum _{i=1}^\infty \frac{x^{i}}{(i+c)^{n_1}}Z(i+o-1;n_2,\ldots ;1,\ldots ,1)Z(i-1;n^{\prime }_2,\... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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"hep-th"
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83fb069b60738a5306527c6ac63cd7efa4d68ef2 | subsection | 63 | 84 | A nested sum evaluation algorithm | After these steps, the expression (REF ) for \tilde{f}^{h}_{h_1h_2h_3} may be readily evaluated in terms of multiple polylogarithms, thanks to the definition\text{Li}_{m_1,\ldots ,m_j}(x_1,\ldots ,x_j)=\sum _{i=1}^\infty \frac{x_1^{i}}{i^{m_1}}Z(i-1;m_2,\ldots ,m_j;x_2,\ldots ,x_j)\,.The procedure we have described for... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
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34f9ea2ea2a13c99d2451e977010813714cffac8 | subsection | 64 | 84 | Evaluation by Fourier-Mellin convolutions | In this section, we will give a brief overview of the main aspects of a convolution-based method to compute amplitudes in MRK, introduced in , and how it can be adapted for computations beyond LLA. | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
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"Claude Duhr",
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521be819ede4adeeba9a9c71794509e61bde1f88 | subsection | 65 | 84 | Evaluation by Fourier-Mellin convolutions | For proofs and a more detailed review, we refer the reader to .The dispersion integral (REF ), describing the nontrivial part of \begin{}R\end{}_{h_1h_2 h_3}, corresponds to the two-fold application of an inverse Fourier-Mellin transform,\begin{}F\end{}[F(\nu ,n)] = \sum _{n=-\infty }^{\infty }\left(\frac{z}{{\bar{z}}}... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
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"James Drummond",
"Claude Duhr",
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0e3b40a304801696b886cc0c9df077a4b1185d4b | subsection | 66 | 84 | Evaluation by Fourier-Mellin convolutions | At NLLA, (i.e. for i_1 + i_2 = \ell - 2), we write the perturbative coefficients as{\tilde{g}}_{h_1 h_2 h_{3}}^{(\ell ;i_1,i_{2})}(z_1,z_{2}) &= \sum _{j=1}^{2} i_j {\tilde{g}}_{h_1 h_2 h_{3}}^{j;(\ell ;i_1, i_{2})}(z_1,z_{2}) + \sum _{j=1}^{3} {\tilde{g}}_{j;h_1h_2 h_{3}}^{(\ell ;i_1, i_{2})}(z_1,z_{2}) \,, \text{ and... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
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11e742184d0a4db86ba486427a32fff94de71392 | subsection | 67 | 84 | Evaluation by Fourier-Mellin convolutions | Then these corrected perturbative coefficients are given by\begin{split}
{\tilde{g}}_{h_1h_2 h_3}^{j;(\ell ;i_1,i_2)}(z_1,z_2) =& \,
\frac{1}{2} \,\begin{}F\end{}_2 \left[\varpi _7 \, E^{i_1-\delta _{1j}}_1E^{i_2-\delta _{2j}}_2 E^{(1)}_j \right],\\
{\tilde{g}}_{1;h_1h_2 h_3}^{(\ell ;i_1,i_2)}(z_1,z_2) =& \,
\frac{1}{2... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
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d86c58b35a3da3b531ea6ce169bc7ed2389e5d7c | subsection | 68 | 84 | Evaluation by Fourier-Mellin convolutions | Using the convolution theorem for Fourier-Mellin transforms (REF ), we can identify simple relations between the perturbative coefficients at different loop orders. At six points and at LLA, for example, we have g_{++}^{(\ell ;\ell -1)} = -\frac{1}{2}\begin{}F\end{}\left[ \chi _{0,1}^+ E_1^{\ell -1} \chi _{0,1}^- \righ... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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a5a38149c2de60d751585ff35ef86fc18c2dfcab | subsection | 69 | 84 | Evaluation by Fourier-Mellin convolutions | Close to these singularities, f can be written asf(z) &\,= \sum _{k,m,n}\,c^{a_i}_{k,m,n}\,\log ^k\left|1-\frac{z}{a_i}\right|^2\,(z-a_i)^m\,({\bar{z}}-\bar{a}_i)^n\,, \quad z\rightarrow a_i\,,\\
f(z) &\,= \sum _{k,m,n}\,c^{\infty }_{k,m,n}\,\log ^k\frac{1}{|z|^2}\,\frac{1}{z^m}\,\frac{1}{{\bar{z}}^n}\,, \quad z\righta... | {
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... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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a4f4cd35f93618b74328cf425210a5b44c06862d | subsection | 70 | 84 | Evaluation by Fourier-Mellin convolutions | Its Fourier-Mellin transform reads\begin{split}
\begin{}H\end{}(z) &= \begin{}F\end{}[H(\nu ,n)] \\
&= -\frac{z}{(1-z)^2} + \frac{a}{4} \left(\begin{}G\end{}_1(z)+\frac{z }{(1-z)}\begin{}G\end{}_0(z)+\frac{z }{(1-z)^2} \begin{}G\end{}_{0,0}(z)\right) + \begin{}O\end{}(a^2)\\
&= \Biggl (-\frac{z}{(1-z)^2}\Biggr ) * \Big... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
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d64e3456bbb6df59b84914775d63b3cce730078d | subsection | 71 | 84 | Evaluation by Fourier-Mellin convolutions | Expanding the extra terms in a only yields powers of logarithms \begin{}G\end{}_0(z_i)^k, with 0 \le k \le \ell and six-point perturbative coefficients \tilde{g} and \tilde{h} at any given order \ell , and though NLLA, we can limit our analysis to k \le 2. When convoluted with \begin{}E\end{} and \begin{}H\end{}, these... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
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30edd3286a723f570b8830fe38fb947385d73a5a | subsection | 72 | 84 | Evaluation by Fourier-Mellin convolutions | Expanding the term in a, we find\begin{split}
\frac{|z_1|^{2\pi i \Gamma }}{|z_2|^{2\pi i \Gamma }} &= 1 + \frac{a}{2} i \pi (\begin{}G\end{}_0(z_1)-\begin{}G\end{}_0(z_2)) - \frac{a^2}{12} i \pi ^3 (\begin{}G\end{}_0(z_1)-\begin{}G\end{}_0(z_2)) \\
&+ \frac{a^2}{8} \pi ^2 \left( \begin{}G\end{}_0(z_1)^2 - 2 \begin{}G\... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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2413bb4c01db5937c8554e743465be964133e14c | subsection | 73 | 84 | Evaluation by Fourier-Mellin convolutions | Since Fourier-Mellin convolutions are also suited for the computation of hexagon NLLA amplitudes, convolutions in z_2 will behave in the desired fashion. Once again, we have to ensure that convolutions in z_1 do not spoil our results, which means that convolutions with \begin{}E\end{}(z_1) and \begin{}H\end{}^{(1)}(z_1... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
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588547fa9b866048811798a10b9aba336d4ece20 | subsection | 74 | 84 | Conclusions | In this paper, we took a decisive step towards the description of \begin{}N\end{}=4 super-Yang Mills amplitudes with more than six legs in the multi-Regge limit, to arbitrary logarithmic accuracy. Focusing on the 2\rightarrow 5 amplitude, we first succeeded in describing it in this limit in terms of the all-loop disper... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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17c56744496eed9141bd45f379c5a319c7c8a38d | subsection | 75 | 84 | Conclusions | Using the method described in REF , we have already extracted the NNLO correction to the central emission vertex, up to transcendental constants, from the 3-loop heptagon symbol , and although beyond the scope of this paper, it is a straightforward extension to do the same at N^3LO, from the corresponding 4-loop symbol... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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883bb27d219ef080148b0183e18ef3c085a1219a | subsection | 76 | 84 | A convenient basis | As we have reviewed in the main text, the seven-particle amplitude in MRK is expressible in terms of single-valued A_2 polylogarithms. A generating set for their holomorphic parts may be chosen as\mathcal {L}=\Big \lbrace G_{\vec{a}}(\rho _1)|a_i\in \lbrace 0,1\rbrace \Big \rbrace \cup \Big \lbrace G_{\vec{a}}(1/\rho _... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
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935604906093afaadc2d8face32bcf2ce5f40f3f | subsection | 77 | 84 | A convenient basis | Although the representation obtained depends on the choice of the contour \gamma , the actual function obtained only depends on the choice of base point (in this case the origin in (\rho _1,\hat{\rho }_2) coordinates. This is the statement of homotopy invariance and holds provided our initial symbol is integrable. Diff... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
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92b866cd68c15498a8822f89c93bc07da9272935 | subsection | 78 | 84 | A convenient basis | Particularly at two loops, discussed in subsection REF , the symmetries of f, \tilde{f}, (REF ), as well as (REF ), implies that the function g defined there should separately obeyg(\rho _1,\rho _2)=g(1/\rho _2,1/\rho _1)\,.Enforcing the target-projectile symmetry requires knowledge of how the elements of \begin{}L\end... | {
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logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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e229f91f4666f1b6eb63a070a6a78433b14807f2 | subsection | 79 | 84 | Exchange identities | Here we discuss in more detail the \rho _1\leftrightarrow \hat{\rho }_2 exchange identities of the basis A_2 polylogarithms we constructed in appendix REF .Up to weight 3, the only nontrivial identity needed isG_{1,1/\hat{\rho }_2}(\rho _1)=G_{1}(\hat{\rho }_2) G_{1}(\rho _1)+G_{0,1/\rho _1}(\hat{\rho }_2)-G_{1,1/\rho ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/jhep08(2012)043",
"end": 580,
"openalex_id": "https://openalex.org/W3123109576",
"raw": "C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043, [1203.0454].",
"source_ref_... | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
677f22642ca3877652ac070a8a93e809748c3275 | subsection | 80 | 84 | Exchange identities | As an example, we list all nontrivial exchange identities at weight 4 below.G_{0,1,1/\hat{\rho }_2}(\rho _1)=& G_{1}(\hat{\rho }_2) G_{0,1}(\rho _1)-G_{1}(\hat{\rho }_2) G_{0,1/\rho _1}(\hat{\rho }_2)+G_{0,0,1/\rho _1}(\hat{\rho }_2)\\
&+G_{0,1,1/\rho _1}(\hat{\rho }_2)+G_{0,1/\rho _1,1}(\hat{\rho }_2)\\
G_{0,1/\hat{\r... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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