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Jet Quenching Liliana Apolinrio 26th February 2019 COST THOR - PowerPoint PPT Presentation

May 04, 2023 •128 likes •1.31k views

Jet Quenching Liliana Apolinrio 26th February 2019 COST THOR School, Lund, Sweden Proton-Proton Collision pp collision: a simple (few particles) system Hard scattering: Process evolution can be described with Quantum Field

  1. Jet Coherence q 2 2 q 2 ✦ Soft gluon radiation from a quark - anti-quark pair: θ 1 + ω , k ⊥ θ θ ω , k ⊥ θ 1 ✦ color singlet configuration: q 1 q 1 dI ✦ No radiation outside of the cone Integrating in azimuthal angle… = R q + R ¯ q − 2 J (1 ) = R coh + 2 d Ω k ➡ Angular ordering d ω sin θ d θ dN ω → 0 1 − cos θ Θ (cos θ 1 − cos θ ) ∼ α s C R q ω ✦ color octet configuration: ✦ Radiation outside of the cone re-interpreted as from the initial gluon ➡ Angular ordering preserved L. Apolinário � 11 COST THOR School, “Jet Quenching” � 11

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sha1_base64="dfa3/IBQ0dJ+5glvlabNZFNmx0=">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</latexit> <latexit sha1_base64="c56MuzvQYv82OYy5Cln41rByPRQ=">ACJnicbVBNS0JBFJ1nH5p9WUGbNkMSKJS8V0RtAiGKlgr5ASoyb7zq4Lx5j5l5gr38M62C+i3tItq1iv5Cu0ZtkdqBC4dz7oVzjxtwprRtv1uxhcWl5XhiJbm6tr6xmdraLis/lBRK1Oe+rLpEAWcCSpDtVAvFcDhW3dznyK32QivniVg8CaHikI1ibUaKN1EztBvgCZzLO0V0WXx3ieh9oFAyzVTaztlj4Hni/J0Hhe/PhPx0Iz9V1v+T0QGjKiVI1xw50IyJSM8phmKyHCgJCe6QDNUMF8UA1onH+IT4wSgu3fWlGaDxW/15ExFNq4Llm0yO6q2a9kfivp0yULrSMdw0D4RGBha/Hn08H0u3zRsREGoQdJKnHXKsfTzqDLeYBKr5wBCJTMvYdolklBtmk2arpzZuZJ+TjnOTsoiktjyZIoD20jzLIQWcoj25QAZUQRfoAT2hZ+vRerFerbfJasz6vdlBU7A+fgD6CqZE</latexit> Vacuum formation time See: Basics of Perturbative QCD, 1991 ✦ Formation time of an emission: Uncertainty principle: ∆ E ∆ t = 1 k = ( ! = zE, ~ k ) E 1 ∆ E = m virtual ⇒ τ form = θ m virtual m virtual ∆ E p = ((1 − z ) E, ~ p ) τ form = ∆ t γ boost = ∆ t m virtual k ⊥ ∼ ω sin θ ∼ ωθ zE θ 2 = ω = λ ⊥ 1 where: m 2 virtual = 2 p · k = 2 z (1 � z ) E 2 (1 � cos θ ) ' z (1 � z ) E 2 θ 2 ) τ form ' λ ⊥ ∼ k − 1 k 2 θ ⊥ ⊥ L. Apolinário � 12 COST THOR School, “Jet Quenching”

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sha1_base64="dfa3/IBQ0dJ+5glvlabNZFNmx0=">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</latexit> Vacuum formation time See: Basics of Perturbative QCD, 1991 ✦ Formation time of an emission: Uncertainty principle: ∆ E ∆ t = 1 k = ( ! = zE, ~ k ) E 1 ∆ E = m virtual ⇒ τ form = θ m virtual m virtual ∆ E p = ((1 − z ) E, ~ p ) τ form = ∆ t γ boost = ∆ t m virtual k ⊥ ∼ ω sin θ ∼ ωθ zE θ 2 = ω = λ ⊥ 1 where: m 2 virtual = 2 p · k = 2 z (1 � z ) E 2 (1 � cos θ ) ' z (1 � z ) E 2 θ 2 ) τ form ' λ ⊥ ∼ k − 1 k 2 θ ⊥ ⊥ ✦ As for subsequent emissions: During this time, the previous “antenna” separated: = θ 1 r ⊥ = θ 1 τ form, 2 ⇔ r ⊥ λ ⊥ , 2 θ 2 θ 1 τ form, 2 = λ ⊥ , 2 θ 2 θ 2 L. Apolinário � 12 COST THOR School, “Jet Quenching”

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sha1_base64="WNRDO85s7dKmfaoTc1zJBvmbDU=">ACL3icbVDLSsQwFE19O75GXSoYHAQ3Dq0udDkgiO4UnAfYsaTprRMmSUuSCqXM0v9w70rQbxE34tZPcGemMwtfBwIn59wL54w5Uwb131Jianpmdm5+YrC4tLyvV1bWTjJFoUkTnqhOSDRwJqFpmOHQSRUQEXJoh/3jod+BaVZIi9NnkJXkBvJYkaJsVJQ3fK5HY5I4KegUuxrJnB/9Lku9rxBUK25dbcE/ku8Mak1avdncRFsngfVTz9KaCZAGsqJ1lem5puQZRhlMOg4mcaUkL75AauLJVEgO4W5SEDvGOVCMeJsk8aXKrfNwoitM5FaCcFMT392xuK/3raRulBZL0TyKUgEsvElBX8DGTio27BZJoZkHSUJ84NgkelocjpoAanltCqGL2JEx7RBFqbMUV25X3u5m/pLVf9w7q7oUtrYFGmEMbaBvtIg8dogY6ReoiSi6Qw/oCT07j86L8+a8j0YnPHOvoB5+ML2pCsDg=</latexit> <latexit sha1_base64="/DSxw3KR/WeaRziltg5FSh+9WeM=">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</latexit> Vacuum formation time See: Basics of Perturbative QCD, 1991 ✦ Formation time of an emission: Uncertainty principle: ∆ E ∆ t = 1 k = ( ! = zE, ~ k ) E 1 ∆ E = m virtual ⇒ τ form = θ m virtual m virtual ∆ E p = ((1 − z ) E, ~ p ) τ form = ∆ t γ boost = ∆ t m virtual k ⊥ ∼ ω sin θ ∼ ωθ zE θ 2 = ω = λ ⊥ 1 where: m 2 virtual = 2 p · k = 2 z (1 � z ) E 2 (1 � cos θ ) ' z (1 � z ) E 2 θ 2 ) τ form ' λ ⊥ ∼ k − 1 k 2 θ ⊥ ⊥ ✦ As for subsequent emissions: During this time, the previous “antenna” separated: = θ 1 r ⊥ = θ 1 τ form, 2 ⇔ r ⊥ λ ⊥ , 2 θ 2 θ 1 τ form, 2 = λ ⊥ , 2 θ 2 θ 2 > θ 1 ⇒ r T < 휆 T → Sensitive to the “antenna” charge θ 2 θ 2 > θ 1 ⇒ r T > 휆 T → Sensitive to the “leg” charge L. Apolinário � 12 COST THOR School, “Jet Quenching”

  6. Now back to Heavy-Ions

  7. First Considerations ✦ A jet (parton shower) is a perturbative object ⇒ calculable within pQCD L. Apolinário � 14 COST THOR School, “Jet Quenching”

  8. First Considerations ✦ A jet (parton shower) is a perturbative object ⇒ calculable within pQCD ✦ The QGP is a strongly coupled fluid ✦ It is non perturbative… but we will assume a pQCD description for the jet-medium interaction… ➡ Described by a classical field A μa (x) (recoil effects are neglected) L. Apolinário � 14 COST THOR School, “Jet Quenching”

  9. First Considerations ✦ A jet (parton shower) is a perturbative object ⇒ calculable within pQCD ✦ The QGP is a strongly coupled fluid ✦ It is non perturbative… but we will assume a pQCD description for the jet-medium interaction… ➡ Described by a classical field A μa (x) (recoil effects are neglected) ✦ High-energy particles propagating through a medium: ✦ Particle propagation time < timescale for changes in the medium fields ➡ Medium can be considered in a static configuration ➡ Only transverse momentum exchange L. Apolinário � 14 COST THOR School, “Jet Quenching”

  10. <latexit sha1_base64="+K7EXG8JLgdLG/ZWrO4kD740OKM=">ACNnicbVDLSsNAFJ34rPVdelmUJSKWhJFdCNUBHGpYFvBhjCZ3NjBySTMTMRQ/AO/xpWgP+LGnY+lW3dOUhG1HpjhcM69cO7xE86Utu1Ha2BwaHhktDRWHp+YnJquzMw2VZxKCg0a81ie+kQBZwIamkOp4kEvkcWv7Ffu63LkEqFosTnSXgRuRcsJBRo3kVZb3qlfe6hq+8tbzr52ATFbwLv6We4pXWbRrdgHcT5wvsljHx28vpdGtI6/y0Q5imkYgNOVEqTPHTrTbJVIzyuG63E4VJIRekHM4M1SQCJTbLe65xktGCXAYS/OExoX6c6NLIqWyDeTEdEd9dfLxX89ZaJ0IDeAWQiIgKLWBdN/A6kwx23y0SahC0lydMOdYxzjvEAZNANc8MIVQycxKmHSIJ1abpsunK+dtMP2lu1JzNmn1sSqujHkpoHi2gKnLQNqjQ3SEGoiG3SL7tGDdWc9Wc/Wa290wPramUO/YL1/ArF2rCs=</latexit> <latexit sha1_base64="wCXsMiHwtYTL4rpTHFSZlNQIsDo=">ACJHicbVBNS0JBFJ1nH5p9vT52bYkMAh5z4jaBEIQLRXyA1Qe8arDs6b95iZF5r4X1oF9VvaRYs2LaO/0K5RW6R24MLhnHvh3ONHnCntO9WYml5ZTWZWkuvb2xubds7uxUVxpJCmY8lDWfKOBMQFkzaEWSCBz6Hq967GfvUOpGKhuNWDCJoB6QjWZpRoI3n2ft9rRCAjfImzfc89wX0vf+zZGSfnTIAXiftLMgVc+vpMJc+Knv3daIU0DkBoyolSdeJdHNIpGaUwyjdiBVEhPZIB+qGChKAag4n6Uf4yCgt3A6lGaHxRP17MSBUoPAN5sB0V01743Ffz1lonShZbxrGIiACxCPfl7NpBuXzSHTESxBkGnedoxzrE48Zwi0mgmg8MIVQy8xKmXSIJ1abXtOnKnW9mkVTyOfc05RMaQU0RQodoEOURS46RwV0g4qojCi6Rw/oCT1bj9aL9Wq9TVcT1u/NHpqB9fEDsEmKw=</latexit> <latexit sha1_base64="yh3oc51Wlo20vUosX31DYfikSI=">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</latexit> Light Cone Gauge ✦ Particle moving in the x 3 direction: Light Cone Gauge A + = 0 ✦ Light Cone Coordinates: p + >> p T >> p - x ± = x 0 ± x 3 and ✦ x ⊥ = ( x 1 , x 2 ) √ 2 ✦ Due to Lorentz contraction one can further assume A ( x + , x − , x ⊥ ) = A ( x + , x ⊥ ) + x n x 2 T x 1 T x n T L. Apolinário � 15 COST THOR School, “Jet Quenching”

  11. <latexit sha1_base64="fAdia8kEyAL9A8hJxRtvTGjTpg=">ACMXicbVDLSgMxFM34tr6qLt1ERXFVZnShG6FEJcV+hBqHe6kt20wkxmSjFCGrgU/RXAltN/Snbj1C9yZPgStHgicnHMvnHuCWHBtXHfgzMzOzS8sLi1nVlbX1jeym1sVHSWKYZlFIlI3AWgUXGLZcCPwJlYIYSCwGtxfDP3qAyrNI1kynRjrIbQkb3IGxkp+dpe3aOEOaOkO/LRAC/5xl57Tofj9bP7bs4dgf4l3oTs5w+fegv958Oin/28bUQsCVEaJkDrmufGp6CMpwJ7GZuE40xsHtoYc1SCSHqejo6pUsPrNKgzUjZJw0dqT83Ugi17oSBnQzBtPW0NxT/9bSN0saG9S6xI0OQVEZmVMLvQKZ5Vk+5jBODko3zNBNBTUSH9dEGV8iM6FgCTHF7EmVtUMCMLTlju/Km/lLKsc57yTnXtvS8mSMJbJD9sgR8cgpyZMrUiRlwsgjeSE90ndenYHz5ryPR2ecyc42+QXn4wv4Part</latexit> <latexit sha1_base64="+K7EXG8JLgdLG/ZWrO4kD740OKM=">ACNnicbVDLSsNAFJ34rPVdelmUJSKWhJFdCNUBHGpYFvBhjCZ3NjBySTMTMRQ/AO/xpWgP+LGnY+lW3dOUhG1HpjhcM69cO7xE86Utu1Ha2BwaHhktDRWHp+YnJquzMw2VZxKCg0a81ie+kQBZwIamkOp4kEvkcWv7Ffu63LkEqFosTnSXgRuRcsJBRo3kVZb3qlfe6hq+8tbzr52ATFbwLv6We4pXWbRrdgHcT5wvsljHx28vpdGtI6/y0Q5imkYgNOVEqTPHTrTbJVIzyuG63E4VJIRekHM4M1SQCJTbLe65xktGCXAYS/OExoX6c6NLIqWyDeTEdEd9dfLxX89ZaJ0IDeAWQiIgKLWBdN/A6kwx23y0SahC0lydMOdYxzjvEAZNANc8MIVQycxKmHSIJ1abpsunK+dtMP2lu1JzNmn1sSqujHkpoHi2gKnLQNqjQ3SEGoiG3SL7tGDdWc9Wc/Wa290wPramUO/YL1/ArF2rCs=</latexit> <latexit sha1_base64="yh3oc51Wlo20vUosX31DYfikSI=">ACNHicbZBSxtBFMdnta0x1nZrj16GxhZPYTc56CUQKJQeFYwKblhmJ2+TITOz25m3krDsF+j36L0nwX6TQqEH8SR4762TxIMmPhj4z/Hrz3S3IpLAbBb29t/cXLVxu1zfrW6+03b/13O6c2KwyHs9kZs4TZkEKDT0UKOE8N8BUIuEsGX+e5WeXYKzI9AlOc+grNtQiFZyhs2J/bxJHuaIdGqWG8XISB3T2n8TtqozsN4Nlq6pivxE0g3nRVRE+iEa3cyf/fhr/OIr9f9Eg4UCjVwyay/CIMd+yQwKLqGqR4WFnPExG8KFk5opsP1yfk1FPzpnQNPMuKeRzt3HEyVT1k5V4joVw5Fdzmbms5l1q4xg4LIvMNWKaoznHN4uhCmh/1S6LxA0HyxT1pIihmdEaQDYCjnDrBuBHuJMpHzNFDx7nuWIXLZFbFasZtpvBsYPWJYuqkV3ygeyTkByQLvlKjkiPcPKd/CTX5Jd35f3xbrzbReua9zDznjwp7/4/anGvKw=</latexit> <latexit sha1_base64="wCXsMiHwtYTL4rpTHFSZlNQIsDo=">ACJHicbVBNS0JBFJ1nH5p9vT52bYkMAh5z4jaBEIQLRXyA1Qe8arDs6b95iZF5r4X1oF9VvaRYs2LaO/0K5RW6R24MLhnHvh3ONHnCntO9WYml5ZTWZWkuvb2xubds7uxUVxpJCmY8lDWfKOBMQFkzaEWSCBz6Hq967GfvUOpGKhuNWDCJoB6QjWZpRoI3n2ft9rRCAjfImzfc89wX0vf+zZGSfnTIAXiftLMgVc+vpMJc+Knv3daIU0DkBoyolSdeJdHNIpGaUwyjdiBVEhPZIB+qGChKAag4n6Uf4yCgt3A6lGaHxRP17MSBUoPAN5sB0V01743Ffz1lonShZbxrGIiACxCPfl7NpBuXzSHTESxBkGnedoxzrE48Zwi0mgmg8MIVQy8xKmXSIJ1abXtOnKnW9mkVTyOfc05RMaQU0RQodoEOURS46RwV0g4qojCi6Rw/oCT1bj9aL9Wq9TVcT1u/NHpqB9fEDsEmKw=</latexit> Light Cone Gauge ✦ Particle moving in the x 3 direction: Light Cone Gauge A + = 0 ✦ Light Cone Coordinates: p + >> p T >> p - x ± = x 0 ± x 3 and ✦ x ⊥ = ( x 1 , x 2 ) √ 2 ✦ Due to Lorentz contraction one can further assume A ( x + , x − , x ⊥ ) = A ( x + , x ⊥ ) p 1 p 2 + x n x 2 Additional notation: T A 1 A 2 x 1 T a igA a T a AA 2 = igA AA 2 x n T L. Apolinário � 15 COST THOR School, “Jet Quenching”

  12. Eikonal Approximation ✦ Consider a high energetic particle propagating through a collection of static scattering centres: ✦ Result will be only a color phase rotation: ➡ In-medium propagator: Wilson Line … … ⇤ L + � ⇥ W ( x 0+ , L + ; x ⊥ ) = P exp dx + A − ( x + , x ⊥ ) ig x 0+ Path-ordering Transverse coordinate Medium colour field � � � � L. Apolinário � 16 COST THOR School, “Jet Quenching” �

  13. Eikonal Approximation ✦ Consider a high energetic particle propagating through a collection of static scattering centres: ✦ Result will be only a color phase rotation: ➡ In-medium propagator: Wilson Line … … ⇤ L + � ⇥ W ( x 0+ , L + ; x ⊥ ) = P exp dx + A − ( x + , x ⊥ ) ig x 0+ Path-ordering Transverse coordinate Medium colour field Let’s re-derive this propagator! Starting by two scatterings… � � � � L. Apolinário � 16 COST THOR School, “Jet Quenching” �

  14. <latexit sha1_base64="n71poO5S7cxk/leixwTU1cLoTU=">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</latexit> <latexit sha1_base64="HaXWZXwTkAkAzdBZpURTn4qbKdc=">ACNnicbVDLSgMxFM34tr6qLt0EXwjSMlMXuhEKBXFnC7YKbR0y6W0bzGRCkhHKMH/gR/gNrgQFv8ONO3Hr1p3ptAtfB0IO59wL5AcqaN674E5NT0zOzc/O5hcWl5ZX86lpDR7GiUKcRj9RlQDRwJqBumOFwKRWQMOBwEVxXhv7FDSjNInFuBhLaIekJ1mWUGCv5+V3ps6sSPsYlLP2E7afZV0hxISMtCUqmVyU/v+UW3Qz4L/HGZKuMt8/uapXnqp/bHUiGocgDOVE6bnStNOiDKMckhzrViDJPSa9KBpqSAh6HaS3ZPiHat0cDdS9gmDM/X7RkJCrQdhYCdDYvr6tzcU/W0jdKHjvVOYCBCIrCITNbEz0Cme9ROmJCxAUFHeboxybCw5xhymghg8sIVQxexKmfaINbpnO3K+93MX9IoFb2DoluzpZXRCHNoA2iPeShQ1RGp6iK6oiW3SPHtGT8+C8Om/O+2h0whnvrKMfcD6+APnrXw=</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: See backup slides for more details! M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ (2 π ) 4 A 1 A 2 A i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 x 1 x 2 Consider only the leading terms: u ( p ) / p 2 / p 2 i = 2 p i + p i − − p 2 ¯ A AA 2 / p 2 = 2 p 2 · A AA 2 ¯ u ( p ) − ¯ u ( p ) / A AA 2 i ⊥ L. Apolinário � 17 COST THOR School, “Jet Quenching”

  15. <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="n71poO5S7cxk/leixwTU1cLoTU=">AClHicbVHbitRAEO3Ey47jLSoI6kvhIqzgDsmI6IPCDIvik+yCs7uwGUKnU7PTbKc7dFeEIeTNn/QHRPwD3+zJDLqXKWg4nFNVXUqr5R0FMc/gvDa9Rs3t3q3+rfv3L13P3rw8NCZ2gqcCKOMPc65QyU1TkiSwuPKIi9zhUf52d5SP/qG1kmjv9KiwmnJT7WcScHJU1n0Pc25beoWdqXkDrF3RyLZtxmzRjG2bCF9NV/uloSH2AIHkAqCkM+51/iuUa7sLFtV7/hkyzajgdxF3AVJGuwPYKDXz97W2/2s+hPWhRl6hJ+G7uJIkrmjbckhQK235aO6y4OneOKh5iW6adO51cILzxQwM9Y/TdCx5ysaXjq3KHOfWXKau8vaktyorRfz2idc6Jr0IY6ny8ORLN30bqibUYjXPrFZABpYXgkJaFKQWHnBhpV8JxJxbLsjfse+9Si47cxUcDgfJ60F84E0bsVX02DP2nO2whL1lI/aZ7bMJE+x3EAVPgqfh4/B9uBd+XKWGwbrmEbsQ4Ze/odbJKQ=</latexit> <latexit sha1_base64="HaXWZXwTkAkAzdBZpURTn4qbKdc=">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</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: See backup slides for more details! M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ (2 π ) 4 A 1 A 2 A i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 x 1 x 2 Consider only the leading terms: sub-leading u ( p ) / p 2 / p 2 i = 2 p i + p i − − p 2 ¯ A AA 2 / p 2 = 2 p 2 · A AA 2 ¯ u ( p ) − ¯ u ( p ) / A AA 2 i ⊥ L. Apolinário � 17 COST THOR School, “Jet Quenching”

  16. <latexit sha1_base64="uidGkuOWptDwnyjpqFZbpBtBknU=">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</latexit> <latexit sha1_base64="n71poO5S7cxk/leixwTU1cLoTU=">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</latexit> <latexit sha1_base64="HaXWZXwTkAkAzdBZpURTn4qbKdc=">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</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">ADTHicfVLdatRAFJ6kWuv6t62XjhYhITqkokFCyo0CtIboaLbFprdOJmd3R2aTIaZSeky5GW89z28EvTO9/BGiuAkG9TdFg8kc+Z853znZ04qMqZ0EHx3JUrV1evrV3v3Lh56/ad7vrGgSpKSWifFkhj1KsaMY47WumM3okJMV5mtHD9ORVjR+eUqlYwd/rmaCDHE84GzOCtTUl686zd0kIX8CYcQ1Hw214lqD2DGE8lpiY+iYSVBnPWgTzh9vVAhLWyB8gxdKUFfSED+nQsIbHE4+tmw+3IGv4PXubW2yAyrCa0pGJqsREMLJ0DtLTLhVPYL1GQsqReW3KdnfgDpzVf+HYcMcn2JhWJZwSvIJkvMtozINtFwo5Yb/YcbNdzoMu43ybRuAflJdzPoBY3Aiwpqlc3d6PmnD/fP9/aT7s94VJAyp1wTm0kdo0DogcFSM5LRqhOXigpMTvCEHluV45yqgWleuYIPrWUEx4W0n32sxvpvhMG5UrM8tZ451lO1jNXGS7G2aYu9pjOeYw5oZv9WCxIj3cGhnFRasrJvJ5xmUFdwHqz4IhJSnQ2swomktmWIJliO1ht969jZ4WJ3NROQh76EkveGuH9hLMZQ3cAw+ABxB4CnbBHtgHfUCcj85n56vzf3i/nDP3V9zV9dpY+6CBVlZ/Q3nmQoV</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: See backup slides for more details! M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ (2 π ) 4 A 1 A 2 A i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 x 1 x 2 Consider only the leading terms: sub-leading u ( p ) / p 2 / p 2 i = 2 p i + p i − − p 2 ¯ A AA 2 / p 2 = 2 p 2 · A AA 2 ¯ u ( p ) − ¯ u ( p ) / A AA 2 ' 2 p 2+ ( A − ) AA 2 ¯ u ( p ) i ⊥ L. Apolinário � 17 COST THOR School, “Jet Quenching”

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sha1_base64="k9KbNB25Qp+IkbKDE9yXDEMqjLY=">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</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: See backup slides for more details! M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ (2 π ) 4 A 1 A 2 A i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 x 1 x 2 Consider only the leading terms: sub-leading u ( p ) / p 2 / p 2 i = 2 p i + p i − − p 2 ¯ A AA 2 / p 2 = 2 p 2 · A AA 2 ¯ u ( p ) − ¯ u ( p ) / A AA 2 ' 2 p 2+ ( A − ) AA 2 ¯ u ( p ) i ⊥ Residues theorem in p i- coordinates will bring the “ordering” of the fields: Z dp 2 − i 2 π e ip 2 − ( x 1 − x 2 ) + p 2 − − ( − i ε ) = θ ( x 2 − x 1 ) + L. Apolinário � 17 COST THOR School, “Jet Quenching”

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sha1_base64="k9KbNB25Qp+IkbKDE9yXDEMqjLY=">ACgXicbVFdaxNBFJ1sta3xo1EfFblYhARJ2F0fFEQJFMTHCqYtdOM6O3u3GTo7M87MloZhf6f4A/wRPghONhFs64ULh3PugXvPLbTg1sXxj160dev29s7unf7de/cf7A0ePjqyqjEMZ0wJZU4KalFwiTPHncATbZDWhcDj4vxgpR9foLFcyc9uqXFe0zPJK86oC1Q+JZx6SCrDGW+1LlPx23r0zFvCL57CmYHiZJzCGyzwd5S/bzTxv/V9DMxcMguqEFtuVBy1MI7yNwCHV25086dBHc+2I8ncVdwEyQbsD9/TgZ/rs62E+JWVijU1SscEtfY0ibWbe2ocZwLbftZY1JSd0zM8DVDSGu3cd9G08CIwJVTKhA6Hduy/Dk9ra5d1ESZr6hb2urYi/6vZsMoCy6B9wKWsqQSpXBfq1YVc9WbudSNQ8nW+1SNAKdg9Q4ouUHmxDIAygwPJwFb0JCuC0/rh6yS68ncBEfpJHk1iT+F0KZkXbvkCXlOhiQhr8mUfCSHZEY+U5+97Z7O9FWNIriKF2PRr2N5zG5UtHbP7djwWM=</latexit> <latexit sha1_base64="LuEZ1RGYLVNz5YJ+Nx2hKhXjiYU=">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</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: See backup slides for more details! M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ (2 π ) 4 A 1 A 2 A i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 x 1 x 2 Consider only the leading terms: sub-leading u ( p ) / p 2 / p 2 i = 2 p i + p i − − p 2 ¯ A AA 2 / p 2 = 2 p 2 · A AA 2 ¯ u ( p ) − ¯ u ( p ) / A AA 2 ' 2 p 2+ ( A − ) AA 2 ¯ u ( p ) i ⊥ Residues theorem in p i- coordinates will bring the “ordering” of the fields: Integrals on p it will just set: Z dp 2 − i ⇒ x 1 ⊥ = x 2 ⊥ = 0 2 π e ip 2 − ( x 1 − x 2 ) + p 2 − − ( − i ε ) = θ ( x 2 − x 1 ) + L. Apolinário � 17 COST THOR School, “Jet Quenching”

  19. <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="PoqgjQ23518Uj+bnV6iYCLtYWIs=">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</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="PoqgjQ23518Uj+bnV6iYCLtYWIs=">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</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ (2 π ) 4 A 1 A 2 A i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 After some work… x 1 x 2 Z dx 1+ dx 2+ ig ( A − ) AA 2 ( x 2+ , 0 ⊥ ) θ ( x 2 � x 1 ) + ig ( A − ) A 2 A 1 ( x 1+ , 0 ⊥ ) θ ( x 1 ) + u ' ) + ¯ u ( p ) M h ( p ) L. Apolinário � 18 COST THOR School, “Jet Quenching”

  20. <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="PoqgjQ23518Uj+bnV6iYCLtYWIs=">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</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="PoqgjQ23518Uj+bnV6iYCLtYWIs=">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</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ (2 π ) 4 A 1 A 2 A i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 After some work… x 1 x 2 Z dx 1+ dx 2+ ig ( A − ) AA 2 ( x 2+ , 0 ⊥ ) θ ( x 2 � x 1 ) + ig ( A − ) A 2 A 1 ( x 1+ , 0 ⊥ ) θ ( x 1 ) + u ' ) + ¯ u ( p ) M h ( p ) No interaction term L. Apolinário � 18 COST THOR School, “Jet Quenching”

  21. <latexit sha1_base64="PoqgjQ23518Uj+bnV6iYCLtYWIs=">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</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="3nqzmOSmO2EeNs3phugT08u6g=">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</latexit> <latexit sha1_base64="vT8tRmJ4DiNOSzpltZkBgx/z21A=">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</latexit> <latexit sha1_base64="PoqgjQ23518Uj+bnV6iYCLtYWIs=">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</latexit> <latexit sha1_base64="8oOPgAI/wYlyDmiv5s4BGHoLjWU=">ACEnicbVDLSgMxFM34rPVdekmtAiKUGZ0ocuiIC4r2Ae0Q8lkbtvQJDMkGWEo/QVXBf0Wd+LWH+inuDOdmFbD1w4nHMvnHuCmDNtXHfirK1vbG5t53byu3v7B4eFo+O6jhJFoUYjHqlmQDRwJqFmOHQjBUQEXBoBIP7qd94AaVZJ9NGoMvSE+yLqPETKW410CiW37GbAq8Sbk1Kl2L4cTyptVP4aYcRTQRIQznRuW5sfGHRBlGOYzy7URDTOiA9KBlqSQCtD/Mso7wmVC3I2UHWlwpv69GBKhdSoCuymI6etlbyr+62kbpQ+h9R4glYJILCOTfbkYyHRv/SGTcWJA0lmebsKxifC0HxwyBdTw1BJCFbMvYdonilBjW8zbrzlZlZJ/arsXZfdJ1vaHZoh05REZ0jD92gCnpEVRDFPXRK3pD787Y+XA+na/Z6pozvzlBC3C+fwHcUaC8</latexit> Eikonal Approximation See: arXiv:0712.3443 ✦ Consider a high energetic particle propagating interacting twice with the medium: M h ( p 1 ) d 4 p 1 d 4 p 2 Z p 1 p 2 p d 4 x 1 d 4 x 2 u ( p ) e ix 2 ( p − p 2 )+ ix 1 ( p 2 − p 2 ) A p 1 ) S 2 = (2 π ) 4 ¯ ∑ ( ) (2 π ) 4 A 1 A 2 A … i / p 2 i / p 1 ) / 2 + i ε ig / p 1 ) A AA 2 ( x 2+ , x 2 ⊥ ) A A 2 A 1 ( x 1+ , x 1 ⊥ ) 1 + i ε M h ( p 2 p 2 a 1 a 2 After some work… x 1 x 2 Z dx 1+ dx 2+ ig ( A − ) AA 2 ( x 2+ , 0 ⊥ ) θ ( x 2 � x 1 ) + ig ( A − ) A 2 A 1 ( x 1+ , 0 ⊥ ) θ ( x 1 ) + u ' ) + ¯ u ( p ) M h ( p ) summing over all ’n’ interaction terms No interaction term ( ) Z x n + P exp dx + A − ( x + , x ⊥ = 0) = W ( x n + , x 1+ ; x ⊥ = 0) ig x 1+ L. Apolinário � 18 COST THOR School, “Jet Quenching”

  22. Non-eikonal corrections See: arXiv:0712.3443 ✦ Consider a (not so) high energetic particle propagating through a collection of static scattering centres: ✦ Should include a small “kick” in the transverse plane: ⌦ r ⇤ ( L + )= x ⇤ ⌦ L + ⌥ ⇥ 2 � ip + � d r ⇧ G ( x 0+ , x 0 ⇧ ; L + , x ⇧ | p + ) = D r ⇧ ( ξ ) exp d ξ 2 d ξ r ⇤ ( x 0+ )= x 0 ⇤ x 0+ × W ( x 0+ , L + ; r ⇧ ( ξ )) , Initial/Final coordinates (3.10) L. Apolinário � 19 COST THOR School, “Jet Quenching”

  23. <latexit sha1_base64="HaXWZXwTkAkAzdBZpURTn4qbKdc=">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</latexit> Non-eikonal corrections See: arXiv:0712.3443 ✦ Consider a (not so) high energetic particle propagating through a collection of static scattering centres: ✦ Should include a small “kick” in the transverse plane: ⌦ r ⇤ ( L + )= x ⇤ ⌦ L + ⌥ ⇥ 2 � ip + � d r ⇧ G ( x 0+ , x 0 ⇧ ; L + , x ⇧ | p + ) = D r ⇧ ( ξ ) exp d ξ 2 d ξ r ⇤ ( x 0+ )= x 0 ⇤ x 0+ How to get this in-medium propagator? × W ( x 0+ , L + ; r ⇧ ( ξ )) , Initial/Final coordinates (3.10) sub-leading terms in denominator p 2 i = 2 p i + p i − − p 2 i ⊥ L. Apolinário � 19 COST THOR School, “Jet Quenching”

  24. <latexit sha1_base64="HaXWZXwTkAkAzdBZpURTn4qbKdc=">ACNnicbVDLSgMxFM34tr6qLt0EXwjSMlMXuhEKBXFnC7YKbR0y6W0bzGRCkhHKMH/gR/gNrgQFv8ONO3Hr1p3ptAtfB0IO59wL5AcqaN674E5NT0zOzc/O5hcWl5ZX86lpDR7GiUKcRj9RlQDRwJqBumOFwKRWQMOBwEVxXhv7FDSjNInFuBhLaIekJ1mWUGCv5+V3ps6sSPsYlLP2E7afZV0hxISMtCUqmVyU/v+UW3Qz4L/HGZKuMt8/uapXnqp/bHUiGocgDOVE6bnStNOiDKMckhzrViDJPSa9KBpqSAh6HaS3ZPiHat0cDdS9gmDM/X7RkJCrQdhYCdDYvr6tzcU/W0jdKHjvVOYCBCIrCITNbEz0Cme9ROmJCxAUFHeboxybCw5xhymghg8sIVQxexKmfaINbpnO3K+93MX9IoFb2DoluzpZXRCHNoA2iPeShQ1RGp6iK6oiW3SPHtGT8+C8Om/O+2h0whnvrKMfcD6+APnrXw=</latexit> <latexit sha1_base64="zm464FrBnarbOyHjdHRmvsaQdnw=">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</latexit> Non-eikonal corrections See: arXiv:0712.3443 ✦ Consider a (not so) high energetic particle propagating through a collection of static scattering centres: ✦ Should include a small “kick” in the transverse plane: ⌦ r ⇤ ( L + )= x ⇤ ⌦ L + ⌥ ⇥ 2 � ip + � d r ⇧ G ( x 0+ , x 0 ⇧ ; L + , x ⇧ | p + ) = D r ⇧ ( ξ ) exp d ξ 2 d ξ r ⇤ ( x 0+ )= x 0 ⇤ x 0+ How to get this in-medium propagator? × W ( x 0+ , L + ; r ⇧ ( ξ )) , Initial/Final coordinates (3.10) sub-leading terms in denominator p 2 i = 2 p i + p i − − p 2 Z dp i − p 2 i ⊥ i i ⊥ Integral in p i- : 2 p + ( x i − 1 − x i ) − i 2 π e ip i − ( x i − 1 − x i ) + = θ ( x i − x i − 1 ) + e p i − − p 2 2 p + + i ε i ⊥ L. Apolinário � 19 COST THOR School, “Jet Quenching”

  25. <latexit sha1_base64="zm464FrBnarbOyHjdHRmvsaQdnw=">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</latexit> <latexit sha1_base64="HaXWZXwTkAkAzdBZpURTn4qbKdc=">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</latexit> <latexit sha1_base64="RAD6Is/jgwJFYJhBFUT50NpfJ9Q=">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</latexit> Non-eikonal corrections See: arXiv:0712.3443 ✦ Consider a (not so) high energetic particle propagating through a collection of static scattering centres: ✦ Should include a small “kick” in the transverse plane: ⌦ r ⇤ ( L + )= x ⇤ ⌦ L + ⌥ ⇥ 2 � ip + � d r ⇧ G ( x 0+ , x 0 ⇧ ; L + , x ⇧ | p + ) = D r ⇧ ( ξ ) exp d ξ 2 d ξ r ⇤ ( x 0+ )= x 0 ⇤ x 0+ How to get this in-medium propagator? × W ( x 0+ , L + ; r ⇧ ( ξ )) , Initial/Final coordinates (3.10) sub-leading terms in denominator p 2 i = 2 p i + p i − − p 2 Z dp i − p 2 i ⊥ i i ⊥ Integral in p i- : 2 p + ( x i − 1 − x i ) − i 2 π e ip i − ( x i − 1 − x i ) + = θ ( x i − x i − 1 ) + e p i − − p 2 2 p + + i ε i ⊥ Z d 2 p i ⊥ p 2 ( x i − 1 − x i ) 2 ⇢ � p + ip + 2 p + ( x i − 1 − x i ) − ip i ⊥ ( x i − 1 − x i ) ⊥ = i ⊥ Integral in p T : i ⊥ exp = (2 π ) 2 e 2 π i ( x i − 1 − x i ) + 2 ( x i − 1 − x i ) + Z x i ⊥ Z x i + ( ◆ 2 ) ✓ dx ⊥ ip + D ( x ⊥ ) exp = G 0 ( x i + , x i ⊥ ; x ( i − 1)+ , x ( i − 1) ⊥ | p + ) 2 dx + x ( i − 1) ⊥ x ( i − 1)+ L. Apolinário � 19 COST THOR School, “Jet Quenching”

  26. In-medium gluon emission Physical picture ✦ Medium-induced gluon radiation: Mathematical picture High energy approximation: g ⇒ Decomposition with a S in fixed number of propagators q I II III ⇒ 3 different regions q S in† g x 0+ x 1+ x 2+ L + L. Apolinário � 20 COST THOR School, “Jet Quenching”

  27. <latexit sha1_base64="ivmy640BZzWP45y4zcC8gPmbjfw=">ACSHicbVA9axtBEN1THMdWYkdJyjSLjcFu5Ds7EJUKAZNSAcky+JRjbm/utHhv9idCwjhH5ImdX5IKoPduPK/SBOMO68kF/56MPB4bwbevLRS0lEYXgWNF0svl1+trDZfv1lbf9t69/7QmdoKHAijD1KwaGSGgckSeFRZRHKVOEwPfk684c/0TpdJ8mFY5KLTMpQDyUtLq9C2PFeYUK9CFQj5MvBtmxQ7fPgjzqAo0M6UcIfHVhZjiu1iL2lthu1wDv6URHdks/vp/+Uv82e3l7Ru4syIukRNQoFzx1FY0WgKlqRQeNqMa4cViBMo8NhTDSW60XT+4Snf8krGc2P9aOJz9f7FErnJmXqN0ugsXvszcRnPejDHz3gFOdAma0Pzbh4Gorwzmkpd1YRaLPLkteJk+KxVnkmLgtTExBW+pe4GIMFQb7pu8qetzMU3K41472+F3X1qXLbDCPrINts0i9pl12TfWYwMm2G/2l52zi+As+BdcBzeL1UZwd/OBPUCjcQsjZ7S+</latexit> In-medium gluon emission Physical picture ✦ Medium-induced gluon radiation: Mathematical picture High energy approximation: g ⇒ Decomposition with a S in fixed number of propagators q I II III ⇒ 3 different regions q S in† g x 0+ x 1+ x 2+ L + Emission Kernel D E W A ( r g ) W † A (0) Tr L. Apolinário � 20 COST THOR School, “Jet Quenching”

  28. <latexit sha1_base64="G4WhvcNXPsSdpS3JBuC1uGmhuw=">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</latexit> <latexit sha1_base64="ivmy640BZzWP45y4zcC8gPmbjfw=">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</latexit> In-medium gluon emission Physical picture ✦ Medium-induced gluon radiation: Mathematical picture High energy approximation: g ⇒ Decomposition with a S in fixed number of propagators q I II III ⇒ 3 different regions q S in† g x 0+ x 1+ x 2+ L + Emission Kernel Classical Broadening D E D E W A ( r g ) W † W A ( r g ) W † A (0) A (¯ r g ) Tr Tr L. Apolinário � 20 COST THOR School, “Jet Quenching”

  29. <latexit sha1_base64="ivmy640BZzWP45y4zcC8gPmbjfw=">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</latexit> <latexit sha1_base64="G4WhvcNXPsSdpS3JBuC1uGmhuw=">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</latexit> In-medium gluon emission Physical picture ✦ Medium-induced gluon radiation: Mathematical picture High energy approximation: g ⇒ Decomposition with a S in fixed number of propagators q I II III ⇒ 3 different regions q S in† g And, finally, some numerics: x 0+ x 1+ x 2+ L + Emission Kernel Classical Broadening D E D E W A ( r g ) W † W A ( r g ) W † A (0) A (¯ r g ) Tr Tr L. Apolinário � 20 COST THOR School, “Jet Quenching”

  30. <latexit sha1_base64="ivmy640BZzWP45y4zcC8gPmbjfw=">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</latexit> <latexit sha1_base64="G4WhvcNXPsSdpS3JBuC1uGmhuw=">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</latexit> In-medium gluon emission Physical picture ✦ Medium-induced gluon radiation: Mathematical picture High energy approximation: g ⇒ Decomposition with a S in fixed number of propagators q I II III ⇒ 3 different regions q S in† g And, finally, some numerics: x 0+ x 1+ x 2+ L + Emission Kernel Classical Broadening LPM (QCD) D E D E W A ( r g ) W † W A ( r g ) W † A (0) A (¯ r g ) Tr Tr suppression L. Apolinário � 20 COST THOR School, “Jet Quenching”

  31. LPM effect k = z p ✦ Heuristic discussion on single gluon emission spectrum (BDMPS): Transport coefficient: q = (1-z) p q = < k 2 ⊥ > ˆ λ mfp L. Apolinário � 21 COST THOR School, “Jet Quenching”

  32. <latexit sha1_base64="nJlLnbSmhqnvbh+wMFb05wD+lfQ=">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</latexit> <latexit sha1_base64="qlnYWHvb1+VhiJ6O5+o3UjfZhw=">ACQ3icbVBNSxBEO0x8WsTdROPXpqIadlRg/xuBAIOXhQcHXBXpea3pqdZrt7Jt01wjLsKb8i/8Sbp0DyE/IbvIkXD4K9ux6i5kHB470qePXSUitPcfw3Wnj1enFpeW18ebt2vpG8937E19UTmJHFrpw3RQ8amWxQ4o0dkuHYFKNp+noy9Q/vUDnVWGPaVxiz8DQqkxJoCD1m3tCY0ZCgx1q5KO+KNGV57tcODXMSbi5LrwyXORA9fcJP+g3t+NWPAN/SZJHst3+eKl+HUvD/vNOzEoZGXQktTg/VkSl9SrwZGSGicNUXksQY5giGeBWjDoe/XsuQnfCcqAZ4ULY4nP1H8vajDej0aNg1Q7p97U/G/ng9RchwE7yuOrQHLbUGzWp4Gomy/VytbVoRWzvNkleZU8GmhfKAcStLjQEA6FV7iMgcHkLtjdBV8ryZl+Rkt5XsteKjUFqbzbHCtgH9okl7DNrs2/skHWYZD/ZFfvN/kS/ouvoJrqdry5Ejzeb7Ami+wehe7UZ</latexit> LPM effect k = z p ✦ Heuristic discussion on single gluon emission spectrum (BDMPS): Transport coefficient: q = (1-z) p q = < k 2 ⊥ > r ω ω ω ˆ ) τ form = k 2 ⌦ ↵ τ form = ⊥ i = qL ∼ ˆ λ mfp h k 2 ⊥ q τ form q ˆ ˆ Soft gluons have shorter formation times L. Apolinário � 21 COST THOR School, “Jet Quenching”

  33. <latexit sha1_base64="QBOpAkQsB+Df/DHGsf9b/XkOYIo=">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</latexit> <latexit sha1_base64="8TE7GL+Ujpzhn/b9dBqzALCmZWI=">ACIXicbVC7SgNBFJ31bXxFLQUZFMEq7GqhZUAQC4sIRgUTwuzsTJkHsvMXSUsqfwOCxH0W+zETvySdE4SC2M8MHA4517m3BOnUjgMw89ganpmdm5+YbGwtLyulZc37hyJrMcqtxIY29i5kAKDVUKOEmtcBULOE67pwM/Os7sE4YfYndFOqKtbRoCs7QS43ieg1Z1sibxqoerUlJzxvF3bAUDkEnSfRDds7Tw/97cRUGsV+LTE8U6CRS+bcbRSmWM+ZRcEl9Aq1zEHKeIe14NZTzRS4ej6M3qN7Xkmo/94/jXSo/t7ImXKuq2I/qRi23V9vIP7rOR+lDYn3TqGrFdNUGxwePR4Im8f1XOg0Q9B8lKeZSYqGDuqibDAUXY9YdwKfxLlbWYZR19qwXcV/W1mklwdlKLDUnjhSyuTERbIFtkh+yQiR6RMzkiFVAkn9+SRvJDX4Dl4C96Dj9HoVPCzs0nGEHx9A9IrpxY=</latexit> <latexit sha1_base64="hMb2OHp1GUYNJxw4QIf+PYlIfQ=">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</latexit> <latexit sha1_base64="nJlLnbSmhqnvbh+wMFb05wD+lfQ=">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</latexit> <latexit sha1_base64="qlnYWHvb1+VhiJ6O5+o3UjfZhw=">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</latexit> <latexit sha1_base64="BTfoq/V+lBtgjz4IQ/9EYeqPISM=">ACIXicbVC7SgNBFJ31bXwlWgoyRASrsKuFlgFBLCwUjAaSEGZnb5LBeSwzd5WwpPI7LETQb7ETO/FL7JwkFho9MHA4517m3BOnUjgMw/dganpmdm5+YbGwtLyulYsrV86k1kONW6ksfWYOZBCQw0FSqinFpiKJVzF10dD/+oGrBNGX2A/hZiXS06gjP0UrtYaiL2nHWDWgzW6XnraL2ElHIH+JdE32a6WH+4+txJz1i5+NhPDMwUauWTONaIwxVbOLAouYVBoZg5Sxq9ZFxqeaqbAtfJR9AHd8UpC/f+aQj9edGzpRzfRX7ScWw5ya9ofiv53yUHiTeO4a+VkxTbXB09O9A2Dls5UKnGYLm4zydTFI0dFgXTYQFjrLvCeNW+JMo7zHLOPpSC76raLKZv+RyrxLtV8JzX1qVjLFANkmZ7JKIHJAqOSFnpEY4uSX35Ik8B4/BS/AavI1Hp4LvnQ3yC8HF8Fkpw=</latexit> LPM effect k = z p ✦ Heuristic discussion on single gluon emission spectrum (BDMPS): Transport coefficient: q = (1-z) p q = < k 2 ⊥ > r ω ω ω ˆ ) τ form = k 2 ⌦ ↵ τ form = ⊥ i = qL ∼ ˆ λ mfp h k 2 ⊥ q τ form q ˆ ˆ Soft gluons have shorter formation times L Number of emitted gluons during L: N g ∝ τ form Multiple soft emissions r ω c � ω dI τ form ⌧ L � ∼ α s C R (incoherent scatterings) � d ω ω � ω ≤ ω c Hard gluon spectrum is suppressed � ω dI ⌘ 2 ⇣ ω c � ∼ α s C R τ form � L � (scattering centres act as a whole) d ω ω � ω > ω c ω c ( τ form = L ) = 1 Critical energy: qL 2 2 ˆ L. Apolinário � 21 COST THOR School, “Jet Quenching”

  34. Intra-jet (de)coherence? [Mehtar-Tani, Salgado, Tywoniuk (2010-2011)] [Casalderrey-Solana, Iancu (2011)] ✦ Considering the in-medium singlet quark - antiquark antenna in the simplest case: ➡ Soft gluon emission outside medium q 2 2 2 q 2 θ 1 + + ω , k ⊥ θ θ ω , k ⊥ θ 1 q 1 q 1 dI Eikonal approximation: = R q + R ¯ q − 2 J (1 − ∆ med ) = R coh + 2 J ∆ med d Ω k q 1+ R q ∼ α s C F ( k · q 1 ) q 2+ R ¯ q ∼ α s C F ( k · q 2 )  � k + ( q 1 · q 2 ) q 1+ q 2+ 2 J ∼ α s C F ( k · q 1 ) + ( k · q 2 ) − ( k · q 1 )( k · q 2 ) L. Apolinário � 22 COST THOR School, “Jet Quenching”

  35. Intra-jet (de)coherence? [Mehtar-Tani, Salgado, Tywoniuk (2010-2011)] [Casalderrey-Solana, Iancu (2011)] ✦ Considering the in-medium singlet quark - antiquark antenna in the simplest case: ➡ Soft gluon emission outside medium q 2 2 2 q 2 θ 1 + + ω , k ⊥ θ θ ω , k ⊥ θ 1 q 1 q 1 dI Eikonal approximation: = R q + R ¯ q − 2 J (1 − ∆ med ) = R coh + 2 J ∆ med d Ω k new in-medium contribution: q 1+ 1 R q ∼ α s C F D E W A ( x q ) W † 1 − ∆ med = Tr A ( x ¯ q ) ( k · q 1 ) N 2 c q 2+ R ¯ q ∼ α s C F ( k · q 2 ) k + ( q 1 · q 2 )  � k + ( q 1 · q 2 ) q 1+ q 2+ as in vacuum: R coh ∼ α s C F 2 J ∼ α s C F ( k · q 1 ) + ( k · q 2 ) − ( k · q 1 )( k · q 2 ) ( k · q 1 )( k · q 2 ) L. Apolinário � 22 COST THOR School, “Jet Quenching”

  36. Intra-jet (de)coherence? [Mehtar-Tani, Salgado, Tywoniuk (2010-2011)] [Casalderrey-Solana, Iancu (2011)] ✦ Considering the in-medium singlet quark - antiquark antenna in the simplest case: ➡ Soft gluon emission outside medium Integrating over azimuthal angle (soft limit): sin θ d θ d ω 12 Q 2 s r 2 ∆ med ≈ 1 − e − 1 dN ω → 0 1 − cos θ [ Θ (cos θ 1 − cos θ ) + ∆ med Θ (cos θ − cos θ 1 )] ∼ α s C R ⊥ q ω L. Apolinário � 23 COST THOR School, “Jet Quenching”

  37. Intra-jet (de)coherence? [Mehtar-Tani, Salgado, Tywoniuk (2010-2011)] [Casalderrey-Solana, Iancu (2011)] ✦ Considering the in-medium singlet quark - antiquark antenna in the simplest case: ➡ Soft gluon emission outside medium Integrating over azimuthal angle (soft limit): sin θ d θ d ω 12 Q 2 s r 2 ∆ med ≈ 1 − e − 1 dN ω → 0 1 − cos θ [ Θ (cos θ 1 − cos θ ) + ∆ med Θ (cos θ − cos θ 1 )] ∼ α s C R ⊥ q ω Angular ordering Antenna Transverse Q − 1 r ⊥ resolution: r ⟂ = θ L s Medium Transverse Scale: Q s-1 = √(q L) -1 ^ ∆ med → 0 L. Apolinário � 23 COST THOR School, “Jet Quenching”

  38. Intra-jet (de)coherence? [Mehtar-Tani, Salgado, Tywoniuk (2010-2011)] [Casalderrey-Solana, Iancu (2011)] ✦ Considering the in-medium singlet quark - antiquark antenna in the simplest case: ➡ Soft gluon emission outside medium Integrating over azimuthal angle (soft limit): sin θ d θ d ω 12 Q 2 s r 2 ∆ med ≈ 1 − e − 1 dN ω → 0 1 − cos θ [ Θ (cos θ 1 − cos θ ) + ∆ med Θ (cos θ − cos θ 1 )] ∼ α s C R ⊥ q ω Anti-Angular ordering Angular ordering Antenna Transverse Q − 1 r ⊥ resolution: r ⟂ = θ L s r ⊥ Q − 1 s Medium Transverse Scale: Q s-1 = √(q L) -1 ^ ∆ med → 0 ∆ med → 1 L. Apolinário � 23 COST THOR School, “Jet Quenching”

  39. More th developments t* ✦ Single gluon radiation beyond eikonal limit: 휏 form ✦ Small 훕 form : parton shower can as incoherent sum ⇒ Rate equations L of gluon radiation [J.-P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar-Tani (12), Jeon, Moore (05)] L. Apolinário � 24 COST THOR School, “Jet Quenching”

  40. More th developments t* ✦ Single gluon radiation beyond eikonal limit: 휏 form ✦ Small 훕 form : parton shower can as incoherent sum ⇒ Rate equations L of gluon radiation [J.-P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar-Tani (12), Jeon, Moore (05)] ✦ Non-eikonal corrections to QCD antenna [LA, N. Armesto, G. Milhano, C. Salgado (17)] dI ⇒ Interplay Coherence/Decoherence = ∆ 0 q ) − 2(1 − ∆ 0 coh ( R q + R ¯ med ) J d Ω q d Ω ¯ q d Ω k L. Apolinário � 24 COST THOR School, “Jet Quenching”

  41. More th developments t* ✦ Single gluon radiation beyond eikonal limit: 휏 form ✦ Small 훕 form : parton shower can as incoherent sum ⇒ Rate equations L of gluon radiation [J.-P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar-Tani (12), Jeon, Moore (05)] ✦ Non-eikonal corrections to QCD antenna [LA, N. Armesto, G. Milhano, C. Salgado (17)] dI ⇒ Interplay Coherence/Decoherence = ∆ 0 q ) − 2(1 − ∆ 0 coh ( R q + R ¯ med ) J d Ω q d Ω ¯ q d Ω k ✦ Hard emission from In-medium antenna: [Domínguez, Salgado and Vila (18) ] ⇒ Effective emitters in the parton shower L. Apolinário � 24 COST THOR School, “Jet Quenching”

  42. More th developments t* ✦ Single gluon radiation beyond eikonal limit: 휏 form ✦ Small 훕 form : parton shower can as incoherent sum ⇒ Rate equations L of gluon radiation [J.-P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar-Tani (12), Jeon, Moore (05)] ✦ Non-eikonal corrections to QCD antenna [LA, N. Armesto, G. Milhano, C. Salgado (17)] dI ⇒ Interplay Coherence/Decoherence = ∆ 0 q ) − 2(1 − ∆ 0 coh ( R q + R ¯ med ) J d Ω q d Ω ¯ q d Ω k ✦ Hard emission from In-medium antenna: [Domínguez, Salgado and Vila (18) ] ⇒ Effective emitters in the parton shower ✦ Effects of the medium on vacuum-like emissions: ✦ First radiation outside the medium can violate angular-ordering ⇒ Medium effects on vacuum emissions [Caucal, Iancu, Mueller, Soyez (18) & QM18] L. Apolinário � 24 COST THOR School, “Jet Quenching”

  43. See K. Tywoniuk (Thursday) More th developments t* ✦ Single gluon radiation beyond eikonal limit: 휏 form ✦ Small 훕 form : parton shower can as incoherent sum ⇒ Rate equations L of gluon radiation [J.-P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar-Tani (12), Jeon, Moore (05)] ✦ Non-eikonal corrections to QCD antenna [LA, N. Armesto, G. Milhano, C. Salgado (17)] dI ⇒ Interplay Coherence/Decoherence = ∆ 0 q ) − 2(1 − ∆ 0 coh ( R q + R ¯ med ) J d Ω q d Ω ¯ q d Ω k ✦ Hard emission from In-medium antenna: [Domínguez, Salgado and Vila (18) ] ⇒ Effective emitters in the parton shower ✦ Effects of the medium on vacuum-like emissions: ✦ First radiation outside the medium can violate angular-ordering ⇒ Medium effects on vacuum emissions [Caucal, Iancu, Mueller, Soyez (18) & QM18] L. Apolinário � 24 COST THOR School, “Jet Quenching”

  44. Resulting picture [See also: 1801.09703] [1401.8293] ✦ Resulting picture of a medium-modified parton shower: Finite size “pp-like” structures (Angular Ordering) Medium-induced radiation (not collinear) L. Apolinário � 25 COST THOR School, “Jet Quenching”

  45. Resulting picture [See also: 1801.09703] [1401.8293] ✦ Resulting picture of a medium-modified parton shower: hard structure: unmodified Finite size “pp-like” structures (Angular Ordering) Medium-induced radiation (not collinear) L. Apolinário � 25 COST THOR School, “Jet Quenching”

  46. Resulting picture [See also: 1801.09703] [1401.8293] ✦ Resulting picture of a medium-modified parton shower: hard structure: unmodified Finite size “pp-like” structures (Angular Ordering) increase of soft fragments Medium-induced radiation (not collinear) L. Apolinário � 25 COST THOR School, “Jet Quenching”

  47. Resulting picture arXiv:1803.00042 ✦ Resulting picture of a medium-modified parton shower: Finite size “pp-like” structures Soft fragments (Angular Ordering) radiated up to large angles Medium-induced radiation (not collinear) L. Apolinário � 26 COST THOR School, “Jet Quenching”

  48. Resulting picture arXiv:1803.00042 ✦ Resulting picture of a medium-modified parton shower: Finite size “pp-like” structures Soft fragments (Angular Ordering) radiated up to large angles Medium-induced radiation (not collinear) Everything seems to work, we can go home! L. Apolinário � 26 COST THOR School, “Jet Quenching”

  49. Not so fast… [L.Apolinário, QCD Forward Physics 2014] = 0 Q-PYTHIA 2.76TeV ptgen = 70 GeV 4 ✦ Quantitative comparisons show some disagreement… q (r) q = 1 GeV 2 fm -1 � 2 -1 q = 4 GeV fm / 0 3.5 2 -1 � q = 8 GeV fm q (r) � ✦ Monte Carlo approaches based on in-medium single gluon 3 radiation fail to describe some of the intra-jet features! 2.5 2 1.5 p track ∑ T ρ ( r ) = 1 1 tracks ∈ [ r a , r b ) 1 N jet ∑ p jet δ r 0.5 jets T 0 0.05 0.1 0.15 0.2 0.25 0.3 R ∫ ∫ -1 -1 pp, L dt = 5.3 pb PbPb, L dt = 150 b CMS, s = 2.76 TeV µ NN 10 10 10 10 10 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 anti-k jets: R = 0.3 PbPb 1.5 1.5 1.5 1.5 1.5 pp (r) ρ / 1 1 1 1 1 PbPb (r) ρ 70-100% 50-70% 30-50% 10-30% 0-10% -1 -1 -1 -1 10 10 10 10 0.5 0.5 0.5 0.5 0.5 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 r r r r r L. Apolinário � 27 COST THOR School, “Jet Quenching”

  50. Not so fast… [L.Apolinário, QCD Forward Physics 2014] = 0 Q-PYTHIA 2.76TeV ptgen = 70 GeV 4 ✦ Quantitative comparisons show some disagreement… q (r) q = 1 GeV 2 fm -1 � 2 -1 q = 4 GeV fm / 0 3.5 2 -1 � q = 8 GeV fm q (r) � ✦ Monte Carlo approaches based on in-medium single gluon 3 radiation fail to describe some of the intra-jet features! 2.5 2 1.5 p track ∑ T ρ ( r ) = 1 1 tracks ∈ [ r a , r b ) 1 N jet ∑ p jet δ r 0.5 jets T 0 0.05 0.1 0.15 0.2 0.25 0.3 R ∫ ∫ -1 -1 pp, L dt = 5.3 pb PbPb, L dt = 150 b CMS, s = 2.76 TeV µ NN 10 10 10 10 10 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 anti-k jets: R = 0.3 PbPb 1.5 1.5 1.5 1.5 1.5 Qualitative disagreement… pp (r) ρ / 1 1 1 1 1 PbPb (r) ρ 70-100% 50-70% 30-50% 10-30% 0-10% -1 -1 -1 -1 10 10 10 10 0.5 0.5 0.5 0.5 0.5 Let’s go back again… 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 r r r r r L. Apolinário � 27 COST THOR School, “Jet Quenching”

  51. What is a jet? ✦ A jet in pp: ✦ Defined with a jet clustering algorithm based (or not) in QCD principles: anti-k T , k T , C/A,… ➡ Have an object that can be related to the parton shower ➡ Have an object that can be equally treated at parton, particle or calorimetric level L. Apolinário � 28 COST THOR School, “Jet Quenching”

  52. What is a jet? ✦ A jet in heavy-ions: ✦ Defined with a jet clustering algorithm based (or not) in QCD principles: anti-k T , k T , C/A,… ➡ Have an object that can be related to the parton shower ➡ Have an object that can be equally treated at parton, particle or calorimetric level L. Apolinário � 29 COST THOR School, “Jet Quenching”

  53. What is a jet? ✦ A jet in heavy-ions: ✦ Defined with a jet clustering algorithm based (or not) in QCD principles: anti-k T , k T , C/A,… ➡ Have an object that can be related to the parton shower ??? ➡ Have an object that can be equally treated at parton, particle or calorimetric level L. Apolinário � 29 COST THOR School, “Jet Quenching”

  54. How to define a jet in a heavy-ion?

  55. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: L. Apolinário � 31 COST THOR School, “Jet Quenching”

  56. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: In-medium parton shower Expanding Medium L. Apolinário � 31 COST THOR School, “Jet Quenching”

  57. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: In-medium parton shower Medium-induced energy loss Expanding Medium L. Apolinário � 31 COST THOR School, “Jet Quenching”

  58. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: In-medium parton shower Medium-induced energy loss Collisional energy loss Expanding Medium L. Apolinário � 31 COST THOR School, “Jet Quenching”

  59. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: In-medium parton shower Medium-induced energy loss Collisional energy loss Medium recoils Expanding Medium L. Apolinário � 31 COST THOR School, “Jet Quenching”

  60. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: In-medium parton shower Medium effects on Hadronization? Medium-induced energy loss Collisional energy loss Medium recoils Expanding Medium L. Apolinário � 31 COST THOR School, “Jet Quenching”

  61. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: In-medium parton shower Q 2 P T Initial Parton Medium effects on Hadronization? pQCD Scattering from Point-Like Bare Color Charges Medium-induced energy loss What scale sets this transition? Scattering from Thermal Mass Gluons? What scale sets this transition? Collisional energy loss pQCD Scattering Strong Coupling From Quasiparticles No Quasiparticles ? ! with size ~ µ Debye µ Debye ! 0 " ? Medium recoils AdS/CFT µ D T c Expanding Medium L. Apolinário � 31 COST THOR School, “Jet Quenching”

  62. Jet Quenching v2.0 ✦ The way we see and define a jet should include all momentum scales: In-medium parton shower Q 2 P T Initial Parton Medium effects on Hadronization? pQCD Scattering from Point-Like Bare Color Charges Medium-induced energy loss What scale sets this transition? Scattering from Thermal Mass Gluons? What scale sets this transition? Collisional energy loss pQCD Scattering Strong Coupling From Quasiparticles No Quasiparticles ? ! with size ~ µ Debye µ Debye ! 0 " ? Medium recoils AdS/CFT µ D T c Expanding Medium L. Apolinário � 31 COST THOR School, “Jet Quenching”

  63. <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> Medium response LBT: [Cao, Luo, Qin, Wang (16) He, Luo, Wang, Zhu (17)] MARTINI: [Schenke, Gale, Jeon (09)] ✦ QGP part that become correlated with the jet (not to be subtracted!) JEWEL: [Elayavalli, Zapp (17)] ✦ Seen as (pQCD approach): ✦ Recoils from jet-medium interactions with a thermal/3D hydro particle distribution ➡ Recoiled particle makes part of the jet: JEWEL C R 2 πα 2 d ˆ σ s, | ˆ s E.g: JEWEL t (ˆ t | ) ' d ˆ ( | ˆ t | + µ 2 D ) 2 L. Apolinário � 32 COST THOR School, “Jet Quenching”

  64. <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> <latexit sha1_base64="I/OYFlN0XUTYdjD2YSNwKZEgo8s=">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</latexit> Medium response LBT: [Cao, Luo, Qin, Wang (16) He, Luo, Wang, Zhu (17)] MARTINI: [Schenke, Gale, Jeon (09)] ✦ QGP part that become correlated with the jet (not to be subtracted!) JEWEL: [Elayavalli, Zapp (17)] ✦ Seen as (pQCD approach): ✦ Recoils from jet-medium interactions with a thermal/3D hydro particle distribution ➡ Recoiled particle makes part of the jet: JEWEL ➡ Recoiled particle can further interact with medium constituents: MARTINI, LBT C R 2 πα 2 E.g: MARTINI d ˆ σ s, | ˆ s E.g: JEWEL t (ˆ t | ) ' d ˆ ( | ˆ t | + µ 2 D ) 2 L. Apolinário � 32 COST THOR School, “Jet Quenching”

  65. There are still problems… ✦ Magnitude of the medium recoil component to the jet varies from model to model… ✦ Coupled Jet-Fluid (Analytical approach) ✦ MARTINI (Monte Carlo approach based on in-medium gluon radiation) Couple Jet-Fluid: [arXiv: 1701.07951] [MARINI: Park QM18] Recoil component Best observed in the “jet radial profile” without recoil without recoil L. Apolinário � 33 COST THOR School, “Jet Quenching”

  66. And more problems… ✦ Within the same model, not so easy to simultaneously describe jet radial profile and jet mass: [JEWEL:1707.01539] [JEWEL:1707.01539] L. Apolinário � 34 COST THOR School, “Jet Quenching”

  67. And more problems… [MARTINI:1807.06550] ✦ Within the same model, not so easy to simultaneously describe jet radial profile and jet mass: [JEWEL:1707.01539] [JEWEL:1707.01539] [MARTINI:1807.06550] L. Apolinário � 34 COST THOR School, “Jet Quenching”

  68. And more problems… [MARTINI:1807.06550] ✦ Within the same model, not so easy to simultaneously describe jet radial profile and jet mass: [JEWEL:1707.01539] [JEWEL:1707.01539] [MARTINI:1807.06550] See K. Zapp (Thursday) for news about JEWEL L. Apolinário � 34 COST THOR School, “Jet Quenching”

  69. And even more problems… [Jet collaboration (13)] [Andrés, Armesto, Luzum, Salgado, Zurita (17)] ✦ Quenching parameter (qhat) at RHIC larger is than at LHC [Casalderrey-Solana, Hulcher, Milhano, Pablos, Rajagopal (18)] ✦ Not clearly understood so far… both pQCD and non-pQCD approaches! L. Apolinário � 35 COST THOR School, “Jet Quenching”

  70. JQ Phenomenology ✦ Theoretical developments that address elementary jet processes: ✓ Able to build up a clear qualitative picture X Not suited for describing medium recoil component (essential to withdraw QGP properties) L. Apolinário � 36 COST THOR School, “Jet Quenching”

  71. JQ Phenomenology ✦ Theoretical developments that address elementary jet processes: ✓ Able to build up a clear qualitative picture X Not suited for describing medium recoil component (essential to withdraw QGP properties) Being highly developed within Monte Carlo approaches (see references at backup slides)! L. Apolinário � 36 COST THOR School, “Jet Quenching”

  72. JQ Phenomenology ✦ Theoretical developments that address elementary jet processes: ✓ Able to build up a clear qualitative picture X Not suited for describing medium recoil component (essential to withdraw QGP properties) Being highly developed within Monte Carlo approaches (see references at backup slides)! Based on phenomenological assumptions… But Jet Quenching must feed (and be fed) with jet phenomenology! L. Apolinário � 36 COST THOR School, “Jet Quenching”

  73. JQ Phenomenology ✦ Theoretical developments that address elementary jet processes: ✓ Able to build up a clear qualitative picture X Not suited for describing medium recoil component (essential to withdraw QGP properties) Being highly developed within Monte Carlo approaches (see references at backup slides)! Based on phenomenological assumptions… But Jet Quenching must feed (and be fed) with jet phenomenology! ✦ To ultimately assess QGP properties, also need new jet observables (from pp and new for heavy-ions): ✦ Sensitive to selected jet quenching effects (establish a baseline): [LA, Milhano, Ploskon, Zhang (17)] Initial times: [Andres, Armesto, Niemi, Paatelainen, Salgado (19)] ✦ Probing different QGP timescales: Final times: [LA, Milhano, Salgado, Salam (18)] L. Apolinário � 36 COST THOR School, “Jet Quenching”

  74. JQ Phenomenology ✦ Theoretical developments that address elementary jet processes: ✓ Able to build up a clear qualitative picture X Not suited for describing medium recoil component (essential to withdraw QGP properties) Being highly developed within Monte Carlo approaches (see references at backup slides)! Based on phenomenological assumptions… But Jet Quenching must feed (and be fed) with jet phenomenology! ✦ To ultimately assess QGP properties, also need new jet observables (from pp and new for heavy-ions): ✦ Sensitive to selected jet quenching effects (establish a baseline): [LA, Milhano, Ploskon, Zhang (17)] Initial times: [Andres, Armesto, Niemi, Paatelainen, Salgado (19)] ✦ Probing different QGP timescales: Final times: [LA, Milhano, Salgado, Salam (18)] See L. Apolinário (Thursday) L. Apolinário � 36 COST THOR School, “Jet Quenching”

  75. Wrapping-up

  76. Summary ✦ This was a brief lecture/overview of “jet quenching”: ✦ Things I didn’t cover: Heavy-quarks, AdS/CFT approaches, Monte Carlo approaches,… ✦ Several developments towards the understanding of what is a jet in a heavy-ion environment! ✦ But no consistent picture has emerged yet… ✦ Not clear the role of medium recoil effects, missing in-medium evolution equation (coupled to vacuum radiation), … ➡ Several developments in building new observables (particular QGP effect, a particular type of probe (q/g-jets) and/or different QGP timescales ⇒ Jet substructure!) See J. Barata (Thursday) L. Apolinário � 38 COST THOR School, “Jet Quenching”

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