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4384858203839c9f83ce909b67e2cb08b2a76787 | subsection | 61 | 114 | A conformal Ward Identity | An important ingredient in the construction of the minimal surface
governing the four-gluon scattering amplitude was the existence of a
dual SO(2,4) symmetry.In principle this symmetry is
unrelated to the original conformal symmetry. It has been suggested
that, at the level of the worldsheet sigma-model, the symmetrie... | {
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"raw": "R. Ricci, A. A. Tseytlin and M. Wolf, “On T-Duality and Integrability for Strings on AdS Backgrounds,” JHEP 0712, 082 (2007) [arXiv:0711.0707 [hep-th]].",
"source_ref_id": "a07cde0b183635a4f10... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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d0a3a52112fa76c271603cd09dee5a13f52553fd | subsection | 62 | 114 | A conformal Ward Identity | By requiring the area to
be invariant under the action of special conformal transformations
generated by \mathbb {K}^\mu\mathbb {K}^\mu A^{reg}_n=\left(\sum _{i=1}^n 2 x_i^\mu (x_i \cdot \partial _{x_i}+r(x_i) \partial _{r(x_i)})-x_i^2 \partial _{x_i^\mu }
\right) A^{reg}_n=0one may derive an equation for the finite pa... | {
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"raw": "J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, “Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes,” arXiv:0712.1223 [hep-th].",
"source_re... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0ee8e9f282670dc5d494a3f7f314fdce1f85755f | subsection | 63 | 114 | Processes involving asymptotic gluons and local operators | It is natural to extend the discussion in section
REF to cover the decay of fields \phi _i which
couple to the {\cal N}=4 SYM fields through some gauge-invariant
operators Such couplings are toy models for effective
interactions arising from integrating out heavy fields.{\cal L}=L_{YM}+\phi _i O_i~~.While these fields ... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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2528248d6b9c8769241252eaec02911704a4c83a | subsection | 64 | 114 | Processes involving asymptotic gluons and local operators | In fact we have \sum _{i=1}^n k_i^\mu = q^\mu where
q^\mu is the momentum of the operator. It is convenient to
formally think of the coordinate along q^\mu as compact and to
consider a closed string as winding that coordinate. This is
equivalent to saying that we consider an infinite periodic
superposition of the set o... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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f212f50bc0891f6c61e0ae1ab94619711b6cc0ba | subsection | 65 | 114 | Processes involving asymptotic gluons and local operators | The action isi S=-\frac{R^2}{2\pi \alpha ^{\prime }}\int dy dr
\frac{\sqrt{1-(\partial _y t)^2-(\partial _r t)^2}}{r^2}The equations of motion coming from this action should be supplemented
with the appropriate boundary conditionst(r=\infty ,y)=0,\hspace{14.45377pt}t(r=0,y)=y~~\mbox{for}~ |y|\le \frac{\kappa }{4},\hspa... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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7068ece2eea96b98a88d0cd57c96e9c76eb74dcc | subsection | 66 | 114 | Processes involving asymptotic gluons and local operators | For a protected operator such as the stress tensor, whose
dimension equal to four, this is indeed the case. We expect to obtain
logarithmic terms when we go to higher order in the 1/\sqrt{\lambda }
expansion.We can now consider the small r region. The analysis of this
region is the same as the analysis in the small r r... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
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3d28823e436c156d321dbe2ce3a184e4bc0bc828 | subsection | 67 | 114 | Processes involving a mesonic operator and final
quark and antiquarks | In this subsection we consider a small variant of the configuration
considered above. We consider a large N theory with flavors and we
insert a mesonic operator, which contains a quark and an antiquark
field.Here and elsewhere quarks denote fields in the
fundamental representation of the gauge group. For SU(N) their co... | {
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"source_ref_id": "63... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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467a37a8240f691bfbf7c79a74d87eca865a9240 | subsection | 68 | 114 | Processes involving a mesonic operator and final
quark and antiquarks | Due to the
collinear nature of the function g, in the case that we have a cusp
that joins a quark and a gluon we expect to have the average of the
above two formulas In another words, g should measure the
contribution coming from the region closed to the edges joined by the
cusp.. Similarly, we can consider asymptotic ... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
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906f60c24edd7db578a8ccb0acf30c391c81b651 | subsection | 69 | 114 | Processes involving quarks and gluons | Scattering of quarks and gluons were considered in
and . As already
mentioned, we can have fundamental matter by adding extra D-branes
in the bulk. More precisely, we add N_f D7-branes that wrap an
S^3\subset S^5 in the AdS_5 \times S^5. Such configuration
preserves {\cal N}=2 SUSY. As long as N_f \ll N the back
react... | {
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"source_ref_id": "d2245bf3015782f1c378cf5cd74... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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c7064d15ea0b7f5ff0bba47cb702efd31604553e | subsection | 70 | 114 | Processes involving quarks and gluons | The part of the worldsheet at x=0 is
invariant under such transformation and it can be shown that the
radial coordinate satisfies Neumann boundary conditions there.
Using the known expression for the divergent piece of such
amplitude one obtains\log {A_{\bar{q}ggq}}_{div}=
-\frac{f(\lambda )}{8}\left(\log ^2\left(\frac... | {
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... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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1da9f3090b1cb47e61ffbe6ca1de509c3b1654e7 | subsection | 71 | 114 | Further generalizations | In the following we briefly describe many generalizations of the
results presented in this section. We refer the reader to the original
literature for the details.According to the discussion in section REF , the
calculation of the scattering amplitudes in {\cal N}=4 SYM theory at
strong coupling is equivalent to findin... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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074d7f106566b4ffe6c000967aa4efb4c19c482c | subsection | 72 | 114 | Further generalizations | A difficulty in the numerical evaluation of
the area, is the need of a regulator. This causes large errors in the
computation of the finite piece of the area. However, given the
difficulties in constructing analytical solutions, a numerical
approach may be appropriate. Besides, the study of numerical
solutions can give... | {
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"Luis F. Alday",
"Radu Roiban"
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0.0936889... |
34191a7a645c020c111e4870fa61204968f93b85 | subsection | 73 | 114 | Further generalizations | The gravity dual of the {\cal N}=1 theory was
constructed in while that of the
nonsupersymmetric deformations in . In particular,
the authors have considered deformations that break supersymmetry down
to {\cal N}=1 and {\cal N}=0.
It was known that, for real deformation parameter
\beta , that planar scattering amplitud... | {
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"raw": "O. Lunin and J. M. Maldacena, “Deforming field theories with U(1) x U(1) global symmetry and their gravity duals,” JHEP 0505, 033 (2005) [arXiv:hep-th/0502086].",
"source_ref_id": "2d682fd6441... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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bc2bb945e53c7bbc2445208fc0ae77e5d5e09d72 | subsection | 74 | 114 | Scattering Amplitudes vs. Wilson loops at
weak coupling | The discussion in the previous section connects (to leading order in
the strong coupling expansion) two apparently different quantities:
scattering amplitudes and the expectation value of a special type of
Wilson loops. Without additional specifications this relation is
restricted to MHV amplitudes which, as discussed ... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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967b0dec2008ef4412a84a2f44c36815080fe718 | subsection | 75 | 114 | The general statement | While the arguments discussed in the previous section are phrased
in the string theory dual to {\cal N}=4 SYM theory, the final statement
appears to be independent of the coupling constant. Despite the fact
that neither scattering amplitudes nor this particular type of Wilson
loops are BPS quantities, one may conjectur... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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92a71373307dbf20c868dff32af499b1b97b7399 | subsection | 76 | 114 | The general statement | For this reason we will interpret in the following the Wilson loops
as defined directly in the configuration space with coordinates x
related to particle momenta by 2\pi k_i=x_i-x_{i+1}. The
corresponding conformal group is that acting in momentum space,
i.e. the dual conformal group.It was mentioned briefly in section... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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7275de29646c49c3a1b384997ac5da7cef105946 | subsection | 77 | 114 | The MHV amplitudes – Wilson loop relation at
one-loop | The perturbative evaluation of the expectation value of the Wilson
loops (REF ) (and, generally, of any Wilson loop) is
quite straightforward: one expands the exponent while keeping track of
the path ordering and then one evaluates the expectation value in
equation (REF ) by Wick-contracting the resulting
fields either... | {
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"source_... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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a73d7350b9af198b9caf975dafd93271540ad905 | subsection | 78 | 114 | Four-sided polygon | The simplest example of Wilson loop of the type described in
section is a four-sided polygon. The contour
C_4 consists of four light-like segments C_4={\cal C}_1 \cup {\cal C}_2 \cup {\cal C}_3 \cup {\cal C}_4; They are parametrized
in terms of affine parameters \tau _i as{\cal C}_i=\lbrace x^\mu (\tau _i)=x_i^\mu +\ta... | {
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"source_... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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0e453d2befd7a325faa6afb8fefd0dad4a6322d8 | subsection | 79 | 114 | Four-sided polygon | Consider for example I_{12}; it yields
I_{12}=-\int _0^1 d\tau _1 \int _0^1 d\tau _2 \frac{(k_1 \cdot k_2)\Gamma (1-\epsilon _{UV})(\pi \tilde{\mu }^2)^{\epsilon _{UV}}}{\left(-2(k_1 \cdot k_2)(1-\tau _1)\tau _2\right)^{1-\epsilon _{UV}}}=(-s \pi \tilde{\mu }^2
)^{\epsilon _{UV}}\frac{\Gamma (1-\epsilon _{UV})}{2\epsi... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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] | 2,008 | en | Physics | [
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0bd304190fad46426a6c77edbb22b8d9115b63f9 | subsection | 80 | 114 | Four-sided polygon | Inserting
(REF ), (REF ), (REF ) and their relabeled versions into
(REF ) leads to\ln W_4=
\frac{g^2 N}{8\pi ^2}\left({{\rm Div}}_4+\frac{1}{2} \left(\ln \frac{s}{t}\right)^2+2\zeta _2
+{\cal O}(\epsilon _{UV})\right)+{\cal O}(g^4)where the divergent part, denoted by {{\rm Div}}, arises entirely from
integrals of type ... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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a4b2c72e2065fabc719d764d177df074ac3d2a64 | subsection | 81 | 114 | Higher polygons | The calculation of the expectation value of Wilson loops constructed
on higher polygons bears a certain similarity with the expectation
value of the four-sided loop. The curve C_n is now given by
C_n=C_1\cup \dots \cup C_n where each segment C_i (i=1,\dots ,n)
is parametrized as before
C_i=\lbrace x^\mu (\tau _i)=x_i^\... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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378196f146ef60232951ddd47fdb87c0063a114b | subsection | 82 | 114 | Higher polygons | Denoting it by P\equiv k_{i+1,\dots ,j-1} and also introducing
Q as the sum of momenta on the other side of the edges connected
by the gluon, Q\equiv k_{j+1,\dots ,i-1} (such that
P+Q+k_i+k_j=0), s=(k_i+P)^2 and t=(P+k_j)^2, it is not hard
to find that(x(\tau _i)-x(\tau _j))^2=P^2+(s-P^2)(1-\tau _i)+(t-P^2)\tau _j+
(-s... | {
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"source_ref_id": "cab25f78aec... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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82e7d7b9677e3cad49091c3897738afeae3dea89 | subsection | 83 | 114 | Higher polygons | Then,
the complete contribution to the expectation value of the Wilson
loop comes from the integrals REF (c) each of
which, in this gauge, equals one easy two-mass box function.It is interesting that in Feynman gauge, used in the preceeding
calculation, there is such a clean separation of the divergent and the
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"source_r... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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"hep-th"
] | 2,008 | en | Physics | [
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26dfa6e1e8345d3173e9506abfdd8e160df301a3 | subsection | 84 | 114 | A conformal Ward identity | Wilson loops are generically not invariant under coordinate
transformations, since the latter changes the contour defining
them; if the Lagrangian is invariant and if the Wilson loop is
well-defined (finite), then one finds instead that\langle W(\tilde{C}) \rangle = \langle W(C)\ranglewhere \tilde{C} is the image of cu... | {
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"source_r... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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bd1f3881eb175b70c7c4b2463e900a93273feaa4 | subsection | 85 | 114 | General Properties of Cusp Singularities | A general feature of the polygonal Wilson loops conjecturally
related to MHV scattering amplitudes is the presence of cusps. As
we have seen in explicit calculations, their presence combined
with the fact that the edges are light-like leads to an
\epsilon ^{-2} short distance singularity at
one-loop.We remind the reade... | {
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"raw": "G. P. Korchemsky and A. V. Radyushkin, “Renormalization of the Wilson Loops Beyond the Leading Order,” Nucl. Phys. B 283, 342 (1987).",
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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723b721dc550d96b476a1e88f82d824f9cdebe7a | subsection | 86 | 114 | General Properties of Cusp Singularities | In other words, divergences may
be subtracted recursively and, after subtraction of all
subdivergences, the remaining overall divergence is local and can
also be subtracted by a counterterm, i.e.\langle W_{\rm cusp}\rangle =Z_{\rm cusp} F_{\rm cusp}~~,where F_{\rm cusp} is finite as the regulator is removed while
Z_{\r... | {
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"raw": "I. A. Korchemskaya and G. P. Korchemsky, “On light-like Wilson loops,” Phys. Lett. B 287 (1992) 169.",
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"start": 585
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... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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] | 2,008 | en | Physics | [
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ad9c01b720a86450417ffb9176add53e0dca7f5e | subsection | 87 | 114 | General Properties of Cusp Singularities | It has been argued in
that \Gamma (a) and G(a) are related by
a scheme-independent multiple of the first subleading term in the
large spin expansion of the anomalous dimension of twist-two
operators.The structure of cusp divergences implies that the logarithm of the
expectation value of the light-like Wilson loops W_n... | {
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739e16d2dfadec807f7615d255904438f14dab00 | subsection | 88 | 114 | Conformal properties of light-like Wilson loops | On general grounds, when a symmetry is broken by a regulator it
may develop anomalies at the quantum level. They typically appear
in the form of the product between a factor vanishing as the
regulator is removed and a factor that would diverge in the same
limit. Wilson loops in dimensional regularization could (and, as... | {
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Correspondence | [
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"Radu Roiban"
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de59b7d0278cc063ec858b45a680c98fa53af0ee | subsection | 89 | 114 | Conformal properties of light-like Wilson loops | It thus follows that
the Lagrangian transforms homogeneously with weight four:\mathbb {D}:{\cal L}(x)\mapsto \Lambda ^4 {\cal L}(\Lambda x)Thus, under infinitesimal dilatations, the regularized action
transforms asIn the presence of the regulator the measure
transforms as \mathbb {D}:d^Dx\mapsto \Lambda ^{-4+2\epsilon ... | {
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} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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cc0392f38a730dc1dcfda27a0656a41da19d8a4c | subsection | 90 | 114 | Conformal properties of light-like Wilson loops | Similarly to (REF ), the presence
of divergences prevents at this stage setting \epsilon \rightarrow 0 on the right hand side of the equation above; these terms are
the origin of the conformal boost anomaly .The renormalization properties of Wilson loops (REF )
suggest that it would be convenient to organize the equati... | {
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Correspondence | [
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29455e5444af38fc177f9a02006fccd4b4649f68 | subsection | 91 | 114 | Conformal properties of light-like Wilson loops | The wigglyline represents the gluon propagator and the blob the insertionpoint.]The final result for this expectation value,
obtained in , is\frac{2i}{g^2 \mu ^{2\epsilon }}\frac{\langle {\cal L}(x) W_n
\rangle }{\langle W_n \rangle }=- {\rm a}\sum _{i=1}^n
(-x^2_{i-1,i+1}\mu ^2)^\epsilon \left(\frac{1}{\epsilon ^2}\de... | {
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f276cc74d5e2745053288c8d00b2c4702763834c | subsection | 92 | 114 | Conformal properties of light-like Wilson loops | Thus, collecting
everything, it is easy to find that\mathbb {D}\ln \langle W_n \rangle &=&-{\rm a}\frac{1}{\epsilon }\sum _{i=1}^n
(-x^2_{i-1,i+1}\mu ^2)^\epsilon +{\cal O}({\rm a}^2)\\
\mathbb {K}^\mu \ln \langle W_n \rangle &=&-{\rm a}\frac{1}{\epsilon }\sum _{i=1}^n x_i^\mu (-x^2_{i-1,i+1}\mu ^2)^\epsilon +{\cal O}(... | {
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1f79f6e90541653deeeb7d59d396138b187da8ee | subsection | 93 | 114 | An all-loop generalization of the conformal Ward identity | Unlike axial anomalies, the anomalies of the dual conformal
symmetry are not one-loop exact. It is possible, though not
completely straightforward, to generalize to higher loops the
calculation described in the previous section. A key point which
makes the calculations tractable is that, as repeatedly mentioned
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687997bf6bf2e26fb844e454bca830bad64501bf | subsection | 94 | 114 | An all-loop generalization of the conformal Ward identity | The argument is based on dimensional
analysis, the scheme independence of \Upsilon
and the fact that the form of \Upsilon is, up to its arguments,
independent of the cusp it originates from.Independently of
these arguments, it is possible to show explicitly that
\int d^Dx \Upsilon ^{(l)}(x|x_{i-1},x_i,x_{i+1})=0~~,
a... | {
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76f503bff37134ee924b6e23393a667fd0142213 | subsection | 95 | 114 | An all-loop generalization of the conformal Ward identity | The
consequences of the conformal boost Ward identity will be
discussed in the next section; not surprisingly, the results
reproduce the structure of constraints on scattering amplitudes
following from dual conformal invariance. | {
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47088529790063f4e6cde6c6465028edec15fb14 | subsection | 96 | 114 | Constraints on expectation values of Wilson loops | As we have seen previously, the equation (REF ) implies a
relatively simple constraint on the finite part F_n of the
Wilson loop. The Ward identity for conformal boosts requires
further analysis.The notation may be slightly simplified by making use of the maximal
nonabelian exponentiation theorem , , which
states that ... | {
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08b97900b07070bce3de6df8df2d7c42d5174226 | subsection | 97 | 114 | Constraints on expectation values of Wilson loops | This
strategy is extensively applied in two-dimensional conformal field
theories; in four dimensional theories it was initially applied, in a
related context, in .As we briefly mentioned in section REF , inversion
transformations act asI:x_i^\mu \mapsto \frac{x_i^\mu }{x_i^2}
~~~~~~{\rm and}~~~~~~
I:x_{ij}^2\mapsto \fr... | {
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]... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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473d9775fb8e79eb878de873d637fbd979528831 | subsection | 98 | 114 | Constraints on expectation values of Wilson loops | A similar function may be defined for Wilson loops with an arbitrary
number of edges{\rm F}_n=\frac{1}{2}F^{(1)}_{n}(0)+{R}_{Wn}({\bf u};{\rm a})and represents the Wilson loop analog of the remainder function
{R}_{An} capturing the departure of scattering amplitudes
from the BDS ansatz (cf. section
REF ).The first argu... | {
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} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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00820ccf0cea6218ff72ece031ba9c7c59c41f90 | subsection | 99 | 114 | Higher-loop tests of the amplitude/Wilson loop relation | The discussion in previous sections exposes a relation between two
apparently different quantities: MHV gluon scattering amplitudes in
{\cal N}=4 SYM theory and the expectation value of Wilson loops
constructed in a special way from light-like segments. Manipulations
using T-duality transformations for strings in AdS_5... | {
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8cdcc1d12477d4a550585d95ef33cc15761eb9a3 | subsection | 100 | 114 | Higher-loop tests of the amplitude/Wilson loop relation | It is also important to
mention that this separation of the divergent part of the logarithm of
the Wilson loop expectation value from its finite part is consistent
with the strong coupling analysis described in section
.If both the amplitude/Wilson loop relation as well as the BDS ansatz
were indeed to hold, then toget... | {
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} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
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1942ecbeda1cf976d4546773f287d6c69f2589bd | subsection | 101 | 114 | Rectangular configuration with a large number of gluons | To test the relation between Wilson loops and MHV amplitude at strong
coupling in the form of the equation (REF ) it appears
necessary, at least at first sight, to find the minimal surface
corresponding to some n-sided polygonal boundary conditions. As seen
in section , such a construction is challenging
for any number... | {
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Correspondence | [
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d15ac314b08ec487a3f9935d9f1c7cf45bbde104 | subsection | 102 | 114 | Two loops and beyond | The arguments in the previous sections, leading to the conjectured
relation between scattering amplitudes and light-like cusped Wilson
loops, are very compelling. Dual conformal symmetry SO(2,4) is,
however, yet to be proven to be a symmetry of the (MHV) amplitudes. Thus,
its presence and consequences need to be tested... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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] | 2,008 | en | Physics | [
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650fc6282039626646fc7497c61fc654282516ca | subsection | 103 | 114 | Two loops and beyond | Indeed, to this order, the two-gluon
term above combines with two three-field terms or one four-field
term from the Lagrangian, while the three-gluon term above
combines with a three-field term in {\cal L}.As in the one-loop computation, all terms may be conveniently
represented in terms of Feynman diagrams. Depending ... | {
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c9099f7be005c4d04d69f67d0c80a21bf760d958 | subsection | 104 | 114 | Two loops and beyond | To the first few orders in the loop expansion, l=1,2,3, the maximally
nonabelian factor is c^{(l)}=C_F N^{l -1}, but starting from
four loops it is not expressible in terms of the Casimirs C_F
and C_A , .The diagrams with non-maximal color factor factorize in products of
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Correspondence | [
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"Radu Roiban"
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a37b0bc7094f3515fd7ee87664063328ca35f399 | subsection | 105 | 114 | Polygon with four cusps | The basic ingredients of two loops computations already appear when
studying two loops corrections to the four cusps Wilson loop, so we
begin by reviewing this calculation in some detail .
The complete set of diagrams, not making use of the nonabelian
exponentiation theorem, is shown in figure REF .
[Figure: Diagrams c... | {
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c93d38ab666aa2eec2a9235fa5518916ad265305 | subsection | 106 | 114 | Polygon with four cusps | Therefore,
this diagram does not contribute to w^{(2)}. In fact, it combines
with the terms proportional to C_F^2 ignored in the previous item to
yield unrestricted integrals over the end-points of the two gluon
propagators; these terms are nothing but the finite part of
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"Luis F. Alday",
"Radu Roiban"
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6d4031b01c7c3f9c49ff8b4c9302d2438fcca99c | subsection | 107 | 114 | Polygon with four cusps | The subleading poles in the same equation,
evaluated for the universal scaling function and G-function of
{\cal N}=4 SYM, are also reproduced after a further
\lambda -dependent redefinition of \mu .
The double-logarithm in the finite term F_4 reproduces the kinematic
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"Radu Roiban"
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db6fb15c211331354ea95762b08b83b5df2a4543 | subsection | 108 | 114 | Polygon with six cusps | As discussed at length in section REF , dual
conformal symmetry is not sufficiently powerful to completely fix the
expectation value of light-like Wilson loops with at least six
cusps. The expectation value is instead fixed up to the addition of
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0cd214eed5d84dcca83a0d8c0e734e8f77f78c94 | subsection | 109 | 114 | Polygon with six cusps | An immediate test of the result is that
indeed the Wilson loop remainder function depends only on the
three conformal cross-ratios, as the initial construction implies.
Besides this consequence of dual conformal invariance, it is also
possible to identify other properties of the
{R}_{W6}^{(2)}(u_1,u_2,u_3) and compare ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1529,
"openalex_id": "",
"raw": "J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, “The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude,” arXiv:0712.4138 [hep-th].",
"source_ref_id": "43796de8f8c... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
168894,
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16750,
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100708,
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0.1337890625,
0.163330078125,
0.088012695... |
d4d48c5ab2986d63c82c3dddf5bd4efc327edcce | subsection | 110 | 114 | Polygon with six cusps | A convenient one is the symmetric point
K^{(0)} in table
REF for which the conformal cross-ratios are(u_1,u_2,u_3)=\left(\frac{1}{4},\,\frac{1}{4},\,\frac{1}{4}\right)~~.The differences between the amplitude and Wilson loop remainder
functions at the points K^{(i)} and at K^{(0)} (denoted by
{R}_{A6}^{0} and {R}_{W6}^{... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
62,
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0.1075439453125,
0.1380615234375,
0.1824951... |
dea4a1f158c9429daa9ad8275698cd56b4a1f868 | subsection | 111 | 114 | Outlook | Scattering amplitudes remain one of the basic ingredients in our
understanding of quantum field theories. They are usually evaluated
order by order in a weakly-coupled perturbation theory and it is
rarely the case that the resulting series can be constructed and
resummed to all orders in perturbation theory, even only ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 868,
"openalex_id": "",
"raw": "Z. Bern, L. J. Dixon and D. A. Kosower, “On-Shell Methods in Perturbative QCD,” Annals Phys. 322, 1587 (2007) [arXiv:0704.2798 [hep-ph]].",
"source_ref_id": "bb3f439ff021909abc0a86526cb81a4663... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
159,
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90,
47143,
70,
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0.009735107421875,
0.026916503906... |
e90d98c43bcb2c98f929a768e65cba9b4a956bb9 | subsection | 112 | 114 | Outlook | Minimal surfaces with
this interpretation are not known even in the simplest cases.
Besides their obvious interpretation in terms of decay amplitudes,
understanding in detail such processes may also lead to
understanding the calculations of anomalous dimensions of short
operators on the string theory side of the AdS/CF... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 859,
"openalex_id": "",
"raw": "J. McGreevy and A. Sever, “Planar scattering amplitudes from Wilson loops,” arXiv:0806.0668 [hep-th].",
"source_ref_id": "56b09a8f6834c61ba87849b002b1e0ca39386463",
"start": 772
},
... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0.0226593017578125,
0.1492919921875,
... |
f8408266afdc2d08df5be6df46075ec55face6c6 | subsection | 113 | 114 | Outlook | The analysis described there also exposes
a close relation between the generators of the dual (super)conformal
group and the hidden (non-local) integrals of motion of the world
sheet theory in the original AdS_5\times S_5 (i.e. the worldsheet
theory prior to the T-duality transformations relating scattering
amplitudes ... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0.05078125,
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0.03338623046875,
0.08837890625,... |
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