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1ade1330b14e7369a3f5f86e9f61eaced7465129 | subsection | 37 | 115 | Old-fashioned Perturbation Theory | () can be shown to provide the fermion instantaneous interaction-iV_{2}&=
-\dfrac{1}{2}e^{2}\delta _{\mathbb {C}0}\int d^{2}\mathbf {x}^{\perp }dx^{-} \bar{\tilde{\psi }}(0,\mathbf {x}^{\perp },x^{-})\gamma ^{i}\tilde{A}_{i}(0,\mathbf {x}^{\perp },x^{-})\\
&\times \frac{\gamma ^+}{\partial _-} \tilde{A}_{j}(0,\mathbf {... | {
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8563d3329ab8cc2b25780fcade9edc04d808c6d3 | subsection | 38 | 115 | Old-fashioned Perturbation Theory | () written out in full is-iV_{3}
&=\frac{1}{2}ie^{2}
\int d^{2}\mathbf {x}^{\perp }dx^{\widehat{-}} \bar{\psi }(0,\mathbf {x}^{\perp },x^{\widehat{-}})\gamma ^{\widehat{+}}\psi (0,\mathbf {x}^{\perp },x^{\widehat{-}})\\
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014779a8cd04e335fc6550f5df8ad1f9a59bf530 | subsection | 39 | 115 | Old-fashioned Perturbation Theory | Using&\int d^{2}\mathbf {X}^{\prime \perp }dX^{\prime \widehat{-}} \frac{ e^{-i\left[\sqrt{\mathbb {C}} \mathbf {q}_{\perp }\cdot (\mathbf {X}^{\perp }-\mathbf {X}^{\prime \perp })+q_{\widehat{-}}(X^{\widehat{-}}-X^{\prime \widehat{-}})\right]}}{4\pi \sqrt{(\mathbf {X}^{\perp }-\mathbf {X}^{\prime \perp })^{2}+(X^{\wid... | {
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... | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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920c64c94fa3d44d8f1ad9692d60d3aba0f40113 | subsection | 40 | 115 | Toy calculation of | Having laid out the foundation of interpolating QED, we can now make some calculations. The first simple heuristic example we consider is e^+ e^- annihilation producing two scalar particles. In the next section, we will consider the typical QED process of e^+e^-\rightarrow \gamma \gamma , as well as e\gamma \rightarrow... | {
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"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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32f6a6ca7fa6e926767df98b1df5a171027c21c0 | subsection | 41 | 115 | Toy calculation of | The interpolating time-ordered diagrams are also Figs. REF and REF , and the propagators of the intermediate virtual fermion for each time-ordering are given by\Sigma _a&=\frac{1}{2Q^{\widehat{+}}}\;\frac{{Q}_a+m}{q_{\widehat{+}}-Q_{a\widehat{+}}}\\
\Sigma _b&=\frac{1}{2Q^{\widehat{+}}}\;\frac{-{Q}_b+m}{-q_{\widehat{+}... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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4690d3d482ad22cba9f3997e34e16d640bc762da | subsection | 42 | 115 | Toy calculation of | This can be now shown explicitly as follows:&\Sigma _{b,\delta \rightarrow \frac{\pi }{4}}=\lim \limits _{\mathbb {C}\rightarrow 0}\left( \frac{1}{2Q^{\widehat{+}}}\;\frac{{Q}_b-m}{q_{\widehat{+}}+\frac{\mathbb {S}q_{\widehat{-}}+Q^{\widehat{+}}}{\mathbb {C}}}\right) \\
&=\lim \limits _{\mathbb {C}\rightarrow 0} \frac{... | {
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"Chueng-Ryong Ji",
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81bc7f118ecf4d27f863f6ccf516d488f0b688da | subsection | 43 | 115 | Toy calculation of | (REF ) as discussed in the introduction (Sec. )
as well as in the formal derivation (Sec. ).Let's now compute the time-ordered amplitudes for the e^+ e^- annihilation into two scalar particles using the interpolating formulation, which
are given by\mathcal {M}_a^{\lambda _1, \lambda _2} =\bar{v}_{\lambda _2}(p_2)\cdot ... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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b8e0fc62d3f69a3693711804518ea619dc5e459d | subsection | 44 | 115 | Toy calculation of | Similarly, if q=p_1 - p_4, then we may denote them as the u-channel amplitudes \mathcal {M}_{a,u}^{\lambda _1, \lambda _2} and \mathcal {M}_{b,u}^{\lambda _1, \lambda _2}, respectively.The spinors in the interpolation form were studied in Ref. and the results were given byu_H^{(+1/2)}(P)&=\left( \begin{array}{c}
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abb62fe35b6bb47e9f0acbfc4b123b8a3cf65dd6 | subsection | 45 | 115 | Toy calculation of | We choose the initial reference frame to be the e^+e^- center of mass frame (CMF), and study the whole landscape of the amplitude change under the boost operation in the \hat{z}-direction as well as the change of the interpolation angle \delta . The moving direction of the incoming electron is chosen as the +\hat{z} -d... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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970e9f596eda92bbc22c1995592acec02ff58a81 | subsection | 46 | 115 | Collinear Scattering/Annihilation, | Before we discuss the angular dependence of the interpolating helicity amplitudes,
we first consider the collinear amplitude taking the the center of mass angle \theta between the moving direction of incoming electron (particle 1) and outgoing photon (particle 3) as \pi , i.e. the collinear back-to-back scattering/anni... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
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7ba9b1d791f04af9cd359783132d15eb3961a0a6 | subsection | 47 | 115 | Collinear Scattering/Annihilation, | The red solid line in the middle of all the figures is given by
[Figure: e^+e^- pair annihilation process at angle \theta in center of mass frame][Figure: Annihilation amplitudes for e^+ e^- to two scalars t channel time-orderingprocess-a : for (a) helicity ++ , (b) helicity +- , (c) helicity -+ and (d) helicity -- .][... | {
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84a7cf98f37a6525bdab9c4aba3116454ef29075 | subsection | 48 | 115 | Collinear Scattering/Annihilation, | Such swap of the helicity between the IFD and LFD
for the particle moving in the -\hat{z} direction has been extensively discussed in Ref. and the application in the deeply virtual Compton scattering has been
reviewed in Ref.. We find indeed that the behavior of the angle between the momentum direction and the spin dir... | {
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... | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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5528656a92da7aef8643afaf119f9dc982d05f23 | subsection | 49 | 115 | Collinear Scattering/Annihilation, | (REF ).
One may note that the amplitude \bar{v}\gamma ^+ u vanishes for the helicity non-flip case, i.e. ++ and --, while it survives for the helicity flip case, i.e. +- and -+.
This demonstrates that the LFD (\delta =\pi /4) results of ++ and -- helicity amplitudes, \mathcal {M}_{b,t}^{+,+}, \mathcal {M}_{b,t}^{-,-}, ... | {
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"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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88e791d948cd18c026b8130e9146fd5c116157fe | subsection | 50 | 115 | Collinear Scattering/Annihilation, | REF , REF , REF , REF and REF ,
we note that the IFD results in P^z \rightarrow +\infty appear to yield the corresponding LFD results as one can see the smooth connection of each and every amplitude in the
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
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63995bb12eeba63b01c2a0caad640ac3aef7d1ac | subsection | 51 | 115 | Non-collinear Scattering/Annihilation, | Now, the non-collinear helicity amplitudes can be computed by varying the center of mass angle \theta in the scattering/annihilation process.
As discussed earlier, for the non-collinear kinematics, the same amplitude can correspond to either the
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2c86db1be5e73702a0882800bc36448c0696ea12 | subsection | 52 | 115 | Non-collinear Scattering/Annihilation, | Due to the sign change of the intermediate fermion momentum q_b=-q_a=-q for the other time-ordered amplitude \mathcal {M}_{b,t}^{+,+},
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257da911ea2cb5c022bd3fc255d4961dace684c2 | subsection | 53 | 115 | Non-collinear Scattering/Annihilation, | Since the values of the amplitude \mathcal {M}_{a,t}^{+,+} dramatically change around the critical angle \theta _{c,t} from 0.0 on the left (\theta < \theta _{c,t}) to around 2.0 on the right immediately passing the critical angle \theta _{c,t} as depicted in Fig. REF , we should be able to see the corresponding dramat... | {
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... |
cc1e7c5388a75778c191f7aa72ef11fe6a42bb50 | subsection | 54 | 115 | Non-collinear Scattering/Annihilation, | As discussed earlier, the angle regions for the “instantaneous fermion contribution" and the “on-mass-shell propagating contribution" swap in \mathcal {M}_{b,t}^{+,+} with respect to
\mathcal {M}_{a,t}^{+,+} due to the sign change of the intermediate fermion momentum q_b=-q_a=-q for
\mathcal {M}_{b,t}^{+,+}, i.e. the r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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c8b5a34dc36f771cddc299e3a402cf74815e663a | subsection | 55 | 115 | Non-collinear Scattering/Annihilation, | REF ,
the helicity amplitude \mathcal {M}_{b,t}^{+,+} doesn't change much except its value at \delta =\pi /4 or at LFD.
[Figure: ++ annihilation helicity amplitudes for: (a) \mathcal {M}^{+,+}_{a,t} ,(b) \mathcal {M}^{+,+}_{a,t} + \mathcal {M}^{+,+}_{b,t} , and (c) \mathcal {M}^{+,+}_{a,u} + \mathcal {M}^{+,+}_{b,u} .]... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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34f4e1f6840eff8976bad67c401c10a1462c3a42 | subsection | 56 | 115 | Non-collinear Scattering/Annihilation, | REF where we present the +- helicity amplitudes in LFD (\delta = \pi /4)
(a) \mathcal {M}_{a,t}^{+,-}, (b) \mathcal {M}_{b,t}^{+,-}, (c) \mathcal {M}_{a,t}^{+,-} + \mathcal {M}_{b,t}^{+,-}
and (d) \mathcal {M}_{a,u}^{+,-} + \mathcal {M}_{b,u}^{+,-}. In contrast to \mathcal {M}_{a,t}^{+,+} discussed above, the “instanta... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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5b665c6f6e7e497e629f002ed523ca6a4734f9cf | subsection | 57 | 115 | Non-collinear Scattering/Annihilation, | REF , REF and REF ,
we provide the whole landscape of the interpolation angle (\delta ) dependence for the angular distributions of the helicity ++ and +- amplitudes at CMF (i.e. P^z=0).
In each and every figure, the critical interpolation angle \delta _c which separates the IFD side and the LFD side of helicity branch... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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d78a8a3c2b0da5022713bb68adbb76d940cdffc5 | subsection | 58 | 115 | Non-collinear Scattering/Annihilation, | Since the incident e^- e^+ annihilation takes place along the z-axis and the positron (e^+) is moving in the -\hat{z} direction,
the swap of the helicity between the IFD and LFD for the positron can be understood as we see the IFD/LFD profile correspondence in Fig. REF .To examine the frame dependence of the whole land... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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bdcc20f51fd2759c80623a3ebc557ae7f7c417e3 | subsection | 59 | 115 | Non-collinear Scattering/Annihilation, | REF (a) \mathcal {M}_{a,t}^{+,-} (b) \mathcal {M}_{b,t}^{+,-} (c) \mathcal {M}_{a,u}^{+,-} (d) \mathcal {M}_{b,u}^{+,-}.
The profiles of the “instantaneous fermion contribution" and the “on-mass-shell propagating contribution" at \delta = \pi /4 (LFD) discussed at CMF (P^z=0)
survive invariantly although significant ch... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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b6204966809ea3b2a0a434f91cfabb2e23854524 | subsection | 60 | 115 | Non-collinear Scattering/Annihilation, | Except the LFD profiles, the whole landscapes of angular distributions
are dynamically varied both for 0 \le \delta < \delta _{c,e^-} \approx 0.55062 and \delta _{c,e^-} \approx 0.55062 < \delta < \delta _{c, e^+} \approx 0.784165
depending on the reference frames (P^z = +15 m_e, 0, -15 m_e).
The net results adding bot... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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de6fa55b14e76f0dca4b55e8fccbc1748fccc250 | subsection | 61 | 115 | Non-collinear Scattering/Annihilation, | However, the sum of helicity amplitude squares is completely independent of not only the interpolating angle \delta but also the reference frames as it should be.
The boost-invariant physical quantity must be of course completely independent of the interpolation angle, regardless of IFD, LFD or any other dynamics in be... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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63136fe416dc4087fb90f8dba6f910f7c551c935 | subsection | 62 | 115 | Summary of | As we have shown in all of these results, the LFD results are completely independent of the reference frame due to the boost invariance
while the IFD results are dependent on the reference frame. As discussed in the collinear case (see Figs. REF , REF , REF , REF and REF ), the LFD results are outside the spin-flip bou... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
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49490dddb2317316b59b66568eece83ab390bf18 | subsection | 63 | 115 | Summary of | REF is in exact agreement
with the analytic result of the total amplitude square for the scalar particle pair production in e^+ e^- annihilation given by&\left| \mathcal {M} \right|^2_{\rm {scalar}} \equiv \sum _{\lambda _1,\lambda _2}|\mathcal {M}_{a,t}^{\lambda _1,\lambda _2}+\mathcal {M}_{b,t}^{\lambda _1,\lambda _2... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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ff1b8a2046ae83a889f42bee5fb08a1570d3ecaa | subsection | 64 | 115 | Body | Having discussed all the helicity amplitudes of the pair production of scalar particles in e^+ e^- annihilation,
we now look into the two photon production process in the same initial state of e^+ e^- annihilation.
While there must be some similarity inherited from the same initial state, there must be also some differ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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1fa674587b8659ca3ddf63994057f37059bc1fd0 | subsection | 65 | 115 | Body | (REF ).The QED helicity amplitudes \mathcal {M}_t^{\lambda _1,\lambda _2,\lambda _3,\lambda _4} and \mathcal {M}_u^{\lambda _1,\lambda _2,\lambda _3,\lambda _4}
with the two initial lepton helicities \lambda _1 and \lambda _2
and the final two photon helicities \lambda _3 and \lambda _4 in t and u channels, respectivel... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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f0326836836ecb5d9f8ac07084446f83b01f3c52 | subsection | 66 | 115 | Body | Note that this interpolating polarization vector \epsilon _{\widehat{\mu }}^\lambda (P) respects the gauge condition A^{\widehat{+}}=0 and \partial _{\widehat{-}}A_{\widehat{-}}+\partial _{\perp }\cdot \mathbf {A}_{\perp }\mathbb {C}=0, which
links the light-front gauge A^{+}=0 in the LFD and the Coulomb gauge \nabla \... | {
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"doi": "... | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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6ec1e804830aca5b74a3c0ef5e7bede692813139 | subsection | 67 | 115 | Body | (REF ) and (), the time-ordered amplitudes in t-channel can be written in short-hand notations
without specifying the helicities as\mathcal {M}_{a,t}=\bar{v}(p_2) {\epsilon }(p_4)^* \left( \frac{1}{2Q_t^{\widehat{+}}}\frac{{Q}_{a,t}+m}{q_{t\widehat{+}}-Q_{a,t\widehat{+}}}\right) {\epsilon }(p_3)^* u(p_1),and\mathcal {M... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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24e1a1c84099bdafe4b450f67d55f2a6e590e218 | subsection | 68 | 115 | Body | Consequently, the interpolating on-mass-shell energy of the
intermediate propagagting fermion Q_{a,u\widehat{+}} and Q_{b,u\widehat{+}} for the two time-ordered amplitudes are also given by
replacing q_{a,t} and q_{b,t} by q_{a,u} and q_{b,u}, respectively, in Eqs. (REF ) and ()
together with the replacement of Q_t^{\w... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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e42b376ca92ef3743cf9291675ef04ac61e988af | subsection | 69 | 115 | Body | REF show the helicity amplitudes \mathcal {M}^{+,+,\lambda _3,\lambda _4}_{a,t} and \mathcal {M}^{+,-,\lambda _3,\lambda _4}_{a,t} with the final four helicity configurations
of the photon pairs \lbrace \lambda _3,\lambda _4\rbrace = \lbrace +,+\rbrace , \lbrace +,-\rbrace , \lbrace -,+\rbrace , \lbrace -,-\rbrace but ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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ac39dd68650d1f2779faa02d4b259a42ace5701a | subsection | 70 | 115 | Body | REF shows the helicity amplitudes \mathcal {M}^{+,+,\lambda _3,\lambda _4}_{a,t} with the final four helicity configurations
of the photon pairs \lbrace \lambda _3,\lambda _4\rbrace = \lbrace +,+\rbrace , \lbrace +,-\rbrace , \lbrace -,+\rbrace , \lbrace -,-\rbrace but with the same initial ++ helicity configuration of... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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8be69d70329fdc303abafa53f555b551d3fd5804 | subsection | 71 | 115 | Body | For \theta \approx \pi , one should note that p_3^+\approx 0 and the corresponding photon's polarization component \epsilon ^+_+ yields the singular behavior
exhibited in the LFD result of \mathcal {M}^{+,+,+,+}_{a,t}. This light-front singularity in \mathcal {M}^{+,+,+,+}_{a,t} turns out to be cancelled by the
same wi... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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1d34aed111d7c9e03d59fd549500307c9733bc5a | subsection | 72 | 115 | Body | However, one should also note that the survival of this singular behavior depends on the time-ordering of the process as well as the helicities of the particles in the process
as not only the longitudinal component but also the transverse component of the polarization vector also matters in affecting the removal or sur... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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f6f1eeb1b7d64a85eb913779a681daa66f7e51f1 | subsection | 73 | 115 | Body | REF .Having discussed the helicity amplitudes \mathcal {M}^{+,+,\lambda _3,\lambda _4}_{a,t}+\mathcal {M}^{+,+,\lambda _3,\lambda _4}_{b,t}+\mathcal {M}^{+,+,\lambda _3,\lambda _4}_{a,u}+\mathcal {M}^{+,+,\lambda _3,\lambda _4}_{b,u} in Fig. REF , we note here the IFD/LFD profile correspondence similar to what we have ... | {
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... | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
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"Alfredo Takashi Suzuki"
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805ca6c9720455e888b67deeb8e91613707e5ad0 | subsection | 74 | 115 | Body | REF , the correspondence between the profile of the total amplitude
\mathcal {M}^{+,-,+,-}_{a,t} + \mathcal {M}^{+,-,+,-}_{b,t} + \mathcal {M}^{+,-,+,-}_{a,u} + \mathcal {M}^{+,-,+,-}_{b,u} in LFD and
the profile of the total amplitude \mathcal {M}^{+,+,+,-}_{a,t} + \mathcal {M}^{+,+,+,-}_{b,t} + \mathcal {M}^{+,+,+,-}... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
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"Alfredo Takashi Suzuki"
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8e8887aca43ac8b8107ca3e9b590d5e38c50200a | subsection | 75 | 115 | Body | In this work, although we keep in mind of the treacherous LF helicity identification
at the exact boundary values, we present our work focusing on the region 0< \theta < \pi without involving the exact boundary values of \theta =0 and \theta = \pi .
[Figure: (a) Profile of the t-channel (a) time-ordered annihilation am... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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0e2b0cbe23672b7c32a5f583c99d9d2edb778be5 | subsection | 76 | 115 | Body | Depending on the final photon helicities, however, the amplitude can still vanish as in the case of \mathcal {M}^{+,-,-,+}_{a,t} and \mathcal {M}^{+,-,-,-}_{a,t}. Moreover, it is interesting to note the dramatic rise of the amplitude \mathcal {M}^{+,-,+,+}_{a,t} as the scattering/annihilation process becomes collinear ... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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8093e1ec02eda419b848194dea084d108d15c1d2 | subsection | 77 | 115 | Body | REF can be understood by realizing the symmetry under the exchange of the outgoing pair of the photons as well as the forward-backward correspondence \theta \leftrightarrow \pi -\theta . It may not be too difficult to see the \theta \leftrightarrow \pi -\theta correspondence between
\mathcal {M}^{+,-,\pm ,\pm }_{a,u} a... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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12b071d98a5a7ce8fcf5779e16b539d178c1020e | subsection | 78 | 115 | Body | (REF ), the same correspondence applies to
\mathcal {M}^{-,\pm ,\lambda _3,\lambda _4}_{a,t} + \mathcal {M}^{-,\pm ,\lambda _3,\lambda _4}_{b,t}+\mathcal {M}^{-,\pm ,\lambda _3,\lambda _4}_{a,u} + \mathcal {M}^{-,\pm ,\lambda _3,\lambda _4}_{b,u}
and \mathcal {M}^{-,\pm ,\lambda _4,\lambda _3}_{a,t} + \mathcal {M}^{-,\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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ad1087863297b926476cfe43bf3f0fcf4bba9e83 | subsection | 79 | 115 | Body | REF summing the final helicities, \sum \limits _{\lambda _3,\lambda _4}|\mathcal {M}^{+,\pm ,\lambda _3,\lambda _4}_{a,t} + \mathcal {M}^{+,\pm ,\lambda _3,\lambda _4}_{b,t}+\mathcal {M}^{+,\pm ,\lambda _3,\lambda _4}_{a,u} + \mathcal {M}^{+,\pm ,\lambda _3,\lambda _4}_{b,u}|^2, exhibit the
swap of the helicity between... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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a3169f5eab1916ae8f1db7b37357e28b8e179ca3 | subsection | 80 | 115 | Body | By adding the two initial helicity states as well, we may now compare our total result
with the well-known manifestly Lorentz invariant result given by\left| \mathcal {M} (e^+e^- \rightarrow \gamma \gamma ) \right|^2
&\equiv \sum _{\lambda _1,\lambda _2,\lambda _3,\lambda _4}
|\mathcal {M}^{\lambda _1,\lambda _2,\lambd... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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5a33936ecb73a11aab4dcf9d794e464d4299ea70 | subsection | 81 | 115 | Body | (REF ) is apparently symmetric under t \leftrightarrow u exchange as it must be and
gets reduced to the well-known textbook result in the massless limit (m \rightarrow 0) given by\left| \mathcal {M} (e^+e^- \rightarrow \gamma \gamma ) \right|^2 =8\left(\frac{u}{t}+\frac{t}{u} \right) .It may be interesting to compare ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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bb5767f79cb7e4be84a4141f0fe6269bb076cf26 | subsection | 82 | 115 | Compton Scattering | Another important physical scattering processes in QED which involves the fermion propagator in the lowest order
is the Compton scattering e \gamma \rightarrow e \gamma . Similar to the e^+ e^- \rightarrow \gamma \gamma process shown in
Fig. REF which we have extensively discussed in the previous subsection,
the lowest... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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2d16e20b913c9d9285d15a7d5bc4fc39df925ad8 | subsection | 83 | 115 | Compton Scattering | Then, the time-ordered amplitudes of the s-channel Compton scattering can be written in short-hand notations
without specifying the helicities as\mathcal {M}_{a,s}=\bar{u}(p_3) {\epsilon }(p_4)^* \left( \frac{1}{2Q_s^{\widehat{+}}}\frac{{Q}_{a,s}+m}{q_{s\widehat{+}}-Q_{a,s\widehat{+}}}\right) {\epsilon }(p_2) u(p_1),an... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
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"Alfredo Takashi Suzuki"
] | [
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10e3f07d4a118d7687685dcc7dec08bc4d066a0d | subsection | 84 | 115 | Compton Scattering | There is no need to figure out the critical scattering angles as we have obtained in the case of the e^+ e^- \rightarrow \gamma \gamma process such as Eqs. (REF ) and (REF ). Regardless of kinematics the Compton scattering, the positivity of q_s^+ >0 allows the use of Eqs. (REF ) and (REF ) to identify immediately the ... | {
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"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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59021760908e761c6f79ad733f0877fca451fd30 | subsection | 85 | 115 | Compton Scattering | Any further discussion such as the angular distribution,
the energy (E_0) dependence, etc. in CMF will be presented together with the discussion of the target rest frame
elsewhere as mentioned earlier.
[Figure: Compton Scattering Amplitudes — s channel, time-ordering (a)][Figure: Compton Scattering Amplitudes — s chann... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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58062a8a4fbf61be2f06209ea72be41c7eba0e58 | subsection | 86 | 115 | Compton Scattering | We note that this triviality of the LFD results here is due to the fact that
the initial photon is incident in the -\hat{z} direction in the kinematics chosen for this calculation
(see Eq. (REF )) and thus gets only the zero-mode p_2^+ = 0 and \mathbf {p}_{2\perp } = 0.
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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fd03105c6385fd4224de471b3ebfce21ab22edaa | subsection | 87 | 115 | Compton Scattering | Due to \lbrace \gamma ^{+},\gamma ^{+}\rbrace = {\gamma ^+}^2 = 0, the only non-vanishing “instantaneous contribution"
to the s-channel helicity amplitudes in the light-front gauge A^+ = 0 are provided by only the transverse components of the photon polarization vectors for the helicity non-flip matrix elements between... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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2f97b59e58dcc83ddc63c6711e2b380a9d69b4a3 | subsection | 88 | 115 | Compton Scattering | REF is nontrivial in contrast to the trivial s-channel (a) time-ordered result.
However, the “instantaneous contribution" in LFD corresponding to the u-channel (b) time-ordered process shown in Fig. REF gets again effectively only one helicity amplitude in LFD due to \lbrace \gamma ^{+},\gamma ^{+}\rbrace = {\gamma ^+}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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0b493ab2b826d5603b7179026db9324f14e0d356 | subsection | 89 | 115 | Compton Scattering | Indeed, this result is in complete agreement with the well-known manifestly Lorentz invariant result given by\left| \mathcal {M} (e\gamma \rightarrow e\gamma ) \right|^2
&\equiv \sum _{\lambda _1,\lambda _2,\lambda _3,\lambda _4}
|\mathcal {M}^{\lambda _1,\lambda _2,\lambda _3,\lambda _4}_{a,s} + \mathcal {M}^{\lambda ... | {
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5ec6343a93de4baa31d3193d9105169d85edae49 | subsection | 90 | 115 | Summary and Conclusion | In this work, we have completed the interpolation of Quantum Electrodynamics between the instant form and the front form proposed by Dirac in 1949. We started from the QED Lagrangian and presented the interpolating Hamiltonian formulation
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"end"... | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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ebf9b4f3a018912868a5fa96c4544d5f6d201b1c | subsection | 91 | 115 | Summary and Conclusion | On the other hand, all the helicity amplitudes in LFD are independent of the reference frame, and certain simplifications to the theory (e.g. suppression of vacuum fluctuations, vanishing of a number of diagrams, etc. ) can be realized even in the rest frame of the system. Since the helicity definition in LFD is frame-... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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a80c6e656d7aa443ffa47a62d85a5ec1a3283cbe | subsection | 92 | 115 | Fermion propagator in the position space | The Feynman propagator in the position space is given by\Delta _{\rm F}(x) & = i\int \frac{d^4 q}{(2\pi )^4}\,\frac{{\rm e}^{-iq_{\mu }x^{\mu }}}{\left(q^2-m^2+i\varepsilon \right)}.In the interpolation form, it can be written as\Delta _{\rm F}(x)
&=i\int \frac{d^2{\bf q}_{\perp }dq_{\widehat{-}}dq_{\widehat{+}}}{(2\pi... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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232b241ee5079b4a6a50132cabd424e18ac4071d | subsection | 93 | 115 | Fermion propagator in the position space | (REF ) and (), we see that for any sign of q_{\widehat{-}}, {\mathcal {A}}_{\widehat{+}} is always positive and corresponds to the positive energy solution, while -{\mathcal {B}}_{\widehat{+}} is always negative and corresponds to the negative energy solution. Therefore, we see the pole structure in the q_{\widehat{+}}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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853bcd2880ecb1e0c1638de8c4e925ae3b8d4019 | subsection | 94 | 115 | Fermion propagator in the position space | We thus have for this case:& \oint \frac{dq_{\widehat{+}}}{(2\pi )}\frac{{\rm e}^{-iq_{\widehat{+}}x^{\widehat{+}}}}{\mathbb {C}\left(q_{\widehat{+}} - {\cal A}_{\widehat{+}} + i\varepsilon ^{\prime }\right)\left( q_{\widehat{+}} + {\cal B}_{\widehat{+}} - i\varepsilon ^{\prime }\right)} \\
=& \lim _{R\rightarrow \inft... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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2deb1d2f13d01bdd87f348fc555349948b5123c5 | subsection | 95 | 115 | Fermion propagator in the position space | Since the arc contribution in the limit R \rightarrow \infty goes to zero, in this limit we have& \int _{-\infty }^{+\infty }\frac{dq_{\widehat{+}}}{(2\pi )}\frac{{\rm e}^{-iq_{\widehat{+}}x^{\widehat{+}}}}{\mathbb {C}\left(q_{\widehat{+}} - {\cal A}_{\widehat{+}} + i\varepsilon ^{\prime }\right)\left( q_{\widehat{+}} ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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70ee68b989cb2bc3524fc202043c679f9a58e453 | subsection | 96 | 115 | Fermion propagator in the position space | We thus have for this case:& \oint \frac{dq_{\widehat{+}}}{(2\pi )}\frac{{\rm e}^{-iq_{\widehat{+}}x^{\widehat{+}}}}{\mathbb {C}\left(q_{\widehat{+}} - {\cal A}_{\widehat{+}} + i\varepsilon ^{\prime }\right)\left( p_{\widehat{+}} + {\cal B}_{\widehat{+}} - i\varepsilon ^{\prime }\right)} \\
= & \lim _{R\rightarrow \inf... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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... |
c379267bc02b908d15508ab2447611333d7ff7a1 | subsection | 97 | 115 | Fermion propagator in the position space | Since the arc contribution in the limit R \rightarrow \infty goes to zero, in this limit we now have&\int _{-\infty }^{+\infty }\frac{dq_{\widehat{+}}}{(2\pi )}\frac{{\rm e}^{-iq_{\widehat{+}}x^{\widehat{+}}}}{\mathbb {C}\left(q_{\widehat{+}} - {\cal A}_{\widehat{+}} + i\varepsilon ^{\prime }\right)\left( q_{\widehat{+... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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aff97fa55990d53c15675d4cd68b1cb8fc1fcf12 | subsection | 98 | 115 | Fermion propagator in the position space | Therefore, closing the contour from below, that is, with C_R in the clockwise direction. | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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fb68070be5915b1bff9c6313718114f0e5ec5ce6 | subsection | 99 | 115 | Fermion propagator in the position space | This encloses the pole q_{\widehat{+}}={\mathcal {A}}_{\widehat{+}}-i\varepsilon ^{\prime } and we get&\int \limits _{-\infty }^{+\infty }\frac{dq_{\widehat{+}}}{(2\pi )}\frac{1}{\mathbb {C}\left(q_{\widehat{+}} - {\cal A}_{\widehat{+}} + i\varepsilon ^{\prime }\right)\left( q_{\widehat{+}} + {\cal B}_{\widehat{+}} - i... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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30e6a93e4f0d95bd7a83cd447179ca841a888d9b | subsection | 100 | 115 | Fermion propagator in the position space | Similarly, when q^+<0 , the pole is located in the second quadrant of the q^- complex plane, and to make sure the arc contribution is zero, when x^+>0 , one has to close the contour from below, which again gives no contribution because there is no pole in the lower half plane, while when x^+<0 , one needs to close the ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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1d57355c58a195a4f461a1e7753a6422c53f5ade | subsection | 101 | 115 | Fermion propagator in the position space | (REF ), we can combine the results and write as follows\Delta _{\rm F}(x)&=\int \frac{d^2{\bf q}_{\perp }}{(2\pi )^2} \int \limits _{-\infty }^{+\infty }\frac{dq_{\widehat{-}}}{(2\pi )} \frac{1}{2Q^{\widehat{+}}}\widehat{\Theta }(q_{\widehat{-}})\left\lbrace {\Theta (x^{\widehat{+}}){\rm e}^{-i{\mathcal {A}}_{\widehat{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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98132db2c3287384a5bba4e9fc0903a54dcac092 | subsection | 102 | 115 | Fermion propagator in the position space | Note here that the argument of the Hankel function is imaginary.To derive the fermion propagator we need to apply the Dirac operator on it,S_{\rm F} (x)& = \left(i\gamma ^{\widehat{+}}\partial _{\widehat{+}}+i\gamma ^{\widehat{-}}\partial _{\widehat{-}}+i{{ \gamma }}^{\perp }\cdot {\partial }_{\perp }+m\right)\Delta _{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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acaef1abca2357f080dd54f281bcb36b9d4f039f | subsection | 103 | 115 | Fermion propagator in the position space | We finally get&\Sigma _{\rm F}(q)\equiv iS_{\rm F}(q)=i\int d^4 x \ S_{\rm F}(x)\ {\rm e}^{iq_{\widehat{\mu }}x^{\widehat{\mu }}}\\
&=
\begin{}
\frac{1}{2Q^{\widehat{+}}}\left( \frac{{Q}_a+m}{q_{\widehat{+}}-Q_{a\widehat{+}}}+\frac{-{Q}_b+m}{-q_{\widehat{+}}-Q_{b\widehat{+}}}\right) , \qquad (\mathbb {C}\ne 0),\end{}\\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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3945e25da1a8200aa5232fd8ac87710977849345 | subsection | 104 | 115 | Derivation of Interpolating QED Hamiltonian | In this Appendix, we show how the Hamiltonian in subsection (REF ) is derived, and how the consistency with the LFD formulation presented by Kogut and Soper can be seen.We start from the interpolating QED Hamiltonian density, as given in Eq. (REF ),\mathcal {H}&=\bar{\psi }\left(-i\gamma ^{j}\partial _{j}-i \gamma ^{\w... | {
"cite_spans": [
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"raw": "J. Kogut and D. Soper, Phys. Rev. D 1, 2901(1970).",
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"start": 0
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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13fe4ecb7ab20e51c93c8a36a856c671d3a8a78c | subsection | 105 | 115 | Derivation of Interpolating QED Hamiltonian | \bar{\psi }\left[ \left( i\partial _j-eA_j\right)\gamma ^j-m \right] \psi =-2\bar{\psi }\left( i\partial _-\gamma ^-\right) \psi .Recalling in the light front we can separate the fermion field into the free one and constrained one \psi =\psi _++\psi _- =\tilde{\psi }_++\psi _- with \gamma ^+\psi _-=\gamma ^-\psi _+=0 ,... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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21df7974af4805e18c5ab8ba8ea7ee77db511594 | subsection | 106 | 115 | Derivation of Interpolating QED Hamiltonian | (REF ) as\mathcal {H}_{\mathrm {f}}&= \bar{\tilde{\psi }}\left( -i\partial _{\widehat{-}}\gamma ^{\widehat{-}}-i\partial _{j}\gamma ^{j}+m\right) \tilde{\psi } \\
&+\delta _{\mathbb {C}0}\left[ \bar{\tilde{\psi }}\left( -i\partial _-\gamma ^-\right)\Upsilon +\bar{\Upsilon }\left( -i\partial _-\gamma ^-\right)\tilde{\ps... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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140dc0e53576f8b5b27247cf630f926269e73076 | subsection | 107 | 115 | Derivation of Interpolating QED Hamiltonian | (REF ) reduces to\mathcal {H}_{\mathrm {f}}&= \bar{\tilde{\psi }}\left( -i\partial _{\widehat{-}}\gamma ^{\widehat{-}}-i\partial _{j}\gamma ^{j}+m\right) \tilde{\psi }+e\tilde{A}_{\widehat{\mu }} \bar{\tilde{\psi }}\gamma ^{\widehat{\mu }} \tilde{\psi }\\
&+\delta _{\mathbb {C}0}\bar{\Upsilon }\left( i\partial _-\gamma... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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eb47fa830985afcd40ecba2a25ede8aab426562d | subsection | 108 | 115 | Derivation of Interpolating QED Hamiltonian | (REF ).Using A_{\widehat{\mu }}=\tilde{A}_{\widehat{\mu }}+{g_{\widehat{\mu }}}^{\widehat{+}}\phi and A^{\widehat{\mu }}=\tilde{A}^{\widehat{\mu }}+g^{\widehat{\mu }\widehat{+}}\phi , we find&\mathcal {H}_{\mathrm {g}}^{\mathrm {constraint}}=\mathcal {H}_{\mathrm {g}}-\mathcal {H}_{\mathrm {g}}^{\mathrm {free}}\\
&=\fr... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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e53c1f9c04547e404571f2288f8c57cda09e3df6 | subsection | 109 | 115 | Derivation of Interpolating QED Hamiltonian | (REF ).Adding two pieces together, we can identify the free and interaction Hamiltonian\mathcal {H}=\mathcal {H}_{\mathrm {f}}+\mathcal {H}_{\mathrm {g}}=\mathcal {H}_{0}+\mathcal {V},where\mathcal {H}_{0}&=\bar{\tilde{\psi }}\left( -i\partial _{\widehat{-}}\gamma ^{\widehat{-}}-i\partial _{j}\gamma ^{j}+m\right) \tild... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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92d2cb27622b0cb321000a9457bd10791fff33dd | subsection | 110 | 115 | Sum of the Interpolating Time-Ordered Fermion Propagators | In this Appendix, we show how the addition of the two time-ordered propagators gives correctly the covariant one. We start with the expressions given in Eqs. | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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56610eac6d5b26ffda24608d0c603cfe7ca80d5d | subsection | 111 | 115 | Sum of the Interpolating Time-Ordered Fermion Propagators | (REF ) and ().&\Sigma _a+\Sigma _b\\
&=\frac{1}{2Q^{\widehat{+}}}\left( \frac{{Q}_a+m}{q_{\widehat{+}}-Q_{a\widehat{+}}}-\frac{-{Q}_b+m}{q_{\widehat{+}}+Q_{b\widehat{+}}}\right)\\
&=\frac{1}{2Q^{\widehat{+}}}\left(\frac{\mathbb {C}{Q}_a+\mathbb {C}m}{\mathbb {C}q_{\widehat{+}}+\mathbb {S}q_{\widehat{-}}-Q^{\widehat{+}}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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9b47a8032568d0449332f4440d12342401fb5c40 | subsection | 112 | 115 | Apparent Angle Distribution of Interpolating Helicity Amplitudes for the two scalar particle production in | In this Appendix, we present the angular distribution shown in Sec. re-plotted in terms of the apparent angle of the scattering/annihilation process in a moving frame viewed from the lab frame, \theta _{\mathrm {app}} , as well as the interpolation angle \delta .By boosting the system with total momentum P^z , we get\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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0480d44def6296b00615221cab7df9822a5b16df | subsection | 113 | 115 | Boosted | In this Appendix, similar to what was done in Sec. REF , we examine the frame dependence of the whole landscape of all the angular distributions of the helicity amplitudes discussed in Sec. REF by computing them with non-zero center of momentum (P^z=+15m_e and p^z=-15m_e ).
In Figs. REF , REF , REF ,
REF and REF ,
w... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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b9f1bd3d2db723c9311a9ae1bae9d867514285ac | subsection | 114 | 115 | Boost Dependence in | In this Appendix, we plot the helicity amplitudes of e^+ e^- \rightarrow \gamma \gamma , as given by Eq. (REF ) and (), in terms of both the interpolation angle \delta and the total momentum P^z . As was done in Sec. REF , we take m=m_e , E_0=2m_e , and instead of looking at the angular distribution, we fix the angle \... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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fe9b7ce2c0526d90ef3e43fa62ead93ed62b7100 | abstract | 0 | 59 | Abstract | The cross or soft anomalous dimension matrix describes the renormalization of
Wilson loops with a self-intersection and is an important object in the study
of infrared divergences of scattering amplitudes. In this paper it is studied
for the Maldacena--Wilson loop in N=4 supersymmetric Yang--Mills theory and
Euclidean ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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] | 2,018 | en | Physics | [
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94486e287a87afda4c43fccfcc6c907a5a9b3d6a | subsection | 1 | 59 | Introduction | Recent years have witnessed much progress in our understanding of the soft and collinear singularities of gauge theory
scattering amplitudes. This progress is of phenomenological relevance in QCD, where our control over these singularities
allows for the resummation of logarithmically enhanced contributions. The key el... | {
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"doi": "10.1103/physrevlett.102.162001",
"end": 779,
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"raw": "T. Becher and M. Neubert, “Infrared singularities of scattering amplitudes in perturbative QCD”, Phys. Rev. Lett. 102, 162001 (2009), ar... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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15d1d1871938951e0bfae4ae602ea4bd296a7284 | subsection | 2 | 59 | Introduction | The first of these two cases has been studied in QCD in
refs. , up to the two-loop level.Let us briefly explain how this paper is structured. We begin by reviewing the renormalization properties of
self-intersecting Wilson loops in section .
The minimal surfaces appearing in the strong-coupling description of the Malda... | {
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"raw": "G. P. Korchemsky, “On Near forward high-energy scattering in QCD”, Phys. Lett. B325, 459 (1994), hep-ph/9311294.",
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... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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f6b8ce80fabb78ba7f31d26dcbf3df149b6b6fdf | subsection | 3 | 59 | The Cross Anomalous Dimension | The renormalization of Wilson loops with self-intersections was described in ref. building on the renormalization for smooth and cusped Wilson loops , . It was found that the renormalization requires a mixing between Wilson loops with different path-orderings at the intersection point. In the case of a single intersect... | {
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"doi": "10.1103/physrevd.24.879",
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"raw": "R. A. Brandt, F. Neri and M.-a. Sato, “Renormalization of Loop Functions for All Loops”, Phys. Rev. D24, 879 (1981).",
"source_ref_id": "... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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1601c0be6a4069d391cc648e311d161d25de871b | subsection | 4 | 59 | The Cross Anomalous Dimension | The relevant curves are parametrized by( x(\sigma ) , n(\sigma ) ) = {\left\lbrace \begin{array}{ll}
(v_1 \sigma \, , \, n_1 )
\quad &\text{for} \quad C_1 \, , \\
(v_2 \sigma \, , \, n_2 )
\quad &\text{for} \quad C_2 \, , \\
(\theta (-\sigma ) \, v_1 \sigma + \theta (\sigma ) \, v_2 \sigma \, , \,
\theta (-\sigma ) \, ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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87fc3e4972451755400153372da4aa4a596d5a18 | subsection | 5 | 59 | The Cross Anomalous Dimension | We are hence considering the expectation values\mathcal {W}_1 {} _{i j i^\prime j^\prime } &= \left\langle {\mathcal {P} \exp }{\left( i \hspace*{-2.84526pt} \int _{C_1} \hspace*{-7.11317pt} {\mathrm {d}}\tau \left( A_\mu \dot{x}^\mu + i \Phi _I n^I
\vert \dot{x} \vert \right) \right)}{} _{j j^\prime } \;
{\mathcal {P}... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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091e2ef4824690581b5b5e466fcc245e83a8073c | subsection | 6 | 59 | The Cross Anomalous Dimension | At the lowest order, the expressions for the functions \mathcal {W}_1 and \mathcal {W}_2 are trivial, we find the basic color structures\mathcal {W}_1 {} _{i j i^\prime j^\prime }
= \delta _{i i^\prime } \, \delta _{j j^\prime } =: \left|1 \right\rangle , \qquad \mathcal {W}_2 {} _{i j i^\prime j^\prime }
= \delta _{i ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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004890fdee504cb759cdd10e1249f17ea83bea99 | subsection | 7 | 59 | The Planar Limit and Strong Coupling | It is instructive consider the planar limit of sending N \rightarrow \infty and g \rightarrow 0 while keeping the 't Hooft coupling constant
\lambda = g^2 N fixed. In this limit, the leading contributions to the cross anomalous dimension \widehat{\Gamma }_{\mathrm {cross}} can be expressed in terms of
the cusp anomalou... | {
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Theory | [
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ab8a07e505afc79f48b57fedf24a9c6ba7257a07 | subsection | 8 | 59 | The Planar Limit and Strong Coupling | Contracting the renormalization group equation (REF )
with the color structures \vert 1 \rangle and \vert 2 \rangle , then allows to conclude that \widehat{\Gamma }_{\mathrm {cross}}(\phi , \rho , \lambda )
= \begin{pmatrix}
\mathrm {O}\left(N^{-2}\right) & \mathrm {O}\left(N^{-1}\right) \\
\mathrm {O}\left(N^{-1}\righ... | {
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Theory | [
"Hagen Münkler"
] | [
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e21d9a8cd915b3fb62b969a265a5be973967ba49 | subsection | 9 | 59 | The Planar Limit and Strong Coupling | Combining this insight with the finding (REF ) about the cross anomalous dimension in the planar limit, we find that\Gamma _{\mathrm {cross}}(\phi , \rho , \lambda )
\overset{\lambda \gg 1}{=}
\frac{\sqrt{\lambda }}{2 \pi }
\begin{pmatrix}
0 & 2 \Gamma _{\mathrm {cusp}}^{(\infty )} (\phi , \rho ) \\
0 & 2 \Gamma _{\mat... | {
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Theory | [
"Hagen Münkler"
] | [
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74d7dd716008978605398e3b2277004fa621dbae | subsection | 10 | 59 | The Planar Limit and Strong Coupling | After subtracting the linear divergence, the area of the minimal surface has
a logarithmic divergence which corresponds to the anomalous dimension at strong coupling,A_{\mathrm {ren}} &= - \Gamma _{\mathrm {cusp}}^{(\infty )} \, \ln \left( {\textstyle \frac{L}{\varepsilon }} \right) , &
\Gamma _{\mathrm {cusp}}^{(\inft... | {
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fd76e3546640dd5e989b86434c2f9b005d94d163 | subsection | 11 | 59 | The Planar Limit and Strong Coupling | \end{array}\right.}In other words, we consider the sequence of lines(v_1 , n_1) \; \; \rightarrow \; \;
(- v_2 ,- n_2) \; \; \rightarrow \; \;
(- v_1 ,- n_1) \; \; \rightarrow \; \;
(v_2 , n_2) \, ,We denote the respective Wilson loop correlators by \widetilde{W}_i and
the associated cross anomalous dimension by \widet... | {
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0.019576508551836014,
0.00... |
dd13bc8a65da6cadc1cbba745907095319931259 | subsection | 12 | 59 | The Planar Limit and Strong Coupling | This gives the following result for the cross anomalous dimension
\widetilde{\Gamma }_{\mathrm {cross}} in the limit of infinite coupling:\widetilde{\Gamma }_{\mathrm {cross}}(\phi , \rho , \lambda ) \overset{\lambda \gg 1}{=}
\frac{\sqrt{\lambda }}{\pi } \begin{pmatrix}
\Gamma _{\mathrm {cusp}}^{(\infty )} (\pi - \phi... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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01f104f9ddedd172b3599ecc5b5de20286ef8686 | subsection | 13 | 59 | Weak Coupling – One Loop | We now turn to the weak-coupling calculation of the cross anomalous dimension, beginning with the one-loop level. Following ref. , we employ dimensional reduction to regularize divergences. Dimensional reduction is a version of dimensional regularization in which \mathcal {N} \! = 4 supersymmetric Yang–Mills theory in ... | {
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{
"arxiv_id": "",
"doi": "",
"end": 189,
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"raw": "J. K. Erickson, G. W. Semenoff and K. Zarembo, “Wilson loops in N=4 supersymmetric Yang-Mills theory”, Nucl. Phys. B582, 155 (2000), hep-th/0003055.",
"source_r... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0d18dd36924199316f65678c8dc95846877584c8 | subsection | 14 | 59 | Weak Coupling – One Loop | For the scalar propagator, we then have D(x) = g^2 \mu ^{2 \epsilon } \, \frac{\Gamma (1- \epsilon )}{4 \pi ^{2-\epsilon } } \,
\frac{1}{ \left( x^2 \right) ^{1- \epsilon } } \, ,and the two-point functions\left\langle A_\mu ^a (x_1) \, A_\nu ^b (x_2) \right\rangle =
\delta _{\mu \nu } \, \delta ^{a b} D(x_1 - x_2)
\, ... | {
"cite_spans": [
{
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"doi": "",
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"raw": "J. K. Erickson, G. W. Semenoff and K. Zarembo, “Wilson loops in N=4 supersymmetric Yang-Mills theory”, Nucl. Phys. B582, 155 (2000), hep-th/0003055.",
"source_r... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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15f1fa94b18a09e59b26cb416ffb115b6142fedb | subsection | 15 | 59 | Weak Coupling – One Loop | The convention to include a factor of N^{-1} is a consequence of expanding in the 't Hooft coupling constant \lambda = g^2 N. For diagram 1, we find the color factorsC^1_{1,1} &= T^a_{j j^\prime } \, T^a_{i i^\prime }
= {\textstyle \frac{1}{2}} \left( \delta _{i j^\prime } \, \delta _{j i^\prime }
- {\textstyle \frac{1... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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3e937ad84d723201bc1719f6bae3fa6c31c7c928 | subsection | 16 | 59 | Weak Coupling – One Loop | For F^1_2, we findF^1 _2 &= - \frac{\Gamma (1 - \epsilon )}{4 \pi ^{2 - \epsilon } } \,g^2 \mu ^{2 \epsilon }
\int \limits _{-L} ^0 {\mathrm {d}}\tau _1 \,
\int \limits _0 ^L {\mathrm {d}}\tau _2 \,
\frac{\cos \phi - \cos \rho }{\left[ \tau _1 ^2 + \tau _2 ^2 - 2 \tau _1 \tau _2 \cos \phi \right]^{1- \epsilon } } \\
&=... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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7b38f7b5f655b92cca269c05425660da66cfbfe2 | subsection | 17 | 59 | Weak Coupling – One Loop | Then we see thatF^1 _3 &= F^1 _1 \, , &
F^1 _4 &= F^1 _2 \, .The cross anomalous dimension at the one-loop level is thus encoded in the function I_0 (\phi ), which also describes the cusp anomalous dimension.We then turn to the evaluation of I(\phi ) = I_0 (\phi ) + \epsilon I_1 (\phi ) + \mathrm {O}( \epsilon ^2) . Fo... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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64ee6661676accb3c3c8a1e2b1e4c2d81ad8b9f8 | subsection | 18 | 59 | Weak Coupling – One Loop | Using the color factors given in appendix , we find\begin{pmatrix}
\mathcal {W}_1 ^{\, (1)} \\
\mathcal {W}_2 ^{\, (1)}
\end{pmatrix} =
\frac{\mu ^{2 \epsilon } ( \cos \phi - \cos \rho )}{N^2 \, \epsilon }
\begin{pmatrix}
- (I(\phi ) + I(\pi - \phi )) & N (I(\phi ) + I(\pi - \phi )) \\
N \, I(\pi - \phi ) & N^2 \, I(\p... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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49463b893e365e0a0e9599be527d8d83f0c32c75 | subsection | 19 | 59 | Weak Coupling – One Loop | Combining this factor with the color factor \operatorname{tr}( T^a T^a) = {\textstyle \frac{1}{2}}(N^2 - 1) gives the cusp anomalous dimension at the one-loop level,\Gamma _{\mathrm {cusp}}^{(1)} (\phi , \rho ) = \frac{N^2-1}{N^2}
\left( \cos \phi - \cos \rho \right) I_0(\phi )
= \frac{N^2-1}{N^2} \, \gamma _{\mathrm {... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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... |
5d1248915a6417af9517d073cb4899ac89c7f4c7 | subsection | 20 | 59 | Weak Coupling – One Loop | Consider e.g. the kinematic factor associated to the first diagram,\widetilde{F}_1 ^1 = - \frac{\Gamma ( 1 - \epsilon ) }{4 \pi ^{2 - \epsilon }} g^2 \mu ^{2 \epsilon }
\int \limits _0 ^L {\mathrm {d}}\tau _1 \, {\mathrm {d}}\tau _2 \,
\frac{- \cos \phi + \cos \rho }{\left[ \tau _1 ^2 + \tau _2 ^2 - 2 \tau _1 \tau _2 \... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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166e13c252ca09f4584ffe45915a1f49ac23e5cb | subsection | 21 | 59 | Weak Coupling – Two Loops | Before we turn to the two-loop calculation of the cross anomalous dimension, let us comment on some organizing principles
that can be applied. In the calculation of both the cross and the cusp anomalous dimension, one acquires a factor
\xi = (\cos \phi - \cos \rho ) for each connection of one Wilson line to another. It... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0550-3213(87)90277-x",
"end": 1405,
"openalex_id": "https://openalex.org/W2026692681",
"raw": "G. P. Korchemsky and A. V. Radyushkin, “Renormalization of the Wilson Loops Beyond the Leading Order”, Nucl. Phys. B283, 342 (1987).",
... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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