Limits on Electroweak
Instanton-induced Processes
Kazuki Sakurai(University of Warsaw)
In collaboration withAndreas Ringwald, Bryan WebberarXiv:1809.10833 [hep-ph]

Beyond BSM conference 2018/10/02 @ Ikaho• In pure SU(2) YM theory, there are infinitely many gauge equivalent vacua.

Aaµ T a = 0

gauge trans.
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They are as many as.
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U ∂µ Ug• In pure SU(2) YM theory, there are infinitely many gauge equivalent vacua.
gauge trans.

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sha1_base64="py4hyT4yOTmakPPlv+6cMh1ppe0=">AAACDnicZVDLSsNAFJ34rPUVdelmsBZclcSN7iy6EdxUMG2hqWEymaRDJw9mJkIJ+QFXbv0L3QiK4ta9K//CHxCctEVse2DgcM59zD1uwqiQhvGlzc0vLC4tl1bKq2vrG5v61nZTxCnHxMIxi3nbRYIwGhFLUslIO+EEhS4jLbd/VvitG8IFjaMrOUhIN0RBRH2KkVSSo+/bPkc4o3kW5NCCdoK4pIg5dphC69r2UBAQ7ugVo2YMAWeJOSaV+snj68/F933D0T9tL8ZpSCKJGRKiYxqJ7GbFbMxIXrZTQRKE+yggHUUjFBLRzYbX5LCqFA/6MVcvknCo/u/IUCjEIHRVZYhkT0x7hfjnVSdWSf+4m9EoSSWJ8GiTnzIoY1hkAz3KCZZsoAjCnKrPQtxDKh+pEiyrFMzpm2dJ87BmGjXzUsVxCkYogV2wBw6ACY5AHZyDBrAABrfgATyDF+1Oe9LetPdR6Zw27tkBE9A+fgGHMaDD</latexit>
sha1_base64="Ebh5sb20W4LCLV0nAf2Lp2zo2GE=">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</latexit>i†
U ∂µ Ug• Some of those vacuum configurations are not continuously connected.
topological gauge

Uij (x)<latexit sha1_base64="J+tH0g69F9MUoe4YnvahTWwgtPI=">AAAB/3icZVBNS8NAEN34WetXVDx5CZaCp5J4UI9FLx4r2A9oQ9lsJ+nS3U3Y3QglVPCneBIUxKu/w5P/xm2aiq0PBh7vzTAzL0gYVdp1v62V1bX1jc3SVnl7Z3dv3z44bKk4lQSaJGax7ARYAaMCmppqBp1EAuYBg3Ywupn67QeQisbiXo8T8DmOBA0pwdpIffu4J7mj48Q0RkZkjxFOI+jbFbfm5nD+E68gFVSg0be/eoOYpByEJgwr1fXcRPsZlpoSBpNyL1WQYDLCEXQNFZiD8rP8/IlTNcrACWNpSmgnV/9OZJgrNeaB6eRYD9WyNxV/verCKh1e+RkVSapBkNmmMGXmYWcahjOgEohmY0MwkdQc65AhlphoE1k5T+Fi/vwymafQOq95bs27cyv16yKPEjpBp+gMeegS1dEtaqAmIihDz+gVvVlP1ov1bn3MWlesYuYILcD6/AFM85Yy</latexit>
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U (∞) → 1<latexit sha1_base64="A2+2hE1r715HNWBv6j59lSwQg2U=">AAACFnicZVDLSgMxFM34rPU16tJNsBQqSJlxoS6LblxWcNpCW0omvdPGZh4kGbEM8xOu/BRXgoK4FVz5N6bTqdh6IHA4596c5LgRZ1JZ1rextLyyurZe2Chubm3v7Jp7+w0ZxoKCQ0MeipZLJHAWgKOY4tCKBBDf5dB0R1cTv3kPQrIwuFXjCLo+GQTMY5QoLfXMk6STXZII6KfY6SXsLk0reKa6PIYUd1wPP6T4uGeWrKqVAf8ndk5KKEe9Z351+iGNfQgU5UTKtm1FqpsQoRjlkBY7sYSI0BEZQFvTgPggu0kWneKyVvrYC4U+gcKZ+ncjIb6UY9/Vkz5RQ7noTcRfrzwXpbyLbsKCKFYQ0GmSF3OsQjzpCPeZAKr4WBNCBdOPxXRIBKFKN1nMWjibfX6RzFponFZtq2rfWKXaZd5HAR2iI1RBNjpHNXSN6shBFD2iZ/SK3own48V4Nz6mo0tGvnOA5mB8/gBfcJ9e</latexit>
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sha1_base64="+9aXW/aQK0aBBIm9ev3XTpRvn70=">AAACFnicZVDLSgMxFM34rPVVdekmWAsVpMy40WXRjcsK9gFtKZn0ThubeZBkxDLOJ7hxpf6GG1eCgrgSBFf+jem0im0PBA7n3JuTHDvgTCrT/DZmZufmFxZTS+nlldW19czGZkX6oaBQpj73Rc0mEjjzoKyY4lALBBDX5lC1eycDv3oJQjLfO1f9AJou6XjMYZQoLbUy+1EjuSQS0I5xuRWxizjO41/V5iHEuGE7+CrGe61M1iyYCfA0sUYkW9y9v/l4vH4otTJfjbZPQxc8RTmRsm6ZgWpGRChGOcTpRighILRHOlDX1CMuyGaURMc4p5U2dnyhj6dwov7fiIgrZd+19aRLVFdOegPxz8uNRSnnqBkxLwgVeHSY5IQcKx8POsJtJoAq3teEUMH0YzHtEkGo0k2mdQvW5J+nSeWgYJkF60zXcYyGSKFttIPyyEKHqIhOUQmVEUW36Am9oFfjzng23oz34eiMMdrZQmMwPn8A6AOjbw==</latexit>
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SU(2)<latexit sha1_base64="J10CVrOIEnAv53LwEdDPR6aoboQ=">AAAB/3icZVBNS8NAEJ3Ur1q/ouLJy2Ip1EtJPKjHohePFUxbaELZbDft0s0m7G6EEnrwp3gSFMSrv8OT/8btl9j6YODx3gwz88KUM6Ud59sqrK1vbG4Vt0s7u3v7B/bhUVMlmSTUIwlPZDvEinImqKeZ5rSdSorjkNNWOLyd+K1HKhVLxIMepTSIcV+wiBGsjdS1T7yqz0SkR+fI1wnK/TBC7rhrl52aMwX6T9w5KcMcja795fcSksVUaMKxUh3XSXWQY6kZ4XRc8jNFU0yGuE87hgocUxXk0/PHqGKUHooSaUpoNFX/TuQ4VmoUh6YzxnqgVr2J+OtVllbp6DrImUgzTQWZbYoyjsynkzBQj0lKNB8Zgolk5lhEBlhiok1kpWkKl4vnV8kiheZFzXVq7r1Trt/M8yjCKZxBFVy4gjrcQQM8IJDDM7zCm/VkvVjv1sestWDNZ45hCdbnDx+SlMs=</latexit>
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sha1_base64="UuBO8EVW8t/++59hJv9Au73pc5w=">AAAB/3icZVBNS8NAEJ3Ur1q/qqIXPSwthXopiRc9Fr14rGBboQlls920SzebsLsRQujBXyKeBAXRo7/Dk//G7Qdi2wcDj/dmmJnnx5wpbds/Vm5ldW19I79Z2Nre2d0r7h+0VJRIQpsk4pG897GinAna1Exzeh9LikOf07Y/vB777QcqFYvEnU5j6oW4L1jACNZG6haPmlWXiUCnZ8jVEcpcP0DOqFss2zV7ArRMnBkp10uQezr+PG10i99uLyJJSIUmHCvVcexYexmWmhFORwU3UTTGZIj7tGOowCFVXjY5f4QqRumhIJKmhEYT9f9EhkOl0tA3nSHWA7XojcU/rzK3SgeXXsZEnGgqyHRTkHBkPh2HgXpMUqJ5aggmkpljERlgiYk2kRVMCs7iz8ukdV5z7Jpza+K4ginycAIlqIIDF1CHG2hAEwhk8Ayv8GY9Wi/Wu/Uxbc1Zs5lDmIP19QunuJai</latexit>
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sha1_base64="TY+FXtqjzuHvE4Nu8/bgH1ZQNro=">AAACAXicZVDLSgMxFL1TX7W+qoIudBFaCnVTZrpQl0U3LivaB3SGkknTNph5kGSEMszKD3HhSlBQt36GK//GdNqKrQcCh3PuzUmOG3ImlWl+G5ml5ZXVtex6bmNza3snv7vXlEEkCG2QgAei7WJJOfNpQzHFaTsUFHsupy337nLst+6pkCzwb9UopI6HBz7rM4KVlrr5w9hOL4kF7SXIFh66aZSrJ0k3XzQrZgr0n1hTUqwVIPN48H5c7+a/7F5AIo/6inAsZccyQ+XEWChGOE1ydiRpiMkdHtCOpj72qHTiNDtBJa30UD8Q+vgKperfjRh7Uo48V096WA3lojcWf73SXJTqnzsx88NIUZ9MkvoRRypA4zpQjwlKFB9pgolg+rGIDLHAROnScmkLp7PPL5JZC81qxTIr1rWu4wImyMIRFKAMFpxBDa6gDg0gkMATvMCr8WA8G2/Gx2Q0Y0x39mEOxucP6b+X+Q==</latexit>

π3 (S ) = Z
3

SU(2) ≃ S
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3

map<latexit sha1_base64="3teZbPx2Qnz4MIcCXYzN6GO7KVc=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrQT0WvXisYD+gXUo2zbahSXabZIWy9Ed4EhTEq7/Hk//GdLsVWx8MPN6bYWZeEHOmjet+O4WNza3tneJuaW//4PCofHzS0lGiCG2SiEeqE2BNOZO0aZjhtBMrikXAaTsY38399hNVmkXy0Uxj6gs8lCxkBBsrtXtKIIHjfrni1twM6D/xclKBHI1++as3iEgiqDSEY627nhsbP8XKMMLprNRLNI0xGeMh7VoqsaDaT7NzZ6hqlQEKI2VLGpSpfydSLLSeisB2CmxGet2bi79edWWVCW/8lMk4MVSSxaYw4chEaP48GjBFieFTSzBRzB6LyAgrTIyNqJSlcLV8fp0sU2hd1jy35j24lfptnkcRzuAcLsCDa6jDPTSgCQTG8Ayv8OZMnBfn3flYtBacfOYUVuB8/gCLs4+P</latexit>
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sha1_base64="YZB7GzavOfoGcVQvCm2hBr3u+Cs=">AAAB73icZVBNSwMxEJ2tX7V+VT3qIVgKnsquB/VY9OKxgv2AdinZNNuGJtk1yQpl6Y/wJCiIV3+PJ/+N6W4rtj4YeLw3w8y8IOZMG9f9dgpr6xubW8Xt0s7u3v5B+fCopaNEEdokEY9UJ8CaciZp0zDDaSdWFIuA03Ywvp357SeqNIvkg5nE1Bd4KFnICDZWaveUQALH/XLFrbkZ0H/izUmlfgoZGv3yV28QkURQaQjHWnc9NzZ+ipVhhNNpqZdoGmMyxkPatVRiQbWfZudOUdUqAxRGypY0KFP/TqRYaD0Rge0U2Iz0qjcTf73q0ioTXvspk3FiqCT5pjDhyERo9jwaMEWJ4RNLMFHMHovICCtMjI2olKVwuXh+lSxSaF3UPLfm3ds4bvI4oAgncAbn4MEV1OEOGtAEAmN4hld4cx6dF+fd+chbC8585hiW4Hz+AAkSj+s=</latexit>

3

S ≃R<latexit sha1_base64="pm0iZ/cFcXBQProsLu0B+aK6mdM=">AAAB9HicZVDLSgMxFM34rPVVdekmWAquSlKhj13Rjcsq9gGdsWTSTBuaeZhkCmXob7gSFMStP+PKvzEzrWL1wIXDOfdy7z1uJLjSCH1aa+sbm1vbuZ387t7+wWHh6LijwlhS1qahCGXPJYoJHrC25lqwXiQZ8V3Buu7kKvW7UyYVD4M7PYuY45NRwD1OiTaSbSvusweY3M7vLwaFIiojhDDGMCW4VkWGNBr1Cq5DnFoGRbBEa1D4sIchjX0WaCqIUn2MIu0kRGpOBZvn7VixiNAJGbG+oQHxmXKS7OY5LBllCL1Qmgo0zNTfEwnxlZr5run0iR6rv14q/nillVXaqzsJD6JYs4AuNnmxgDqEaQJwyCWjWswMIVRycyykYyIJ1SanfJZCNXsV/iffKXQqZYzK+AYVm5fLPHLgFJyBc4BBDTTBNWiBNqAgAo/gGbxYU+vJerXeFq1r1nLmBKzAev8CeNKR1A==</latexit>
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sha1_base64="U9SKBTu7M4ZOHmcGFjBfcOkwngM=">AAAB/HicZVDLSgMxFL1TX7W+xrp0EywFV2VGF7osunFZ0T6gHUsmzbShmcyQZMQy1E9xJSiIW3/ElX9jOh3EtgcCh3PuzUmOH3OmtOP8WIW19Y3NreJ2aWd3b//APiy3VJRIQpsk4pHs+FhRzgRtaqY57cSS4tDntO2Pr2d++5FKxSJxrycx9UI8FCxgBGsj9e1y2ssuSX2e0Cm6ezif9u2KU3MyoFXi5qQCORp9+7s3iEgSUqEJx0p1XSfWXoqlZoTTaamXKBpjMsZD2jVU4JAqL81ip6hqlAEKImmO0ChT/2+kOFRqEvpmMsR6pJa9mfjnVReidHDppUzEiaaCzJOChCMdoVkTaMAkJZpPDMFEMvNYREZYYqJNXyXTgrv851XSOqu5Ts29dSr1q7yPIhzDCZyCCxdQhxtoQBMIPMELvMG79Wy9Wh/W53y0YOU7R7AA6+sXn9mUnQ==</latexit>3• In pure SU(2) YM theory, there are infinitely many gauge equivalent vacua.
gauge trans.

Aaµ T a = 0<latexit sha1_base64="HGQza32dhbg4pAFKiU/N5sYqa50=">AAAB+nicZVBNS8NAEN34WetX1KOXxVLwFBIP6rHoxWMF+wFtKZvtJF262YTdTaHE+ks8CQri1X/iyX/jNk3F1gcDj/dmmJnnJ5wp7brf1tr6xubWdmmnvLu3f3BoHx03VZxKCg0a81i2faKAMwENzTSHdiKBRD6Hlj+6nfmtMUjFYvGgJwn0IhIKFjBKtJH6tt2VEQ5JGsKTlkQop29XXMfNgf8TryAVVKDet7+6g5imEQhNOVGq47mJ7mVEakY5TMvdVEFC6IiE0DFUkAhUL8svn+KqUQY4iKUpoXGu/p3ISKTUJPJNZ0T0UK16M/HXqy6t0sF1L2MiSTUIOt8UpBzrGM9ywAMmgWo+MYRQycyxmA6JJFSbtMp5CpeL51fJIoXmheO5jnfvVmo3RR4ldIrO0Dny0BWqoTtURw1E0Rg9o1f0Zj1aL9a79TFvXbOKmRO0BOvzB/gkk7k=</latexit>
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sha1_base64="nl6KV1NIoYsveOybKdzXdSD1i0g=">AAACInicZVDLSsNAFJ34rPUVHzs3g6UgKCVxYzdC1Y3LCn1BE8NkOm2HziRhZiKUkA9x46qf4kpQkK4EP8ZpWsW2By4czrl37p3jR4xKZVlfxsrq2vrGZm4rv72zu7dvHhw2ZBgLTOo4ZKFo+UgSRgNSV1Qx0ooEQdxnpOkP7iZ+84kIScOgpoYRcTnqBbRLMVJa8sxy4mSPtEXPdxOrZGW4WCLpjefw+BHBmq5raKWeWfj14DKxZ6RQOX9O1PFoXPXMsdMJccxJoDBDUrZtK1JugoSimJE078SSRAgPUI+0NQ0QJ9JNsuNSWNRKB3ZDoStQMFP/TySISznkvu7kSPXlojcR/7zi3CrVLbsJDaJYkQBPN3VjBlUIJ3nBDhUEKzbUBGFB9bEQ95FAWOlU8zoFe/HPy6RxWbKtkv2g47gFU+TACTgFZ8AGV6AC7kEV1AEGL+AVvIMPY2S8GZ/GeNq6YsxmjsAcjO8fcUWiLQ==</latexit>
sha1_base64="ZOzS7yfnq7D+rVGuXCRnpyJBJL4=">AAACInicZVDLSsNAFJ3UV62v+Ni5GSwFQSmJC+1GqLpxWaEvaGKYTCft0MmDmYlQQj7Ejat+iitBQboS/BinaSvWHrhwOOfeuXeOGzEqpGF8abmV1bX1jfxmYWt7Z3dP3z9oijDmmDRwyELedpEgjAakIalkpB1xgnyXkZY7uJv4rSfCBQ2DuhxGxPZRL6AexUgqydEriZU90uE9106MspHhfImkN47lx48I1lVdQyN19OLcg8vEnJFi9ew5kUejcc3Rx1Y3xLFPAokZEqJjGpG0E8QlxYykBSsWJEJ4gHqko2iAfCLsJDsuhSWldKEXclWBhJn6dyJBvhBD31WdPpJ98d+biL9eaWGV9Cp2QoMoliTA001ezKAM4SQv2KWcYMmGiiDMqToW4j7iCEuVaiFL4XL++f9knkLzomwaZfNBxXELpsiDY3ACToEJrkAV3IMaaAAMXsAreAcf2kh70z618bQ1p81mDsECtO8fd8+iPA==</latexit><latexit sha1_base64="nGrso9br3Sou9LDGmFDW4J+wzVs=">AAACDnicZVDLSsNAFJ3UV62vqEs3g7XgqiQu1GXRjcsKpi00MUwmk3To5MHMRCghP+DKT3ElKIhb9678G6dpKrYeGDiccx9zj5cyKqRhfGu1ldW19Y36ZmNre2d3T98/6Ikk45hYOGEJH3hIEEZjYkkqGRmknKDIY6Tvja+nfv+BcEGT+E5OUuJEKIxpQDGSSnL1EzvgCOe0yMMCWtBOEZcUMdeOMmjd2z4KQ8JdvWm0jRLwPzEr0gQVuq7+ZfsJziISS8yQEEPTSKWTT2djRoqGnQmSIjxGIRkqGqOICCcvrylgSyk+DBKuXixhqf7tyFEkxCTyVGWE5Egse1Px12strJLBpZPTOM0kifFsU5AxKBM4zQb6lBMs2UQRhDlVn4V4hFQ+UiXYKFM4nx+/TOYp9M7aptE2b41m56rKow6OwDE4BSa4AB1wA7rAAhg8gmfwCt60J+1Fe9c+ZqU1reo5BAvQPn8AKsecFQ==</latexit>
sha1_base64="5g2gUsxDCJTlgVTesc+IfvbKddg=">AAACDnicZVDLSsNAFL3xWesr6tLNYC24Kokb3Vl047KCaQtNDZPpJB06eTAzEUrID7jyU3QjKBS37l35N04fiG0PDBzOuY+5x085k8qyfoyV1bX1jc3SVnl7Z3dv3zw4bMokE4Q6JOGJaPtYUs5i6iimOG2nguLI57TlD27GfuuRCsmS+F4NU9qNcBizgBGstOSZp24gMMlZkYcFcpCbYqEY5p4bZch5cHs4DKnwzIpVsyZAy8SekUr96nUEGg3P/HZ7CckiGivCsZQd20pVNx/PJpwWZTeTNMVkgEPa0TTGEZXdfHJNgapa6aEgEfrFCk3U/x05jqQcRr6ujLDqy0VvLP551blVKrjs5ixOM0VjMt0UZBypBI2zQT0mKFF8qAkmgunPItLHOh+lEyzrFOzFm5dJ87xmWzX7TsdxDVOU4BhO4AxsuIA63EIDHCDwBC/wDh/Gs/FmjIzPaemKMes5gjkYX7/AnJ3z</latexit>
sha1_base64="py4hyT4yOTmakPPlv+6cMh1ppe0=">AAACDnicZVDLSsNAFJ34rPUVdelmsBZclcSN7iy6EdxUMG2hqWEymaRDJw9mJkIJ+QFXbv0L3QiK4ta9K//CHxCctEVse2DgcM59zD1uwqiQhvGlzc0vLC4tl1bKq2vrG5v61nZTxCnHxMIxi3nbRYIwGhFLUslIO+EEhS4jLbd/VvitG8IFjaMrOUhIN0RBRH2KkVSSo+/bPkc4o3kW5NCCdoK4pIg5dphC69r2UBAQ7ugVo2YMAWeJOSaV+snj68/F933D0T9tL8ZpSCKJGRKiYxqJ7GbFbMxIXrZTQRKE+yggHUUjFBLRzYbX5LCqFA/6MVcvknCo/u/IUCjEIHRVZYhkT0x7hfjnVSdWSf+4m9EoSSWJ8GiTnzIoY1hkAz3KCZZsoAjCnKrPQtxDKh+pEiyrFMzpm2dJ87BmGjXzUsVxCkYogV2wBw6ACY5AHZyDBrAABrfgATyDF+1Oe9LetPdR6Zw27tkBE9A+fgGHMaDD</latexit>
sha1_base64="Ebh5sb20W4LCLV0nAf2Lp2zo2GE=">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</latexit>i†
U ∂µ Ug• Some of those vacuum configurations are not continuously connected.
topological gauge

Uij (x)<latexit sha1_base64="J+tH0g69F9MUoe4YnvahTWwgtPI=">AAAB/3icZVBNS8NAEN34WetXVDx5CZaCp5J4UI9FLx4r2A9oQ9lsJ+nS3U3Y3QglVPCneBIUxKu/w5P/xm2aiq0PBh7vzTAzL0gYVdp1v62V1bX1jc3SVnl7Z3dv3z44bKk4lQSaJGax7ARYAaMCmppqBp1EAuYBg3Ywupn67QeQisbiXo8T8DmOBA0pwdpIffu4J7mj48Q0RkZkjxFOI+jbFbfm5nD+E68gFVSg0be/eoOYpByEJgwr1fXcRPsZlpoSBpNyL1WQYDLCEXQNFZiD8rP8/IlTNcrACWNpSmgnV/9OZJgrNeaB6eRYD9WyNxV/verCKh1e+RkVSapBkNmmMGXmYWcahjOgEohmY0MwkdQc65AhlphoE1k5T+Fi/vwymafQOq95bs27cyv16yKPEjpBp+gMeegS1dEtaqAmIihDz+gVvVlP1ov1bn3MWlesYuYILcD6/AFM85Yy</latexit>
sha1_base64="4c4fnyeb85e47i8ujtriYLvZjLo=">AAAB/3icZVBNSwMxEJ2tX7V+rYonPQRLwVPZ9aLHohePFewHtEvJpuk2NMkuSVYoSwV/iidBQbz6Ozz5b0y3RWz7YODx3gwz88KEM20878cprK1vbG4Vt0s7u3v7B+7hUVPHqSK0QWIeq3aINeVM0oZhhtN2oigWIaetcHQ79VuPVGkWywczTmggcCTZgBFsrNRzT7pKIBMntjGyIn+KcBrRnlv2ql4OtEr8OSnXziBHved+d/sxSQWVhnCsdcf3EhNkWBlGOJ2UuqmmCSYjHNGOpRILqoMsP3+CKlbpo0GsbEmDcvX/RIaF1mMR2k6BzVAve1Pxz6ssrDKD6yBjMkkNlWS2aZBy+zCahoH6TFFi+NgSTBSzxyIyxAoTYyMr2RT85Z9XSfOy6ntV/97GcTOLA4pwCudwAT5cQQ3uoA4NIJDBC7zBu/PsvDofzuesteDMZ45hAc7XL8O5ln8=</latexit>
sha1_base64="Yw4vlUbULgGeAeKeHmAVEXwDB/4=">AAAB/3icZVBNS8NAEJ3Ur1q/ouJJD8FS8FQSD+qx6MVjBfsBbSib7SZdursJuxuhhAr+FE+Cgnj1d3jy37hNW7H1wcDjvRlm5gUJo0q77rdVWFldW98obpa2tnd29+z9g6aKU4lJA8cslu0AKcKoIA1NNSPtRBLEA0ZawfBm4rceiFQ0Fvd6lBCfo0jQkGKkjdSzj7qSOzpOTGNkRPYYoTQiPbvsVt0czn/izUi5dgI56j37q9uPccqJ0JghpTqem2g/Q1JTzMi41E0VSRAeooh0DBWIE+Vn+fljp2KUvhPG0pTQTq7+ncgQV2rEA9PJkR6oZW8i/nqVhVU6vPIzKpJUE4Gnm8KUmYedSRhOn0qCNRsZgrCk5lgHD5BEWJvISnkKF/Pnl8k8heZ51XOr3p2J43oaBxThGE7hDDy4hBrcQh0agCGDZ3iFN+vJerHerY9pa8GazRzCAqzPH8pDlo4=</latexit>

U (∞) → 1<latexit sha1_base64="A2+2hE1r715HNWBv6j59lSwQg2U=">AAACFnicZVDLSgMxFM34rPU16tJNsBQqSJlxoS6LblxWcNpCW0omvdPGZh4kGbEM8xOu/BRXgoK4FVz5N6bTqdh6IHA4596c5LgRZ1JZ1rextLyyurZe2Chubm3v7Jp7+w0ZxoKCQ0MeipZLJHAWgKOY4tCKBBDf5dB0R1cTv3kPQrIwuFXjCLo+GQTMY5QoLfXMk6STXZII6KfY6SXsLk0reKa6PIYUd1wPP6T4uGeWrKqVAf8ndk5KKEe9Z351+iGNfQgU5UTKtm1FqpsQoRjlkBY7sYSI0BEZQFvTgPggu0kWneKyVvrYC4U+gcKZ+ncjIb6UY9/Vkz5RQ7noTcRfrzwXpbyLbsKCKFYQ0GmSF3OsQjzpCPeZAKr4WBNCBdOPxXRIBKFKN1nMWjibfX6RzFponFZtq2rfWKXaZd5HAR2iI1RBNjpHNXSN6shBFD2iZ/SK3own48V4Nz6mo0tGvnOA5mB8/gBfcJ9e</latexit>
sha1_base64="RU5DSePB09v4Ia5pqHd653iE7Ik=">AAACFnicZVDLSgMxFL3j2/qqunQTrIUKUmbc6FJ047KCfUBbSia900YzD5KMWIb5BDeu9E9ciQriVnDl35g+FNseCBzOuTcnOW4kuNK2/W3NzM7NLywuLWdWVtfWN7KbWxUVxpJhmYUilDWXKhQ8wLLmWmAtkkh9V2DVvT7r+9UblIqHwaXuRdj0aSfgHmdUG6mVPUgag0sSie2UlFsJv0rTAvlVXRFjShquR25Tst/K5uyiPQCZJs6I5E72Hu9eAKDUyn412iGLfQw0E1SpumNHuplQqTkTmGYascKIsmvawbqhAfVRNZNBdEryRmkTL5TmBJoM1P8bCfWV6vmumfSp7qpJry/+efmxKO0dNxMeRLHGgA2TvFgQHZJ+R6TNJTIteoZQJrl5LGFdKinTpsmMacGZ/PM0qRwWHbvoXJg6TmGIJdiBXSiAA0dwAudQgjIwuIcneIU368F6tt6tj+HojDXa2YYxWJ8/cfihmQ==</latexit>
sha1_base64="+9aXW/aQK0aBBIm9ev3XTpRvn70=">AAACFnicZVDLSgMxFM34rPVVdekmWAsVpMy40WXRjcsK9gFtKZn0ThubeZBkxDLOJ7hxpf6GG1eCgrgSBFf+jem0im0PBA7n3JuTHDvgTCrT/DZmZufmFxZTS+nlldW19czGZkX6oaBQpj73Rc0mEjjzoKyY4lALBBDX5lC1eycDv3oJQjLfO1f9AJou6XjMYZQoLbUy+1EjuSQS0I5xuRWxizjO41/V5iHEuGE7+CrGe61M1iyYCfA0sUYkW9y9v/l4vH4otTJfjbZPQxc8RTmRsm6ZgWpGRChGOcTpRighILRHOlDX1CMuyGaURMc4p5U2dnyhj6dwov7fiIgrZd+19aRLVFdOegPxz8uNRSnnqBkxLwgVeHSY5IQcKx8POsJtJoAq3teEUMH0YzHtEkGo0k2mdQvW5J+nSeWgYJkF60zXcYyGSKFttIPyyEKHqIhOUQmVEUW36Am9oFfjzng23oz34eiMMdrZQmMwPn8A6AOjbw==</latexit>
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SU(2)<latexit sha1_base64="J10CVrOIEnAv53LwEdDPR6aoboQ=">AAAB/3icZVBNS8NAEJ3Ur1q/ouLJy2Ip1EtJPKjHohePFUxbaELZbDft0s0m7G6EEnrwp3gSFMSrv8OT/8btl9j6YODx3gwz88KUM6Ud59sqrK1vbG4Vt0s7u3v7B/bhUVMlmSTUIwlPZDvEinImqKeZ5rSdSorjkNNWOLyd+K1HKhVLxIMepTSIcV+wiBGsjdS1T7yqz0SkR+fI1wnK/TBC7rhrl52aMwX6T9w5KcMcja795fcSksVUaMKxUh3XSXWQY6kZ4XRc8jNFU0yGuE87hgocUxXk0/PHqGKUHooSaUpoNFX/TuQ4VmoUh6YzxnqgVr2J+OtVllbp6DrImUgzTQWZbYoyjsynkzBQj0lKNB8Zgolk5lhEBlhiok1kpWkKl4vnV8kiheZFzXVq7r1Trt/M8yjCKZxBFVy4gjrcQQM8IJDDM7zCm/VkvVjv1sestWDNZ45hCdbnDx+SlMs=</latexit>
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sha1_base64="lZ+dYV75sge0JA03KGRJ/skBqyE=">AAAB/3icZVDLSgNBEOyNrxhfq6IXPQwJgXgJux7UY9CLxwjmAdkQZiezyeDs7DIzK4QlB79EPAkKoke/w5N/4+QlJhY0FFXddHf5MWdKO863lVlaXlldy67nNja3tnfs3b26ihJJaI1EPJJNHyvKmaA1zTSnzVhSHPqcNvy7q5HfuKdSsUjc6kFM2yHuCRYwgrWROvZBreQxEejBCfJ0hFLPD5A77NgFp+yMgf4Td0oKlTxkHg8/jqsd+8vrRiQJqdCEY6VarhPrdoqlZoTTYc5LFI0xucM92jJU4JCqdjo+f4iKRumiIJKmhEZj9e9EikOlBqFvOkOs+2rRG4m/XnFulQ4u2ikTcaKpIJNNQcKR+XQUBuoySYnmA0Mwkcwci0gfS0y0iSw3TuFs9vwimaVQPy27Ttm9MXFcwgRZOII8lMCFc6jANVShBgRSeIIXeLUerGfrzXqftGas6cw+zMH6/AGuQpax</latexit><latexit sha1_base64="g0DhUefArJGpvl00S15W09O9U6s=">AAACAXicZVBPS8MwHE3nvzn/VT148BIcg3kZ7Q7qcejF40S7DdYy0jTbwpK2JKkwSk9+FE+Cgnj1Y3jy25h1nbj5IPB47/fLS54fMyqVZX0bpbX1jc2t8nZlZ3dv/8A8POrIKBGYODhikej5SBJGQ+IoqhjpxYIg7jPS9Sc3M7/7SISkUfigpjHxOBqFdEgxUloamCepm1+SChJk0BUc3jv15nk2MKtWw8oB/xO7IFVQoD0wv9wgwgknocIMSdm3rVh5KRKKYkayiptIEiM8QSPS1zREnEgvzbMzWNNKAIeR0CdUMFf/bqSISznlvp7kSI3lqjcTf73aUpQaXnkpDeNEkRDPk4YJgyqCszpgQAXBik01QVhQ/ViIx0ggrHRplbyFi8XnV8mihU6zYVsN+86qtq6LPsrgFJyBOrDBJWiBW9AGDsAgA8/gFbwZT8aL8W58zEdLRrFzDJZgfP4AWw+WEw==</latexit>
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sha1_base64="TY+FXtqjzuHvE4Nu8/bgH1ZQNro=">AAACAXicZVDLSgMxFL1TX7W+qoIudBFaCnVTZrpQl0U3LivaB3SGkknTNph5kGSEMszKD3HhSlBQt36GK//GdNqKrQcCh3PuzUmOG3ImlWl+G5ml5ZXVtex6bmNza3snv7vXlEEkCG2QgAei7WJJOfNpQzHFaTsUFHsupy337nLst+6pkCzwb9UopI6HBz7rM4KVlrr5w9hOL4kF7SXIFh66aZSrJ0k3XzQrZgr0n1hTUqwVIPN48H5c7+a/7F5AIo/6inAsZccyQ+XEWChGOE1ydiRpiMkdHtCOpj72qHTiNDtBJa30UD8Q+vgKperfjRh7Uo48V096WA3lojcWf73SXJTqnzsx88NIUZ9MkvoRRypA4zpQjwlKFB9pgolg+rGIDLHAROnScmkLp7PPL5JZC81qxTIr1rWu4wImyMIRFKAMFpxBDa6gDg0gkMATvMCr8WA8G2/Gx2Q0Y0x39mEOxucP6b+X+Q==</latexit>

π3 (S ) = Z
3

SU(2) ≃ S
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sha1_base64="NiY5Otgkwg1biEIfE6xxrx7OMFA=">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</latexit>
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3

map<latexit sha1_base64="3teZbPx2Qnz4MIcCXYzN6GO7KVc=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrQT0WvXisYD+gXUo2zbahSXabZIWy9Ed4EhTEq7/Hk//GdLsVWx8MPN6bYWZeEHOmjet+O4WNza3tneJuaW//4PCofHzS0lGiCG2SiEeqE2BNOZO0aZjhtBMrikXAaTsY38399hNVmkXy0Uxj6gs8lCxkBBsrtXtKIIHjfrni1twM6D/xclKBHI1++as3iEgiqDSEY627nhsbP8XKMMLprNRLNI0xGeMh7VoqsaDaT7NzZ6hqlQEKI2VLGpSpfydSLLSeisB2CmxGet2bi79edWWVCW/8lMk4MVSSxaYw4chEaP48GjBFieFTSzBRzB6LyAgrTIyNqJSlcLV8fp0sU2hd1jy35j24lfptnkcRzuAcLsCDa6jDPTSgCQTG8Ayv8OZMnBfn3flYtBacfOYUVuB8/gCLs4+P</latexit>
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sha1_base64="YZB7GzavOfoGcVQvCm2hBr3u+Cs=">AAAB73icZVBNSwMxEJ2tX7V+VT3qIVgKnsquB/VY9OKxgv2AdinZNNuGJtk1yQpl6Y/wJCiIV3+PJ/+N6W4rtj4YeLw3w8y8IOZMG9f9dgpr6xubW8Xt0s7u3v5B+fCopaNEEdokEY9UJ8CaciZp0zDDaSdWFIuA03Ywvp357SeqNIvkg5nE1Bd4KFnICDZWaveUQALH/XLFrbkZ0H/izUmlfgoZGv3yV28QkURQaQjHWnc9NzZ+ipVhhNNpqZdoGmMyxkPatVRiQbWfZudOUdUqAxRGypY0KFP/TqRYaD0Rge0U2Iz0qjcTf73q0ioTXvspk3FiqCT5pjDhyERo9jwaMEWJ4RNLMFHMHovICCtMjI2olKVwuXh+lSxSaF3UPLfm3ds4bvI4oAgncAbn4MEV1OEOGtAEAmN4hld4cx6dF+fd+chbC8585hiW4Hz+AAkSj+s=</latexit>

3

S ≃R

3

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sha1_base64="U9SKBTu7M4ZOHmcGFjBfcOkwngM=">AAAB/HicZVDLSgMxFL1TX7W+xrp0EywFV2VGF7osunFZ0T6gHUsmzbShmcyQZMQy1E9xJSiIW3/ElX9jOh3EtgcCh3PuzUmOH3OmtOP8WIW19Y3NreJ2aWd3b//APiy3VJRIQpsk4pHs+FhRzgRtaqY57cSS4tDntO2Pr2d++5FKxSJxrycx9UI8FCxgBGsj9e1y2ssuSX2e0Cm6ezif9u2KU3MyoFXi5qQCORp9+7s3iEgSUqEJx0p1XSfWXoqlZoTTaamXKBpjMsZD2jVU4JAqL81ip6hqlAEKImmO0ChT/2+kOFRqEvpmMsR6pJa9mfjnVReidHDppUzEiaaCzJOChCMdoVkTaMAkJZpPDMFEMvNYREZYYqJNXyXTgrv851XSOqu5Ts29dSr1q7yPIhzDCZyCCxdQhxtoQBMIPMELvMG79Wy9Wh/W53y0YOU7R7AA6+sXn9mUnQ==</latexit>• These disconnected vacua are characterised by the integer NCS.
E

Aµ (x)<latexit sha1_base64="0z4J+kfh0hlSCQBVQcvBpnafrss=">AAAB9nicZVBNS8NAEJ3Ur1q/qh69LJZCvZTEg3qsevFYwX5AG8tmu2mXbjZhdyMtof/Dk6AgXv0vnvw3btNUbH0w8Hhvhpl5XsSZ0rb9beXW1jc2t/LbhZ3dvf2D4uFRU4WxJLRBQh7KtocV5UzQhmaa03YkKQ48Tlve6Hbmt56oVCwUD3oSUTfAA8F8RrA20uN1rxvElaTr+Wg8PesVS3bVToH+EycjJchQ7xW/uv2QxAEVmnCsVMexI+0mWGpGOJ0WurGiESYjPKAdQwUOqHKT9OopKhulj/xQmhIaperfiQQHSk0Cz3QGWA/VqjcTf73y0irtX7kJE1GsqSDzTX7MkQ7RLAPUZ5ISzSeGYCKZORaRIZaYaJNUIU3hYvH8Klmk0DyvOnbVubdLtZssjzycwClUwIFLqMEd1KEBBCQ8wyu8WWPrxXq3PuatOSubOYYlWJ8/IaiSHw==</latexit>
sha1_base64="RD6F2GN9tk+cbuJoCdEq72lqG3E=">AAAB9nicZVDLSgNBEOyNrxhfUY9eBmMgXsKuFz1GvXiMYB6QXcPsZDYZMju7zMxKwpJP8K4XQUG8+gn+gyf/xskDMUlBQ1HVTXeXH3OmtG3/WJmV1bX1jexmbmt7Z3cvv39QV1EiCa2RiEey6WNFORO0ppnmtBlLikOf04bfvx77jQcqFYvEnR7G1AtxV7CAEayNdH/ZdsOklLp+gAaj03a+YJftCdAycWakUDl5fvwCgGo7/+12IpKEVGjCsVItx461l2KpGeF0lHMTRWNM+rhLW4YKHFLlpZOrR6holA4KImlKaDRR/0+kOFRqGPqmM8S6pxa9sfjnFedW6eDCS5mIE00FmW4KEo50hMYZoA6TlGg+NAQTycyxiPSwxESbpHImBWfx52VSPys7dtm5NXFcwRRZOIJjKIED51CBG6hCDQhIeIJXeLMG1ov1bn1MWzPWbOYQ5mB9/gI0MJRa</latexit>
sha1_base64="VxpoDDI16ywwghLumDzcVxN61W0=">AAAB9nicZVDLSgNBEOz1GeMr6tHLYAzES9j1oseoF48RzAOSNcxOZpMhM7vLzKwkrPkE7+pBUBCvgt/hyZN/4+SBmKSgoajqprvLizhT2rZ/rIXFpeWV1dRaen1jc2s7s7NbUWEsCS2TkIey5mFFOQtoWTPNaS2SFAuP06rXvRj61VsqFQuDa92PqCtwO2A+I1gb6eas2RBxPml4PuoNjpqZrF2wR0DzxJmQbPHw8f7r8+6p1Mx8N1ohiQUNNOFYqbpjR9pNsNSMcDpIN2JFI0y6uE3rhgZYUOUmo6sHKGeUFvJDaSrQaKT+n0iwUKovPNMpsO6oWW8o/nm5qVXaP3UTFkSxpgEZb/JjjnSIhhmgFpOUaN43BBPJzLGIdLDERJuk0iYFZ/bneVI5Ljh2wbkycZzDGCnYhwPIgwMnUIRLKEEZCEh4gBd4tXrWs/VmvY9bF6zJzB5Mwfr4Bao7ljA=</latexit>
sha1_base64="wrKYhqc1MrxjKnnKlz385rJsZFw=">AAAB9nicZVDLSgMxFM3UV62vqks3wVqomzLjQl1W3bisYB/QGUsmzbShSWZIMtIy9hPcqwtBQdwKfocrV/6N6UusHrhwOOde7r3HjxhV2ra/rNTc/MLiUno5s7K6tr6R3dyqqjCWmFRwyEJZ95EijApS0VQzUo8kQdxnpOZ3z4Z+7ZpIRUNxqfsR8ThqCxpQjLSRrk6aLo8LiesHsDfYb2ZzdtEeAf4nzoTkSnv3tx/vNw/lZvbTbYU45kRozJBSDceOtJcgqSlmZJBxY0UihLuoTRqGCsSJ8pLR1QOYN0oLBqE0JTQcqb8nEsSV6nPfdHKkO+qvNxR/vPzMKh0cewkVUayJwONNQcygDuEwA9iikmDN+oYgLKk5FuIOkghrk1RmlMLh9Pm/ZJpC9aDo2EXnwsRxCsZIgx2wCwrAAUegBM5BGVQABhLcgSfwbPWsR+vFeh23pqzJzDaYgfX2DbDFlj8=</latexit>• One can think of transitions between disconnected vacua.
[Klinkhamer, Manton ’84]

Esph ≃ 9 TeV

Esphaleron

unstable solution sitting on
top of the energy barrier<latexit sha1_base64="n5F3YGh1pPixcHkz5oUCogzoVNY=">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</latexit>
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sha1_base64="fm+jwWicEhRhfGpkVcEfa/lTtII=">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</latexit>
sha1_base64="DBAd22kzjAFcTahi1+wepoZXhXs=">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</latexit>

Instanton

quantum tunneling• One can think of transitions between disconnected vacua.
[Klinkhamer, Manton ’84]

Esph ≃ 9 TeV

sphaleronEunstable solution sitting on
top of the energy barrier<latexit sha1_base64="n5F3YGh1pPixcHkz5oUCogzoVNY=">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</latexit>
sha1_base64="p7NCtKLpHmXcLxgQVeT+0zRZ89A=">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</latexit>
sha1_base64="fm+jwWicEhRhfGpkVcEfa/lTtII=">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</latexit>
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quantum tunneling

Instanton• At zero temperature/energy, the instanton rate is exponentially small:⟨NCS |NCS + 1⟩instanton ∼ e<latexit sha1_base64="69c162CA+EEMC1+uOQPKqw2bQ4s=">AAAB/3icZVDLSsNAFJ3UV62vqLhyM1gKLrQkLtRl1Y3LivYBTQyT6aQdOpmEmYlQQhZ+iitBQdz6Ha78G6dpKlYPXDiccy/33uPHjEplWV9GaWFxaXmlvFpZW9/Y3DK3d9oySgQmLRyxSHR9JAmjnLQUVYx0Y0FQ6DPS8UdXE7/zQISkEb9T45i4IRpwGlCMlJY8c8+RNHSOILlPj297F16KWeZmnlm16lYO+J/YBamCAk3P/HT6EU5CwhVmSMqebcXKTZFQFDOSVZxEkhjhERqQnqYchUS6aX5+Bmta6cMgErq4grn6eyJFoZTj0NedIVJD+debiD9ebW6VCs7dlPI4UYTj6aYgYVBFcBIG7FNBsGJjTRAWVB8L8RAJhJWOrJKncDp7/i+ZpdA+qdtW3b6xqo3LIo8y2AcH4BDY4Aw0wDVoghbAIAVP4AW8Go/Gs/FmvE9bS0YxswvmYHx8A/qilVo=</latexit>
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sha1_base64="gDNjhDdQ7lPElREZF56jYsBWWdw=">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</latexit>−S[Acl ]= e<latexit sha1_base64="KEYkcVbl55EEVVw1NgtGGf+aJ5s=">AAACCXicZZDLSsNAFIYn9VbrLerShYOl4EJL0oW6EYpuXFawF2himEwn7dDJJMxMhBKydOWjuBIUxK2P4Mq3cdqmYusPAz//OYcz5/NjRqWyrG+jsLS8srpWXC9tbG5t75i7ey0ZJQKTJo5YJDo+koRRTpqKKkY6sSAo9Blp+8Prcb39QISkEb9To5i4IepzGlCMlI488/DSOYHkPj11AoFwWoNOTLPUQSweIK+dZZ5ZtqrWRPC/sXNTBrkanvnl9CKchIQrzJCUXduKlZsioShmJCs5iSQxwkPUJ11tOQqJdNPJIRms6KQHg0joxxWcpH8nUhRKOQp93RkiNZCLtXH4W6vMrVLBhZtSHieKcDzdFCQMqgiOscAeFQQrNtIGYUH1ZyEeII1EaXilCYWz2fGLZkahVavaVtW+tcr1q5xHERyAI3AMbHAO6uAGNEATYPAInsEreDOejBfj3fiYthaMfGYfzMn4/AFYf5lr</latexit>
sha1_base64="O21EYN148hvDwSS4OCNzQmoxYJ4=">AAACCXicZVDLSsNAFJ3UV62vqEsXDpaCCy1JN7oRim5cVrAPaGKZTCft0MlkmJkIJWTpyk9xJSiIWz/BlX/jtA1i2wMXDufcy733BIJRpR3nxyqsrK6tbxQ3S1vbO7t79v5BS8WJxKSJYxbLToAUYZSTpqaakY6QBEUBI+1gdDPx249EKhrzez0WxI/QgNOQYqSN1LOPr7wzSB7Scy+UCKc16AmapR5iYoh67Szr2WWn6kwBl4mbkzLI0ejZ314/xklEuMYMKdV1HaH9FElNMSNZyUsUEQiP0IB0DeUoIspPp49ksGKUPgxjaYprOFX/T6QoUmocBaYzQnqoFr2J+OdV5lbp8NJPKReJJhzPNoUJgzqGk1hgn0qCNRsbgrCk5liIh8hEok14JZOCu/jzMmnVqq5Tde+ccv06z6MIjsAJOAUuuAB1cAsaoAkweAIv4A28W8/Wq/Vhfc5aC1Y+cwjmYH39AlH1mVw=</latexit>

− α2π

W

∼ e−180<latexit sha1_base64="mEnUFx1qqnuDg1x+Iu8UKmzydn0=">AAAB+XicZVDLSgMxFM34rPXRUZdugqXgQkvGhXZZdOOygn1AZyyZNG1Dk8yQZIQy9EtcCQri1k9x5d+YTqdi64ELh3Pu5d57wpgzbRD6dtbWNza3tgs7xd29/YOSe3jU0lGiCG2SiEeqE2JNOZO0aZjhtBMrikXIaTsc38789hNVmkXywUxiGgg8lGzACDZW6rklXzPhn0P6mF54NTTtuWVURRngf+LlpAxyNHrul9+PSCKoNIRjrbseik2QYmUY4XRa9BNNY0zGeEi7lkosqA7S7PAprFilDweRsiUNzNS/EykWWk9EaDsFNiO96s3EX6+ytMoMakHKZJwYKsl80yDh0ERwFgPsM0WJ4RNLMFHMHgvJCCtMjA2rmKVwtXh+lSxSaF1WPVT17lG5fpPnUQAn4BScAQ9cgzq4Aw3QBAQk4Bm8gjcndV6cd+dj3rrm5DPHYAnO5w+iTZJF</latexit>
sha1_base64="QaGkOdnzvP/Bxx2pmxwBsesmcQU=">AAAB+XicZVBNSwMxEM36WetHVz16CZaCBy1ZL/ZY9OKxgv2Abi3ZdLYNTXaXJCuUpb/Ek6AgXv0pnvw3pu0itn0w8Hhvhpl5QSK4NoT8OBubW9s7u4W94v7B4VHJPT5p6ThVDJosFrHqBFSD4BE0DTcCOokCKgMB7WB8N/Pbz6A0j6NHM0mgJ+kw4iFn1Fip75Z8zaV/ieEpu/JqZNp3y6RK5sDrxMtJGeVo9N1vfxCzVEJkmKBadz2SmF5GleFMwLTopxoSysZ0CF1LIypB97L54VNcscoAh7GyFRk8V/9PZFRqPZGB7ZTUjPSqNxP/vMrSKhPWehmPktRAxBabwlRgE+NZDHjAFTAjJpZQprg9FrMRVZQZG1bRpuCt/rxOWtdVj1S9B1Ku3+Z5FNAZOkcXyEM3qI7uUQM1EUMpekFv6N3JnFfnw/lctG44+cwpWoLz9Qubw5I2</latexit>• One can think of transitions between disconnected vacua.
[Klinkhamer, Manton ’84]

Esph ≃ 9 TeV

sphaleronEunstable solution sitting on
top of the energy barrier<latexit sha1_base64="n5F3YGh1pPixcHkz5oUCogzoVNY=">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</latexit>
sha1_base64="p7NCtKLpHmXcLxgQVeT+0zRZ89A=">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</latexit>
sha1_base64="fm+jwWicEhRhfGpkVcEfa/lTtII=">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</latexit>
sha1_base64="DBAd22kzjAFcTahi1+wepoZXhXs=">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</latexit>

quantum tunneling

Instanton• At zero temperature/energy, the instanton rate is exponentially small:⟨NCS |NCS + 1⟩instanton ∼ e<latexit sha1_base64="69c162CA+EEMC1+uOQPKqw2bQ4s=">AAAB/3icZVDLSsNAFJ3UV62vqLhyM1gKLrQkLtRl1Y3LivYBTQyT6aQdOpmEmYlQQhZ+iitBQdz6Ha78G6dpKlYPXDiccy/33uPHjEplWV9GaWFxaXmlvFpZW9/Y3DK3d9oySgQmLRyxSHR9JAmjnLQUVYx0Y0FQ6DPS8UdXE7/zQISkEb9T45i4IRpwGlCMlJY8c8+RNHSOILlPj297F16KWeZmnlm16lYO+J/YBamCAk3P/HT6EU5CwhVmSMqebcXKTZFQFDOSVZxEkhjhERqQnqYchUS6aX5+Bmta6cMgErq4grn6eyJFoZTj0NedIVJD+debiD9ebW6VCs7dlPI4UYTj6aYgYVBFcBIG7FNBsGJjTRAWVB8L8RAJhJWOrJKncDp7/i+ZpdA+qdtW3b6xqo3LIo8y2AcH4BDY4Aw0wDVoghbAIAVP4AW8Go/Gs/FmvE9bS0YxswvmYHx8A/qilVo=</latexit>
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sha1_base64="GzZW4FBRu2QpQ7pqYiuv2cai51Q=">AAACHXicZZBNS8NAEIYn9avWr6pHL4ulIAgl0YMei714kopWBVPCZrttl242YXcilNgf4kn/iSdBQbx4EP+Nm7aIHwMLD+87w+y8YSKFQdf9dAozs3PzC8XF0tLyyupaeX3jwsSpZrzFYhnrq5AaLoXiLRQo+VWiOY1CyS/DQSP3L2+4NiJW5zhMeDuiPSW6glG0UlDe9yVVPcnJSZA1zkbkdgq7HvH12AkyX0dEKINUYaxGpBSUK27NHRf5D94UKvVtKDwAQDMov/udmKURV8gkNebacxNsZ1SjYJKPSn5qeELZgPb4tUVFI27a2fi4EalapUO6sbZPIRmrPycyGhkzjELbGVHsm79eLn571V+rsHvYzoRKUuSKTTZ1U0kwJnlUpCM0ZyiHFijTwn6WsD7VlKENNE/B+3vzf7jYq3luzTu1cRzBpIqwBduwAx4cQB2OoQktYHAHj/AML8698+S8Om+T1oIzndmEX+V8fAEp8KIA</latexit>
sha1_base64="bChAqkWYU7cVeO0VRf7Mzh9VYQQ=">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</latexit>
sha1_base64="gDNjhDdQ7lPElREZF56jYsBWWdw=">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</latexit>−S[Acl ]= e<latexit sha1_base64="KEYkcVbl55EEVVw1NgtGGf+aJ5s=">AAACCXicZZDLSsNAFIYn9VbrLerShYOl4EJL0oW6EYpuXFawF2himEwn7dDJJMxMhBKydOWjuBIUxK2P4Mq3cdqmYusPAz//OYcz5/NjRqWyrG+jsLS8srpWXC9tbG5t75i7ey0ZJQKTJo5YJDo+koRRTpqKKkY6sSAo9Blp+8Prcb39QISkEb9To5i4IepzGlCMlI488/DSOYHkPj11AoFwWoNOTLPUQSweIK+dZZ5ZtqrWRPC/sXNTBrkanvnl9CKchIQrzJCUXduKlZsioShmJCs5iSQxwkPUJ11tOQqJdNPJIRms6KQHg0joxxWcpH8nUhRKOQp93RkiNZCLtXH4W6vMrVLBhZtSHieKcDzdFCQMqgiOscAeFQQrNtIGYUH1ZyEeII1EaXilCYWz2fGLZkahVavaVtW+tcr1q5xHERyAI3AMbHAO6uAGNEATYPAInsEreDOejBfj3fiYthaMfGYfzMn4/AFYf5lr</latexit>
sha1_base64="O21EYN148hvDwSS4OCNzQmoxYJ4=">AAACCXicZVDLSsNAFJ3UV62vqEsXDpaCCy1JN7oRim5cVrAPaGKZTCft0MlkmJkIJWTpyk9xJSiIWz/BlX/jtA1i2wMXDufcy733BIJRpR3nxyqsrK6tbxQ3S1vbO7t79v5BS8WJxKSJYxbLToAUYZSTpqaakY6QBEUBI+1gdDPx249EKhrzez0WxI/QgNOQYqSN1LOPr7wzSB7Scy+UCKc16AmapR5iYoh67Szr2WWn6kwBl4mbkzLI0ejZ314/xklEuMYMKdV1HaH9FElNMSNZyUsUEQiP0IB0DeUoIspPp49ksGKUPgxjaYprOFX/T6QoUmocBaYzQnqoFr2J+OdV5lbp8NJPKReJJhzPNoUJgzqGk1hgn0qCNRsbgrCk5liIh8hEok14JZOCu/jzMmnVqq5Tde+ccv06z6MIjsAJOAUuuAB1cAsaoAkweAIv4A28W8/Wq/Vhfc5aC1Y+cwjmYH39AlH1mVw=</latexit>

− α2π

W

∼ e−180<latexit sha1_base64="mEnUFx1qqnuDg1x+Iu8UKmzydn0=">AAAB+XicZVDLSgMxFM34rPXRUZdugqXgQkvGhXZZdOOygn1AZyyZNG1Dk8yQZIQy9EtcCQri1k9x5d+YTqdi64ELh3Pu5d57wpgzbRD6dtbWNza3tgs7xd29/YOSe3jU0lGiCG2SiEeqE2JNOZO0aZjhtBMrikXIaTsc38789hNVmkXywUxiGgg8lGzACDZW6rklXzPhn0P6mF54NTTtuWVURRngf+LlpAxyNHrul9+PSCKoNIRjrbseik2QYmUY4XRa9BNNY0zGeEi7lkosqA7S7PAprFilDweRsiUNzNS/EykWWk9EaDsFNiO96s3EX6+ytMoMakHKZJwYKsl80yDh0ERwFgPsM0WJ4RNLMFHMHgvJCCtMjA2rmKVwtXh+lSxSaF1WPVT17lG5fpPnUQAn4BScAQ9cgzq4Aw3QBAQk4Bm8gjcndV6cd+dj3rrm5DPHYAnO5w+iTZJF</latexit>
sha1_base64="QaGkOdnzvP/Bxx2pmxwBsesmcQU=">AAAB+XicZVBNSwMxEM36WetHVz16CZaCBy1ZL/ZY9OKxgv2Abi3ZdLYNTXaXJCuUpb/Ek6AgXv0pnvw3pu0itn0w8Hhvhpl5QSK4NoT8OBubW9s7u4W94v7B4VHJPT5p6ThVDJosFrHqBFSD4BE0DTcCOokCKgMB7WB8N/Pbz6A0j6NHM0mgJ+kw4iFn1Fip75Z8zaV/ieEpu/JqZNp3y6RK5sDrxMtJGeVo9N1vfxCzVEJkmKBadz2SmF5GleFMwLTopxoSysZ0CF1LIypB97L54VNcscoAh7GyFRk8V/9PZFRqPZGB7ZTUjPSqNxP/vMrSKhPWehmPktRAxBabwlRgE+NZDHjAFTAjJpZQprg9FrMRVZQZG1bRpuCt/rxOWtdVj1S9B1Ku3+Z5FNAZOkcXyEM3qI7uUQM1EUMpekFv6N3JnFfnw/lctG44+cwpWoLz9Qubw5I2</latexit>• At temperature around TeV, the sphaleron rate is order 1.

• One can think of transitions between disconnected vacua.
[Klinkhamer, Manton ’84]

Esph ≃ 9 TeV

sphaleronEunstable solution sitting on
top of the energy barrier<latexit sha1_base64="n5F3YGh1pPixcHkz5oUCogzoVNY=">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</latexit>
sha1_base64="p7NCtKLpHmXcLxgQVeT+0zRZ89A=">AAACCXicdZDLSgMxFIbPeLfeqi5dGLyACykztVjdiSK4VLBVcErJpKc2NJkZk4xQhi5diQ/gM7gSFMSV4CO48m3MtCoq+EPg5zvncHL+IBZcG9d9dwYGh4ZHRsfGcxOTU9Mz+dm5qo4SxbDCIhGp04BqFDzEiuFG4GmskMpA4EnQ3svqJ5eoNI/CY9OJsSbpecibnFFjUT2/uF9PfSWJjltd4msu8YJs++s9dozVbj2/7Ba2tzaLpU3iFly37BW9zBTLpY0S8SzJtLyzcnvzAgCH9fyb34hYIjE0TFCtzzw3NrWUKsOZwG7OTzTGlLXpOZ5ZG1KJupb2DumSVUsapBkp+0JDevTnREql1h0Z2E5JTUv/rWXwu7b6a5VpbtVSHsaJwZD1NzUTQUxEslhIgytkRnSsoUxx+1nCWlRRZmx4OZvC16nkf1MtFjy34B3ZOHahrzFYgCVYAw/KsAMHcAgVYHAFd/AAj861c+88Oc/91gHnc2Yefsl5/QCImJu0</latexit>
sha1_base64="fm+jwWicEhRhfGpkVcEfa/lTtII=">AAACCXicdZDNSgMxFIUz/tb6V3XpwqAWXEiZqcW2u6IILhVsFZxSMumtDSYzY5IRytilK/EB9BVciQri1rUrV76NmVbFCh4IHL57Lzf3eCFnStv2hzU0PDI6Np6aSE9OTc/MZubmayqIJIUqDXggjzyigDMfqpppDkehBCI8Dofe6XZSPzwHqVjgH+hOCHVBTnzWYpRogxqZpZ1G7EqBVdjuYlcxAWe47K732AHUuo3Mip0rlzbzhU1s52y76OSdxOSLhY0CdgxJtFJZvbl+e7i43Wtk3t1mQCMBvqacKHXs2KGux0RqRjl0026kICT0lJzAsbE+EaDqce+QLs4a0sStQJrna9yjvydiIpTqCM90CqLb6m8tgT+17MAq3SrVY+aHkQaf9je1Io51gJNYcJNJoJp3jCFUMvNZTNtEEqpNeGmTwvep+H9Ty+ccO+fsmzi2UF8ptIiW0RpyUBFV0C7aQ1VE0SW6Q4/oybqy7q1n66XfOmR9zSygAVmvn/6jnYo=</latexit>
sha1_base64="DBAd22kzjAFcTahi1+wepoZXhXs=">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</latexit>

quantum tunneling

Instanton• At zero temperature/energy, the instanton rate is exponentially small:⟨NCS |NCS + 1⟩instanton ∼ e<latexit sha1_base64="69c162CA+EEMC1+uOQPKqw2bQ4s=">AAAB/3icZVDLSsNAFJ3UV62vqLhyM1gKLrQkLtRl1Y3LivYBTQyT6aQdOpmEmYlQQhZ+iitBQdz6Ha78G6dpKlYPXDiccy/33uPHjEplWV9GaWFxaXmlvFpZW9/Y3DK3d9oySgQmLRyxSHR9JAmjnLQUVYx0Y0FQ6DPS8UdXE7/zQISkEb9T45i4IRpwGlCMlJY8c8+RNHSOILlPj297F16KWeZmnlm16lYO+J/YBamCAk3P/HT6EU5CwhVmSMqebcXKTZFQFDOSVZxEkhjhERqQnqYchUS6aX5+Bmta6cMgErq4grn6eyJFoZTj0NedIVJD+debiD9ebW6VCs7dlPI4UYTj6aYgYVBFcBIG7FNBsGJjTRAWVB8L8RAJhJWOrJKncDp7/i+ZpdA+qdtW3b6xqo3LIo8y2AcH4BDY4Aw0wDVoghbAIAVP4AW8Go/Gs/FmvE9bS0YxswvmYHx8A/qilVo=</latexit>
sha1_base64="yUryVAFDzuayi+tmMriGdeqGSWQ=">AAAB/3icZVDLSsNAFJ3UV62vqLhyM1gKLrQkbnRZdeOyon1AE8NkOmmHzkzCzEQoIQs/xZWgIG79Dlf+jdM2iLUHLhzOuZd77wkTRpV2nG+rtLS8srpWXq9sbG5t79i7e20VpxKTFo5ZLLshUoRRQVqaaka6iSSIh4x0wtH1xO88EqloLO71OCE+RwNBI4qRNlJgH3iKcu8Ekofs9K53GWSY5X4e2FWn7kwBF4lbkCoo0AzsL68f45QToTFDSvVcJ9F+hqSmmJG84qWKJAiP0ID0DBWIE+Vn0/NzWDNKH0axNCU0nKp/JzLElRrz0HRypIfqvzcRf73a3CodXfgZFUmqicCzTVHKoI7hJAzYp5JgzcaGICypORbiIZIIaxNZxaTg/v95kbTP6q5Td2+dauOqyKMMDsEROAYuOAcNcAOaoAUwyMAzeAVv1pP1Yr1bH7PWklXM7IM5WJ8/9BiVSw==</latexit><latexit sha1_base64="LPZw+XumHh7sdkmdabWYCN5O27M=">AAACHXicZZDLSsNAFIYn9VbrrerSzWApCEJJFNRlsRtXUtFeoAlhMp20QyeTMHMilNgHceWjuBIUxI0L8W1M21RsPTDw8f/ncOb8XiS4BtP8NnJLyyura/n1wsbm1vZOcXevqcNYUdagoQhV2yOaCS5ZAzgI1o4UI4EnWMsb1MZ+654pzUN5B8OIOQHpSe5zSiCV3OKpLYjsCYav3aR2O8IPGRxb2FYTx01sFWAuNRAJoRzhglssmRVzUvg/WBmUUFZ1t/hpd0MaB0wCFUTrjmVG4CREAaeCjQp2rFlE6ID0WCdFSQKmnWRy3AiXU6WL/VClTwKeqH8nEhJoPQy8tDMg0NeL3lj89cpzq8C/cBIuoxiYpNNNfiwwhHgcFe5yxSiIYQqEKp5+FtM+UYRCGug0hbPZ8YswS6F5UrHMinVjlqqXWR55dIAO0RGy0DmqoitURw1E0SN6Rq/ozXgyXox342PamjOymX00V8bXD9CQoQs=</latexit>
sha1_base64="GzZW4FBRu2QpQ7pqYiuv2cai51Q=">AAACHXicZZBNS8NAEIYn9avWr6pHL4ulIAgl0YMei714kopWBVPCZrttl242YXcilNgf4kn/iSdBQbx4EP+Nm7aIHwMLD+87w+y8YSKFQdf9dAozs3PzC8XF0tLyyupaeX3jwsSpZrzFYhnrq5AaLoXiLRQo+VWiOY1CyS/DQSP3L2+4NiJW5zhMeDuiPSW6glG0UlDe9yVVPcnJSZA1zkbkdgq7HvH12AkyX0dEKINUYaxGpBSUK27NHRf5D94UKvVtKDwAQDMov/udmKURV8gkNebacxNsZ1SjYJKPSn5qeELZgPb4tUVFI27a2fi4EalapUO6sbZPIRmrPycyGhkzjELbGVHsm79eLn571V+rsHvYzoRKUuSKTTZ1U0kwJnlUpCM0ZyiHFijTwn6WsD7VlKENNE/B+3vzf7jYq3luzTu1cRzBpIqwBduwAx4cQB2OoQktYHAHj/AML8698+S8Om+T1oIzndmEX+V8fAEp8KIA</latexit>
sha1_base64="bChAqkWYU7cVeO0VRf7Mzh9VYQQ=">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</latexit>
sha1_base64="gDNjhDdQ7lPElREZF56jYsBWWdw=">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</latexit>−S[Acl ]= e<latexit sha1_base64="KEYkcVbl55EEVVw1NgtGGf+aJ5s=">AAACCXicZZDLSsNAFIYn9VbrLerShYOl4EJL0oW6EYpuXFawF2himEwn7dDJJMxMhBKydOWjuBIUxK2P4Mq3cdqmYusPAz//OYcz5/NjRqWyrG+jsLS8srpWXC9tbG5t75i7ey0ZJQKTJo5YJDo+koRRTpqKKkY6sSAo9Blp+8Prcb39QISkEb9To5i4IepzGlCMlI488/DSOYHkPj11AoFwWoNOTLPUQSweIK+dZZ5ZtqrWRPC/sXNTBrkanvnl9CKchIQrzJCUXduKlZsioShmJCs5iSQxwkPUJ11tOQqJdNPJIRms6KQHg0joxxWcpH8nUhRKOQp93RkiNZCLtXH4W6vMrVLBhZtSHieKcDzdFCQMqgiOscAeFQQrNtIGYUH1ZyEeII1EaXilCYWz2fGLZkahVavaVtW+tcr1q5xHERyAI3AMbHAO6uAGNEATYPAInsEreDOejBfj3fiYthaMfGYfzMn4/AFYf5lr</latexit>
sha1_base64="O21EYN148hvDwSS4OCNzQmoxYJ4=">AAACCXicZVDLSsNAFJ3UV62vqEsXDpaCCy1JN7oRim5cVrAPaGKZTCft0MlkmJkIJWTpyk9xJSiIWz/BlX/jtA1i2wMXDufcy733BIJRpR3nxyqsrK6tbxQ3S1vbO7t79v5BS8WJxKSJYxbLToAUYZSTpqaakY6QBEUBI+1gdDPx249EKhrzez0WxI/QgNOQYqSN1LOPr7wzSB7Scy+UCKc16AmapR5iYoh67Szr2WWn6kwBl4mbkzLI0ejZ314/xklEuMYMKdV1HaH9FElNMSNZyUsUEQiP0IB0DeUoIspPp49ksGKUPgxjaYprOFX/T6QoUmocBaYzQnqoFr2J+OdV5lbp8NJPKReJJhzPNoUJgzqGk1hgn0qCNRsbgrCk5liIh8hEok14JZOCu/jzMmnVqq5Tde+ccv06z6MIjsAJOAUuuAB1cAsaoAkweAIv4A28W8/Wq/Vhfc5aC1Y+cwjmYH39AlH1mVw=</latexit>

− α2π

W

∼ e−180<latexit sha1_base64="mEnUFx1qqnuDg1x+Iu8UKmzydn0=">AAAB+XicZVDLSgMxFM34rPXRUZdugqXgQkvGhXZZdOOygn1AZyyZNG1Dk8yQZIQy9EtcCQri1k9x5d+YTqdi64ELh3Pu5d57wpgzbRD6dtbWNza3tgs7xd29/YOSe3jU0lGiCG2SiEeqE2JNOZO0aZjhtBMrikXIaTsc38789hNVmkXywUxiGgg8lGzACDZW6rklXzPhn0P6mF54NTTtuWVURRngf+LlpAxyNHrul9+PSCKoNIRjrbseik2QYmUY4XRa9BNNY0zGeEi7lkosqA7S7PAprFilDweRsiUNzNS/EykWWk9EaDsFNiO96s3EX6+ytMoMakHKZJwYKsl80yDh0ERwFgPsM0WJ4RNLMFHMHgvJCCtMjA2rmKVwtXh+lSxSaF1WPVT17lG5fpPnUQAn4BScAQ9cgzq4Aw3QBAQk4Bm8gjcndV6cd+dj3rrm5DPHYAnO5w+iTZJF</latexit>
sha1_base64="QaGkOdnzvP/Bxx2pmxwBsesmcQU=">AAAB+XicZVBNSwMxEM36WetHVz16CZaCBy1ZL/ZY9OKxgv2Abi3ZdLYNTXaXJCuUpb/Ek6AgXv0pnvw3pu0itn0w8Hhvhpl5QSK4NoT8OBubW9s7u4W94v7B4VHJPT5p6ThVDJosFrHqBFSD4BE0DTcCOokCKgMB7WB8N/Pbz6A0j6NHM0mgJ+kw4iFn1Fip75Z8zaV/ieEpu/JqZNp3y6RK5sDrxMtJGeVo9N1vfxCzVEJkmKBadz2SmF5GleFMwLTopxoSysZ0CF1LIypB97L54VNcscoAh7GyFRk8V/9PZFRqPZGB7ZTUjPSqNxP/vMrSKhPWehmPktRAxBabwlRgE+NZDHjAFTAjJpZQprg9FrMRVZQZG1bRpuCt/rxOWtdVj1S9B1Ku3+Z5FNAZOkcXyEM3qI7uUQM1EUMpekFv6N3JnFfnw/lctG44+cwpWoLz9Qubw5I2</latexit>• At temperature around TeV, the sphaleron rate is order 1.
• What about the zero-temperature but high energy?

• One can think of transitions between disconnected vacua.
[Klinkhamer, Manton ’84]

Esph ≃ 9 TeV

sphaleronEunstable solution sitting on
top of the energy barrier<latexit sha1_base64="n5F3YGh1pPixcHkz5oUCogzoVNY=">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</latexit>
sha1_base64="p7NCtKLpHmXcLxgQVeT+0zRZ89A=">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</latexit>
sha1_base64="fm+jwWicEhRhfGpkVcEfa/lTtII=">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</latexit>
sha1_base64="DBAd22kzjAFcTahi1+wepoZXhXs=">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</latexit>

quantum tunneling

Instanton• At zero temperature/energy, the instanton rate is exponentially small:⟨NCS |NCS + 1⟩instanton ∼ e<latexit sha1_base64="69c162CA+EEMC1+uOQPKqw2bQ4s=">AAAB/3icZVDLSsNAFJ3UV62vqLhyM1gKLrQkLtRl1Y3LivYBTQyT6aQdOpmEmYlQQhZ+iitBQdz6Ha78G6dpKlYPXDiccy/33uPHjEplWV9GaWFxaXmlvFpZW9/Y3DK3d9oySgQmLRyxSHR9JAmjnLQUVYx0Y0FQ6DPS8UdXE7/zQISkEb9T45i4IRpwGlCMlJY8c8+RNHSOILlPj297F16KWeZmnlm16lYO+J/YBamCAk3P/HT6EU5CwhVmSMqebcXKTZFQFDOSVZxEkhjhERqQnqYchUS6aX5+Bmta6cMgErq4grn6eyJFoZTj0NedIVJD+debiD9ebW6VCs7dlPI4UYTj6aYgYVBFcBIG7FNBsGJjTRAWVB8L8RAJhJWOrJKncDp7/i+ZpdA+qdtW3b6xqo3LIo8y2AcH4BDY4Aw0wDVoghbAIAVP4AW8Go/Gs/FmvE9bS0YxswvmYHx8A/qilVo=</latexit>
sha1_base64="yUryVAFDzuayi+tmMriGdeqGSWQ=">AAAB/3icZVDLSsNAFJ3UV62vqLhyM1gKLrQkbnRZdeOyon1AE8NkOmmHzkzCzEQoIQs/xZWgIG79Dlf+jdM2iLUHLhzOuZd77wkTRpV2nG+rtLS8srpWXq9sbG5t79i7e20VpxKTFo5ZLLshUoRRQVqaaka6iSSIh4x0wtH1xO88EqloLO71OCE+RwNBI4qRNlJgH3iKcu8Ekofs9K53GWSY5X4e2FWn7kwBF4lbkCoo0AzsL68f45QToTFDSvVcJ9F+hqSmmJG84qWKJAiP0ID0DBWIE+Vn0/NzWDNKH0axNCU0nKp/JzLElRrz0HRypIfqvzcRf73a3CodXfgZFUmqicCzTVHKoI7hJAzYp5JgzcaGICypORbiIZIIaxNZxaTg/v95kbTP6q5Td2+dauOqyKMMDsEROAYuOAcNcAOaoAUwyMAzeAVv1pP1Yr1bH7PWklXM7IM5WJ8/9BiVSw==</latexit><latexit sha1_base64="LPZw+XumHh7sdkmdabWYCN5O27M=">AAACHXicZZDLSsNAFIYn9VbrrerSzWApCEJJFNRlsRtXUtFeoAlhMp20QyeTMHMilNgHceWjuBIUxI0L8W1M21RsPTDw8f/ncOb8XiS4BtP8NnJLyyura/n1wsbm1vZOcXevqcNYUdagoQhV2yOaCS5ZAzgI1o4UI4EnWMsb1MZ+654pzUN5B8OIOQHpSe5zSiCV3OKpLYjsCYav3aR2O8IPGRxb2FYTx01sFWAuNRAJoRzhglssmRVzUvg/WBmUUFZ1t/hpd0MaB0wCFUTrjmVG4CREAaeCjQp2rFlE6ID0WCdFSQKmnWRy3AiXU6WL/VClTwKeqH8nEhJoPQy8tDMg0NeL3lj89cpzq8C/cBIuoxiYpNNNfiwwhHgcFe5yxSiIYQqEKp5+FtM+UYRCGug0hbPZ8YswS6F5UrHMinVjlqqXWR55dIAO0RGy0DmqoitURw1E0SN6Rq/ozXgyXox342PamjOymX00V8bXD9CQoQs=</latexit>
sha1_base64="GzZW4FBRu2QpQ7pqYiuv2cai51Q=">AAACHXicZZBNS8NAEIYn9avWr6pHL4ulIAgl0YMei714kopWBVPCZrttl242YXcilNgf4kn/iSdBQbx4EP+Nm7aIHwMLD+87w+y8YSKFQdf9dAozs3PzC8XF0tLyyupaeX3jwsSpZrzFYhnrq5AaLoXiLRQo+VWiOY1CyS/DQSP3L2+4NiJW5zhMeDuiPSW6glG0UlDe9yVVPcnJSZA1zkbkdgq7HvH12AkyX0dEKINUYaxGpBSUK27NHRf5D94UKvVtKDwAQDMov/udmKURV8gkNebacxNsZ1SjYJKPSn5qeELZgPb4tUVFI27a2fi4EalapUO6sbZPIRmrPycyGhkzjELbGVHsm79eLn571V+rsHvYzoRKUuSKTTZ1U0kwJnlUpCM0ZyiHFijTwn6WsD7VlKENNE/B+3vzf7jYq3luzTu1cRzBpIqwBduwAx4cQB2OoQktYHAHj/AML8698+S8Om+T1oIzndmEX+V8fAEp8KIA</latexit>
sha1_base64="bChAqkWYU7cVeO0VRf7Mzh9VYQQ=">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</latexit>
sha1_base64="gDNjhDdQ7lPElREZF56jYsBWWdw=">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</latexit>−S[Acl ]= e<latexit sha1_base64="KEYkcVbl55EEVVw1NgtGGf+aJ5s=">AAACCXicZZDLSsNAFIYn9VbrLerShYOl4EJL0oW6EYpuXFawF2himEwn7dDJJMxMhBKydOWjuBIUxK2P4Mq3cdqmYusPAz//OYcz5/NjRqWyrG+jsLS8srpWXC9tbG5t75i7ey0ZJQKTJo5YJDo+koRRTpqKKkY6sSAo9Blp+8Prcb39QISkEb9To5i4IepzGlCMlI488/DSOYHkPj11AoFwWoNOTLPUQSweIK+dZZ5ZtqrWRPC/sXNTBrkanvnl9CKchIQrzJCUXduKlZsioShmJCs5iSQxwkPUJ11tOQqJdNPJIRms6KQHg0joxxWcpH8nUhRKOQp93RkiNZCLtXH4W6vMrVLBhZtSHieKcDzdFCQMqgiOscAeFQQrNtIGYUH1ZyEeII1EaXilCYWz2fGLZkahVavaVtW+tcr1q5xHERyAI3AMbHAO6uAGNEATYPAInsEreDOejBfj3fiYthaMfGYfzMn4/AFYf5lr</latexit>
sha1_base64="O21EYN148hvDwSS4OCNzQmoxYJ4=">AAACCXicZVDLSsNAFJ3UV62vqEsXDpaCCy1JN7oRim5cVrAPaGKZTCft0MlkmJkIJWTpyk9xJSiIWz/BlX/jtA1i2wMXDufcy733BIJRpR3nxyqsrK6tbxQ3S1vbO7t79v5BS8WJxKSJYxbLToAUYZSTpqaakY6QBEUBI+1gdDPx249EKhrzez0WxI/QgNOQYqSN1LOPr7wzSB7Scy+UCKc16AmapR5iYoh67Szr2WWn6kwBl4mbkzLI0ejZ314/xklEuMYMKdV1HaH9FElNMSNZyUsUEQiP0IB0DeUoIspPp49ksGKUPgxjaYprOFX/T6QoUmocBaYzQnqoFr2J+OdV5lbp8NJPKReJJhzPNoUJgzqGk1hgn0qCNRsbgrCk5liIh8hEok14JZOCu/jzMmnVqq5Tde+ccv06z6MIjsAJOAUuuAB1cAsaoAkweAIv4A28W8/Wq/Vhfc5aC1Y+cwjmYH39AlH1mVw=</latexit>

− α2π

W

∼ e−180<latexit sha1_base64="mEnUFx1qqnuDg1x+Iu8UKmzydn0=">AAAB+XicZVDLSgMxFM34rPXRUZdugqXgQkvGhXZZdOOygn1AZyyZNG1Dk8yQZIQy9EtcCQri1k9x5d+YTqdi64ELh3Pu5d57wpgzbRD6dtbWNza3tgs7xd29/YOSe3jU0lGiCG2SiEeqE2JNOZO0aZjhtBMrikXIaTsc38789hNVmkXywUxiGgg8lGzACDZW6rklXzPhn0P6mF54NTTtuWVURRngf+LlpAxyNHrul9+PSCKoNIRjrbseik2QYmUY4XRa9BNNY0zGeEi7lkosqA7S7PAprFilDweRsiUNzNS/EykWWk9EaDsFNiO96s3EX6+ytMoMakHKZJwYKsl80yDh0ERwFgPsM0WJ4RNLMFHMHgvJCCtMjA2rmKVwtXh+lSxSaF1WPVT17lG5fpPnUQAn4BScAQ9cgzq4Aw3QBAQk4Bm8gjcndV6cd+dj3rrm5DPHYAnO5w+iTZJF</latexit>
sha1_base64="QaGkOdnzvP/Bxx2pmxwBsesmcQU=">AAAB+XicZVBNSwMxEM36WetHVz16CZaCBy1ZL/ZY9OKxgv2Abi3ZdLYNTXaXJCuUpb/Ek6AgXv0pnvw3pu0itn0w8Hhvhpl5QSK4NoT8OBubW9s7u4W94v7B4VHJPT5p6ThVDJosFrHqBFSD4BE0DTcCOokCKgMB7WB8N/Pbz6A0j6NHM0mgJ+kw4iFn1Fip75Z8zaV/ieEpu/JqZNp3y6RK5sDrxMtJGeVo9N1vfxCzVEJkmKBadz2SmF5GleFMwLTopxoSysZ0CF1LIypB97L54VNcscoAh7GyFRk8V/9PZFRqPZGB7ZTUjPSqNxP/vMrSKhPWehmPktRAxBabwlRgE+NZDHjAFTAjJpZQprg9FrMRVZQZG1bRpuCt/rxOWtdVj1S9B1Ku3+Z5FNAZOkcXyEM3qI7uUQM1EUMpekFv6N3JnFfnw/lctG44+cwpWoLz9Qubw5I2</latexit>• At temperature around TeV, the sphaleron rate is order 1.
• What about the zero-temperature but high energy?
Setting limits on vacuum transitions from the LHCThis talk• The change of NCS is related to the following quantity:

2g∆NCS =
16π 2<latexit sha1_base64="erEnSVUwLyAewBDVwn7GKvjHEek=">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</latexit>
sha1_base64="HiUUt9anQT9u6b8OLojaGt6F4vA=">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</latexit>
sha1_base64="FThUEZZ4zTKK+LOPrt24IUua9ys=">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</latexit>
sha1_base64="MQAf3eMnnJ040vcFvm/95FJg85Y=">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</latexit>!

"#
Tr Fµν F̃ µν d4 x• The change of NCS is related to the following quantity:

2g∆NCS =
16π 2<latexit sha1_base64="erEnSVUwLyAewBDVwn7GKvjHEek=">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</latexit>
sha1_base64="HiUUt9anQT9u6b8OLojaGt6F4vA=">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</latexit>
sha1_base64="FThUEZZ4zTKK+LOPrt24IUua9ys=">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</latexit>
sha1_base64="MQAf3eMnnJ040vcFvm/95FJg85Y=">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</latexit>!

"#
Tr Fµν F̃ µν d4 x

= ∆NF<latexit sha1_base64="fIc9EIilQJw+V3DUx27rYWku9SU=">AAAB6XicZVBNS8NAEJ3Ur1q/qh69BEvBU0k8qBeh6MVjC/YD2lA220m7dLMJuxuhhP4CT4KCePUnefLfuE1TsfXBwOO9GWbm+TFnSjvOt1XY2Nza3inulvb2Dw6PyscnbRUlkmKLRjySXZ8o5ExgSzPNsRtLJKHPseNP7ud+5wmlYpF41NMYvZCMBAsYJdpIzdtBueLUnAz2f+LmpAI5GoPyV38Y0SREoSknSvVcJ9ZeSqRmlOOs1E8UxoROyAh7hgoSovLS7NCZXTXK0A4iaUpoO1P/TqQkVGoa+qYzJHqs1r25+OtVV1bp4MZLmYgTjYIuNgUJt3Vkz9+2h0wi1XxqCKGSmWNtOiaSUG3CKWUpXC2fXyfLFNqXNdepuU2nUr/L8yjCGZzDBbhwDXV4gAa0gALCM7zCmzWxXqx362PRWrDymVNYgfX5AxB+jPc=</latexit><latexit
sha1_base64="oOkSrOTONDP88C/Ly4kWWJmUctQ=">AAAB6XicZVBNS8NAEJ3Ur1q/qh69LJaCp5J40YtQ9OKxBfsBbSib7aRdutmE3Y1QQn+BJ0FBvPqTPPlv3LZBbPtg4PHeDDPzgkRwbVz3xylsbe/s7hX3SweHR8cn5dOzto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmD3O/84xK81g+mWmCfkRHkoecUWOl5t2gXHFr7gJkk3g5qUCOxqD83R/GLI1QGiao1j3PTYyfUWU4Ezgr9VONCWUTOsKepZJGqP1sceiMVK0yJGGsbElDFur/iYxGWk+jwHZG1Iz1ujcX/7zqyioT3voZl0lqULLlpjAVxMRk/jYZcoXMiKkllClujyVsTBVlxoZTsil46z9vkvZ1zXNrXtOt1O/zPIpwAZdwBR7cQB0eoQEtYIDwAm/w7kycV+fD+Vy2Fpx85hxW4Hz9Agn0jOg=</latexit>

anomaly<latexit sha1_base64="w0fw65p6hitbuDWJvYFLEVTAZjk=">AAAB9XicZVDLSgNBEOyNrxhfUY9eBkPAU9j1oB6DXjxGMA9IljA7mSRD5rHOzCrLku/wJCiIVz/Gk3/jZrMREwsaiqpuuruCkDNjXffbKaytb2xuFbdLO7t7+wflw6OWUZEmtEkUV7oTYEM5k7RpmeW0E2qKRcBpO5jczPz2I9WGKXlv45D6Ao8kGzKCbSr5SU8LhKUSmMfTfrni1twM6D/xclKBHI1++as3UCQSVFrCsTFdzw2tn2BtGeF0WupFhoaYTPCIdlMqsaDGT7Kjp6iaKgM0VDotaVGm/p1IsDAmFkHaKbAdm1VvJv561aVVdnjlJ0yGkaWSzDcNI46sQrMI0IBpSiyPU4KJZumxiIyxxsSmQZWyFC4Wz6+SRQqt85rn1rw7t1K/zvMowgmcwhl4cAl1uIUGNIHAAzzDK7w5T86L8+58zFsLTj5zDEtwPn8AjneSdg==</latexit>
sha1_base64="ZD4pztw8WQEZrRdyQ3Fg88Y42RY=">AAAB9XicZVDLSgNBEOz1GeMr6lEPgyHgKex60WPQi8cI5gHJEmYnk2TIPNaZWWVZ8h2eBAXx6sd48m+cbIKYpKChqOqmuyuKOTPW93+8tfWNza3twk5xd2//4LB0dNw0KtGENojiSrcjbChnkjYss5y2Y02xiDhtRePbqd96otowJR9sGtNQ4KFkA0awdVKYdbVAWCqBeTrplcp+1c+BVkkwJ+XaGeSo90rf3b4iiaDSEo6N6QR+bMMMa8sIp5NiNzE0xmSMh7TjqMSCmjDLj56gilP6aKC0K2lRrv6fyLAwJhWR6xTYjsyyNxX/vMrCKju4DjMm48RSSWabBglHVqFpBKjPNCWWp45gopk7FpER1phYF1TRpRAs/7xKmpfVwK8G9y6Om1kcUIBTOIcLCOAKanAHdWgAgUd4gTd49569V+/D+5y1rnnzmRNYgPf1CwVMksM=</latexit>
sha1_base64="cKOnnc9uhs+J13Uf50OHZcnTBbQ=">AAAB9XicZZDLSgMxFIbPeK31VnWpi2ApuCozLtRl0Y3LCvYC7VAyaaYNTTJjklGGoc/hSlAQtz6MK9/GdNqKrT8Efv5zDufkC2LOtHHdb2dldW19Y7OwVdze2d3bLx0cNnWUKEIbJOKRagdYU84kbRhmOG3HimIRcNoKRjeTeuuRKs0ieW/SmPoCDyQLGcHGRn7WVQJhGQnM03GvVHarbi7033gzU66dQK56r/TV7UckEVQawrHWHc+NjZ9hZRjhdFzsJprGmIzwgHaslVhQ7Wf50WNUsUkfhZGyTxqUp38nMiy0TkVgOwU2Q71cm4S/tcrCKhNe+RmTcWKoJNNNYcKRidAEAeozRYnhqTWYKGaPRWSIFSbGgirmFC7mn182cwrN86rnVr07i+N6igMKcAyncAYeXEINbqEODSDwAM/wCm/Ok/PivDsf09YVZzZzBAtyPn8AC9aS0g==</latexit><latexit sha1_base64="7fwLhOhsn20UVfntvaTn62WFcdY=">AAAB8nicZVBNS8NAEN3Ur1q/qh69LJaCp5J4UI9FRTxJBfuBbSib7aRdutmE3YlQSv+FJ0FBvPpvPPlvTNNUbH0w8Hhvhpl5XiSFQdv+tnIrq2vrG/nNwtb2zu5ecf+gYcJYc6jzUIa65TEDUiioo0AJrUgDCzwJTW94NfWbT6CNCNUDjiJwA9ZXwhecYSI9dq5BIqN33ZtusWRX7BT0P3EyUiIZat3iV6cX8jgAhVwyY9qOHaE7ZhoFlzApdGIDEeND1od2QhULwLjj9OIJLSdKj/qhTkohTdW/E2MWGDMKvKQzYDgwy95U/PXKC6vQv3DHQkUxguKzTX4sKYZ0+j/tCQ0c5SghjGuRHEv5gGnGMUmpkKZwNn9+mcxTaJxWHLvi3Nul6mWWR54ckWNyQhxyTqrkltRInXCiyDN5JW8WWi/Wu/Uxa81Z2cwhWYD1+QMmg5Bt</latexit>
sha1_base64="WfMbtMwJY9eEacQ6wXtkrEqb7d4=">AAAB8nicZVDLSgNBEOyNrxhfUfHkZTAEPIVdL3oMKuJJIpgHJkuYnUySIbOzy0yvEELAD/DoRUFBvPo3nvwbJw/EJAUNRVU33V1BLIVB1/1xUkvLK6tr6fXMxubW9k52d69iokQzXmaRjHQtoIZLoXgZBUpeizWnYSB5NehdjPzqA9dGROoO+zH3Q9pRoi0YRSvdNy65REpumlfNbM4tuGOQReJNSa548PL0CAClZva70YpYEnKFTFJj6p4boz+gGgWTfJhpJIbHlPVoh9ctVTTkxh+MLx6SvFVapB1pWwrJWP0/MaChMf0wsJ0hxa6Z90bin5efWYXtM38gVJwgV2yyqZ1IghEZ/U9aQnOGsm8JZVrYYwnrUk0Z2pQyNgVv/udFUjkpeG7Bu7VxnMMEaTiEIzgGD06hCNdQgjIwUPAMb/DuoPPqfDifk9aUM53Zhxk4X7/M/pJY</latexit>
sha1_base64="d+mqtW8gy2uYc26lqBdnhsRX684=">AAAB8nicZVDLSgNBEOyNr5j4iIonL4MxkFPY9aLHoCKeJIJ5YBLC7GSSDJmdXWZ6hRDiF3j0oqAgXv0bT/6NkwdiYkFDUdVNd5cfSWHQdb+dxNLyyupacj2V3tjc2s7s7FZMGGvGyyyUoa751HApFC+jQMlrkeY08CWv+v3zsV+959qIUN3iIOLNgHaV6AhG0Up3jQsukZLr1mUrk3UL7gTkP/FmJFvcf358yB+lS63MV6MdsjjgCpmkxtQ9N8LmkGoUTPJRqhEbHlHWp11et1TRgJvmcHLxiOSs0iadUNtSSCbq34khDYwZBL7tDCj2zKI3Fn+93Nwq7Jw2h0JFMXLFpps6sSQYkvH/pC00ZygHllCmhT2WsB7VlKFNKWVT8BZ//k8qxwXPLXg3No4zmCIJB3AIefDgBIpwBSUoAwMFT/AKbw46L8678zFtTTizmT2Yg/P5Az8Jkq0=</latexit>
sha1_base64="kKEnyCyx0iupiSp+P/0PH5wNsyw=">AAAB8nicZVDLSgNBEJyNr5j4iIonL4MayCnselCPQUU8SQTzwGQJs5NJMmR2dpnpFcISv8CjFwUF8erfePJvnGwSMbGgoajqprvLCwXXYNvfVmphcWl5Jb2aya6tb2zmtrarOogUZRUaiEDVPaKZ4JJVgINg9VAx4nuC1bz++civ3TOleSBvYRAy1yddyTucEjDSXfOCCSD4unXZyh3YRTsB/k+cCTko7T4/PhQOs+VW7qvZDmjkMwlUEK0bjh2CGxMFnAo2zDQjzUJC+6TLGoZK4jPtxsnFQ5w3Sht3AmVKAk7UvxMx8bUe+J7p9An09Lw3En+9/Mwq6Jy6MZdhBEzS8aZOJDAEePQ/bnPFKIiBIYQqbo7FtEcUoWBSyiQpHE+fnyfTFKpHRccuOjcmjjM0RhrtoX1UQA46QSV0hcqogiiS6Am9ojcLrBfr3foYt6asycwOmoH1+QNFk5K8</latexit>

SU(2) charged fermion• The change of NCS is related to the following quantity:

2g∆NCS =
16π 2<latexit sha1_base64="erEnSVUwLyAewBDVwn7GKvjHEek=">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</latexit>
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sha1_base64="cgY//rScT8vilBvlFX5xyJFbzuY=">AAAB+nicZVDLSgMxFM34rPU16tJNUAquhqQ4tSsp6sKFSAXbKm0dMmlag5mHSaZQxv6JK0FBBFf+iSv/xnSqYvXAhcM593LvPX4suNIIfVhT0zOzc/O5hfzi0vLKqr22XldRIimr0UhE8sInigkesprmWrCLWDIS+II1/JvDkd/oM6l4FJ7rQczaAemFvMsp0UbybLt1xIQm8NRLbz181Rt69jZyUMndxWWIHBfhMnYNKboYoSLEDsqwXUHu6744uax69nurE9EkYKGmgijVxCjW7ZRIzalgw3wrUSwm9Ib0WNPQkARMtdPs8iEsGKUDu5E0FWqYqb8nUhIoNQh80xkQfa3+eiPxxytMrNLdcjvlYZxoFtLxpm4ioI7gKAfY4ZJRLQaGECq5ORbSayIJ1SatfJZCKXsV/iffKdSLDkYOPjNxHIAxcmATbIEdgMEeqIBjUAU1QEEf3INH8GTdWQ/Ws/Uybp2yvmY2wASst08ztZYq</latexit>∆Nq1b<latexit sha1_base64="9zUrjkCIs8HNjozd8yxIJaY1zgM=">AAACI3icZVBJSwMxGM241rpVPXoJFsGDlMRDl1tRD56kgl2gHYdMmrahmcUkI5Rhfosnf4onQUG9ePC/mJm24vJB4PHe+5Y8NxRcaYQ+rIXFpeWV1dxafn1jc2u7sLPbUkEkKWvSQASy4xLFBPdZU3MtWCeUjHiuYG13fJbq7TsmFQ/8az0Jme2Roc8HnBJtKKdQi3vZkK4cunaMSgghjPFxCnCljAyo1aonuJr0zpnQBF468a2Db9wkcQrFuR3O7XBuhziVTBXBrBpO4a3XD2jkMV9TQZTqYhRqOyZScypYku9FioWEjsmQdQ30iceUHWfXJfDQMH04CKR5voYZ+7MjJp5SE881To/okfqrpeS3dvhrlR5U7Zj7YaSZT6ebBpGAOoBpYLDPJaNaTAwgVHJzLKQjIgnVJtZ8lkI5+yr8D+YptE5KGJXwFSrWT2d55MA+OABHAIMKqIML0ABNQME9eATP4MV6sJ6sV+t9al2wZj174FdZn1+dV6CD</latexit>
sha1_base64="bwbrxMbM6qAehvy/KbDzAQC1Qbg=">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</latexit>
sha1_base64="eu/WTw1bUjUqetMfYPjJwXjRBAw=">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</latexit>
sha1_base64="R4EFP95Z3sSRb/yz+yJNdT4v/Yo=">AAACI3icZVDLSgMxFM3UV62vqks3wVJwISXpQttdUZGuRMGq0I5DJk1raOZhkhHKMDv/w5V7f8KVYEHduPBfzEyt+LgQOJxz7iPHDQVXGqF3Kzc1PTM7l58vLCwuLa8UV9fOVBBJylo0EIG8cIligvuspbkW7CKUjHiuYOfuYD/Vz2+YVDzwT/UwZLZH+j7vcUq0oZxiPe5kQ9qy79oxqiCEMMbbKcC7O8iAer1WxbWkc8CEJvDIia8dfOkmiVMsTexwYocTO8SpZKrUwNHD4W1zdOwUXzvdgEYe8zUVRKk2RqG2YyI1p4IlhU6kWEjogPRZ20CfeEzZcXZdAsuG6cJeIM3zNczYnx0x8ZQaeq5xekRfqb9aSn5r5V+rdK9mx9wPI818Ot7UiwTUAUwDg10uGdViaAChkptjIb0iklBtYi1kKexkX4X/wSSFs2oFowo+MXHsgXHlwQbYBFsAg13QAE1wDFqAgjvwCJ7ByLq3nqwX621szVlfPevgV1kfn0ito/o=</latexit>∆Nℓ1<latexit sha1_base64="exXUMb/aF8wzK0wsKgo9KFKe1so=">AAAB+3icZVBNS8NAEN3Ur1q/Uj16WSwFTyXxoB6LevAkFewHNCFstpN26WYTdjdKCf0pngQF8eov8eS/MU1TsfXBwOO9GWbm+TFnSlvWt1FaW9/Y3CpvV3Z29/YPzOphR0WJpNCmEY9kzycKOBPQ1kxz6MUSSOhz6Prj65nffQSpWCQe9CQGNyRDwQJGic4kz6w6N8A1wXde6gDnnj31zJrVsHLg/8QuSA0VaHnmlzOIaBKC0JQTpfq2FWs3JVIzymFacRIFMaFjMoR+RgUJQblpfvoU1zNlgINIZiU0ztW/EykJlZqEftYZEj1Sq95M/PXqS6t0cOmmTMSJBkHnm4KEYx3hWRB4wCRQzScZIVSy7FhMR0QSqrO4KnkK54vnV8kihc5Zw7Ya9r1Va14VeZTRMTpBp8hGF6iJblELtRFFT+gZvaI3Y2q8GO/Gx7y1ZBQzR2gJxucPGYeTvw==</latexit>
sha1_base64="A0ocvvDjb3HdTK4ozX8pKpYcz8c=">AAAB+3icZVDJSgNBEK2JW4zbRI9eWkPAU5jxosegHjxJBLNAEoaeTiVp0rPQ3aOEIeCPeBIUxKtf4slP8aSdBTHJg4LHe1VU1fNjwZV2nC8rs7K6tr6R3cxtbe/s7tn5/ZqKEsmwyiIRyYZPFQoeYlVzLbARS6SBL7DuDy7Hfv0epeJReKeHMbYD2gt5lzOqjeTZ+dYVCk3JjZe2UAjPHXl2wSk5E5Bl4s5IoXz08/0IABXP/mx1IpYEGGomqFJN14l1O6VScyZwlGslCmPKBrSHTUNDGqBqp5PTR6RolA7pRtJUqMlE/T+R0kCpYeCbzoDqvlr0xuKfV5xbpbvn7ZSHcaIxZNNN3UQQHZFxEKTDJTIthoZQJrk5lrA+lZRpE1fOpOAu/rxMaqcl1ym5tyaOC5giC4dwDCfgwhmU4RoqUAUGD/AEL/Bqjaxn6816n7ZmrNnMAczB+vgF7fWWiQ==</latexit>
sha1_base64="7D7+jAaeizY8LZkNUdgbGPtFw0E=">AAAB+3icZVDJSgNBEO2JW4zbRI9eWkPAU5jxot6CC3qSCGaBzDD0dCpJk56F7h4lDPMpngQF8eqXePJTPGlnQUzyoODxXhVV9fyYM6ks68vILS2vrK7l1wsbm1vbO2ZxtyGjRFCo04hHouUTCZyFUFdMcWjFAkjgc2j6g4uR33wAIVkU3qthDG5AeiHrMkqUljyz6FwCVwTfeqkDnHt25pklq2KNgReJPSWl6sHPt7i+Oqt55qfTiWgSQKgoJ1K2bStWbkqEYpRDVnASCTGhA9KDtqYhCUC66fj0DJe10sHdSOgKFR6r/ydSEkg5DHzdGRDVl/PeSPzzyjOrVPfUTVkYJwpCOtnUTThWER4FgTtMAFV8qAmhguljMe0TQajScRV0Cvb8z4ukcVyxrYp9p+M4RxPk0T46REfIRieoim5QDdURRY/oCb2gVyMzno03433SmjOmM3toBsbHL+Mnl0A=</latexit>
sha1_base64="fPTtwzdiaWOko7xwjri6hrhycO4=">AAAB+3icZVDJSgNBEO2JW4zbRI9eWkPAU5jx4HILLuhJIpgFMsPQ06kkTXoWunuUMORTPAkK4tUv8eSneNLOJiY+KHi8V0VVPT/mTCrL+jQyC4tLyyvZ1dza+sbmlpnfrskoERSqNOKRaPhEAmchVBVTHBqxABL4HOp+73zo1+9BSBaFd6ofgxuQTsjajBKlJc/MOxfAFcE3XuoA55498MyCVbJGwP+JPSGF8t73l7i6PK145ofTimgSQKgoJ1I2bStWbkqEYpTDIOckEmJCe6QDTU1DEoB009HpA1zUSgu3I6ErVHik/p1ISSBlP/B1Z0BUV857Q/HXK86sUu0TN2VhnCgI6XhTO+FYRXgYBG4xAVTxviaECqaPxbRLBKFKx5UbpXA0fX6eTFOoHZZsq2Tf6jjO0BhZtIv20QGy0TEqo2tUQVVE0QN6RM/oxRgYT8ar8TZuzRiTmR00A+P9B+mxl08=</latexit>× 3 flavour<latexit sha1_base64="sm6pS37f/wSVeCVNll9KMr77e7E=">AAACA3icZVDLSgMxFM3UV62vUVfiJlgKLkqZUVCXRTcuK9gHOEPJpJk2NMkMSaZQhuLKT3ElKIhbv8KVf2M6nYqtFwKHc+69J/cEMaNKO863VVhZXVvfKG6WtrZ3dvfs/YOWihKJSRNHLJKdACnCqCBNTTUjnVgSxANG2sHwZqq3R0QqGol7PY6Jz1Ff0JBipA3VtY88TTlR0Kuee9XUkxyGDI3M9knXLjs1Jyv4H7g5KIO8Gl37y+tFOOFEaMyQUg+uE2s/RVJTzMik5CWKxAgPUZ88GCiQ8fXT7IQJrBimB8NImic0zNi/EyniSo15YDo50gO1rE3JX62yYKXDKz+lIk40EXjmFCYM6ghOA4E9KgnWbGwAwpKaz0I8QBJhbWIrZSlczI9fBvMUWmc116m5d065fp3nUQTH4AScAhdcgjq4BQ3QBBg8gmfwCt6sJ+vFerc+Zq0FK585BAtlff4ACYqXDg==</latexit>
sha1_base64="73vafpkBaaMcdgnpqSwr8pu1nqc=">AAACA3icZVDLSgMxFL3jq7W+Rl2Jm2ApuChlRhe6LLpxWcE+oFNKJs20oZnMkGQKZSi68VNcCQri1q9w5Ye41vSB2PZC4HDOvffkHj/mTGnH+bJWVtfWNzLZzdzW9s7unr1/UFNRIgmtkohHsuFjRTkTtKqZ5rQRS4pDn9O6378e6/UBlYpF4k4PY9oKcVewgBGsDdW2jzzNQqqQVzz3iqknQxRwPDDbR20775ScSaFl4M5Avpz5+X4AgErb/vQ6EUlCKjThWKmm68S6lWKpGeF0lPMSRWNM+rhLmwYKbHxb6eSEESoYpoOCSJonNJqw/ydSHCo1DH3TGWLdU4vamPzTCnNWOrhspUzEiaaCTJ2ChCMdoXEgqMMkJZoPDcBEMvNZRHpYYqJNbDmTgrt48zKonZVcp+TemjiuYFpZOIYTOAUXLqAMN1CBKhC4hyd4gVfr0Xq23qz3aeuKNZs5hLmyPn4BtTiZug==</latexit>
sha1_base64="+UHlZTMXqpL+CQWPx10Bc8Kew7w=">AAACA3icZVDLSgMxFM3UR2t9jboSN8FScFHKjC4U3BTduKxgH9ApJZNm2tAkMySZQhmKC3HrX3QlKIhbv8KVH+Ja0wdi2wuBwzn33pN7/IhRpR3ny0qtrK6tpzMb2c2t7Z1de2+/qsJYYlLBIQtl3UeKMCpIRVPNSD2SBHGfkZrfux7rtT6RiobiTg8i0uSoI2hAMdKGatmHnqacKOgVzrxC4kkOA4b6ZvuwZeecojMpuAzcGciV0j/fD6PLp3LL/vTaIY45ERozpFTDdSLdTJDUFDMyzHqxIhHCPdQhDQMFMr7NZHLCEOYN04ZBKM0TGk7Y/xMJ4koNuG86OdJdtaiNyT8tP2elg4tmQkUUayLw1CmIGdQhHAcC21QSrNnAAIQlNZ+FuIskwtrEljUpuIs3L4PqadF1iu6tieMKTCsDjsAxOAEuOAclcAPKoAIwuAcj8AJerUfr2Xqz3qetKWs2cwDmyvr4BaPamys=</latexit>
sha1_base64="12b1VIrr2SxiDuuc8kxCX3d3W6Q=">AAACA3icZVDLSgMxFM3UR2t9jboSN8FScFHKjIIKbopuXFawD+iUkkkzbWgyMySZQhmKC3HrX3QlKIhbv8KVH+Ja02krtl4IHM65957c44aMSmVZn0ZqaXllNZ1Zy65vbG5tmzu7VRlEApMKDlgg6i6ShFGfVBRVjNRDQRB3Gam5vauxXusTIWng36pBSJocdXzqUYyUplrmvqMoJxI6hROnEDuCQ4+hvt4+bJk5q2glBf8DewpypfT31/3o4rHcMj+cdoAjTnyFGZKyYVuhasZIKIoZGWadSJIQ4R7qkIaGPtK+zTg5YQjzmmlDLxD6+Qom7N+JGHEpB9zVnRyprlzUxuSvlp+zUt55M6Z+GCni44mTFzGoAjgOBLapIFixgQYIC6o/C3EXCYSVji2bpHA6O34RzFKoHhdtq2jf6DguwaQy4AAcgiNggzNQAtegDCoAgzswAs/gxXgwnoxX423SmjKmM3tgroz3H6pkmzo=</latexit>• The change of NCS is related to the following quantity:

2g∆NCS =
16π 2<latexit sha1_base64="erEnSVUwLyAewBDVwn7GKvjHEek=">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</latexit>
sha1_base64="HiUUt9anQT9u6b8OLojaGt6F4vA=">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</latexit>
sha1_base64="FThUEZZ4zTKK+LOPrt24IUua9ys=">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</latexit>
sha1_base64="MQAf3eMnnJ040vcFvm/95FJg85Y=">AAACVXicZVFdaxQxFM2Mtdb1o6s++nKxFHwoy0yR6otYWik+ScVuW9jMDplMZjdskhmSO9Il7C/pc/+Jf0B8EJ/9ET4JZndbaeuBwMk593JvTopGSYdJ8jOK76zcXb23dr/z4OGjx+vdJ0+PXd1aLvq8VrU9LZgTShrRR4lKnDZWMF0ocVJM9uf+yRdhnazNEU4bkWk2MrKSnGGQ8u6EvhcKGXzM/f7nGdCtt3QLaGUZ96Ph9synO0AbGRhQaRA8tRqObLgpUeEADnJPdQvUtEFCqUoBB8NrkpWjMWZQDl/BGeTdjaSXLAD/k/SSbOy+I/D1/OLXYd79Tsuat1oY5Io5N0iTBjPPLEquxKxDWycaxidsJAaBGqaFy/wilBlsBqWEqrbhhM0X6vUOz7RzU12ESs1w7G57c/Gft3ljFFZvMi9N06IwfDmpahVgDfOIoZRWcFTTQBi3MiwLfMxCpBg+orNIYefq8bfJVQrH27006aWfQhx7ZIk18py8IC9JSl6TXfKBHJI+4eQb+R2RKIp+RH/ilXh1WRpHlz3PyA3E638Bbs60ew==</latexit>!

"#
Tr Fµν F̃ µν d4 x=
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sha1_base64="fPTtwzdiaWOko7xwjri6hrhycO4=">AAAB+3icZVDJSgNBEO2JW4zbRI9eWkPAU5jx4HILLuhJIpgFMsPQ06kkTXoWunuUMORTPAkK4tUv8eSneNLOJiY+KHi8V0VVPT/mTCrL+jQyC4tLyyvZ1dza+sbmlpnfrskoERSqNOKRaPhEAmchVBVTHBqxABL4HOp+73zo1+9BSBaFd6ofgxuQTsjajBKlJc/MOxfAFcE3XuoA55498MyCVbJGwP+JPSGF8t73l7i6PK145ofTimgSQKgoJ1I2bStWbkqEYpTDIOckEmJCe6QDTU1DEoB009HpA1zUSgu3I6ErVHik/p1ISSBlP/B1Z0BUV857Q/HXK86sUu0TN2VhnCgI6XhTO+FYRXgYBG4xAVTxviaECqaPxbRLBKFKx5UbpXA0fX6eTFOoHZZsq2Tf6jjO0BhZtIv20QGy0TEqo2tUQVVE0QN6RM/oxRgYT8ar8TZuzRiTmR00A+P9B+mxl08=</latexit>• ΔNCS is related to the change of SU(2) charged fermion numbers.
ĉL

t̂L

ℓe
ĉL

t̂L

∆n = 1

ûL
ĉL
ℓτ

ûL

t̂L

ûL

|ΔNCS| = 1 transition
creates 12 fermionsaltogether!

ℓµParty at collider!

• The LO Matrix Element in the instanton backgroundiM ∼<latexit sha1_base64="D6L+ZaICho9hAxwOlcPt6Dumdys=">AAAB+3icZVDLSsNAFJ3UV62vVJduBkvBhZTEhbosunEjVLAPaEKZTCft0JlJmJkoJeRTXAkK4tYvceXfOE1TsfXAhcM593LvPUHMqNKO822V1tY3NrfK25Wd3b39A7t62FFRIjFp44hFshcgRRgVpK2pZqQXS4J4wEg3mNzM/O4jkYpG4kFPY+JzNBI0pBhpIw3sKoWphxGDdxn0zjxF+cCuOQ0nB/xP3ILUQIHWwP7yhhFOOBEaM6RU33Vi7adIaooZySpeokiM8ASNSN9QgThRfpqfnsG6UYYwjKQpoWGu/p1IEVdqygPTyZEeq1VvJv569aVVOrzyUyriRBOB55vChEEdwVkQcEglwZpNDUFYUnMsxGMkEdYmrkqewsXi+VWySKFz3nCdhnvv1JrXRR5lcAxOwClwwSVoglvQAm2AwRN4Bq/gzcqsF+vd+pi3lqxi5ggswfr8AXCWk1I=</latexit>
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DqDW Dφ q(x1 ) · · · q(x12 )W (y1 ) · · · W (ynW )φ(z1 ) · · · φ(znh ) exp(−SE )LSZ<latexit sha1_base64="F+ngYZ/X9rvBvbF+ALqe09uvdBw=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrQT0WvXjwUNF+YLuUbJptQ5PsmmSFUvojPAkK4tXf48l/Y7rdiq0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUYTWScQj1QqwppxJWjfMcNqKFcUi4LQZDK+mfvOJKs0ieW9GMfUF7ksWMoKNlZodJdDN3UO3WHIrbgr0n3gZKUGGWrf41elFJBFUGsKx1m3PjY0/xsowwumk0Ek0jTEZ4j5tWyqxoNofp+dOUNkqPRRGypY0KFX/Toyx0HokAtspsBnoZW8q/nrlhVUmvPDHTMaJoZLMNoUJRyZC0+dRjylKDB9Zgoli9lhEBlhhYmxEhTSFs/nzy2SeQuO04rkV79YtVS+zPPJwBMdwAh6cQxWuoQZ1IDCEZ3iFN+fReXHenY9Za87JZg5hAc7nDyKKj0o=</latexit>
sha1_base64="zBgX7awZEi4o0/qSRepoKfLUY7k=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrRY9FLx48VLQf2C4lm2bb0CS7JlmhLP0RngQF8erv8eS/Md0uYtsHA4/3ZpiZF8ScaeO6P05hbX1jc6u4XdrZ3ds/KB8etXSUKEKbJOKR6gRYU84kbRpmOO3EimIRcNoOxtczv/1MlWaRfDCTmPoCDyULGcHGSu2eEuj2/rFfrrg1NwNaJV5OKpCj0S9/9wYRSQSVhnCsdddzY+OnWBlGOJ2WeommMSZjPKRdSyUWVPtpdu4UVa0yQGGkbEmDMvX/RIqF1hMR2E6BzUgvezPxz6surDLhpZ8yGSeGSjLfFCYcmQjNnkcDpigxfGIJJorZYxEZYYWJsRGVbAre8s+rpHVe89yad+dW6ld5HkU4gVM4Aw8uoA430IAmEBjDC7zBu/PkvDofzue8teDkM8ewAOfrFxwAjzs=</latexit>• The LO Matrix Element in the instanton backgroundiM ∼<latexit sha1_base64="D6L+ZaICho9hAxwOlcPt6Dumdys=">AAAB+3icZVDLSsNAFJ3UV62vVJduBkvBhZTEhbosunEjVLAPaEKZTCft0JlJmJkoJeRTXAkK4tYvceXfOE1TsfXAhcM593LvPUHMqNKO822V1tY3NrfK25Wd3b39A7t62FFRIjFp44hFshcgRRgVpK2pZqQXS4J4wEg3mNzM/O4jkYpG4kFPY+JzNBI0pBhpIw3sKoWphxGDdxn0zjxF+cCuOQ0nB/xP3ILUQIHWwP7yhhFOOBEaM6RU33Vi7adIaooZySpeokiM8ASNSN9QgThRfpqfnsG6UYYwjKQpoWGu/p1IEVdqygPTyZEeq1VvJv569aVVOrzyUyriRBOB55vChEEdwVkQcEglwZpNDUFYUnMsxGMkEdYmrkqewsXi+VWySKFz3nCdhnvv1JrXRR5lcAxOwClwwSVoglvQAm2AwRN4Bq/gzcqsF+vd+pi3lqxi5ggswfr8AXCWk1I=</latexit>
sha1_base64="4B8KsYoKLanX1MRyMSZVmzDSCig=">AAAB+3icZVBNS8NAEJ3Ur1q/Uj16WSwFD1ISL3osevEiVLAf0ISy2W7apZtN2N0oJeSneBIUxKu/xJP/xm0bxLYPBh7vzTAzL0g4U9pxfqzSxubW9k55t7K3f3B4ZFePOypOJaFtEvNY9gKsKGeCtjXTnPYSSXEUcNoNJrczv/tEpWKxeNTThPoRHgkWMoK1kQZ2laHMI5ij+xx5F55i0cCuOQ1nDrRO3ILUoEBrYH97w5ikERWacKxU33US7WdYakY4zSteqmiCyQSPaN9QgSOq/Gx+eo7qRhmiMJamhEZz9f9EhiOlplFgOiOsx2rVm4l/Xn1plQ6v/YyJJNVUkMWmMOVIx2gWBBoySYnmU0Mwkcwci8gYS0y0iatiUnBXf14nncuG6zTcB6fWvCnyKMMpnME5uHAFTbiDFrSBwDO8wBu8W7n1an1Yn4vWklXMnMASrK9fagyTQw==</latexit>!
<latexit sha1_base64="YVS97ViuOdinheVQWjoW8eRcTag=">AAACt3icZVFda9swFJW9rzb7aLY97kUsFBzogpzQtHkr+4A9dmypO2JjZFmORWXZkeTRzPgn7qlP/TeVHTek3QXBueeee650FRWcKY3QrWU/efrs+Yu9/d7LV6/fHPTfvrtQeSkJnZOc5/IywopyJuhcM83pZSEpziJOvejqS1P3/lCpWC5+6XVBgwwvBUsYwdpQYf+fz4SGlU8wh1/r1RZ5W+QXKfOPYM/kjVsV8ZLWcOVch+4Q+iTOtWqzyh3XQ1hvdZLGNfSc9Y6sySoReru6hVxGQYVGaHo8m6AjNDpG7slsZgBC09PJuJ3v/N1x6QhjlLZGPr0unE8/w2/DXtgfNH1NwP+B24EB6OI87N/4cU7KjApNOFZq4aJCBxWWmhFOjXepaIHJFV7ShYECZ1QFVXvxGh4aJoZJLs0xS2zZ3Y4KZ0qts8goM6xT9bjWkNva4YNROjkNKiaKUlNBNpOSkkOdw+YTYcwkJZqvDcBEMnNZSFIsMdHmqzdbmN4//jG438LFeOSikfsDDc4+d/vYAx/AR+AAF5yAM/AdnIM5INbE+m1FFrFndmgndrqR2lbX8x48CHt1B6e2zvw=</latexit>
sha1_base64="im/AbGSyMn3aCSgxzOlToV7sOfo=">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</latexit>
sha1_base64="XOA//vyPji1bAnO5MCs5E68MStk=">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</latexit>
sha1_base64="XUHzBIdAOtdM3UjoMRdDkF13AXo=">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</latexit>

DqDW Dφ q(x1 ) · · · q(x12 )W (y1 ) · · · W (ynW )φ(z1 ) · · · φ(znh ) exp(−SE )LSZ<latexit sha1_base64="F+ngYZ/X9rvBvbF+ALqe09uvdBw=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrQT0WvXjwUNF+YLuUbJptQ5PsmmSFUvojPAkK4tXf48l/Y7rdiq0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUYTWScQj1QqwppxJWjfMcNqKFcUi4LQZDK+mfvOJKs0ieW9GMfUF7ksWMoKNlZodJdDN3UO3WHIrbgr0n3gZKUGGWrf41elFJBFUGsKx1m3PjY0/xsowwumk0Ek0jTEZ4j5tWyqxoNofp+dOUNkqPRRGypY0KFX/Toyx0HokAtspsBnoZW8q/nrlhVUmvPDHTMaJoZLMNoUJRyZC0+dRjylKDB9Zgoli9lhEBlhhYmxEhTSFs/nzy2SeQuO04rkV79YtVS+zPPJwBMdwAh6cQxWuoQZ1IDCEZ3iFN+fReXHenY9Za87JZg5hAc7nDyKKj0o=</latexit>
sha1_base64="zBgX7awZEi4o0/qSRepoKfLUY7k=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrRY9FLx48VLQf2C4lm2bb0CS7JlmhLP0RngQF8erv8eS/Md0uYtsHA4/3ZpiZF8ScaeO6P05hbX1jc6u4XdrZ3ds/KB8etXSUKEKbJOKR6gRYU84kbRpmOO3EimIRcNoOxtczv/1MlWaRfDCTmPoCDyULGcHGSu2eEuj2/rFfrrg1NwNaJV5OKpCj0S9/9wYRSQSVhnCsdddzY+OnWBlGOJ2WeommMSZjPKRdSyUWVPtpdu4UVa0yQGGkbEmDMvX/RIqF1hMR2E6BzUgvezPxz6surDLhpZ8yGSeGSjLfFCYcmQjNnkcDpigxfGIJJorZYxEZYYWJsRGVbAre8s+rpHVe89yad+dW6ld5HkU4gVM4Aw8uoA430IAmEBjDC7zBu/PkvDofzue8teDkM8ewAOfrFxwAjzs=</latexit>• Evaluate it at the instanton configuration:
2

µa
Winst
≃
<latexit sha1_base64="Zte0JRDJ6xF0UXnxMTNqWO6XH3s=">AAACBXicZVDLSsNAFJ34rPUVddnNYCm4KokLdVl047KCfUBTw2Q6aYfOTOLMRCghC1d+iitBQdz6Ea78G6dpKrYeuHA4517uvSeIGVXacb6tldW19Y3N0lZ5e2d3b98+OGyrKJGYtHDEItkNkCKMCtLSVDPSjSVBPGCkE4yvpn7ngUhFI3GrJzHpczQUNKQYaSP5dqXjp57kkAqls7vU4wlEGfQU5eTet6tO3ckB/xO3IFVQoOnbX94gwgknQmOGlOq5Tqz7KZKaYkayspcoEiM8RkPSM1QgTlQ/zZ/IYM0oAxhG0pTQMFf/TqSIKzXhgenkSI/UsjcVf73awiodXvRTKuJEE4Fnm8KEQR3BaSRwQCXBmk0MQVhScyzEIyQR1ia4cp7C2fz5ZTJPoX1ad526e+NUG5dFHiVQAcfgBLjgHDTANWiCFsDgETyDV/BmPVkv1rv1MWtdsYqZI7AA6/MHXxyYZg==</latexit>
sha1_base64="kkwbPR18VJFdESxLamI7K3yeVFo=">AAACBXicZVC7SgNBFL0bXzG+Vi3TDMaAVdi10TJoYxnBPCAbw+xkkgyZmV1nZoWwpLDyL9JaCQpi60dY+TdOHohJDlw4nHMv994Txpxp43k/TmZtfWNzK7ud29nd2z9wD49qOkoUoVUS8Ug1QqwpZ5JWDTOcNmJFsQg5rYeD64lff6RKs0jemWFMWwL3JOsygo2V2m6+3k4DJRCT2ozu00AkCI9QoJmgD2234JW8KdAq8eekUD4djxEAVNrud9CJSCKoNIRjrZu+F5tWipVhhNNRLkg0jTEZ4B5tWiqxoLqVTp8YoaJVOqgbKVvSoKn6fyLFQuuhCG2nwKavl72J+OcVF1aZ7mUrZTJODJVktqmbcGQiNIkEdZiixPChJZgoZo9FpI8VJsYGl7Mp+Ms/r5Laecn3Sv6tjeMKZshCHk7gDHy4gDLcQAWqQOAJXuAN3p1n59X5cD5nrRlnPnMMC3C+fgG3tpoX</latexit>
sha1_base64="4kU/3S15OoPsUtBOsP4Z8cimfzE=">AAACBXicZVDLSsNAFJ3UV62vqMtuBmvBVUnc6LLoxmUF+4Amhsl00g6dmcSZiVBCFq78BxfduhIUxK0f4cq/cfpAbD1w4XDOvdx7T5gwqrTjfFuFldW19Y3iZmlre2d3z94/aKk4lZg0ccxi2QmRIowK0tRUM9JJJEE8ZKQdDi8nfvueSEVjcaNHCfE56gsaUYy0kQK73A4yT3JIhdL5bebxFKIceopychfYFafmTAH/E3dOKvXj8bhcfCo0AvvL68U45URozJBSXddJtJ8hqSlmJC95qSIJwkPUJ11DBeJE+dn0iRxWjdKDUSxNCQ2n6t+JDHGlRjw0nRzpgVr2JuKvV11YpaNzP6MiSTUReLYpShnUMZxEAntUEqzZyBCEJTXHQjxAEmFtgiuZFNzln/+T1mnNdWrutYnjAsxQBGVwBE6AC85AHVyBBmgCDB7AM3gFb9aj9WK9Wx+z1oI1nzkEC7A+fwCMypq2</latexit>
sha1_base64="oHc8/sQ+S8IN4mkmmmLIuTnw14M=">AAACBXicZVC7SgNBFJ2NrxhfUcs0gzFgFXYt1DJoYxnBPCAbl9nJbDI4M7vOzAph2cLKf7BIayUoiK0fYeXfONkkYuKBC4dz7uXee/yIUaVt+9vKLS2vrK7l1wsbm1vbO8XdvaYKY4lJA4cslG0fKcKoIA1NNSPtSBLEfUZa/u3F2G/dE6loKK71MCJdjvqCBhQjbSSvWGp5iSs5pELp9CZxeQxRCl1FObnzimW7ameA/4kzJeXa4WhUyj/l6l7xy+2FOOZEaMyQUh3HjnQ3QVJTzEhacGNFIoRvUZ90DBWIE9VNsidSWDFKDwahNCU0zNS/EwniSg25bzo50gO16I3FX68yt0oHZ92EiijWRODJpiBmUIdwHAnsUUmwZkNDEJbUHAvxAEmEtQmukKVwMnt+kcxSaB5XHbvqXJk4zsEEeVACB+AIOOAU1MAlqIMGwOABPINX8GY9Wi/Wu/Uxac1Z05l9MAfr8weTVJrF</latexit><latexit sha1_base64="HHgphYVIJ8UC0r0NqGRrDXGYI+Q=">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</latexit>
sha1_base64="fz7ZKXqmwthabBArg3HWYqmwamQ=">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</latexit>
sha1_base64="vhw//dmVCugGrNVUyffdFyKE+8Y=">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</latexit>
sha1_base64="xp/jTakQdZwNC3p/XUxbLuFZibg=">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</latexit>2ρη̄bµν (x − x0 )ν
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φinst (x) ≃ v(x − x0 )2 + ρ2
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• The LO Matrix Element in the instanton backgroundiM ∼<latexit sha1_base64="D6L+ZaICho9hAxwOlcPt6Dumdys=">AAAB+3icZVDLSsNAFJ3UV62vVJduBkvBhZTEhbosunEjVLAPaEKZTCft0JlJmJkoJeRTXAkK4tYvceXfOE1TsfXAhcM593LvPUHMqNKO822V1tY3NrfK25Wd3b39A7t62FFRIjFp44hFshcgRRgVpK2pZqQXS4J4wEg3mNzM/O4jkYpG4kFPY+JzNBI0pBhpIw3sKoWphxGDdxn0zjxF+cCuOQ0nB/xP3ILUQIHWwP7yhhFOOBEaM6RU33Vi7adIaooZySpeokiM8ASNSN9QgThRfpqfnsG6UYYwjKQpoWGu/p1IEVdqygPTyZEeq1VvJv569aVVOrzyUyriRBOB55vChEEdwVkQcEglwZpNDUFYUnMsxGMkEdYmrkqewsXi+VWySKFz3nCdhnvv1JrXRR5lcAxOwClwwSVoglvQAm2AwRN4Bq/gzcqsF+vd+pi3lqxi5ggswfr8AXCWk1I=</latexit>
sha1_base64="4B8KsYoKLanX1MRyMSZVmzDSCig=">AAAB+3icZVBNS8NAEJ3Ur1q/Uj16WSwFD1ISL3osevEiVLAf0ISy2W7apZtN2N0oJeSneBIUxKu/xJP/xm0bxLYPBh7vzTAzL0g4U9pxfqzSxubW9k55t7K3f3B4ZFePOypOJaFtEvNY9gKsKGeCtjXTnPYSSXEUcNoNJrczv/tEpWKxeNTThPoRHgkWMoK1kQZ2laHMI5ij+xx5F55i0cCuOQ1nDrRO3ILUoEBrYH97w5ikERWacKxU33US7WdYakY4zSteqmiCyQSPaN9QgSOq/Gx+eo7qRhmiMJamhEZz9f9EhiOlplFgOiOsx2rVm4l/Xn1plQ6v/YyJJNVUkMWmMOVIx2gWBBoySYnmU0Mwkcwci8gYS0y0iatiUnBXf14nncuG6zTcB6fWvCnyKMMpnME5uHAFTbiDFrSBwDO8wBu8W7n1an1Yn4vWklXMnMASrK9fagyTQw==</latexit>!
<latexit sha1_base64="YVS97ViuOdinheVQWjoW8eRcTag=">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</latexit>
sha1_base64="im/AbGSyMn3aCSgxzOlToV7sOfo=">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</latexit>
sha1_base64="XOA//vyPji1bAnO5MCs5E68MStk=">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</latexit>
sha1_base64="XUHzBIdAOtdM3UjoMRdDkF13AXo=">AAACt3icZVFdb9MwFHUyPkb5KuxxLxbTpFQaldNq3frEtIHE4xB0GWqiyHGcxprjpLYzrUT5JTzzT3jliSee+SM4aVd140qWzj333HPt66jgTGmE/lj21oOHjx5vP+k8ffb8xcvuq9cXKi8loROS81xeRlhRzgSdaKY5vSwkxVnEqRddnTV175pKxXLxRS8KGmR4JljCCNaGCru/fCY0rHyCOXxfz9fIWyO/SJl/ADsmb9yqiJe0hnPnJnR70CdxrlWbVe6g7sF6rZM0rqHnLDZkTVaJ0NvUTeUsCirUR6PD8RAdoP4hco/GYwMQGh0PB+1859uGy4owRmlr5NObwnn7OfzQ64TdvaavCfg/cFdg7+QdgD+///h7HnZ/+3FOyowKTThWauqiQgcVlpoRTo13qWiByRWe0amBAmdUBVV78RruGyaGSS7NMUts2c2OCmdKLbLIKDOsU3W/1pDr2v6dUTo5DiomilJTQZaTkpJDncPmE2HMJCWaLwzARDJzWUhSLDHR5quXWxjdPv4+uN3CxaDvor77yazjFCxjG+yCN8ABLjgCJ+AjOAcTQKyh9dWKLGKP7dBO7HQpta1Vzw64E/b8H2Tm0oI=</latexit>

DqDW Dφ q(x1 ) · · · q(x12 )W (y1 ) · · · W (ynW )φ(z1 ) · · · φ(znh ) exp(−SE )LSZ<latexit sha1_base64="F+ngYZ/X9rvBvbF+ALqe09uvdBw=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrQT0WvXjwUNF+YLuUbJptQ5PsmmSFUvojPAkK4tXf48l/Y7rdiq0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUYTWScQj1QqwppxJWjfMcNqKFcUi4LQZDK+mfvOJKs0ieW9GMfUF7ksWMoKNlZodJdDN3UO3WHIrbgr0n3gZKUGGWrf41elFJBFUGsKx1m3PjY0/xsowwumk0Ek0jTEZ4j5tWyqxoNofp+dOUNkqPRRGypY0KFX/Toyx0HokAtspsBnoZW8q/nrlhVUmvPDHTMaJoZLMNoUJRyZC0+dRjylKDB9Zgoli9lhEBlhhYmxEhTSFs/nzy2SeQuO04rkV79YtVS+zPPJwBMdwAh6cQxWuoQZ1IDCEZ3iFN+fReXHenY9Za87JZg5hAc7nDyKKj0o=</latexit>
sha1_base64="zBgX7awZEi4o0/qSRepoKfLUY7k=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp7LrRY9FLx48VLQf2C4lm2bb0CS7JlmhLP0RngQF8erv8eS/Md0uYtsHA4/3ZpiZF8ScaeO6P05hbX1jc6u4XdrZ3ds/KB8etXSUKEKbJOKR6gRYU84kbRpmOO3EimIRcNoOxtczv/1MlWaRfDCTmPoCDyULGcHGSu2eEuj2/rFfrrg1NwNaJV5OKpCj0S9/9wYRSQSVhnCsdddzY+OnWBlGOJ2WeommMSZjPKRdSyUWVPtpdu4UVa0yQGGkbEmDMvX/RIqF1hMR2E6BzUgvezPxz6surDLhpZ8yGSeGSjLfFCYcmQjNnkcDpigxfGIJJorZYxEZYYWJsRGVbAre8s+rpHVe89yad+dW6ld5HkU4gVM4Aw8uoA430IAmEBjDC7zBu/PkvDofzue8teDkM8ewAOfrFxwAjzs=</latexit>• Evaluate it at the instanton configuration:
2

µa
Winst
≃
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sha1_base64="kkwbPR18VJFdESxLamI7K3yeVFo=">AAACBXicZVC7SgNBFL0bXzG+Vi3TDMaAVdi10TJoYxnBPCAbw+xkkgyZmV1nZoWwpLDyL9JaCQpi60dY+TdOHohJDlw4nHMv994Txpxp43k/TmZtfWNzK7ud29nd2z9wD49qOkoUoVUS8Ug1QqwpZ5JWDTOcNmJFsQg5rYeD64lff6RKs0jemWFMWwL3JOsygo2V2m6+3k4DJRCT2ozu00AkCI9QoJmgD2234JW8KdAq8eekUD4djxEAVNrud9CJSCKoNIRjrZu+F5tWipVhhNNRLkg0jTEZ4B5tWiqxoLqVTp8YoaJVOqgbKVvSoKn6fyLFQuuhCG2nwKavl72J+OcVF1aZ7mUrZTJODJVktqmbcGQiNIkEdZiixPChJZgoZo9FpI8VJsYGl7Mp+Ms/r5Laecn3Sv6tjeMKZshCHk7gDHy4gDLcQAWqQOAJXuAN3p1n59X5cD5nrRlnPnMMC3C+fgG3tpoX</latexit>
sha1_base64="4kU/3S15OoPsUtBOsP4Z8cimfzE=">AAACBXicZVDLSsNAFJ3UV62vqMtuBmvBVUnc6LLoxmUF+4Amhsl00g6dmcSZiVBCFq78BxfduhIUxK0f4cq/cfpAbD1w4XDOvdx7T5gwqrTjfFuFldW19Y3iZmlre2d3z94/aKk4lZg0ccxi2QmRIowK0tRUM9JJJEE8ZKQdDi8nfvueSEVjcaNHCfE56gsaUYy0kQK73A4yT3JIhdL5bebxFKIceopychfYFafmTAH/E3dOKvXj8bhcfCo0AvvL68U45URozJBSXddJtJ8hqSlmJC95qSIJwkPUJ11DBeJE+dn0iRxWjdKDUSxNCQ2n6t+JDHGlRjw0nRzpgVr2JuKvV11YpaNzP6MiSTUReLYpShnUMZxEAntUEqzZyBCEJTXHQjxAEmFtgiuZFNzln/+T1mnNdWrutYnjAsxQBGVwBE6AC85AHVyBBmgCDB7AM3gFb9aj9WK9Wx+z1oI1nzkEC7A+fwCMypq2</latexit>
sha1_base64="oHc8/sQ+S8IN4mkmmmLIuTnw14M=">AAACBXicZVC7SgNBFJ2NrxhfUcs0gzFgFXYt1DJoYxnBPCAbl9nJbDI4M7vOzAph2cLKf7BIayUoiK0fYeXfONkkYuKBC4dz7uXee/yIUaVt+9vKLS2vrK7l1wsbm1vbO8XdvaYKY4lJA4cslG0fKcKoIA1NNSPtSBLEfUZa/u3F2G/dE6loKK71MCJdjvqCBhQjbSSvWGp5iSs5pELp9CZxeQxRCl1FObnzimW7ameA/4kzJeXa4WhUyj/l6l7xy+2FOOZEaMyQUh3HjnQ3QVJTzEhacGNFIoRvUZ90DBWIE9VNsidSWDFKDwahNCU0zNS/EwniSg25bzo50gO16I3FX68yt0oHZ92EiijWRODJpiBmUIdwHAnsUUmwZkNDEJbUHAvxAEmEtQmukKVwMnt+kcxSaB5XHbvqXJk4zsEEeVACB+AIOOAU1MAlqIMGwOABPINX8GY9Wi/Wu/Uxac1Z05l9MAfr8weTVJrF</latexit><latexit sha1_base64="HHgphYVIJ8UC0r0NqGRrDXGYI+Q=">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</latexit>
sha1_base64="fz7ZKXqmwthabBArg3HWYqmwamQ=">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</latexit>
sha1_base64="vhw//dmVCugGrNVUyffdFyKE+8Y=">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</latexit>
sha1_base64="xp/jTakQdZwNC3p/XUxbLuFZibg=">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</latexit>2ρη̄bµν (x − x0 )ν
Uabg(x − x0 )2 [(x − x0 )2 + ρ2 ]positionorientation<latexit sha1_base64="05o0hNAnvxve7NqIWD14tHchKBI=">AAAB+XicZZDLSgMxFIYzXmu9tOrSTbAUXA0Z0druim5cVrAXaIeSSdM2NJMMSUYoQ5/ElaAgbn0UV76NmekoVg8Efr5zzR9EnGmD0Keztr6xubVd2Cnu7u0flMqHRx0tY0Vom0guVS/AmnImaNsww2kvUhSHAafdYHaT5rsPVGkmxb2ZR9QP8USwMSPYWDQslwYqhFIxKkxOKsi9RF6jVoPIRci7qHtWNBp1C6FnSRoVkEdrWP4YjCSJQzuAcKx130OR8ROsDCOcLoqDWNMIkxme0L6VAodU+0l2+AJWLRnBsVT2CQMz+rsjwaHW8zCwlSE2U/03l8KfXHVllRnX/YSJKDZUkOWmccyhkTC1AY6YosTwuRWYKGaPhWSKFSbGmlXMXKhlX4X/xbcLnXPXQ653hyrN69yPAjgBp+AMeOAKNMEtaIE2ICAGj+AZvDiJ8+S8Om/L0jUn7zkGK+G8fwHslZPH</latexit>
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size(x − x0 )2 "1/2
φinst (x) ≃ v(x − x0 )2 + ρ2
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sha1_base64="Y0l3iivuh7NwJDUAQpcJzc8gEtE=">AAAB8HicdVDLSgMxFM3UV62vqks3wVJwVZIubLsrunFZwT6gHUomzbShSWZIMkId+hOuBAVx6++48m/MtFWs4IELh3Pu5d57glhwYxH69HIbm1vbO/ndwt7+weFR8fikY6JEU9amkYh0LyCGCa5Y23IrWC/WjMhAsG4wvc787j3Thkfqzs5i5ksyVjzklFgn9QZaQsMf2LBYQhWEEMYYZgTXLpEjjUa9iusQZ5ZDCazQGhY/BqOIJpIpSwUxpo9RbP2UaMupYPPCIDEsJnRKxqzvqCKSGT9d3DuHZaeMYBhpV8rChfp7IiXSmJkMXKckdmL+epn445XXVtmw7qdcxYllii43hYmANoLZ93DENaNWzBwhVHN3LKQTogm1LqOCS+H7Vfg/6VQrGFXwLSo1r1Z55MEZOAcXAIMaaIIb0AJtQIEAj+AZvHjae/Jevbdla85bzZyCNXjvX7gLkD0=</latexit>• Integration over orientation, position, size and phase-space:!
σ(nW , nh ) ∼|M|2 · dΦPS<latexit sha1_base64="5R1tZq+Lur2S4sxdP8wiSN6p1vU=">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</latexit>
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sha1_base64="LTFC01ovc8pdgo/I8gAzXCZVSHY=">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</latexit>

a time-dependent
tunnelling solution which inte
dominantof totheconsist
instanton
approach
The instanton
is Chern-Simons
,xed
which
is expectedof B and
L con- [16,17].
W(Z)events.
atime-dependent
tunnelling
solutionwhichinterpolates
betweengauge
fieldswhose
numbers
differserving
multi-W(Z)events.
betweengauge
fieldswhose
Chern-Simons
numbe
At
high
energies,
these
pointlike
Sand
L
conckti-W(Z)events.
a time-dependent
tunnelling
solutionwhichinterpolates
The
for
this
processcanbeestimatedbymeans
contains
theamplitude
results
of
our
Monte
Carlo
betweengauge
fieldswhose
Chern-Simons
numbers
differ
by
one
unit
and
passesthroughsphaleron-likefieldsfor
Section3 Element
contains
the inresultsofour andMonte
Carloby one unit
andpasses throughsphaleron-like
f
•
The
LO
Matrixtheinstantonbackground
Owing
toofgaugethechiralanomaly[13-15]theV-Astrucate
unitarity.
Owing
to
the
factoriant
3contains
theresults
our
Monte
Carlo
betweenfieldswhose
Chern-Simons
numbers
differoftheinstanton
approach[16,17].
The
instanton
isesent results for studies.
the simpler
modelresults
at threeby onemodel
unit
andpassesthrough coordinates.
sphaleron-likefieldsforparticularvaluesofits collective
It wasfound
We present
forthesimpleratthreeparticularvaluesofitscollective
coordinates.
It
wa
ture
of
weak
interactions,
B
and
L
are
not
strictlyconser!
plitudes
(6)
with
the
numberofponontepresentforthesimpler
model
at
three
aresults
time-dependent
tunnelling
solutionwhichinterpolates
by
one
unit
and
passesthroughsphaleron-likefieldsfording Carlothat
proposed
for
the
LHC.
The
particularvaluesofitscollective
coordinates.
It
was
found
energies, including that proposedin forthe[1,2],
LHC.byTherefs.
expandingthepathintegral
about
the
in
refs.
[1,2],
byexpandingthepathintegralabved
inEfields
the[16,innumbers
17].
In[1,2],
thepresenceof theviolationof unitarity
in at multicluding
that
proposed
forStandard
the
The
del
at threebetweengauge
whose
Chern-Simons
differrefs.
byexpandingpath
integralabout sets
theparticularvaluesofitscollective
coordinates.
It
was
found
L
O
M
model
requiresaModelhigherenergy
and
so
in
that
case
requiresa higherenergyandsoinLHC.
that
case
DqDW
Dφ
q(x
)
·
·
·
q(x
)W
(y
)
·
·
·
W
(y
· · · φ(zexp(−S
iM∼
instanton,that,inleading-order(LO)
instanton,that,inleading-order
in
the
C~w,
1
12
1
nW )φ(z1 )(LO)nh )couplingE )in the coup
nontrivial
SU(2)
gauge
fieldsthefermionic
quantumrlodelrequiresa40
higherenergyand through
so
in
that
case
at(5)
patton-patton
centreof mass
LHC.
The
unit
and
sphaleron-likefieldsforforweshowresults
only
at con40
TeV.W~,
We
present
ourconinstanton,that,
inabout
leading-order(LO)in thethecoupling
C~w,
ts
onlybyatone
TeV.
Wepassespresent
ourin refs.
[1,2],
byexpandingthepathintegraltheamplitudes
theforprocesses(5)
growwith enemul
theamplitudes
theprocesses
growwith
multiplicity
LSZ
reenumberschangeaccording
sults
only
at 40
TeV.
present
ourdetails
con-toof theclusions
in
Sect.
4.in
Brief
HERBVI
package,particularvaluesofitscollective
coordinates.
Itamplitudes
wasinfoundthesphaleronenergyincase
thefor the processes(5) growwithmultiplicityt.
4.thatBriefdetailsof theWe
HERBVI
package,instanton,that,leading-order(LO)thecoupling
C~w,ofparton-partonandcentreof mass(3).
energy ~ likandparton-parton
centremass
energy
~
like
The
used
toby
obtainmajority
resultsin Sect.
are centre of mass The
Sect.
Brief
detailsthe
HERBVI
package,inrefs.
[1,2],
expandingthepathintegralabout
the3,multiplicity
t theour4.majority
concorresponding
parton levelof
theof
results
intheforSect.
3,processes
areof thetheamplitudes
the(5) growwithandparton-partonenergy~ likes~BVn.,n.co~(nn+nn)!c~(~"+n")/2e2z/O:w• Evaluate
itAL,
atcontained
inthe
appendix.
AL~=the=anAL,= 893,(LO)= in-configuration:
ANcs,
(1)
ain
the
majorityof
ininstanton
Sect.
are
s~BV
instanton,
that,
inresults
leading-orderthecoupling
C~w,+ nn)!c~(~"+n")/2en.,n.co~(nn2z/O:wleading order matrix elemenVIpackage,to thenaseappendix.
andparton-parton
centreofmassenergy~likes~BV~ (nn + nn)!c~(~"+n")/2e 2z/O:w
nonan appendix.
multiplicity
Sect.
3,the
areamplitudes for the processes (5) grow withn.,n.co
ton-induced BLNV (5), is given by
ge,
where
s~BV
x
mw (nB+ ,,,) 2
n.,n.co 2~centre(nn +ofnn)!c~(~"+n")/2e
and parton-parton
mass energy ~ 2z/O:w
like
!
"1/2
(x
−
x
)
x
mw
(nB
+
,,,)
(6)
k
n
s
m
w
/
0
are
2ρ
− x0 )xν
1.1 Instanton-indueed
B η̄
andviolation
~BV
=
-2
n
2n[-F(n+ 103/µabµνL(x
mw
(nB
+
,,,)
(6)
v
k
n
s
m
w
/
s~BV
φ
(x)
≃
W
≃
U
Cf#
2
vI
.
.
. .
n.,n.co~(nn+nn)!c~(~"+n")/2e2z/O:w
inst
abndueed B and inst
L violation
2 + ρ2 S-wave amplitud
k
n
s
m
w
/
2
2
2
At
high
energies,
these
pointlike(x−x)
n...... co[ F(103/12
0,
Wj
Wk
)
n-indueed B and
L violation−−2i9x30and)W.the+ρV-A] struc0 ),,,)(2)Ncs
x echiralq k (xtmw
r (anomaly
F(nB
ix
j +W
k -[(x,x=_~todg3the
(6)
Owing
[13-15]
ate unitarity.
Owingamplitudes
to the factorial
At high energies, these pointlike
S-waveviol- growth ofAt
highenergies,
these
pointlikeS-waveamplitudes
violk
n
s
m
w
/
ture of mw
weak
B and position
L are
not
strictlyconserhiral anomaly[13-15]andthestrucplitudes(6) withgrowth
the number
produced boso
orientation
size (6)ateunitarity.
Owing
to
the
factorial
oftheofd3pi
amx(nBinteractions,+ the,,,) V-A
x
y
~
Eij01
e chiral anomaly[13-15]andV-A
strucate
unitarity.
Owing
to
the
factorial
growth
of
the
amved
in
the
Standard
Model
[16,
17].
In
the
presenceofviolationofunitarity
sets
in
at
multiplicitiesof (9 (C
k
n
s
m
w
/
teractions,
B
and
L
are
not
strictlyconserplitudes
(6)
with
the
numberof
producedbosons,this
At high
energies,
these
pointlikeS-waveamplitudes
violtheChern-Simons
numberof
thegauge
field,
k interactions, Bdenotes
and
L
are
not
strictlyconseri=l~(2(6)quantum
with the numberof produced
bosons,thisnontrivial
SU(2)
gauge
fieldsofW~, theplitudes
fermionicat
patton-patton
centreofmassenergies
of
(5'
(c~
w~
ndard
Model
[16,
17].
In
the
presence
V-A
strucviolationof
unitarity
in
at
of(9
(9(C~w(C~w1)1)and
and
ate
Owing
toS-wave
ofthe
thesets
amStandardAt Model[16,17].change
In thepresenceof factorial
highFij
energies,
these
pointlike
amplitudes
violis unitarity.
the
SU(2)
field
strength,
andC~w-growth92/(470
weak
violationofunitaritysets
insphaleronat multiplicities
multiplicitiesofnumbers
accordingtothe
theenergy
(3).
2)
W~,
the
fermionic
ctly
conseratofofenergiesof (5'(5'parton(c~w~
w~row),
row),i.e.
plitudes
(6)
with
the
numberproducedbosons,•gaugeIntegrationoverorientation,
position,sizeandphase-space:
uc-gaugeU(2)fieldsW~,thefermionic
quantum
atefields
unitarity.
Owing
tothequantum
factorialof theamd3pk~ 4 x(4)3
finestructure
constant.
Asis growthsuggestedby eqs.
(1)centre
and this(2),
atpatton-patton
patton-patton
centreof mass
masscorresponding
energies of
(c~
i.e.
The
level
cross section,geaccordingpresenceof to tothesphaleronenergy(3).
serviolationofinmultiplicitiesof(9 (1)(C~w1)
andx(2zc)3
2Ek(LOME)(2,~i ~, of
AL~
= AL,
= unitarity
AL,= 89 sets
- at
ANcs,
ange
according
plitudes
(6)
with
the
numberof=produced
bosons,! ofthisthesphaleronenergy(3).
to the leading order matrix
element
strong,nonperturbative
gauge
fields,C(9-~),
areicofquantum
corresponding
partoncross
section,
3Lo,by
due
at
ofthe
mass
energiesofand(5'(c~
w~ row),i.e. level
violationof patton-patton
unitarity
sets toin centre
atmultiplicitiesofThe(9 (C~w
1)
ton-induced
BLNVsection,(5), is given
[1,2]
The
corresponding
partonlevel cross
3Lo,
due
2
neededinorderchangeChern-Simons
number,
or,
AL,
=
89
=
ANcs,
(1)
where
σ(n
∼
|M|
· dΦ
um
= AL, =at 89= the- ANcs,(1), nhto)to(5'thematrix(LOME) ofof instaninstanPS element
sphaleronenergy(3).W
patton-patton
centremass
energiesof(c~w~leadingrow),
i.e.ordertheleadingordermatrixelement (LOME)
equivalently,
theoffermionnumbers,byanintegeramount.
) < ( P i n+- 103/12)72-~Pi~1 P j - ~BV=by-- [1,2]
2 n 2n[-F(nton-induced
BLNV
(5),
is
given
the sphaleronenergy(3).
Thecorresponding
partonlevel
cross
section,3Lo,
due
ton-induced
BLNV
(5),
is
given
by
[1,2]
Cf#
2
vI
.
.
.
.
I
.
i=1
j=l
This
is
reflected
by
the
fact
that
there
exists,on
topologicaln......
co,[F(103/12)
J
nR!nn!
=_
2i9
W.
Wj
Wk
)
(1)The corresponding(2) instanlevelsection,3Lo,
due ofNcs
3 x e q kpartont rorder(FijW
kmatrix- cross3element,
to the~ dleading(LOME)
grounds,
anenergybarrier[18,
19]
betweengauge
fields + 103/12)72~BV== ---- 22 nn 2n[-F(n(1) • Result~BV
2n[-F(n103/12)72
11’90, Espinosa
In[Ringwald
this IexpressionnB is the number’90]
to the leading
ordermatrixelement(LOME)
ofinstanton-induced
BLNV
(5),
is
given
by
[1,2]
Cf#
2
vI
.
.
.
.
.
Cf#
2
vI
I
.
x
y
~
d3pi
Eij01
d3pj
2(4E~whose
Chern
Simonsnumbers
differby an , ,integer. Then......
co[F(103/12)
J
nR!nn!
2i9
W.
Wj
Wk
)
n
=
nB
+
nn.
The
effective
couplingn......
co[J
nR!nn!
2i9
W.
Wj
Wk
)
ton-induced
BLNV
(5),
is
given
by
[1,2]
(2)(2)
ke tqrk(tFr (i FjW
3 3 the, barrierdenotesnumberof thethe energy
gauge field,
i=l ~(2n)32Ejm
i jkW- k -minimum, Chern-Simons
heightisgiven
byof a static
~BV
=
-2
n
2n[-F(n+103/12)72
1
Fij
is
field
C~w
weak
σ~BV
nhthe
∼ 2n[-F(n(# = d3pj
1.6x 10-1~ GeV -14,
Cf#
2 strength,vI so.and.called. 1-. 92/(470I . xtheLO (nsaddle-pointW= , -2) nSU(2)+103/12)72
solution,the"sphaleron"
[20,
21],
y
~
d3pi
Eij01
d3pj
2(4E~-m2w)
x
y
~
d3pi
2(4E~-m2w)
nCf#......
co,I . . As
F(103/12)
J (1)nR!nn!
d3pk~ 4 x(4)finestructureand (2),)2 v- constant.
. [.is suggestedI.
- by eqs.
(2)
n...... cowhich, depends[ slightly
F(103/12)
nR!nn!ofmass,hern-Simons
numberofofthethe
gauge
field,
Chern-Simons
numbergauge
field,~~(2n)32Ejm2 ~,ontheJ fields,
Higgs
i=lare(2n)32Ejx(2zc)32Ek m2(2,~istrong,nonperturbativegauge
C(9-~),i=l
(2)
v
=
2mw/9
is the Higgs field vacuu
U(2)
field
strength,
andC~w
C~w-92/(470
92/(470
thed3piweak) field
strength,
and
the
weak
neededin- order
the Chern-Simons
or,x yto~change
Eij01 d3pjnumber,2(4E~-m2w)
and(~ is a numerical factor repr
x
y
~
d3pi
Eij01
d3pj
2(4E~-m2w)
d3pk)
e
constant.
Asissuggested
by
eqs.
(1)
and
(2),
d3pk
~
4
x(4
onstant.
Asissuggested
by
eqs.
(1)
and
(2),
equivalently,
thefermionnumbers,byanintegeramount.
×
gauge field, M~p = 2B(mn/mw)
i=l mw~ -~ 7 - 14 TeV,(2n)32Ej xx
m2) < ( P i n - - ~andP i summing~Pj-~overP k ) " incaveraging(3)(2zc)32Ek(2zc)3
2Ek
(2,~i
~,
eld,
i=l
~
(2n)32Ej
m2
perturbativegauge
C(9-~),
arethere exists, on topological
i=1
j=l
k=l
Thisfields,
isfields,
reflected
the fact thatrturbative
gauge
ofofbyC(9-~),
are
70
the
weak
~W
states of the fermions. The leadingeak
tochangetheChern-Simons
number,or,
grounds,
an energy
barrier[18,
19]
betweengauge fieldsrdertochangetheChern-Simons
number,or,In
thisexpressionnB is the number of vector bos
d3pk
~
4
x(4
)
(1) and (2),
d3pk
~
4
x(4
)
taken
at
the
dominant
multiplicitie
(2),
whose
Chern
Simonsnumbers
differbyaninteger.
The, thefermionnumbers,
byananintegeramount.
hefermionnumbers,byintegeramount.
x(2zc)3
2Ek
(2,~i
~,
n
=
nB
+
nn.
The
effective
coupling
constant f# is
)
<
(
P
i
n
~
P
i
~
P
j
~
P
k
)
"
(7)
where
the
parameterBisrestricted
to
lie
in
the
range
)
<
(
P
i
n
~
P
i
~
P
j
~
P
k
)
"
(7)
C(9-~),arex(2zc)32Ekheight(2,~i ~,The
cross-section
becomes
very
large
for
large
n
and
n
!
W
h
are
minimum
barrier
is
given
by
the
energyofastaticit rises rapidly with
i=1
j=lAs anticipated,
k=l
cted
fact that
there
exists,topological
i=1
j=l
k=l
dnumber,by bythethefact
thereonontopological_< Bexists,_< 2.72.
At low energies ( ~ Msp), therefore,
anor, that1.57(# = 1.6x 10 -1~ GeV -14,<latexit sha1_base64="D6L+ZaICho9hAxwOlcPt6Dumdys=">AAAB+3icZVDLSsNAFJ3UV62vVJduBkvBhZTEhbosunEjVLAPaEKZTCft0JlJmJkoJeRTXAkK4tYvceXfOE1TsfXAhcM593LvPUHMqNKO822V1tY3NrfK25Wd3b39A7t62FFRIjFp44hFshcgRRgVpK2pZqQXS4J4wEg3mNzM/O4jkYpG4kFPY+JzNBI0pBhpIw3sKoWphxGDdxn0zjxF+cCuOQ0nB/xP3ILUQIHWwP7yhhFOOBEaM6RU33Vi7adIaooZySpeokiM8ASNSN9QgThRfpqfnsG6UYYwjKQpoWGu/p1IEVdqygPTyZEeq1VvJv569aVVOrzyUyriRBOB55vChEEdwVkQcEglwZpNDUFYUnMsxGMkEdYmrkqewsXi+VWySKFz3nCdhnvv1JrXRR5lcAxOwClwwSVoglvQAm2AwRN4Bq/gzcqsF+vd+pi3lqxi5ggswfr8AXCWk1I=</latexit>
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10−1

γ

10−2

10−3
0

5

10

15

20
25
30
Boson Multipliticy

O(30) EW gauge bosons are produced!

Figure 1: Normalised distributions of the number of W , Z andFestival at collider!

generated by HERBVI

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y interesting,could
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Figure 1: Illustrationa QFD
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8 in deep-inelastic
16
32electron-protonmW ◆scattering✏ (right).
✓✓◆
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ritten in the22
99 4/399 2 2 33mm
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8/3
M
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Instanton-induced
total cross-sections
for16
hard
parton-parton(p-p)
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(cf.
8
32
✏
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2
8
16
32
m
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W ,nh is the2 quark-quark
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14, 33,W color⇥ [12, 13, 2/3⇤ 34])⇡factor3 orientation
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while
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PP
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23]
while
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[19,
23]
W ,nh
2 [19,pwhere
d ' 0.15. (1.6)
d✏ 9 1 + O ✏p m,
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W
1024
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22
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15/2
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m
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m
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e
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)
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,
typicalaquantumtunnelingprocess.
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E
=
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⇠
W
sph
W
W
W
h thatObviously,
inintheenergyregionŝ ŝ⌧M
cross-section
forfor(B(B++L)WWof
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Obviously,
theenergym⌧⌧,theinclusive
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2regionm
2⌧
2M
as
a
hint
the
processfield
equationstheelectroweak
theory,corresponding
to
the
minim
×
F
−p
ρ
F
−p
ρ
F
−p
ρ
F
−p
ρ
P
(U,
R,
ρ,
ρ̄;
p
·
p
)
.
1
2
1
1
2
2
p
p
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo9
TeV
–
the
staticenergyofaclassicallyunstablesaddle-point
solutionofthestaticbosonicpviolation
due
exponential
suppression
factor,mW small,
tion.
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interpretationbviolation isisMunobservably
unobservably
small,
due totothe
thetremendous
tremendous
exponential
suppression
factor,is also
tween
inequivalent
vacua
with
di↵erent
valuesoftheChern-Simons
topo
⌘
6⇡
'
18.3
TeV
,
(1.4)
nentialis10M
andthus offor
thetheory,orderofthetunnelingenergy.
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mayinterpretthis
0 ⇠arising
0162
field
equationsoftheelectroweak
corresponding
totheexpansion
minimalheightbeHere,the 4⇡/↵
basicgrowth
blocksininstanton-perturbation
theory–sphaleron
the semi-classicalof barrier
162
exp(
)
,
typicalaquantum
process.
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W
↵
exp(
4⇡/↵
)
⇠
10
,
typicalforaquantumtunnelingprocess.
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it
is
W
W
as a hintthat
theintegral
process
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intermediate
virtualfield index
configurathe tween
corresponding
path
expression
about
instanton
solution– are sphaleron-likei) the instantoninequivalent
vacua
with
di↵erentoftheChern-Simons
topological
N
[16,
CS
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expoponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
exposize distribution
D(ρ),ii) the functionΩ, whichtakes into
account
exponentiationof gaugetion.
Thisinterpretationis alsobacked-upby =thefact2thethatcross-section (1.2)
2 the inclusive
parametricallysimilartotheenergyofthesphaleron,
E
f
(m
/m
)⇡m
/↵
⇠
nentialgrowth
is
M
and
thus
of
the
order
of
the
sphaleronenergy.
Oneinterpret this
sph
W
W
hnW
00 and thus of the order of the sphaleron
nentialgrowthandisanM
energy.
Onemay1Bothan instanton
enter here,
since number
we write the(2) asa discontinuity
ofmay
is dominatedbyaanti-instantonsemi-classicallylarge
ofcross-section
Wh(Z) Wand
Higgs
bosons[24], interpret thisthe static energy
of athatclassicallyunstablesaddle-point
solutionthestaticbosonic¯viaantheprocess
proceeds
virtualsphaleron-like
theas
p1 a
pa2 hint
forwardelastic
scattering
amplitude
inthe Ivia
I-background(cf. Fig. 1).ofAlternatively,one
may calculate field
as
hint
that
the
process
proceeds
anintermediate
intermediate
virtualsphaleron-likefieldconfiguraconfigura1✓◆
thetion.
cross-section
by takingthe modulus
ofbacked-up
amplitudes
in bytheminimal
instanton-background.
ations of theelectroweak
theory,corresponding
to4⇡thebarrierbe- cross-section (1.2)
andThis
issquared
the
fact
that
theheight
inclusive
3 single
tion.
This interpretationinterpretation
isalsoalso
backed-up
by
the
fact
4/3
2that the inclusive cross-section (1.2)
1hnW i =
✏ +O ✏,(1.7)
isis dominatedbyasemi-classicallylarge
number
of
W
(Z)
and
Higgs
bosons[24],
quivalent vacua
withdi↵erent
valuesoftheChern-Simons
topological
index
N
[16,
CS
8
dominated by a semi-classically largeof W (Z) and Higgs bosons [24],3 ↵Wnumber
✓✓
◆◆
4⇡
and
in line with the expectation from2
4⇡ 3 34/3

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1

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a lot of e↵ort, the actual size of the cross-sections in the relevant, tens of TeV energy regime

forinstanton-perturbativeelectroweak (B +L)-violatingprocesses,> 10 GeV in order to becross-section
taken ∼
in the semi-classical,
regime[20, 21, 22, 24].
y interesting,could
albeitnot
unsolved
question is whether manifestations of such topologidetermined.
As mentioned
inbe
Sect.
1, axial anomalies [1] force instanton-induced hard scattering processes
toXX
uations might
beNevertheless,directlyinhigh-energy
scattering
atpresentorfuturebe always
associatedobservable
withitanomalous
fermion-number
violation[2],
inparticularB+
Lviolation,
has been established (for reviews,
that
the inclusive
q + q !see7q̄Refs.
+ 3`¯[11,+ 14, 15])
nW,Z
W (Z)
+
nh h,
△Bcross-section=has△Lbeen= −nraised
Q, electroweak
inoriginallythe case of in
QFD= 3 fermion
generations,chirality
This question
latengeneightiesin [12,13], but, anddespitegenfor(Bthe+withL)-violatingprocesses,nnhW,Zviolation,△Qof
2 nfcross-sections
Q, in the case of in
QCD
typically tens
nf = 3oflightquark
flavors.
[Khoze, Ringwald’91]
5 =↵ort, the actualsize
thethewithrelevant,
TeV
energyregime• The inclusive cross-section can be estimated using the dispersion
relations (optical theorem).
X

Xt be determined.
q+q written! 7q̄ + in
3e`¯the+ formnW,Z W (Z) +
nh h,
(1.1)
can
be
p been established (for reviews, see Refs. [11,rtheless, it has
14, 15]) that thenhinclusivenW,Z17]. The ‘holy grail function’ F
X
p1
P (✏)
4⇡ are given bytion for electroweakp1(B + L)-violating processes,
whose
first few terms
BV
BV
IW
I
I
I
ˆ s⌘
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17].beThe‘holyingrail
function’
F in the X
exponent
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can
theWX
form
2 expansion in ✏,p2written
p2 inclis known in nterms
mW
↵W 9
9
nW
,nh
whose
first
few
terms
are
given
by
[18–22]

4/3
¯
q
+
q
!
7q̄
+
3
`
+
n
W
(Z)
+
n
h,
(1.1)
F
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=
1
✏
+
W,Z X
hP (✏)
p17].
p exponent
17]. TheThe ‘holy‘holy grail
grail function’function’theknown4⇡
termsofofananexpansion
expansioninin✏,8✏,
FF ininthe
isisknown
ininterms
1
BV
BVexponent✓◆
ˆ9incl ⌘ 9 ˆnW3,nnhh = 2m2exp
F
(✏)
,
(1.2)
nW,Z
1
where
whose first
first few
fewterms
termsarearegiven
given
by[18–22][18–22]
whose
by
↵
8/3 p
W + O(✏F (✏) of =1 instanton-induced✏4/3 +nW ,n✏h2 process4inm
3W 2h ✏8/3 ln·andconst),(1.5) factor P
Figure 1: Illustrationa QFD
proton-proton
scattering(left)
whilethepre-exponentialŝ
8 in deep-inelastic
16
32electron-protonmW ◆scattering✏ (right).
✓✓◆
of aform
QCD instanton-induced process
ritteninthe22‘holy grail’
99 4/399 2 2 33mm
11 ✏ ⌘
hh
8/3
8/3
✓
4/3
8/3
8/3
M
where FF(✏)
15/2
=
1
✏
+
✏
4
3
O(✏
·
const)
,
(1.5)
(✏)
=
1
✏
+
✏
4
O(✏
·
,
(1.5)
=

0 const)p\13 2 2 ✏ ✏ lnln ++⇡3while theXpre-exponential
factor
P reads[19, 23] m
Instanton-induced
total cross-sections
for16
hard
parton-parton(p-p)
scattering
processes(cf.
function
8
32
✏
1
2
8
16
32
m
✏
P
(✏)
4⇡
p
P
(✏)
=
ŝ WW 1 ¯
BV1024 2
ˆincl
ˆnBV
=
exp
F
(✏)
,
(1.2)
Fig. 1)are ⌘given in termsofanintegralovertheinstanton-anti-instanton(IenergyI) collective
coordi✏ ⌘◆(1.3)
centre-of-mass
ŝ
in
terms
of
the
scale
W ,nh is the2 quark-quark
2
✓
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7/2
m
15/2
M\10(see74also
\1
3 ↵W
nates
(sizesρ,nρ̄,
I I¯ distance R, relativeU)[21]
14, 33,W color⇥ [12, 13, 2/3⇤ 34])⇡factor3 orientation
4⇡
while
the
pre-exponential
factor
PP
reads
23]
while
the
pre-exponential
reads
[19,
23]
W ,nh
2 [19,pwhere
d ' 0.15. (1.6)
d✏ 9 1 + O ✏p m,
"P (✏) =1
W
1024
2◆energy↵W(I)the quark-quark centre-of-mass+p )·Rŝ ◆in◆terms of
the
scale
22
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in the, energy regionσ̂is∼Im d4 R ei (p15/2(2)
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✓
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p p
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7/2
15/2
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2/3
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2/3
9
9
P
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O
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,
(1.6)
where d '∞0.15.
m
ŝ
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=
d
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1
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O
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,
(1.6)
#
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W
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%
&
p↵
162
"∞
"
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2 TeV , n
1024
2M
(1.4)
exp(
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)
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, typical
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1024
2
−0 ⌘
Ω U, 6⇡, ↵,...WW ' 18.3
ρ̄R
W
Obviously,energyregion
m
⌧
ŝ
⌧
M
,
the
inclusive
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for
(B
+
L)W
0
↵
,
,
.
.
.
D(ρ)
D(ρ)
dU
e
× dρ indρtheωU,
M0is parametricallyWsimilarwhichtothe energyponentiallyof the sphaleron,Ethis= f (msph energy
growinginρρ̄ρviolation
small, due to the tremendous exponential suppression factor,p0
0is unobservably
where
dd'
0.15.
where
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0.15.
9
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–+the)for
staticenergyof+ a classicallyunstablesaddle-pointp
nential2 growth
M0 andsolution
thus ofp162
2 itis is
ark-quark centre-of-massenergy
ŝ
in
terms
of
the
scale
)
+
)
+
)
*
*
*
*
exp(
4⇡/↵
)
⇠
10
,
typicalaquantumtunnelingprocess.
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E
=
f
(m
/m
)⇡m
/↵
⇠
W
sph
W
W
W
h thatObviously,
inintheenergyregionŝ ŝ⌧M
cross-section
forfor(B(B++L)WWof
0 ,0theg inclusive
Obviously,
theenergym⌧⌧,theinclusive
cross-sectionL)-proceed2
2regionm
2⌧
2M
as
a
hint
the
processfield
equationstheelectroweak
theory,corresponding
to
the
minim
×
F
−p
ρ
F
−p
ρ
F
−p
ρ
F
−p
ρ
P
(U,
R,
ρ,
ρ̄;
p
·
p
)
.
1
2
1
1
2
2
p
p
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo9
TeV
–
the
staticenergyofaclassicallyunstablesaddle-point
solutionofthestaticbosonicpviolation
due
exponential
suppression
factor,mW small,
tion.
This
interpretationbviolation isisMunobservably
unobservably
small,
due totothe
thetremendous
tremendous
exponential
suppression
factor,is also
tween
inequivalent
vacua
with
di↵erent
valuesoftheChern-Simons
topo
⌘
6⇡
'
18.3
TeV
,
(1.4)
nentialis10M
andthus offor
thetheory,orderofthetunnelingenergy.
One
mayinterpretthis
0 ⇠arising
0162
field
equationsoftheelectroweak
corresponding
totheexpansion
minimalheightbeHere,the 4⇡/↵
basicgrowth
blocksininstanton-perturbation
theory–sphaleron
the semi-classicalof barrier
162
exp(
)
,
typicalaquantum
process.
Nevertheless,itisex-exisdominatedbyasemi-classically
W
↵
exp(
4⇡/↵
)
⇠
10
,
typicalforaquantumtunnelingprocess.
Nevertheless,
it
is
W
W
as a hintthat
theintegral
process
proceeds
via theanvalues
intermediate
virtualfield index
configurathe tween
corresponding
path
expression
about
instanton
solution– are sphaleron-likei) the instantoninequivalent
vacua
with
di↵erentoftheChern-Simons
topological
N
[16,
CS
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expoponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
exposize distribution
D(ρ),ii) the functionΩ, whichtakes into
account
exponentiationof gaugetion.
Thisinterpretationis alsobacked-upby =thefact2thethatcross-section (1.2)
2 the inclusive
parametricallysimilartotheenergyofthesphaleron,
E
f
(m
/m
)⇡m
/↵
⇠
nentialgrowth
is
M
and
thus
of
the
order
of
the
sphaleronenergy.
Oneinterpret this
sph
W
W
hnW
00 and thus of the order of the sphaleron
nentialgrowthandisanM
energy.
Onemay1Bothan instanton
enter here,
since number
we write the(2) asa discontinuity
ofmay
is dominatedbyaanti-instantonsemi-classicallylarge
ofcross-section
Wh(Z) Wand
Higgs
bosons[24], interpret thisthe static energy
of athatclassicallyunstablesaddle-point
solutionthestaticbosonic¯viaantheprocess
proceeds
virtualsphaleron-like
theas
p1 a
pa2 hint
forwardelastic
scattering
amplitude
inthe Ivia
I-background(cf. Fig. 1).ofAlternatively,one
may calculate field
as
hint
that
the
process
proceeds
anintermediate
intermediate
virtualsphaleron-likefieldconfiguraconfigura1✓◆
thetion.
cross-section
by takingthe modulus
ofbacked-up
amplitudes
in bytheminimal
instanton-background.
ations of theelectroweak
theory,corresponding
to4⇡thebarrierbe- cross-section (1.2)
andThis
issquared
the
fact
that
theheight
inclusive
3 single
tion.
This interpretationinterpretation
isalsoalso
backed-up
by
the
fact
4/3
2that the inclusive cross-section (1.2)
1hnW i =
✏ +O ✏,(1.7)
isis dominatedbyasemi-classicallylarge
number
of
W
(Z)
and
Higgs
bosons[24],
quivalent vacua
withdi↵erent
valuesoftheChern-Simons
topological
index
N
[16,
CS
8
dominated by a semi-classically largeof W (Z) and Higgs bosons [24],3 ↵Wnumber
✓✓
◆◆
4⇡
and
in line with the expectation from2
4⇡ 3 34/3

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1

2

1 2

4π
αg

R2 ρ̄
ρρ̄ ρ

fin

1 2

a lot of e↵ort, the actual size of the cross-sections in the relevant, tens of TeV energy regime

forinstanton-perturbativeelectroweak (B +L)-violatingprocesses,> 10 GeV in order to becross-section
taken ∼
in the semi-classical,
regime[20, 21, 22, 24].
y interesting,could
albeitnot
unsolved
question is whether manifestations of such topologidetermined.
As mentioned
inbe
Sect.
1, axial anomalies [1] force instanton-induced hard scattering processes
toXX
uations might
beNevertheless,directlyinhigh-energy
scattering
atpresentorfuturebe always
associatedobservable
withitanomalous
fermion-number
violation[2],
inparticularB+
Lviolation,
has been established (for reviews,
that
the inclusive
q + q !see7q̄Refs.
+ 3`¯[11,+ 14, 15])
nW,Z
W (Z)
+
nh h,
△Bcross-section=has△Lbeen= −nraised
Q, electroweak
inoriginallythe case of in
QFD= 3 fermion
generations,chirality
This question
latengeneightiesin [12,13], but, anddespitegenfor(Bthe+withL)-violatingprocesses,nnhW,Zviolation,△Qof
2 nfcross-sections
Q, in the case of in
QCD
typically tens
nf = 3oflightquark
flavors.
[Khoze, Ringwald’91]
5 =↵ort, the actualsize
thethewithrelevant,
TeV
energyregime• The inclusive cross-section can be estimated using the dispersion
relations (optical theorem).
X

Xt be determined.
q+q written! 7q̄ + in
3e`¯the+ formnW,Z W (Z) +
nh h,
(1.1)
can
be
p been established (for reviews, see Refs. [11,rtheless, it has
14, 15]) that thenhinclusivenW,Z17]. The ‘holy grail function’ F
X
p1
P (✏)
4⇡ are given bytion for electroweakp1(B + L)-violating processes,
whose
first few terms
BV
BV
IW
I
I
I
ˆ s⌘
ˆ sW ,nh =expF (✏) ,
17].beThe‘holyingrail
function’
F in the X
exponent
of an
can
theWX
form
2 expansion in ✏,p2written
p2 inclis known in nterms
mW
↵W 9
9
nW
,nh
whose
first
few
terms
are
given
by
[18–22]

4/3
¯
q
+
q
!
7q̄
+
3
`
+
n
W
(Z)
+
n
h,
(1.1)
F
(✏)
=
1
✏
+
W,Z X
hP (✏)
p17].
p exponent
17]. TheThe ‘holy‘holy grail
grail function’function’theknown4⇡
termsofofananexpansion
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FF ininthe
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1
BV
BVexponent✓◆
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1
where
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fewterms
termsarearegiven
given
by[18–22][18–22]
whose
by
↵
8/3 p
W + O(✏F (✏) of =1 instanton-induced✏4/3 +nW ,n✏h2 process4inm
3W 2h ✏8/3 ln·andconst),(1.5) factor P
Figure 1: Illustrationa QFD
proton-proton
scattering(left)
whilethepre-exponentialŝ
8 in deep-inelastic
16
32electron-protonmW ◆scattering✏ (right).
✓✓◆
of aform
QCD instanton-induced process
ritteninthe22‘holy grail’
99 4/399 2 2 33mm
11 ✏ ⌘
hh
8/3
8/3
✓
4/3
8/3
8/3
M
where FF(✏)
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✏
4
3
O(✏
·
const)
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(✏)
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+
✏
4
O(✏
·
,
(1.5)
=

0 const)p\13 2 2 ✏ ✏ lnln ++⇡3while theXpre-exponential
factor
P reads[19, 23] m
Instanton-induced
total cross-sections
for16
hard
parton-parton(p-p)
scattering
processes(cf.
function
8
32
✏
1
2
8
16
32
m
✏
P
(✏)
4⇡
p
P
(✏)
=
ŝ WW 1 ¯
BV1024 2
ˆincl
ˆnBV
=
exp
F
(✏)
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(1.2)
Fig. 1)are ⌘given in termsofanintegralovertheinstanton-anti-instanton(IenergyI) collective
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centre-of-mass
ŝ
in
terms
of
the
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2
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7/2
m
15/2
M\10(see74also
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3 ↵W
nates
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I I¯ distance R, relativeU)[21]
14, 33,W color⇥ [12, 13, 2/3⇤ 34])⇡factor3 orientation
4⇡
while
the
pre-exponential
factor
PP
reads
23]
while
the
pre-exponential
reads
[19,
23]
W ,nh
2 [19,pwhere
d ' 0.15. (1.6)
d✏ 9 1 + O ✏p m,
"P (✏) =1
W
1024
2◆energy↵W(I)the quark-quark centre-of-mass+p )·Rŝ ◆in◆terms of
the
scale
22
✓
✓
in the, energy regionσ̂is∼Im d4 R ei (p15/2(2)
M⌘
6⇡⇤ Obviously,' 18.3 TeV
✓
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p p
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0
7/2
15/2
3
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3⇥
p7474 ⇥
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33
4⇡
↵⇤W
4⇡
2p
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2
2/3
9
9
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where d '∞0.15.
m
ŝ
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=
d
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1
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,
(1.6)
#
$
W
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%
&
p↵
162
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2 TeV , n
1024
2M
(1.4)
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, typical
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−0 ⌘
Ω U, 6⇡, ↵,...WW ' 18.3
ρ̄R
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Obviously,energyregion
m
⌧
ŝ
⌧
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the
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0
↵
,
,
.
.
.
D(ρ)
D(ρ)
dU
e
× dρ indρtheωU,
M0is parametricallyWsimilarwhichtothe energyponentiallyof the sphaleron,Ethis= f (msph energy
growinginρρ̄ρviolation
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where
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staticenergyof+ a classicallyunstablesaddle-pointp
nential2 growth
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thus ofp162
2 itis is
ark-quark centre-of-massenergy
ŝ
in
terms
of
the
scale
)
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)
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)
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*
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)
⇠
10
,
typicalaquantumtunnelingprocess.
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E
=
f
(m
/m
)⇡m
/↵
⇠
W
sph
W
W
W
h thatObviously,
inintheenergyregionŝ ŝ⌧M
cross-section
forfor(B(B++L)WWof
0 ,0theg inclusive
Obviously,
theenergym⌧⌧,theinclusive
cross-sectionL)-proceed2
2regionm
2⌧
2M
as
a
hint
the
processfield
equationstheelectroweak
theory,corresponding
to
the
minim
×
F
−p
ρ
F
−p
ρ
F
−p
ρ
F
−p
ρ
P
(U,
R,
ρ,
ρ̄;
p
·
p
)
.
√
1
2
1
1
2
2
p
p
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo9
TeV
–
the
staticenergyofaclassicallyunstablesaddle-point
solutionofthestaticbosonicpviolation
due
exponential
suppression
factor,mW small,
tion.
This
interpretationbviolation isisMunobservably
unobservably
small,
due totothe
thetremendous
tremendous
exponential
suppression
factor,is also
F
(
ŝ)
tween
inequivalent
vacua
with
di↵erent
valuesoftheChern-Simons
topo
⌘
6⇡
'
18.3
TeV
,
(1.4)
nentialis10M
andthus offor
thetheory,orderofthetunnelingenergy.
One
mayinterpretthis
0 ⇠arising
0162
field
equationsoftheelectroweak
corresponding
totheexpansion
minimalheightbeHere,the 4⇡/↵
basicgrowth
blocksininstanton-perturbation
theory–sphaleron
the semi-classicalof barrier
162
exp(
)
,
typicalaquantum
process.
Nevertheless,itisex-exisdominatedbyasemi-classically
0.5
W
↵
exp(
4⇡/↵
)
⇠
10
,
typicalforaquantumtunnelingprocess.
Nevertheless,
it
is
W
W
as a hintthat
theintegral
process
proceeds
via theanvalues
intermediate
virtualfield index
configurathe tween
corresponding
path
expression
about
instanton
solution– are sphaleron-likei) the instantoninequivalent
vacua
with
di↵erentoftheChern-Simons
topological
N
[16,
CS
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo√
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
exposize distribution
D(ρ),ii) the functionΩ, whichtakes into
account
exponentiationof gaugetion.
Thisinterpretationis alsobacked-upby =thefact2thethatcross-section (1.2)
2 the inclusive
parametricallysimilartotheenergyofthesphaleron,
E
f
(m
/m
)⇡m
/↵
⇠
0.0
nentialgrowth
is
M
and
thus
of
the
order
of
the
sphaleronenergy.
Oneinterpret this
sph
W
W
hnW
ŝ
[TeV]
W
00 and thus of the order of the sphaleronhnentialgrowthandisanM
energy.
Onemay1
1030(Z) andBoth
an instanton
enter here,
since number
we 20
write the(2)40asa discontinuity
ofmay
is dominatedbyaanti-instantonsemi-classicallylarge
ofcross-sectionW
Higgs
bosons[24], interpret thisthe static energy
of athatclassicallyunstablesaddle-point
solutionthestaticbosonic¯viaantheprocess
proceeds
virtualsphaleron-like
theas
p1 a
pa2 hint
forwardelastic
scattering
amplitude
inthe Ivia
I-background(cf. Fig. 1).ofAlternatively,one
may calculate field
as
hint
that
the
process
proceeds
anintermediate
intermediate
virtualsphaleron-likefieldconfiguraconfigura1✓◆
thetion.
cross-section
by takingthe modulus
ofbacked-up
amplitudes
in bytheminimal
instanton-background.
-0.5
ations of theelectroweak
theory,corresponding
to4⇡thebarrierbe- cross-section (1.2)
andThis
issquared
the
fact
that
theheight
inclusive
3 single
tion.
This interpretationinterpretation
isalsoalso
backed-up
by
the
fact
4/3
2that the inclusive cross-section (1.2)
1hnW i =
✏ +O ✏,(1.7)
isis dominatedbyasemi-classicallylarge
number
of
W
(Z)
and
Higgs
bosons[24],
quivalent vacua
withdi↵erent
valuesoftheChern-Simons
topological
index
N
[16,
CS
8
dominated by a semi-classically largeof W (Z) and Higgs bosons [24],3 ↵Wnumber
-1.0
✓✓
◆◆
4⇡
and
in line with the expectation from2
4⇡ 3 34/3

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1

2

1 2

4π
αg

R2 ρ̄
ρρ̄ ρ

fin

1 2

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a lot of e↵ort, the actual size of the cross-sections in the relevant, tens of TeV energy regime

forinstanton-perturbativeelectroweak (B +L)-violatingprocesses,> 10 GeV in order to becross-section
taken ∼
in the semi-classical,
regime[20, 21, 22, 24].
y interesting,could
albeitnot
unsolved
question is whether manifestations of such topologidetermined.
As mentioned
inbe
Sect.
1, axial anomalies [1] force instanton-induced hard scattering processes
toXX
uations might
beNevertheless,directlyinhigh-energy
scattering
atpresentorfuturebe always
associatedobservable
withitanomalous
fermion-number
violation[2],
inparticularB+
Lviolation,
has been established (for reviews,
that
the inclusive
q + q !see7q̄Refs.
+ 3`¯[11,+ 14, 15])
nW,Z
W (Z)
+
nh h,
△Bcross-section=has△Lbeen= −nraised
Q, electroweak
inoriginallythe case of in
QFD= 3 fermion
generations,chirality
This question
latengeneightiesin [12,13], but, anddespitegenfor(Bthe+withL)-violatingprocesses,nnhW,Zviolation,△Qof
2 nfcross-sections
Q, in the case of in
QCD
typically tens
nf = 3oflightquark
flavors.
[Khoze, Ringwald’91]
5 =↵ort, the actualsize
thethewithrelevant,
TeV
energyregime• The inclusive cross-section can be estimated using the dispersion
relations (optical theorem).
X

Xt be determined.
q+q written! 7q̄ + in
3e`¯the+ formnW,Z W (Z) +
nh h,
(1.1)
can
be
p been established (for reviews, see Refs. [11,rtheless, it has
14, 15]) that thenhinclusivenW,Z17]. The ‘holy grail function’ F
X
p1
P (✏)
4⇡ are given bytion for electroweakp1(B + L)-violating processes,
whose
first few terms
BV
BV
IW
I
I
I
ˆ s⌘
ˆ sW ,nh =expF (✏) ,
17].beThe‘holyingrail
function’
F in the X
exponent
of an
can
theWX
form
2 expansion in ✏,p2written
p2 inclis known in nterms
mW
↵W 9
9
nW
,nh
whose
first
few
terms
are
given
by
[18–22]

4/3
¯
q
+
q
!
7q̄
+
3
`
+
n
W
(Z)
+
n
h,
(1.1)
F
(✏)
=
1
✏
+
W,Z X
hP (✏)
p17].
p exponent
17]. TheThe ‘holy‘holy grail
grail function’function’theknown4⇡
termsofofananexpansion
expansioninin✏,8✏,
FF ininthe
isisknown
ininterms
1
BV
BVexponent✓◆
ˆ9incl ⌘ 9 ˆnW3,nnhh = 2m2exp
F
(✏)
,
(1.2)
nW,Z
1
where
whose first
first few
fewterms
termsarearegiven
given
by[18–22][18–22]
whose
by
↵
8/3 p
W + O(✏F (✏) of =1 instanton-induced✏4/3 +nW ,n✏h2 process4inm
3W 2h ✏8/3 ln·andconst),(1.5) factor P
Figure 1: Illustrationa QFD
proton-proton
scattering(left)
whilethepre-exponentialŝ
8 in deep-inelastic
16
32electron-protonmW ◆scattering✏ (right).
✓✓◆
of aform
QCD instanton-induced process
ritteninthe22‘holy grail’
99 4/399 2 2 33mm
11 ✏ ⌘
hh
8/3
8/3
✓
4/3
8/3
8/3
M
where FF(✏)
15/2
=
1
✏
+
✏
4
3
O(✏
·
const)
,
(1.5)
(✏)
=
1
✏
+
✏
4
O(✏
·
,
(1.5)
=

0 const)p\13 2 2 ✏ ✏ lnln ++⇡3while theXpre-exponential
factor
P reads[19, 23] m
Instanton-induced
total cross-sections
for16
hard
parton-parton(p-p)
scattering
processes(cf.
function
8
32
✏
1
2
8
16
32
m
✏
P
(✏)
4⇡
p
P
(✏)
=
ŝ WW 1 ¯
BV1024 2
ˆincl
ˆnBV
=
exp
F
(✏)
,
(1.2)
Fig. 1)are ⌘given in termsofanintegralovertheinstanton-anti-instanton(IenergyI) collective
coordi✏ ⌘◆(1.3)
centre-of-mass
ŝ
in
terms
of
the
scale
W ,nh is the2 quark-quark
2
✓
◆
✓
7/2
m
15/2
M\10(see74also
\1
3 ↵W
nates
(sizesρ,nρ̄,
I I¯ distance R, relativeU)[21]
14, 33,W color⇥ [12, 13, 2/3⇤ 34])⇡factor3 orientation
4⇡
while
the
pre-exponential
factor
PP
reads
23]
while
the
pre-exponential
reads
[19,
23]
W ,nh
2 [19,pwhere
d ' 0.15. (1.6)
d✏ 9 1 + O ✏p m,
"P (✏) =1
W
1024
2◆energy↵W(I)the quark-quark centre-of-mass+p )·Rŝ ◆in◆terms of
the
scale
22
✓
✓
in the, energy regionσ̂is∼Im d4 R ei (p15/2(2)
M⌘
6⇡⇤ Obviously,' 18.3 TeV
✓
◆
✓
p p
7/2
0
7/2
15/2
3
2 p1 · p2
3⇥
p7474 ⇥
⇡⇡
33
4⇡
↵⇤W
4⇡
2p
2/3
violationis unobservably
small,
2
2/3
9
9
P
(✏)
=
d
✏
1
+
O
✏
,
(1.6)
where d '∞0.15.
m
ŝ
P
(✏)
=
d
✏
1
+
O
✏
,
(1.6)
#
$
W
'(
%
&
p↵
162
"∞
"
"
2 TeV , n
1024
2M
(1.4)
exp(
4⇡/↵
)
⇠
10
, typical
✏
⌘
(1.3)
1024
2
−0 ⌘
Ω U, 6⇡, ↵,...WW ' 18.3
ρ̄R
W
Obviously,energyregion
m
⌧
ŝ
⌧
M
,
the
inclusive
cross-section
for
(B
+
L)W
0
↵
,
,
.
.
.
D(ρ)
D(ρ)
dU
e
× dρ indρtheωU,
M0is parametricallyWsimilarwhichtothe energyponentiallyof the sphaleron,Ethis= f (msph energy
growinginρρ̄ρviolation
small, due to the tremendous exponential suppression factor,p0
0is unobservably
where
dd'
0.15.
where
'
0.15.
9
TeV
–+the)for
staticenergyof+ a classicallyunstablesaddle-pointp
nential2 growth
M0 andsolution
thus ofp162
2 itis is
ark-quark centre-of-massenergy
ŝ
in
terms
of
the
scale
)
+
)
+
)
*
*
*
*
exp(
4⇡/↵
)
⇠
10
,
typicalaquantumtunnelingprocess.
Nevertheless,exwhichisparametricallysimilartotheenergyofthesphaleron,
E
=
f
(m
/m
)⇡m
/↵
⇠
W
sph
W
W
W
h thatObviously,
inintheenergyregionŝ ŝ⌧M
cross-section
forfor(B(B++L)WWof
0 ,0theg inclusive
Obviously,
theenergym⌧⌧,theinclusive
cross-sectionL)-proceed2
2regionm
2⌧
2M
as
a
hint
the
processfield
equationstheelectroweak
theory,corresponding
to
the
minim
×
F
−p
ρ
F
−p
ρ
F
−p
ρ
F
−p
ρ
P
(U,
R,
ρ,
ρ̄;
p
·
p
)
.
√
1
2
1
1
2
2
p
p
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo9
TeV
–
the
staticenergyofaclassicallyunstablesaddle-point
solutionofthestaticbosonicpviolation
due
exponential
suppression
factor,mW small,
tion.
This
interpretationbviolation isisMunobservably
unobservably
small,
due totothe
thetremendous
tremendous
exponential
suppression
factor,is also
F
(
ŝ)
tween
inequivalent
vacua
with
di↵erent
valuesoftheChern-Simons
topo
⌘
6⇡
'
18.3
TeV
,
(1.4)
nentialis10M
andthus offor
thetheory,orderofthetunnelingenergy.
One
mayinterpretthis
0 ⇠arising
0162
field
equationsoftheelectroweak
corresponding
totheexpansion
minimalheightbeHere,the 4⇡/↵
basicgrowth
blocksininstanton-perturbation
theory–sphaleron
the semi-classicalof barrier
162
exp(
)
,
typicalaquantum
process.
Nevertheless,itisex-exisdominatedbyasemi-classically
0.5
W
↵
exp(
4⇡/↵
)
⇠
10
,
typicalforaquantumtunnelingprocess.
Nevertheless,
it
is
W
W
as a hintthat
theintegral
process
proceeds
via theanvalues
intermediate
virtualfield index
configurathe tween
corresponding
path
expression
about
instanton
solution– are sphaleron-likei) the instantoninequivalent
vacua
with
di↵erentoftheChern-Simons
topological
N
[16,
CS
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo√
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
exposize distribution
D(ρ),ii) the functionΩ, whichtakes into
account
exponentiationof gaugetion.
Thisinterpretationis alsobacked-upby =thefact2thethatcross-section (1.2)
2 the inclusive
parametricallysimilartotheenergyofthesphaleron,
E
f
(m
/m
)⇡m
/↵
⇠
0.0
nentialgrowth
is
M
and
thus
of
the
order
of
the
sphaleronenergy.
Oneinterpret this
sph
W
W
hnW
ŝ
[TeV]
W
00 and thus of the order of the sphaleronhnentialgrowthandisanM
energy.
Onemay1
1030(Z) andBoth
an instanton
enter here,
since number
we 20
write the(2)40asa discontinuity
ofmay
is dominatedbyaanti-instantonsemi-classicallylarge
ofcross-sectionW
Higgs
bosons[24], interpret thisthe static energy
of athatclassicallyunstablesaddle-point
solutionthestaticbosonic¯viaantheprocess
proceeds
virtualsphaleron-like
theas
p1 a
pa2 hint
forwardelastic
scattering
amplitude
inthe Ivia
I-background(cf. Fig. 1).ofAlternatively,one
may calculate field
as
hint
that
the
process
proceeds
anintermediate
intermediate
virtualsphaleron-likefieldconfiguraconfigura1✓◆
thetion.
cross-section
by takingthe modulus
ofbacked-up
amplitudes
in bytheminimal
instanton-background.
-0.5
ations of theelectroweak
theory,corresponding
to4⇡thebarrierbe- cross-section (1.2)
andThis
issquared
the
fact
that
theheight
inclusive
3 single
tion.
This interpretationinterpretation
isalsoalso
backed-up
by
the
fact
4/3
2that the inclusive cross-section (1.2)
1hnW i =
✏ +O ✏,(1.7)
isis dominatedbyasemi-classicallylarge
number
of
W
(Z)
and
Higgs
bosons[24],
quivalent vacua
withdi↵erent
valuesoftheChern-Simons
topological
index
N
[16,
CS
8
dominated by a semi-classically largeof W (Z) and Higgs bosons [24],3 ↵Wnumber
-1.0
✓✓
◆◆
4⇡
and
in line with the expectation from2
4⇡ 3 34/3

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1

2

1 2

4π
αg

R2 ρ̄
ρρ̄ ρ

fin

1 2

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a lot of e↵ort, the actual size of the cross-sections in the relevant, tens of TeV energy regime

forinstanton-perturbativeelectroweak (B +L)-violatingprocesses,> 10 GeV in order to becross-section
taken ∼
in the semi-classical,
regime[20, 21, 22, 24].
y interesting,could
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toXX
uations might
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scattering
atpresentorfuturebe always
associatedobservable
withitanomalous
fermion-number
violation[2],
inparticularB+
Lviolation,
has been established (for reviews,
that
the inclusive
q + q !see7q̄Refs.
+ 3`¯[11,+ 14, 15])
nW,Z
W (Z)
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This question
latengeneightiesin [12,13], but, anddespitegenfor(Bthe+withL)-violatingprocesses,nnhW,Zviolation,△Qof
2 nfcross-sections
Q, in the case of in
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typically tens
nf = 3oflightquark
flavors.
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5 =↵ort, the actualsize
thethewithrelevant,
TeV
energyregime• The inclusive cross-section can be estimated using the dispersion
relations (optical theorem).
X

Xt be determined.
q+q written! 7q̄ + in
3e`¯the+ formnW,Z W (Z) +
nh h,
(1.1)
can
be
p been established (for reviews, see Refs. [11,rtheless, it has
14, 15]) that thenhinclusivenW,Z17]. The ‘holy grail function’ F
X
p1
P (✏)
4⇡ are given bytion for electroweakp1(B + L)-violating processes,
whose
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BV
BV
IW
I
I
I
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17].beThe‘holyingrail
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9
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first
few
terms
are
given
by
[18–22]

4/3
¯
q
+
q
!
7q̄
+
3
`
+
n
W
(Z)
+
n
h,
(1.1)
F
(✏)
=
1
✏
+
W,Z X
hP (✏)
p17].
p exponent
17]. TheThe ‘holy‘holy grail
grail function’function’theknown4⇡
termsofofananexpansion
expansioninin✏,8✏,
FF ininthe
isisknown
ininterms
1
BV
BVexponent✓◆
ˆ9incl ⌘ 9 ˆnW3,nnhh = 2m2exp
F
(✏)
,
(1.2)
nW,Z
1
where
whose first
first few
fewterms
termsarearegiven
given
by[18–22][18–22]
whose
by
↵
8/3 p
W + O(✏F (✏) of =1 instanton-induced✏4/3 +nW ,n✏h2 process4inm
3W 2h ✏8/3 ln·andconst),(1.5) factor P
Figure 1: Illustrationa QFD
proton-proton
scattering(left)
whilethepre-exponentialŝ
8 in deep-inelastic
16
32electron-protonmW ◆scattering✏ (right).
✓✓◆
of aform
QCD instanton-induced process
ritteninthe22‘holy grail’
99 4/399 2 2 33mm
11 ✏ ⌘
hh
8/3
8/3
✓
4/3
8/3
8/3
M
where FF(✏)
15/2
=
1
✏
+
✏
4
3
O(✏
·
const)
,
(1.5)
(✏)
=
1
✏
+
✏
4
O(✏
·
,
(1.5)
=

0 const)p\13 2 2 ✏ ✏ lnln ++⇡3while theXpre-exponential
factor
P reads[19, 23] m
Instanton-induced
total cross-sections
for16
hard
parton-parton(p-p)
scattering
processes(cf.
function
8
32
✏
1
2
8
16
32
m
✏
P
(✏)
4⇡
p
P
(✏)
=
ŝ WW 1 ¯
BV1024 2
ˆincl
ˆnBV
=
exp
F
(✏)
,
(1.2)
Fig. 1)are ⌘given in termsofanintegralovertheinstanton-anti-instanton(IenergyI) collective
coordi✏ ⌘◆(1.3)
centre-of-mass
ŝ
in
terms
of
the
scale
W ,nh is the2 quark-quark
2
✓
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7/2
m
15/2
M\10(see74also
\1
3 ↵W
nates
(sizesρ,nρ̄,
I I¯ distance R, relativeU)[21]
14, 33,W color⇥ [12, 13, 2/3⇤ 34])⇡factor3 orientation
4⇡
while
the
pre-exponential
factor
PP
reads
23]
while
the
pre-exponential
reads
[19,
23]
W ,nh
2 [19,pwhere
d ' 0.15. (1.6)
d✏ 9 1 + O ✏p m,
"P (✏) =1
W
1024
2◆energy↵W(I)the quark-quark centre-of-mass+p )·Rŝ ◆in◆terms of
the
scale
22
✓
✓
in the, energy regionσ̂is∼Im d4 R ei (p15/2(2)
M⌘
6⇡⇤ Obviously,' 18.3 TeV
✓
◆
✓
p p
7/2
0
7/2
15/2
3
2 p1 · p2
3⇥
p7474 ⇥
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33
4⇡
↵⇤W
4⇡
2p
2/3
violationis unobservably
small,
2
2/3
9
9
P
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=
d
✏
1
+
O
✏
,
(1.6)
where d '∞0.15.
m
ŝ
P
(✏)
=
d
✏
1
+
O
✏
,
(1.6)
#
$
W
'(
%
&
p↵
162
"∞
"
"
2 TeV , n
1024
2M
(1.4)
exp(
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)
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, typical
✏
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1024
2
−0 ⌘
Ω U, 6⇡, ↵,...WW ' 18.3
ρ̄R
W
Obviously,energyregion
m
⌧
ŝ
⌧
M
,
the
inclusive
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for
(B
+
L)W
0
↵
,
,
.
.
.
D(ρ)
D(ρ)
dU
e
× dρ indρtheωU,
M0is parametricallyWsimilarwhichtothe energyponentiallyof the sphaleron,Ethis= f (msph energy
growinginρρ̄ρviolation
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where
dd'
0.15.
where
'
0.15.
9
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staticenergyof+ a classicallyunstablesaddle-pointp
nential2 growth
M0 andsolution
thus ofp162
2 itis is
ark-quark centre-of-massenergy
ŝ
in
terms
of
the
scale
)
+
)
+
)
*
*
*
*
exp(
4⇡/↵
)
⇠
10
,
typicalaquantumtunnelingprocess.
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E
=
f
(m
/m
)⇡m
/↵
⇠
W
sph
W
W
W
h thatObviously,
inintheenergyregionŝ ŝ⌧M
cross-section
forfor(B(B++L)WWof
0 ,0theg inclusive
Obviously,
theenergym⌧⌧,theinclusive
cross-sectionL)-proceed2
2regionm
2⌧
2M
as
a
hint
the
processfield
equationstheelectroweak
theory,corresponding
to
the
minim
×
F
−p
ρ
F
−p
ρ
F
−p
ρ
F
−p
ρ
P
(U,
R,
ρ,
ρ̄;
p
·
p
)
.
√
1
2
1
1
2
2
p
p
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo9
TeV
–
the
staticenergyofaclassicallyunstablesaddle-point
solutionofthestaticbosonicpviolation
due
exponential
suppression
factor,mW small,
tion.
This
interpretationbviolation isisMunobservably
unobservably
small,
due totothe
thetremendous
tremendous
exponential
suppression
factor,is also
F
(
ŝ)
tween
inequivalent
vacua
with
di↵erent
valuesoftheChern-Simons
topo
⌘
6⇡
'
18.3
TeV
,
(1.4)
nentialis10M
andthus offor
thetheory,orderofthetunnelingenergy.
One
mayinterpretthis
0 ⇠arising
0162
field
equationsoftheelectroweak
corresponding
totheexpansion
minimalheightbeHere,the 4⇡/↵
basicgrowth
blocksininstanton-perturbation
theory–sphaleron
the semi-classicalof barrier
162
exp(
)
,
typicalaquantum
process.
Nevertheless,itisex-ex0.5
isdominatedbyasemi-classically
W
↵
exp(
4⇡/↵
)
⇠
10
,
typicalforaquantumtunnelingprocess.
Nevertheless,
it
is
W
W
as a hintthat
theintegral
process
proceeds
via theanvalues
intermediate
virtualfield index
configurathe tween
corresponding
path
expression
about
instanton
solution– are sphaleron-likei) the instantoninequivalent
vacua
with
di↵erentoftheChern-Simons
topological
N
[16,
CS
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo√
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
exposize distribution
D(ρ),ii) the functionΩ, whichtakes into
account
exponentiationof gaugetion.
Thisinterpretationis alsobacked-upby =thefact2thethatcross-section (1.2)
2 the inclusive
parametricallysimilartotheenergyofthesphaleron,
E
f
(m
/m
)⇡m
/↵
⇠
0.0
nentialgrowth
is
M
and
thus
of
the
order
of
the
sphaleronenergy.
One
may interpret
this
sph
W
W
hnW
ŝ
[TeV]
W 40energy. One
00 and thus of the order of the sphaleronhnentialgrowthand
isanM
1
10
30(Z) andBoth
an instanton
enter here,
since number
we 20
write the(2) asa discontinuity
is dominatedbyaanti-instantonsemi-classicallylarge
ofcross-sectionW
Higgs
bosons ofmay[24], interpret thisthe static energy
of athatclassicallyunstablesaddle-point
solutionthestaticbosonic¯viaantheprocess
proceeds
virtualsphaleron-like
theas
p1 a
pa2 hint
forwardelastic
scattering
amplitude
inthe Ivia
I-background(cf. Fig. 1).ofAlternatively,one
may calculate field
as
hint
that
the
process
proceeds
anintermediate
intermediate
virtualsphaleron-likefieldconfiguraconfigura1✓◆
thetion.
cross-section
by takingthe modulus
ofbacked-up
amplitudes
in bytheminimal
instanton-background.
-0.5
ations of theelectroweak
theory,corresponding
to4⇡thebarrierbe- cross-section (1.2)
andThis
issquared
the
fact
that
theheight
inclusive
3 single
tion.
This interpretationinterpretation
isalsoalso
backed-up
by
the
fact
4/3
2that the inclusive cross-section (1.2)
1hnW i =
✏ +O ✏,(1.7)
isis dominatedbyasemi-classicallylarge
number
of
W
(Z)
and
Higgs
bosons[24],
quivalent vacua
withdi↵erent
valuesoftheChern-Simons
topological
index
N
[16,
CS
8
dominated by a semi-classically largeof W (Z) and Higgs bosons [24],3 ↵Wnumber
-1.0
✓✓
◆◆
4⇡
and
in line with the expectation from2
4⇡ 3 34/3

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1

2

1 2

4π
αg

R2 ρ̄
ρρ̄ ρ

fin

1 2

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a lot of e↵ort, the actual size of the cross-sections in the relevant, tens of TeV energy regime

forinstanton-perturbativeelectroweak (B +L)-violatingprocesses,> 10 GeV in order to becross-section
taken ∼
in the semi-classical,
regime[20, 21, 22, 24].
y interesting,could
albeitnot
unsolved
question is whether manifestations of such topologidetermined.
As mentioned
inbe
Sect.
1, axial anomalies [1] force instanton-induced hard scattering processes
toXX
uations might
beNevertheless,directlyinhigh-energy
scattering
atpresentorfuturebe always
associatedobservable
withitanomalous
fermion-number
violation[2],
inparticularB+
Lviolation,
has been established (for reviews,
that
the inclusive
q + q !see7q̄Refs.
+ 3`¯[11,+ 14, 15])
nW,Z
W (Z)
+
nh h,
△Bcross-section=has△Lbeen= −nraised
Q, electroweak
inoriginallythe case of in
QFD= 3 fermion
generations,chirality
This question
latengeneightiesin [12,13], but, anddespitegenfor(Bthe+withL)-violatingprocesses,nnhW,Zviolation,△Qof
2 nfcross-sections
Q, in the case of in
QCD
typically tens
nf = 3oflightquark
flavors.
[Khoze, Ringwald’91]
5 =↵ort, the actualsize
thethewithrelevant,
TeV
energyregime• The inclusive cross-section can be estimated using the dispersion
relations (optical theorem).
X

Xt be determined.
q+q written! 7q̄ + in
3e`¯the+ formnW,Z W (Z) +
nh h,
(1.1)
can
be
p been established (for reviews, see Refs. [11,rtheless, it has
14, 15]) that thenhinclusivenW,Z17]. The ‘holy grail function’ F
X
p1
P (✏)
4⇡ are given bytion for electroweakp1(B + L)-violating processes,
whose
first few terms
BV
BV
IW
I
I
I
ˆ s⌘
ˆ sW ,nh =expF (✏) ,
17].beThe‘holyingrail
function’
F in the X
exponent
of an
can
theWX
form
2 expansion in ✏,p2written
p2 inclis known in nterms
mW
↵W 9
9
nW
,nh
whose
first
few
terms
are
given
by
[18–22]

4/3
¯
q
+
q
!
7q̄
+
3
`
+
n
W
(Z)
+
n
h,
(1.1)
F
(✏)
=
1
✏
+
W,Z X
hP (✏)
p17].
p exponent
17]. TheThe ‘holy‘holy grail
grail function’function’theknown4⇡
termsofofananexpansion
expansioninin✏,8✏,
FF ininthe
isisknown
ininterms
1
BV
BVexponent✓◆
ˆ9incl ⌘ 9 ˆnW3,nnhh = 2m2exp
F
(✏)
,
(1.2)
nW,Z
1
where
whose first
first few
fewterms
termsarearegiven
given
by[18–22][18–22]
whose
by
↵
8/3 p
W + O(✏F (✏) of =1 instanton-induced✏4/3 +nW ,n✏h2 process4inm
3W 2h ✏8/3 ln·andconst),(1.5) factor P
Figure 1: Illustrationa QFD
proton-proton
scattering(left)
whilethepre-exponentialŝ
8 in deep-inelastic
16
32electron-protonmW ◆scattering✏ (right).
✓✓◆
of aform
QCD instanton-induced process
ritteninthe22‘holy grail’
99 4/399 2 2 33mm
11 ✏ ⌘
hh
8/3
8/3
✓
4/3
8/3
8/3
M
where FF(✏)
15/2
=
1
✏
+
✏
4
3
O(✏
·
const)
,
(1.5)
(✏)
=
1
✏
+
✏
4
O(✏
·
,
(1.5)
=

0 const)p\13 2 2 ✏ ✏ lnln ++⇡3while theXpre-exponential
factor
P reads[19, 23] m
Instanton-induced
total cross-sections
for16
hard
parton-parton(p-p)
scattering
processes(cf.
function
8
32
✏
1
2
8
16
32
m
✏
P
(✏)
4⇡
p
P
(✏)
=
ŝ WW 1 ¯
BV1024 2
ˆincl
ˆnBV
=
exp
F
(✏)
,
(1.2)
Fig. 1)are ⌘given in termsofanintegralovertheinstanton-anti-instanton(IenergyI) collective
coordi✏ ⌘◆(1.3)
centre-of-mass
ŝ
in
terms
of
the
scale
W ,nh is the2 quark-quark
2
✓
◆
✓
7/2
m
15/2
M\10(see74also
\1
3 ↵W
nates
(sizesρ,nρ̄,
I I¯ distance R, relativeU)[21]
14, 33,W color⇥ [12, 13, 2/3⇤ 34])⇡factor3 orientation
4⇡
while
the
pre-exponential
factor
PP
reads
23]
while
the
pre-exponential
reads
[19,
23]
W ,nh
2 [19,pwhere
d ' 0.15. (1.6)
d✏ 9 1 + O ✏p m,
"P (✏) =1
W
1024
2◆energy↵W(I)the quark-quark centre-of-mass+p )·Rŝ ◆in◆terms of
the
scale
22
✓
✓
in the, energy regionσ̂is∼Im d4 R ei (p15/2(2)
M⌘
6⇡⇤ Obviously,' 18.3 TeV
✓
◆
✓
p p
7/2
0
7/2
15/2
3
2 p1 · p2
3⇥
p7474 ⇥
⇡⇡
33
4⇡
↵⇤W
4⇡
2p
2/3
violationis unobservably
small,
2
2/3
9
9
P
(✏)
=
d
✏
1
+
O
✏
,
(1.6)
where d '∞0.15.
m
ŝ
P
(✏)
=
d
✏
1
+
O
✏
,
(1.6)
#
$
W
'(
%
&
p↵
162
"∞
"
"
2 TeV , n
1024
2M
(1.4)
exp(
4⇡/↵
)
⇠
10
, typical
✏
⌘
(1.3)
1024
2
−0 ⌘
Ω U, 6⇡, ↵,...WW ' 18.3
ρ̄R
W
Obviously,energyregion
m
⌧
ŝ
⌧
M
,
the
inclusive
cross-section
for
(B
+
L)W
0
↵
,
,
.
.
.
D(ρ)
D(ρ)
dU
e
× dρ indρtheωU,
M0is parametricallyWsimilarwhichtothe energyponentiallyof the sphaleron,Ethis= f (msph energy
growinginρρ̄ρviolation
small, due to the tremendous exponential suppression factor,p0
0is unobservably
where
dd'
0.15.
where
'
0.15.
9
TeV
–+the)for
staticenergyof+ a classicallyunstablesaddle-pointp
nential2 growth
M0 andsolution
thus ofp162
2 itis is
ark-quark centre-of-massenergy
ŝ
in
terms
of
the
scale
)
+
)
+
)
*
*
*
*
exp(
4⇡/↵
)
⇠
10
,
typicalaquantumtunnelingprocess.
Nevertheless,exwhichisparametricallysimilartotheenergyofthesphaleron,
E
=
f
(m
/m
)⇡m
/↵
⇠
W
sph
W
W
W
h thatObviously,
inintheenergyregionŝ ŝ⌧M
cross-section
forfor(B(B++L)WWof
0 ,0theg inclusive
Obviously,
theenergym⌧⌧,theinclusive
cross-sectionL)-proceed2
2regionm
2⌧
2M
as
a
hint
the
processfield
equationstheelectroweak
theory,corresponding
to
the
minim
×
F
−p
ρ
F
−p
ρ
F
−p
ρ
F
−p
ρ
P
(U,
R,
ρ,
ρ̄;
p
·
p
)
.
√
1
2
1
1
2
2
p
p
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo9
TeV
–
the
staticenergyofaclassicallyunstablesaddle-point
solutionofthestaticbosonicp
exponential
suppression
violation
due
exponential
suppression
factor,mW small,
tion.
This
interpretationbviolation isisMunobservably
unobservably
small,
due totothe
thetremendous
tremendous
exponential
suppression
factor,is also
F
(
ŝ)
tween
inequivalent
vacua
with
di↵erent
valuesoftheChern-Simons
topo
⌘
6⇡
'
18.3
TeV
,
(1.4)
nentialis10M
andthus offor
thetheory,orderofthetunnelingenergy.
One
mayinterpretthis
0 ⇠arising
0162
field
equationsoftheelectroweak
corresponding
totheexpansion
minimalheightbeHere,the 4⇡/↵
basicgrowth
blocksininstanton-perturbation
theory–sphaleron
the semi-classicalof barrier
162
exp(
)
,
typicalaquantum
process.
Nevertheless,itisex-ex0.5
isdominatedbyasemi-classicallydisappears
at
E
~
E
?
W
sph
↵
exp(
4⇡/↵
)
⇠
10
,
typicalforaquantumtunnelingprocess.
Nevertheless,
it
is
W
W
as a hintthat
theintegral
process
proceeds
via theanvalues
intermediate
virtualfield index
configurathe tween
corresponding
path
expression
about
instanton
solution– are sphaleron-likei) the instantoninequivalent
vacua
with
di↵erentoftheChern-Simons
topological
N
[16,
CS
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
expo√
ponentiallygrowinginthisenergyregion.
Intriguingly,
thecharacteristic
scale
for
the
exposize distribution
D(ρ),ii) the functionΩ, whichtakes into
account
exponentiationof gaugetion.
Thisinterpretationis alsobacked-upby =thefact2thethatcross-section (1.2)
2 the inclusive
parametricallysimilartotheenergyofthesphaleron,
E
f
(m
/m
)⇡m
/↵
⇠
0.0
nentialgrowth
is
M
and
thus
of
the
order
of
the
sphaleronenergy.
Oneinterpret this
sph
W
W
hnW
ŝ
[TeV]
W
00 and thus of the order of the sphaleronhnentialgrowthandisanM
energy.
Onemay1
1030(Z) andBoth
an instanton
enter here,
since number
we 20
write the(2)40asa discontinuity
ofmay
is dominatedbyaanti-instantonsemi-classicallylarge
ofcross-sectionW
Higgs
bosons[24], interpret thisthe static energy
of athatclassicallyunstablesaddle-point
solutionthestaticbosonic¯viaantheprocess
proceeds
virtualsphaleron-like
theas
p1 a
pa2 hint
forwardelastic
scattering
amplitude
inthe Ivia
I-background(cf. Fig. 1).ofAlternatively,one
may calculate field
as
hint
that
the
process
proceeds
anintermediate
intermediate
virtualsphaleron-likefieldconfiguraconfigura1✓◆
thetion.
cross-section
bytakingthe modulus
ofbacked-up
amplitudes
in bytheminimal
instanton-background.
-0.5
ations of theelectroweak
theory,corresponding
to4⇡thebarrierbe- cross-section (1.2)
andThisinterpretationissquared
also
the
fact
that
theheight
inclusive
3 singletion.
Thisinterpretationisalso
backed-up
by
the
fact
4/3
2that the inclusive cross-section (1.2)
1hnW i =
✏ +O ✏
,
(1.7)
grows
isis dominatedbyasemi-classicallylarge
number
of
W
(Z)
and
Higgs
bosons[24],
quivalentCross-section
vacua
with
di↵erent
valuesoftheChern-Simons
topological
index
N
[16,
CS
8
dominated by a semi-classically largeof W (Z) and Higgs bosons [24],3 ↵Wnumber
rapidly with energy! -1.0
✓✓
◆◆
4⇡
and
in line with the expectation from2
4⇡ 3 34/3

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1

2

1 2

4π
αg

R2 ρ̄
ρρ̄ ρ

fin

1 2

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The Fourier coefficients ak, bk, Ck and hk provide us with the energy distribution of the outgoing wave (eq. (29)) which we interpret according to E(k)N(k)w(k) as multiplicity distribution of the Higgs particles and gauge bosons
plotted in fig. 4. We find an average multiplicity of 42.3 W-bosons and 7.6 Higgs
particles. The Higgs bosons carry about 11.5% of the available energy. The• The time evolution of the sphaleron field configuration has been simulated
numerically, and it found O(30) gauge bosons in the final state.
=

M. Helimund, J. Kripfganz / Decay ofthe sphaleron

754
2
E(r)

N

[Hellmund, Kripfganz ’91]

r
20.

1
t=0
0.8

15.

t=1 0

gauge bosons
10.

~

4I0tEM~~’I
5.
Higgs
k[M]
2
3
5
6
W
2N(k) of the final-state particles over the momentum k in units
Fig. 4. Multiplicity distribution 4irk
of M~.
I-

1

Fig. 1. Radial energy distribution (in units of M~)of the decaying sphaleron at different times (in unitsof M~’).

It takes a time of order 4.5 M~ until the sphaleron radiation shows free-field
behavior. In order to demonstrate this, we gauge transform the solution of the
field equations into unitary gauge K(r, t) 0, Fourier transform in r according toeqs. (24)—(27) and check the free-field time dependence of the Fourier coefficientsak, bk, Ck and hk.
Fig. 3 shows this time development of the k lMw component of the Fouriertransformed Higgs field hk fitted with the harmonic
2 k2 time+ M~dependence of a freefield. It nicely satisfies the dispersion relation w
1.
=

=

=

,CS—charge

110

2’O

3’O

~

4

The Fourier coefficients ak, bk, Ck and hk provide us with the energy distribution of the outgoing wave (eq. (29)) which we interpret according to E(k)N(k)w(k) as multiplicity distribution of the Higgs particles and gauge bosons
plotted in fig. 4. We find an average multiplicity of 42.3 W-bosons and 7.6 Higgs
particles. The Higgs bosons carry about 11.5% of the available energy. The• The time evolution of the sphaleron field configuration has been simulated
numerically, and it found O(30) gauge bosons in the final state.
=

M. Helimund, J. Kripfganz / Decay ofthe sphaleron

754
2
E(r)

N

[Hellmund, Kripfganz ’91]

r
20.

1
t=0
0.8

15.

t=1 0

gauge bosons
10.

~

4I0tEM~~’I
5.
Higgs
k[M]
2
3
5
6
W
2N(k) of the final-state particles over the momentum k in units
Fig. 4. Multiplicity distribution 4irk
of M~.
I-

1

Fig. 1. Radial energy distribution (in units of M~)of the decaying sphaleron at different times (in unitsof M~’).

It takes a time of order 4.5 M~ until the sphaleron radiation shows free-field
behavior. In order to demonstrate this, we gauge transform the solution of the
field equations into unitary gauge K(r, t) 0, Fourier transform in r according toeqs. (24)—(27) and check the free-field time dependence of the Fourier coefficientsak, bk, Ck and hk.
Fig. 3 shows this time development of the k lMw component of the Fouriertransformed Higgs field hk fitted with the harmonic
2 k2 time+ M~dependence of a freefield. It nicely satisfies the dispersion relation w
1.

4

3

• These studies
seem
thatprocess
many W-bosons
produced in theod approxithat the(Bto+ suggestL)-violating
remains to are
be exposo  instanton
is notnentially
suppressed
for incoming 2-bodyenergies
BG, whichproducesevena sphaleron-likefield configuration
at E ~ Esph,
mateandthe vacuum
waytransitions
above the sphaleronenergy.
occur semi-classicallyin some sense.
energy run
distribution
und (3), we
gnitude dis===

,CS—charge

, which ission factor.
der of mag-

110

2’O

3’O

~

• More recently (2015), Tye and Wong claims that at zero temperature
instanton rate overcomes the exponential suppression for E > Esph ~ 9
TeV, if the periodicity of the EW potential is taken into account, due toresonant tunnelling.
Resonant tunneling:Diﬀerent paths coherently interfere at particular energies,
forming a conducing band structureE• More recently (2015), Tye and Wong claims that at zero temperature
instanton rate overcomes the exponential suppression for E > Esph ~ 9
TeV, if the periodicity of the EW potential is taken into account, due toresonant tunnelling.
Resonant tunneling:Diﬀerent paths coherently interfere at particular energies,
forming a conducing band structureE• This triggers many theoretical and phenomenological studies:[Ellis, KS ’16], [Brooĳmans, Schichtel, Spannowsky ’16],[Ellis, KS, Spannowsky ’16], [Spannowsky, Tamarit ’17],[Cerdeno, Reimitz, KS, Tamarit ’18], [Y.Jho, S.C.Park ’18], …

Thereis, however, a remarkabledi↵erence
betweeon-inducedatthe LHC [41].
old energy.
We11,
and q/` processes
represent
quark/leptonoranti-quark/anti-lepton
depending
immediately dropped
considerably
andthe sphaleron-likeresearch inanalysisandthevirtualprocesses
discu
our simulationer,aremarkabledi↵erence
betweentheprocesses
considered
in
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CMS
⇤ the
lines
are
contractedwithane↵ectivevertexininstantonbackground[CMS-EXO-17-023]
this direction
stoppedessentially.
However, aevents.
recent pa• CMS has publishedtheirsearchforsphaleron-likebarrierheightmarisedabove.
Following
earlierstudies
[35]
and
for
tual sphaleron-likeprocesses
discussedintheliteratureandbrieflysumrge
NCS = ±1.
In
the
CMS
analysis,theevent
generator
BaryoGEN[42]
per by Tye and Wong [32], presenting qualitative arguTeV [16]. In
boson
processments
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elowing
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omparing
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CMS+++++++++++++++++++++++++++++++++++
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ed
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partonic
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re postulate the following partonic
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II. EVENTSIMULATION
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the rate
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parton0.4.
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ed
that
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rate
sphaleron-likeprocessescanbepeaviside theta function; ⇥(x) = 1 for x 0 and 0 otherwise, and Esph isof-massasenergy,ŝ [34].
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therefore centre-of-mass
postulate the folquired to p
be is
p
p
thresholdbehaviourafunctionoftheenergy,psph
old energy. We takeas the default value for our ˆsimulation
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lowingEpartonic
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ore postulate the following partonic cross-section:m

[Ringwald, KS, Webber 1809.10833]

HERBVI
BaryoGEN

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, andthescalar sumof>thepT andof allreconstructedobjectswith
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5

35.9 fb (13 TeV)

Events/0.1 TeV

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Data
Background shape
Systematic uncertainties
Normalization region
B1: MD=4 TeV, MBH=10 TeV, n=6
B1: MD=4 TeV, MBH=9 TeV, n=6
B1: MD=4 TeV, MBH=8 TeV, n=6

4
3
2

4

Data

Background shape3

Systematic uncertainties
Normalization region

2

Sphaleron, E= 10 TeV, PEF = 0.2

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2
0

−2

2
0

−2

e
n
35.9 fb (13nTeV)
s
e
e
t
i
os ﬃc l ev
m l e naa sig
nsig onitlim

T(Nmin , STmin [TeV])⇤
✏

104

8.5

CMS
N ≥9

(11, 34.2) (11, 4.2)

[%]

10

97.5

max(a⇤ )
Nobs

10
3.0

(Nmin , STmin [TeV])⇤

103

10

-1

CMS
N ≥ 10Data(11, 4.2)102 (11, 4.2)

99.2

99.6

10

99.9

3.0

3.0

3.0

1

3.0

(9,10−5.4)
1(9, 5.6)(9, 5.6)(8, 6.2)10−1 (8, 6.2)

2
37.7

40.5

45.3

50.5

4.0 5.84.5

5.8
5.5

3.0
6.5

sph

Background shape

Systematic uncertainties
Normalization region
Sphaleron, E

sph

1

⇤

(Data-Fit)
Unc.

✏(a ) [%]

14

(11,64.2)
6

Data
Background shape
Systematic uncertainties
Normalization region
Sphaleron, E = 9 TeV, PEF = 0.02
sph
Sphaleron, E = 8 TeV, PEF = 0.002

94.8
102

Zero Boson

935.9 fb (13 TeV)9.5

max(a⇤ )
Nobs

0

−2
6.9
3.5

5.0

(Data-Fit)
Unc.

multi-boson

(a⇤ )

8

Events/0.1 TeV

Events/0.1 TeV

Esph [TeV]

T

-1

2
0
−2

6.0
ST [TeV]

57.5
3.0 4.0

3.5

4.5

BH

D

BH

D

BH

Events/0.1 TeV

DEvents/0.1 TeV

Events/0.1 TeV⇤)
fb-1 (1335.9 fb-135.9(13 TeV)min3 ⇤
Table 1: The most 5 sensitive signal region (a = (Nmin , ST 10 ) ),CMS
the signal efficiency (✏(aTeV)
)
CMS
CMS
⇤
Data
10
max(a )Data
Data
104 N ≥ 8 N ≥ 11⇤
Backgroundupper
shape
Nthe
≥ 7 observed
at signal region a⇤10
, and
limit
for
signalregiona(N) are shown
4
2
Backgroundshape
Backgroundobs
shape
Systematic uncertainties
3 10
Normalization region
uncertainties
Systematic Systematic
uncertainties
10
for di↵erent values10of3 the partonic
E
B1: Mthreshold=4 TeV, M =10 TeV,energy
n=6
sph .
Normalization
regionNormalization
region
B1: M =4 TeV, M =9 TeV, n=6
102 10Sphaleron, E= 10 TeV,
0.2 PEF = 0.02
Sphaleron,
E PEF= 9=TeV,
B1: M =4 TeV, M =8 TeV, n=610235.9 fb-1 (13⇤ TeV)

sph

sph

10
10
1
1
1
10−1 is also found in the photon multiplicity
A large di↵erence
distributions, shown in the
−1
−1 10
−2
10
10
(Data-Fit)
Unc.

(Data-Fit)
Unc.

(Data-Fit)
Unc.

bottom panel of Fig. 2. For the zero-boson events, photons are produced only from QED
22
20
radiation or decays 0 of hadrons, while for the multi-boson
0process, high-pT photons may−2−2
2
be produced also fromFor 3.5−thezero-boson4.5case,3
4the5 primary67
8 hard
9 10 interaction.
11 12 13
4.0
6.05.0⇠6.5 85%7.0 of
3.5 4.5 4.05.0 5.5
5.5
6.0
ST [TeV]= 9 TeV, PEF = 0.02

ST [TeV]S [TeV]
T

5.0

5.5

6.0
6.5
ST [TeV]It has been suggested that the steeply-rising event rate of sphaler
modelled
by a sharp threshold behaviour as a function of the partonpŝ [43]. We therefore postulate the following partonic cross-sectionp
psph pˆ ( ŝ) = 2 ⇥( ŝ Esph ),mW

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psph

where ⇥(x) is the Heaviside theta function; ⇥(x) = 1 for x 0 andthe partonic threshold energy. We take Esph = 9 TeV as the default(later varying it by ±1 TeV), motivated by the fact that the sphal
mh = 125 GeV is given by ⇠ 9.1 TeV [16]. In Eq. (2.1), the cross1/m2W up to the dimensionless parameter psph , which controls the osame parametrisation is used in the CMS analysis [41].
The event samples for zero- and multi-boson final states are gen
and HERBVI [26], respectively, for pp collisions at centre-of-mass ene
BaryoGEN only generates partonic final states from the hard process
Pythia 8 [44] to simulate the parton shower and hadronization. H
generator interfaced to the Fortran HERWIG generator [45]. For th
Fortran version HERWIG 6.521 [46].
Esph [TeV]The above hadronic samples are then passed to Delphes [47] t
response.
Weobtained
tune
thebyparametersin Delphes
in sucha way tha
FigureFigure6: 6:TheThe
95 %
95 CL
% CL
exclusion
exclusion
limitslimitsobtainedbyrecastingrecasting
CMS
CMS
black
black
hole-sphaleron−1hole-sphaleron
σvs
<
0.8
fbreconstructedanalysis [41]as accurately as possible.
Jets
areusing th
• The
onthe
thesphmulti-boson
cross-section:sph
analysisanalysis[41]limit[41]
in the
in
(E
(E
vs
vs
)
[left
)
[left
panel]
panel]
and
and
(E
(E
vs
p
p
)
[right)[rightpanel]
panel]
planes.
planes.
TheThesph
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ight-red
light-redandandlight-bluelight-blue
regionsregions
areareexcluded
excluded
for forHERBVIHERBVI
andandBaryoGEN,
BaryoGEN,
respectively.
respectively.

• We found that their analysis can be more tuned for multi-boson analysis by
3
extending their range of Nmin cut for higher energy or luminosity
options.
(a⇤ )(a⇤ )Conclusions• The rate of zero temperature, high energy instanton-induced process
is still an open question.
• Many studies suggest that such process may have observably large
rate at collider if multiple gauge bosons are produced together withfermions.
• We recast CMS analysis and found the limit, 0.8 fb-1
• Their analysis can be more tuned for multi-boson final state byextending the range of Nmin cut for higher energy or lumi options.

Normalized Events

Normalized Events
Normalized Events

0.22
0.070.2

HERBVIHERBVI
BaryoGEN
BaryoGEN

0.18
0.06
0.16
0.05
0.14
0.12
0.04
0.1
0.03
0.08
0.02
0.06
0.04
0.01
0.02
0 0
4

0.18
HERBVI

0.16

BaryoGEN

0.14
0.12
0.1
0.08
0.06
0.04
0.02

5

5

10

6

15 7 70 20 8259
ST(N(p ) >= 11)N(p
[TeV]
> 70)
T

T

0
0

50

100

150

200

250

300

350 400
p10 [GeV]TThe ST distribution for the N = 11 multiplicity bin. The red-solid and blueFigure 4: toTheinclusive
multiplicity
containing jets, leptons and photons with pT > 70 GeV.
ograms correspond
HERBVIand BaryoGEN
event generators.

The transverse momentum distributions of the hardest (left panel) and 10th hardest (rightpanel) jets. The red-solid and blue-dashed histograms correspond to HERBVI and BaryoGEN
eventsignal
generators.
multi-boson
efficiencies are always & 95% in the most sensitive signal

min ⇤n , ST )= (11, 4.2 TeV). The larger the Nmin , the lower the Standard Model, and Nmin = 11 is therefore selected throughout Esph in the most sensitive signal
he multi-boson case.
region throughout
the Nrangeof We
Esphseefor
case, whilst Nmin = 8, 9
look at signal
the ST distributionsin Fig. 5 with
thatthethe multi-boson
valuesmin = 11.
min The distribution peaks around 6 TeV for the multi-boson
enerally andquiteShigh.
2 [5.4, 6.2] are chosen for the most sensitive signal regions depending on Esph forT
d 7 TeV for the zero-boson one. This is expected because the ST variable isthe zero-boson case. The tendency is clear that the multi-boson process favours the highreflect the total energy of the event, Esph ⇠ 9 TeV. One can also see that ST ismultiplicity
ST toregions,process prefers the low multiplicity
wer in themulti-bosonandcase small
comparedthe zerowhileboson thecase.zero-boson
This is because