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Hiren Patel [email protected] Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections to Polarized Moller Scattering in collaboration with Ayres Freitas Yong Du, Michael Ramsey-Musolf

Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel [email protected] Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

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Page 1: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren [email protected]

Santa Cruz Institute for Particle Physics

S C I P P

Sept 16, 2019

Two Loop Electroweak Corrections to Polarized Moller Scattering

in collaboration with Ayres Freitas Yong Du, Michael Ramsey-Musolf

Page 2: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!2

- Fixed target polarized Moller scattering experiment - To measure the electron ALR (fractional accuracy 2.4%) - Goal to precisely determine QW(e) and sin2(θW) far below the Z pole

Liquid Hydrogen target

80% polarizationElab = 11 GeV

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Detector

5 to 17 mrad scattering angle

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e�<latexit sha1_base64="Zv8TUPeYhlasrdZ3VxqCmhV7ohQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURwR4LXjxWtB/QxrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzCRBP6JDyUPOqLHSPT5e9MsVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc1Yc3PuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuq55b9e6uKvVaHkcRTuAUzsGDa6jDLTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDkGI1+</latexit><latexit sha1_base64="Zv8TUPeYhlasrdZ3VxqCmhV7ohQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURwR4LXjxWtB/QxrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzCRBP6JDyUPOqLHSPT5e9MsVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc1Yc3PuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuq55b9e6uKvVaHkcRTuAUzsGDa6jDLTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDkGI1+</latexit><latexit sha1_base64="Zv8TUPeYhlasrdZ3VxqCmhV7ohQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURwR4LXjxWtB/QxrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzCRBP6JDyUPOqLHSPT5e9MsVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc1Yc3PuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuq55b9e6uKvVaHkcRTuAUzsGDa6jDLTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDkGI1+</latexit><latexit sha1_base64="Zv8TUPeYhlasrdZ3VxqCmhV7ohQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURwR4LXjxWtB/QxrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzCRBP6JDyUPOqLHSPT5e9MsVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc1Yc3PuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuq55b9e6uKvVaHkcRTuAUzsGDa6jDLTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDkGI1+</latexit>

ALR =�R � �L

�R + �L<latexit sha1_base64="vSJxpowyQ2pUPt4v2/NCo2LUsao=">AAACIXicbZDLSsNAFIYnXmu9RV26GSyCIJZEBLsRKm5cdFGLvUATwmQ6aYdOJmFmIpSQV3Hjq7hxoUh34ss4bQPV1h8Gfr5zDmfO78eMSmVZX8bK6tr6xmZhq7i9s7u3bx4ctmSUCEyaOGKR6PhIEkY5aSqqGOnEgqDQZ6TtD+8m9fYTEZJG/FGNYuKGqM9pQDFSGnlm5dZLa40M3kAnEAinjqT9EHkNeAFzW8vm8HwOPbNkla2p4LKxc1MCueqeOXZ6EU5CwhVmSMqubcXKTZFQFDOSFZ1EkhjhIeqTrrYchUS66fTCDJ5q0oNBJPTjCk7p74kUhVKOQl93hkgN5GJtAv+rdRMVVNyU8jhRhOPZoiBhUEVwEhfsUUGwYiNtEBZU/xXiAdJJKR1qUYdgL568bFqXZdsq2w9XpWolj6MAjsEJOAM2uAZVcA/qoAkweAav4B18GC/Gm/FpjGetK0Y+cwT+yPj+Ad4XorA=</latexit><latexit sha1_base64="vSJxpowyQ2pUPt4v2/NCo2LUsao=">AAACIXicbZDLSsNAFIYnXmu9RV26GSyCIJZEBLsRKm5cdFGLvUATwmQ6aYdOJmFmIpSQV3Hjq7hxoUh34ss4bQPV1h8Gfr5zDmfO78eMSmVZX8bK6tr6xmZhq7i9s7u3bx4ctmSUCEyaOGKR6PhIEkY5aSqqGOnEgqDQZ6TtD+8m9fYTEZJG/FGNYuKGqM9pQDFSGnlm5dZLa40M3kAnEAinjqT9EHkNeAFzW8vm8HwOPbNkla2p4LKxc1MCueqeOXZ6EU5CwhVmSMqubcXKTZFQFDOSFZ1EkhjhIeqTrrYchUS66fTCDJ5q0oNBJPTjCk7p74kUhVKOQl93hkgN5GJtAv+rdRMVVNyU8jhRhOPZoiBhUEVwEhfsUUGwYiNtEBZU/xXiAdJJKR1qUYdgL568bFqXZdsq2w9XpWolj6MAjsEJOAM2uAZVcA/qoAkweAav4B18GC/Gm/FpjGetK0Y+cwT+yPj+Ad4XorA=</latexit><latexit sha1_base64="vSJxpowyQ2pUPt4v2/NCo2LUsao=">AAACIXicbZDLSsNAFIYnXmu9RV26GSyCIJZEBLsRKm5cdFGLvUATwmQ6aYdOJmFmIpSQV3Hjq7hxoUh34ss4bQPV1h8Gfr5zDmfO78eMSmVZX8bK6tr6xmZhq7i9s7u3bx4ctmSUCEyaOGKR6PhIEkY5aSqqGOnEgqDQZ6TtD+8m9fYTEZJG/FGNYuKGqM9pQDFSGnlm5dZLa40M3kAnEAinjqT9EHkNeAFzW8vm8HwOPbNkla2p4LKxc1MCueqeOXZ6EU5CwhVmSMqubcXKTZFQFDOSFZ1EkhjhIeqTrrYchUS66fTCDJ5q0oNBJPTjCk7p74kUhVKOQl93hkgN5GJtAv+rdRMVVNyU8jhRhOPZoiBhUEVwEhfsUUGwYiNtEBZU/xXiAdJJKR1qUYdgL568bFqXZdsq2w9XpWolj6MAjsEJOAM2uAZVcA/qoAkweAav4B18GC/Gm/FpjGetK0Y+cwT+yPj+Ad4XorA=</latexit><latexit sha1_base64="vSJxpowyQ2pUPt4v2/NCo2LUsao=">AAACIXicbZDLSsNAFIYnXmu9RV26GSyCIJZEBLsRKm5cdFGLvUATwmQ6aYdOJmFmIpSQV3Hjq7hxoUh34ss4bQPV1h8Gfr5zDmfO78eMSmVZX8bK6tr6xmZhq7i9s7u3bx4ctmSUCEyaOGKR6PhIEkY5aSqqGOnEgqDQZ6TtD+8m9fYTEZJG/FGNYuKGqM9pQDFSGnlm5dZLa40M3kAnEAinjqT9EHkNeAFzW8vm8HwOPbNkla2p4LKxc1MCueqeOXZ6EU5CwhVmSMqubcXKTZFQFDOSFZ1EkhjhIeqTrrYchUS66fTCDJ5q0oNBJPTjCk7p74kUhVKOQl93hkgN5GJtAv+rdRMVVNyU8jhRhOPZoiBhUEVwEhfsUUGwYiNtEBZU/xXiAdJJKR1qUYdgL568bFqXZdsq2w9XpWolj6MAjsEJOAM2uAZVcA/qoAkweAav4B18GC/Gm/FpjGetK0Y+cwT+yPj+Ad4XorA=</latexit>

Difference in measured rates for incident electrons of different polarization normalized by total rate

Definition:

Q2 ⇠ (0.05 GeV)2<latexit sha1_base64="op26xGfC/S44qfra+jrXTFLy4Vo=">AAACBXicbVDJSgNBEO1xjXGLetRDYxDiJcwExRwDHvSYgFkgGz2dStKkZ6G7RgxDLl78FS8eFPHqP3jzb+wkc9DEBwWP96qoqueGUmi07W9rZXVtfWMztZXe3tnd288cHNZ0ECkOVR7IQDVcpkEKH6ooUEIjVMA8V0LdHV1P/fo9KC0C/w7HIbQ9NvBFX3CGRupmTiqdAm1p4dGcnbcvWwgPGNMbqE3OO4VuJmvEGegycRKSJQnK3cxXqxfwyAMfuWRaNx07xHbMFAouYZJuRRpCxkdsAE1DfeaBbsezLyb0zCg92g+UKR/pTP09ETNP67Hnmk6P4VAvelPxP68ZYb/YjoUfRgg+ny/qR5JiQKeR0J5QwFGODWFcCXMr5UOmGEcTXNqE4Cy+vExqhbxj553KRbZUTOJIkWNySnLEIVekRG5JmVQJJ4/kmbySN+vJerHerY9564qVzByRP7A+fwAZ0pZe</latexit><latexit sha1_base64="op26xGfC/S44qfra+jrXTFLy4Vo=">AAACBXicbVDJSgNBEO1xjXGLetRDYxDiJcwExRwDHvSYgFkgGz2dStKkZ6G7RgxDLl78FS8eFPHqP3jzb+wkc9DEBwWP96qoqueGUmi07W9rZXVtfWMztZXe3tnd288cHNZ0ECkOVR7IQDVcpkEKH6ooUEIjVMA8V0LdHV1P/fo9KC0C/w7HIbQ9NvBFX3CGRupmTiqdAm1p4dGcnbcvWwgPGNMbqE3OO4VuJmvEGegycRKSJQnK3cxXqxfwyAMfuWRaNx07xHbMFAouYZJuRRpCxkdsAE1DfeaBbsezLyb0zCg92g+UKR/pTP09ETNP67Hnmk6P4VAvelPxP68ZYb/YjoUfRgg+ny/qR5JiQKeR0J5QwFGODWFcCXMr5UOmGEcTXNqE4Cy+vExqhbxj553KRbZUTOJIkWNySnLEIVekRG5JmVQJJ4/kmbySN+vJerHerY9564qVzByRP7A+fwAZ0pZe</latexit><latexit sha1_base64="op26xGfC/S44qfra+jrXTFLy4Vo=">AAACBXicbVDJSgNBEO1xjXGLetRDYxDiJcwExRwDHvSYgFkgGz2dStKkZ6G7RgxDLl78FS8eFPHqP3jzb+wkc9DEBwWP96qoqueGUmi07W9rZXVtfWMztZXe3tnd288cHNZ0ECkOVR7IQDVcpkEKH6ooUEIjVMA8V0LdHV1P/fo9KC0C/w7HIbQ9NvBFX3CGRupmTiqdAm1p4dGcnbcvWwgPGNMbqE3OO4VuJmvEGegycRKSJQnK3cxXqxfwyAMfuWRaNx07xHbMFAouYZJuRRpCxkdsAE1DfeaBbsezLyb0zCg92g+UKR/pTP09ETNP67Hnmk6P4VAvelPxP68ZYb/YjoUfRgg+ny/qR5JiQKeR0J5QwFGODWFcCXMr5UOmGEcTXNqE4Cy+vExqhbxj553KRbZUTOJIkWNySnLEIVekRG5JmVQJJ4/kmbySN+vJerHerY9564qVzByRP7A+fwAZ0pZe</latexit><latexit sha1_base64="op26xGfC/S44qfra+jrXTFLy4Vo=">AAACBXicbVDJSgNBEO1xjXGLetRDYxDiJcwExRwDHvSYgFkgGz2dStKkZ6G7RgxDLl78FS8eFPHqP3jzb+wkc9DEBwWP96qoqueGUmi07W9rZXVtfWMztZXe3tnd288cHNZ0ECkOVR7IQDVcpkEKH6ooUEIjVMA8V0LdHV1P/fo9KC0C/w7HIbQ9NvBFX3CGRupmTiqdAm1p4dGcnbcvWwgPGNMbqE3OO4VuJmvEGegycRKSJQnK3cxXqxfwyAMfuWRaNx07xHbMFAouYZJuRRpCxkdsAE1DfeaBbsezLyb0zCg92g+UKR/pTP09ETNP67Hnmk6P4VAvelPxP68ZYb/YjoUfRgg+ny/qR5JiQKeR0J5QwFGODWFcCXMr5UOmGEcTXNqE4Cy+vExqhbxj553KRbZUTOJIkWNySnLEIVekRG5JmVQJJ4/kmbySN+vJerHerY9564qVzByRP7A+fwAZ0pZe</latexit>

MOLLER Experiment

Page 3: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!3

Motivation

K.K.

Deviation from Standard Model prediction signals new physics --

�LBSM =c2±,±2⇤2

(e�µP±e)(e�µP±e)

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�ALR

ALR⇠

c2±±GF⇤2

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Roughly ⇤ ⇠ O(10 TeV)<latexit sha1_base64="y9G3P//pNllvyV9WEtq/mFNfydk=">AAACEHicbVC7SgNBFJ2NrxhfUUubwSDGJuyKYMqAjYVghLwgG8Ls5CYZMvtg5q4YlnyCjb9iY6GIraWdf+NskkITDwwczjmXufd4kRQabfvbyqysrq1vZDdzW9s7u3v5/YOGDmPFoc5DGaqWxzRIEUAdBUpoRQqY70loeqOr1G/eg9IiDGo4jqDjs0Eg+oIzNFI3f+remHCPUVcLn7o+wyFnMrmdFB3bRXjAhNagMTnr5gt2yZ6CLhNnTgpkjmo3/+X2Qh77ECCXTOu2Y0fYSZhCwSVMcm6sIWJ8xAbQNjRgPuhOMj1oQk+M0qP9UJkXIJ2qvycS5ms99j2TTDfWi14q/ue1Y+yXO4kIohgh4LOP+rGkGNK0HdoTCjjKsSGMK2F2pXzIFONoOsyZEpzFk5dJ47zk2CXn7qJQKc/ryJIjckyKxCGXpEKuSZXUCSeP5Jm8kjfryXqx3q2PWTRjzWcOyR9Ynz9h35wg</latexit><latexit sha1_base64="y9G3P//pNllvyV9WEtq/mFNfydk=">AAACEHicbVC7SgNBFJ2NrxhfUUubwSDGJuyKYMqAjYVghLwgG8Ls5CYZMvtg5q4YlnyCjb9iY6GIraWdf+NskkITDwwczjmXufd4kRQabfvbyqysrq1vZDdzW9s7u3v5/YOGDmPFoc5DGaqWxzRIEUAdBUpoRQqY70loeqOr1G/eg9IiDGo4jqDjs0Eg+oIzNFI3f+remHCPUVcLn7o+wyFnMrmdFB3bRXjAhNagMTnr5gt2yZ6CLhNnTgpkjmo3/+X2Qh77ECCXTOu2Y0fYSZhCwSVMcm6sIWJ8xAbQNjRgPuhOMj1oQk+M0qP9UJkXIJ2qvycS5ms99j2TTDfWi14q/ue1Y+yXO4kIohgh4LOP+rGkGNK0HdoTCjjKsSGMK2F2pXzIFONoOsyZEpzFk5dJ47zk2CXn7qJQKc/ryJIjckyKxCGXpEKuSZXUCSeP5Jm8kjfryXqx3q2PWTRjzWcOyR9Ynz9h35wg</latexit><latexit sha1_base64="y9G3P//pNllvyV9WEtq/mFNfydk=">AAACEHicbVC7SgNBFJ2NrxhfUUubwSDGJuyKYMqAjYVghLwgG8Ls5CYZMvtg5q4YlnyCjb9iY6GIraWdf+NskkITDwwczjmXufd4kRQabfvbyqysrq1vZDdzW9s7u3v5/YOGDmPFoc5DGaqWxzRIEUAdBUpoRQqY70loeqOr1G/eg9IiDGo4jqDjs0Eg+oIzNFI3f+remHCPUVcLn7o+wyFnMrmdFB3bRXjAhNagMTnr5gt2yZ6CLhNnTgpkjmo3/+X2Qh77ECCXTOu2Y0fYSZhCwSVMcm6sIWJ8xAbQNjRgPuhOMj1oQk+M0qP9UJkXIJ2qvycS5ms99j2TTDfWi14q/ue1Y+yXO4kIohgh4LOP+rGkGNK0HdoTCjjKsSGMK2F2pXzIFONoOsyZEpzFk5dJ47zk2CXn7qJQKc/ryJIjckyKxCGXpEKuSZXUCSeP5Jm8kjfryXqx3q2PWTRjzWcOyR9Ynz9h35wg</latexit><latexit sha1_base64="y9G3P//pNllvyV9WEtq/mFNfydk=">AAACEHicbVC7SgNBFJ2NrxhfUUubwSDGJuyKYMqAjYVghLwgG8Ls5CYZMvtg5q4YlnyCjb9iY6GIraWdf+NskkITDwwczjmXufd4kRQabfvbyqysrq1vZDdzW9s7u3v5/YOGDmPFoc5DGaqWxzRIEUAdBUpoRQqY70loeqOr1G/eg9IiDGo4jqDjs0Eg+oIzNFI3f+remHCPUVcLn7o+wyFnMrmdFB3bRXjAhNagMTnr5gt2yZ6CLhNnTgpkjmo3/+X2Qh77ECCXTOu2Y0fYSZhCwSVMcm6sIWJ8xAbQNjRgPuhOMj1oQk+M0qP9UJkXIJ2qvycS5ms99j2TTDfWi14q/ue1Y+yXO4kIohgh4LOP+rGkGNK0HdoTCjjKsSGMK2F2pXzIFONoOsyZEpzFk5dJ47zk2CXn7qJQKc/ryJIjckyKxCGXpEKuSZXUCSeP5Jm8kjfryXqx3q2PWTRjzWcOyR9Ynz9h35wg</latexit>

(naive mass reach)

1. 2. To date, Most precise measurement of sin2(θW) is at Z pole (LEP/SLC)

MOLLER will provide determination with similar accuracy

MOLLER 2.4%

δ ( sin2(θW) ) = 0.00036

Erler, Ramsey-Musolf (2005)Erler, Ferro-Hernandez (2017)

Page 4: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!4

Theoretical Input

Tree-level:

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In order to extract sin2(θW) from ALR with high accuracy, need similarly accurate SM prediction. Long history...

Derman, Marciano (1979)

tree-level electron weak chargeQe

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ALR =GµQ2f(Ebeam)p

2⇡↵(1� 4 sin2 ✓W )| {z }

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Because sin2(θW) ~ 0.23, tree-level relation 'accidentally' small

One-loop:Czarnecki, Marciano (1996) Denner, Pozzorini (1998) Aleksejevs et. al. (2010), ...

virtual corrections

Sizable shift due to corrections to Zγ propagator

Interpreted as 'running' of sin2(θW)

QeW �! 1� 4(0) sin2 ✓W

<latexit sha1_base64="OMaB54QSrFyYKGwoCs2lK++sHDA=">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</latexit><latexit sha1_base64="OMaB54QSrFyYKGwoCs2lK++sHDA=">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</latexit><latexit sha1_base64="OMaB54QSrFyYKGwoCs2lK++sHDA=">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</latexit><latexit sha1_base64="OMaB54QSrFyYKGwoCs2lK++sHDA=">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</latexit>

~1.03=> 40% reduction in ALR

MOLLER 2.4%

c.f.

Page 5: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!5

Are radiative corrections stable?Main question essentially keeping the MOLLER experiment from getting the green light.

Rough answer: largest shift to ALR comes from shift in sin2(θW). So, size of higher order corrections to sin2(θW) should provide estimate of higher order corrections to ALR.

But to avoid any surprises, a full two-loop evaluation of the asymmetry must be carried out.

Gong to two-loop:

Aleksejevs et. al. (2011, 2012, 2015)

Yong Du, Ayres Freitas, Hiren Patel, Michael Ramsey-Musolf (2019 TBA)

Currently only two groups working on evaluation

+ Jia Zhou (2020 TBA)

Page 6: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!6

Method

Three challenges in doing calculation:

1. Organization

2. Bookkeeping of O(103) diagrams,

3. Evaluation of two-loop integrals.

Page 7: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!7

1. OrganizationIn addition to powers of GF and e2, we organize the evaluation by counting number of closed loops in the two loop diagrams.

ALR =GF

↵(tree)

⇥1 + ↵(NF a

(1)1 + a(1)0 ) + ↵2(N2

F a(2)2 +NFa

(2)1 + a(2)0 )

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Schematically,one-loop two-loop

-0.39 +0.04

Diagrams drawn by Aleksejevs

One loop Two loop

Reasons: - Diagrams with closed fermion loops are expected to be 'bigger' - Diagrams with fewer closed fermion loops are harder to calculate

1.00we calculated these first

MOLLER 2.4%

c.f.

Two fermion

loops

One fermion loopZero fermion loop (bosonic)

One fermion loop

Zero fermion loop (bosonic)

Page 8: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!8

2. Bookkeeping

Everything done using computer (Mathematica)

Generate diagrams and amplitudes using FeynArts

Two independent codes to proceed with calculation - In house codes developed by A. Freitas - Modified versions of Package-X

Regularization: Naive Dim. Reg. for UV divergences Photon/Electron mass for IR divergences

T. Hahn (2000)

Hiren Patel (2016)

Page 9: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!9

3. Evaluation of Feynman Diagrams1. Take advantage of separation of scales in problem

m2e⌧ s,Q2 ⌧ M2

W,M2

Z,M2

H,M2

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Construct an analytic expansion in large heavy masses (expansion by regions)

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2. Take advantage of closed fermion loops

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Represent fermion sub-loop integral as a dispersion relation

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Hiren Patel SCIPP

Amplitudes and cross section contain soft (photon mass) and collinear (electron mass) singularities

However, soft/collinear factorization ensures that these singularities cancel out in the asymmetry

We are able to confirm exact analytical cancellation of IR singularities, providing a crosscheck on our calculations

!10

Infrared divergences

ALR =�R � �L

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Yennie, Frautschi, Suura (1961) Kinoshita (1962)

Page 11: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!11

Renormalization

Hybrid renormalization scheme:

On shell renormalization of couplings and masses (subtraction point at Z pole),

MS-bar renormalization for sin2(θW).

Page 12: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!12

Preliminary Results

ALR =GF

↵(tree)

⇥1 + ↵(NF a

(1)1 + a(1)0 ) + ↵2(N2

F a(2)2 +NFa

(2)1 + a(2)0 )

⇤<latexit sha1_base64="tuuG+tZYqMrUQgiPD018kVQ7ijM=">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</latexit><latexit sha1_base64="tuuG+tZYqMrUQgiPD018kVQ7ijM=">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</latexit><latexit sha1_base64="tuuG+tZYqMrUQgiPD018kVQ7ijM=">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</latexit><latexit sha1_base64="tuuG+tZYqMrUQgiPD018kVQ7ijM=">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</latexit>

one-loop two-loop

−0.39 +0.04 −0.03 +0.021.00ALR MOLLER

2.4%

c.f.

Individually, NF2 and NF pieces at level of accuracy of MOLLER experiment. But sum largely cancels, and is below sensitivity (good news).

We don't fully understand the cancellation yet (to be analyzed).

These results are preliminary, and we are still performing crosschecks. Paper with results to come out soon (this year).

Next: Complete evaluation of purely bosonic contribution

??(for one representative kinematic point)

Page 13: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!13

Backup

Page 14: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!14

Expansion by RegionsSmirnov (1999)

Big......small=>drop

1

�M2Z

Z(dk)

kµk⌫

[(k + p1)2 +m2e][(k + p2)2 �m2

e][(k + p3)2 �m2� ]

<latexit sha1_base64="kbDJWPPq42OUQ33jaSCeyBKLk8g=">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</latexit><latexit sha1_base64="kbDJWPPq42OUQ33jaSCeyBKLk8g=">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</latexit><latexit sha1_base64="kbDJWPPq42OUQ33jaSCeyBKLk8g=">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</latexit><latexit sha1_base64="kbDJWPPq42OUQ33jaSCeyBKLk8g=">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</latexit>

I =

Z(dk)

kµk⌫

[k2 �M2Z ][(k + p1)2 +m2

e][(k + p2)2 �m2e][(k + p3)2 �m2

� ]<latexit sha1_base64="VsS+rXBOlE2nWP0Hgu5kYpP6oxg=">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</latexit><latexit sha1_base64="VsS+rXBOlE2nWP0Hgu5kYpP6oxg=">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</latexit><latexit sha1_base64="VsS+rXBOlE2nWP0Hgu5kYpP6oxg=">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</latexit><latexit sha1_base64="VsS+rXBOlE2nWP0Hgu5kYpP6oxg=">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</latexit>

Approximation leads to UV divergent integral

�1

M2Z

⇣1✏+ ln

�µ2

�t

�+ . . .

<latexit sha1_base64="//024T8olSMWwwUbQtnNb6sjRAI=">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</latexit><latexit sha1_base64="//024T8olSMWwwUbQtnNb6sjRAI=">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</latexit><latexit sha1_base64="//024T8olSMWwwUbQtnNb6sjRAI=">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</latexit><latexit sha1_base64="//024T8olSMWwwUbQtnNb6sjRAI=">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</latexit>

~

~

However, loop momentum may also be large, and may also give a contribution at same order in 1/MZ.

Page 15: Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel hpatel6@ucsc.edu Santa Cruz Institute for Particle Physics S C I P P Sept 16, 2019 Two Loop Electroweak Corrections

Hiren Patel SCIPP!15

Expansion by Regions

I =

Z(dk)

kµk⌫

[k2 �M2Z ][(k + p1)2 +m2

e][(k + p2)2 �m2e][(k + p3)2 �m2

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Big Big Big

Small

Z(dk)

kµk⌫

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2][k2][k2]<latexit sha1_base64="rADmc2VP5nk552jI1zZomHaGjg8=">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</latexit><latexit sha1_base64="rADmc2VP5nk552jI1zZomHaGjg8=">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</latexit><latexit sha1_base64="rADmc2VP5nk552jI1zZomHaGjg8=">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</latexit><latexit sha1_base64="rADmc2VP5nk552jI1zZomHaGjg8=">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</latexit>

Approximation leads to IR divergent integral

~

Smirnov (1999)

1

M2Z

⇣1✏+ ln

� µ2

M2Z

�+ . . .

<latexit sha1_base64="+9I/jMBxjVO4gm8fDkvc0ViTaNg=">AAACQHicbVBLSwMxGMz6rPW16tFLsAgtQtktgj0WvXgRKtgHdmvJptk2NJssSVYoy/40L/4Eb569eFDEqyfTh0VbBwLDzHxfkvEjRpV2nGdraXlldW09s5Hd3Nre2bX39utKxBKTGhZMyKaPFGGUk5qmmpFmJAkKfUYa/uBi5DfuiVRU8Bs9jEg7RD1OA4qRNlLHbniBRDhx0+Sqc3tXSr1z2svPNI9EijLBU3gCPcY9f2Z6YWzSP0NGL4wjXaHVaEWhY+ecojMGXCTulOTAFNWO/eR1BY5DwjVmSKmW60S6nSCpKWYkzXqxIhHCA9QjLUM5ColqJ+MCUnhslC4MhDSHazhWf08kKFRqGPomGSLdV/PeSPzPa8U6KLcTyqNYE44nFwUxg1rAUZuwSyXBmg0NQVhS81aI+8gUpE3nWVOCO//lRVIvFV2n6F6f5irlaR0ZcAiOQB644AxUwCWoghrA4AG8gDfwbj1ar9aH9TmJLlnTmQPwB9bXN3e+r5k=</latexit><latexit sha1_base64="+9I/jMBxjVO4gm8fDkvc0ViTaNg=">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</latexit><latexit sha1_base64="+9I/jMBxjVO4gm8fDkvc0ViTaNg=">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</latexit><latexit sha1_base64="+9I/jMBxjVO4gm8fDkvc0ViTaNg=">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</latexit>

~

But, after adding both approximations together

1

M2Z

⇣1✏+ ln

� µ2

M2Z

�+ . . .

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�1

M2Z

⇣1✏+ ln

�µ2

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+

(small k, previous slide)

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(large k, this slide)

cancel

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Small Small

obtain correct asymptotic approximation