Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel [email protected] Santa Cruz...
Transcript of Two Loop Electroweak Corrections to Polarized Moller …...Hiren Patel [email protected] Santa Cruz...
![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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/1.jpg)
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
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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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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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/3.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/4.jpg)
Hiren Patel SCIPP!4
Theoretical Input
Tree-level:
ALR = f(sin(✓W ), . . .)<latexit sha1_base64="ZKHJQqNd0hYMymuva4ES3FgITYY=">AAACDHicbVDLSgMxFM34rPVVdekmWIQWpMyIYDdCxY0LF1XsAzpDyaSZNjSTGZI7Qhn6AW78FTcuFHHrB7jzb0zbWWjrgcDhnHO5ucePBddg29/W0vLK6tp6biO/ubW9s1vY22/qKFGUNWgkItX2iWaCS9YADoK1Y8VI6AvW8odXE7/1wJTmkbyHUcy8kPQlDzglYKRuoXjZTW/uxvgCByVXc1lyYcCAdFvlE+yKXgQal03KrthT4EXiZKSIMtS7hS+3F9EkZBKoIFp3HDsGLyUKOBVsnHcTzWJCh6TPOoZKEjLtpdNjxvjYKD0cRMo8CXiq/p5ISaj1KPRNMiQw0PPeRPzP6yQQVL2UyzgBJulsUZAIDBGeNIN7XDEKYmQIoYqbv2I6IIpQMP3lTQnO/MmLpHlaceyKc3tWrFWzOnLoEB2hEnLQOaqha1RHDUTRI3pGr+jNerJerHfrYxZdsrKZA/QH1ucPt5qZdg==</latexit><latexit sha1_base64="ZKHJQqNd0hYMymuva4ES3FgITYY=">AAACDHicbVDLSgMxFM34rPVVdekmWIQWpMyIYDdCxY0LF1XsAzpDyaSZNjSTGZI7Qhn6AW78FTcuFHHrB7jzb0zbWWjrgcDhnHO5ucePBddg29/W0vLK6tp6biO/ubW9s1vY22/qKFGUNWgkItX2iWaCS9YADoK1Y8VI6AvW8odXE7/1wJTmkbyHUcy8kPQlDzglYKRuoXjZTW/uxvgCByVXc1lyYcCAdFvlE+yKXgQal03KrthT4EXiZKSIMtS7hS+3F9EkZBKoIFp3HDsGLyUKOBVsnHcTzWJCh6TPOoZKEjLtpdNjxvjYKD0cRMo8CXiq/p5ISaj1KPRNMiQw0PPeRPzP6yQQVL2UyzgBJulsUZAIDBGeNIN7XDEKYmQIoYqbv2I6IIpQMP3lTQnO/MmLpHlaceyKc3tWrFWzOnLoEB2hEnLQOaqha1RHDUTRI3pGr+jNerJerHfrYxZdsrKZA/QH1ucPt5qZdg==</latexit><latexit sha1_base64="ZKHJQqNd0hYMymuva4ES3FgITYY=">AAACDHicbVDLSgMxFM34rPVVdekmWIQWpMyIYDdCxY0LF1XsAzpDyaSZNjSTGZI7Qhn6AW78FTcuFHHrB7jzb0zbWWjrgcDhnHO5ucePBddg29/W0vLK6tp6biO/ubW9s1vY22/qKFGUNWgkItX2iWaCS9YADoK1Y8VI6AvW8odXE7/1wJTmkbyHUcy8kPQlDzglYKRuoXjZTW/uxvgCByVXc1lyYcCAdFvlE+yKXgQal03KrthT4EXiZKSIMtS7hS+3F9EkZBKoIFp3HDsGLyUKOBVsnHcTzWJCh6TPOoZKEjLtpdNjxvjYKD0cRMo8CXiq/p5ISaj1KPRNMiQw0PPeRPzP6yQQVL2UyzgBJulsUZAIDBGeNIN7XDEKYmQIoYqbv2I6IIpQMP3lTQnO/MmLpHlaceyKc3tWrFWzOnLoEB2hEnLQOaqha1RHDUTRI3pGr+jNerJerHfrYxZdsrKZA/QH1ucPt5qZdg==</latexit><latexit sha1_base64="ZKHJQqNd0hYMymuva4ES3FgITYY=">AAACDHicbVDLSgMxFM34rPVVdekmWIQWpMyIYDdCxY0LF1XsAzpDyaSZNjSTGZI7Qhn6AW78FTcuFHHrB7jzb0zbWWjrgcDhnHO5ucePBddg29/W0vLK6tp6biO/ubW9s1vY22/qKFGUNWgkItX2iWaCS9YADoK1Y8VI6AvW8odXE7/1wJTmkbyHUcy8kPQlDzglYKRuoXjZTW/uxvgCByVXc1lyYcCAdFvlE+yKXgQal03KrthT4EXiZKSIMtS7hS+3F9EkZBKoIFp3HDsGLyUKOBVsnHcTzWJCh6TPOoZKEjLtpdNjxvjYKD0cRMo8CXiq/p5ISaj1KPRNMiQw0PPeRPzP6yQQVL2UyzgBJulsUZAIDBGeNIN7XDEKYmQIoYqbv2I6IIpQMP3lTQnO/MmLpHlaceyKc3tWrFWzOnLoEB2hEnLQOaqha1RHDUTRI3pGr+jNerJerHfrYxZdsrKZA/QH1ucPt5qZdg==</latexit>
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
W<latexit sha1_base64="+xDLEaOBAU4TOoQCRRt1wwjRId4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48tmFZoY9lsJ+3SzSbsboQS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fiko5NMMfRZIhL1EFKNgkv0DTcCH1KFNA4FdsPJ7dzvPqHSPJH3ZppiENOR5BFn1FjJbw+6jzio1ty6uwBZJ15BalCgNah+9YcJy2KUhgmqdc9zUxPkVBnOBM4q/UxjStmEjrBnqaQx6iBfHDsjF1YZkihRtqQhC/X3RE5jradxaDtjasZ61ZuL/3m9zESNIOcyzQxKtlwUZYKYhMw/J0OukBkxtYQyxe2thI2poszYfCo2BG/15XXSuap7bt1rX9eajSKOMpzBOVyCBzfQhDtogQ8MODzDK7w50nlx3p2PZWvJKWZO4Q+czx96tY5s</latexit><latexit sha1_base64="+xDLEaOBAU4TOoQCRRt1wwjRId4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48tmFZoY9lsJ+3SzSbsboQS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fiko5NMMfRZIhL1EFKNgkv0DTcCH1KFNA4FdsPJ7dzvPqHSPJH3ZppiENOR5BFn1FjJbw+6jzio1ty6uwBZJ15BalCgNah+9YcJy2KUhgmqdc9zUxPkVBnOBM4q/UxjStmEjrBnqaQx6iBfHDsjF1YZkihRtqQhC/X3RE5jradxaDtjasZ61ZuL/3m9zESNIOcyzQxKtlwUZYKYhMw/J0OukBkxtYQyxe2thI2poszYfCo2BG/15XXSuap7bt1rX9eajSKOMpzBOVyCBzfQhDtogQ8MODzDK7w50nlx3p2PZWvJKWZO4Q+czx96tY5s</latexit><latexit sha1_base64="+xDLEaOBAU4TOoQCRRt1wwjRId4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48tmFZoY9lsJ+3SzSbsboQS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fiko5NMMfRZIhL1EFKNgkv0DTcCH1KFNA4FdsPJ7dzvPqHSPJH3ZppiENOR5BFn1FjJbw+6jzio1ty6uwBZJ15BalCgNah+9YcJy2KUhgmqdc9zUxPkVBnOBM4q/UxjStmEjrBnqaQx6iBfHDsjF1YZkihRtqQhC/X3RE5jradxaDtjasZ61ZuL/3m9zESNIOcyzQxKtlwUZYKYhMw/J0OukBkxtYQyxe2thI2poszYfCo2BG/15XXSuap7bt1rX9eajSKOMpzBOVyCBzfQhDtogQ8MODzDK7w50nlx3p2PZWvJKWZO4Q+czx96tY5s</latexit><latexit sha1_base64="+xDLEaOBAU4TOoQCRRt1wwjRId4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48tmFZoY9lsJ+3SzSbsboQS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fiko5NMMfRZIhL1EFKNgkv0DTcCH1KFNA4FdsPJ7dzvPqHSPJH3ZppiENOR5BFn1FjJbw+6jzio1ty6uwBZJ15BalCgNah+9YcJy2KUhgmqdc9zUxPkVBnOBM4q/UxjStmEjrBnqaQx6iBfHDsjF1YZkihRtqQhC/X3RE5jradxaDtjasZ61ZuL/3m9zESNIOcyzQxKtlwUZYKYhMw/J0OukBkxtYQyxe2thI2poszYfCo2BG/15XXSuap7bt1rX9eajSKOMpzBOVyCBzfQhDtogQ8MODzDK7w50nlx3p2PZWvJKWZO4Q+czx96tY5s</latexit>
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=">AAACJ3icbVBNaxsxENXmq6nz5abHXkRMITnE7BpDciqBXHJMoI4DXmeZlce2sFZapNkUs/jf9NK/kksgKaU95p9EdnxI7D4YeLw3I828NFfSURj+C1ZW19Y3Pmx+rGxt7+zuVT/tXztTWIEtYZSxNyk4VFJjiyQpvMktQpYqbKej86nfvkPrpNHfaZxjN4OBln0pgLyUVL9dJe1b5LEyemDlYEhgrfnBo+NmGc9eLy32JvEI8hwOw6MJj53Ut42YhkiQtJNqLayHM/BlEs1Jjc1xmVQf454RRYaahALnOlGYU7cES1IonFTiwmEOYgQD7HiqIUPXLWebTPhXr/R431hfmvhMfTtRQubcOEt9ZwY0dIveVPyf1ymof9otpc4LQi1eP+oXipPh09B4T1oUpMaegLDS78rFECwI8tFWfAjR4snL5LpRj8J6dNWsnZ3O49hkX9gBO2QRO2Fn7IJdshYT7Ce7Z0/sd/AreAj+BH9fW1eC+cxn9g7B8wsgM6YM</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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/5.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/6.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/7.jpg)
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 )
⇤<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>
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)
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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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/9.jpg)
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)
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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).
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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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/13.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/14.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022071408/61010c03dd2cee1ad0492a10/html5/thumbnails/15.jpg)
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⌫
[k2 �M2Z ][k
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=">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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><latexit sha1_base64="+9I/jMBxjVO4gm8fDkvc0ViTaNg=">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</latexit>
~
But, after adding both approximations together
1
M2Z
⇣1✏+ ln
� µ2
M2Z
�+ . . .
⌘
<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><latexit sha1_base64="+9I/jMBxjVO4gm8fDkvc0ViTaNg=">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</latexit>
�1
M2Z
⇣1✏+ ln
�µ2
�t
�+ . . .
⌘
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+
(small k, previous slide)
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(large k, this slide)
cancel
= � 1
MZ
⇣ln
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�+ . . .
⌘
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Small Small
obtain correct asymptotic approximation