Vacuum decay rate in the standard model and beyond

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1 KEK-PH 2018 Winter, Dec Vacuum decay rate in the standard model and beyond Yutaro Shoji (KMI, Nagoya U.) Phys. Lett. B771(2017)281; M. Endo, T. Moroi, M. M. Nojiri, YS JHEP11(2017)074; M. Endo, T. Moroi, M. M. Nojiri, YS Phys. Rev. Lett. 119 (2017) no.21, ; S. Chigusa, T. Moroi, YS Phys. Rev. D97 (2018) no.11, ; S. Chigusa, T. Moroi, YS Work in progress with S. Oda, D. Takahashi

2 Introduction

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sha1_base64="idf3/gup7hyfal2egvkzdapttos=">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</latexit> <latexit sha1_base64="idf3/gup7hyfal2egvkzdapttos=">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</latexit> Vacuum decay in the standard model Vacuum Energy V ( )= ( ) 2 M 2 ( ) EW vacuum of the SM is meta-stable Standard Model GeV 10 2 GeV Decay GeV Field value

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sha1_base64="wplbb7mjhbuith9bigtd9czt14c=">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</latexit> <latexit sha1_base64="wplbb7mjhbuith9bigtd9czt14c=">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</latexit> Bubble nucleation rate A rate to have a bubble in unit volume = Ae B B = S E ( ) Independent of R (classical scale invariance) = Bounce solution (Fubini-Lipatov instanton) (r) = s 8 R R 2 + r 2 for a negative λ R : Size of the instanton

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sha1_base64="0qqndsvum3bj+g+a06gw7avwt6w=">aaacbhichvg7sgnbfd1z3/evh4ugqjaovuguciqvj8zg0mrojd7yxsc6uc92nwen/octhyvakiiin2hjd1j4cslyknhyelnzebx1djnz5sw9d87m6k4p/ydoiabu1tc0njw3xfvb2js6e13dy75t8gyrmxzt8fk65gtt2iixymauedctmqwbykxfna3ur5sf50vhxgr2xlfuadu2lepdc5hanv7zpzwc3xjbtkqotwekfwi1aileseakrrcgltgwuiifarsbyxmafg4fqcc4zk2jwpzhsib7ageis7bewyizngz3edzmvsfibv5xa/qh2ubtto4ek5myonu6phe6oxt6ppdfa1xcgluvezzrna1wnzsp+7jv/6osngpsfkr+9bygiinqq2tvbshub2hu9ox945fszgaomkwx9mt+z+mbbvkgdvnvufwumrpe+qpu78/9eyypplvkq4tjqamz6cua0y9bjpb7j2mkc1hajs+1cirtnmwelv6lxxmopsqxsnodl6emfwairiy1</latexit> <latexit sha1_base64="0qqndsvum3bj+g+a06gw7avwt6w=">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</latexit> Typical scale? A rate to have a bubble in unit volume = Ae B A det S 00 E bounce det S 00 E false 1/2 Dimensional analysis A M 4 Typical scale What is the typical scale? SM is classically scale invariant

8 Renormalization scale? A rate to have a bubble in unit volume = Ae B B = S E ( ) = λ - Standard Model [ / ] What is the renormalization scale for λ?

9 The pre-factor is important! A rate to have a bubble in unit volume = Ae B A det S 00 E bounce det S 00 E false 1/2 B = S E ( ) = If we calculate A, we do not have such problems We do not need typical scale The renormalization scale uncertainty is

10 Standard one-loop Pole top mass Mt in GeV G. Ishidori, G. Ridolfi, A. Strumia; Instability Meta-stability 1,2,3 s Stability However, Higgs mass M h in GeV There were no established ways to subtract dilatational and gauge zero modes They use an approx. for the gauge zero mode subtraction, which is gauge dependent

11 <latexit sha1_base64="rhmqvj/ljxkxftxn1mn2kyuwlo4=">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</latexit> <latexit sha1_base64="rhmqvj/ljxkxftxn1mn2kyuwlo4=">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</latexit> <latexit sha1_base64="rhmqvj/ljxkxftxn1mn2kyuwlo4=">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</latexit> <latexit sha1_base64="rhmqvj/ljxkxftxn1mn2kyuwlo4=">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</latexit> However, Gauge zero mode [A. Kusenko, K. M. Lee, E. J. Weinberg; 97] There were no established ways to subtract dilatational and gauge zero modes They use an approx. for the gauge zero mode subtraction, which is gauge dependent Higgs Bounce det S 00 E bounce =0 Flat direction SU(2) We have found the correct treatment of the gauge zero mode JHEP 1711 (2017) 074, M. Endo, T. Moroi, M. M. Nojiri, YS

12 Contents Introduction Gauge invariance and analytic formulas Models beyond the standard model Summary

13 Gauge invariance and analytic formulas

14 <latexit sha1_base64="bs8qllljyyde7wbnfuyzjjm+zmk=">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</latexit> <latexit sha1_base64="rgphyuqiltd7uoxfxlzxrmvgyvs=">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</latexit> <latexit sha1_base64="rgphyuqiltd7uoxfxlzxrmvgyvs=">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</latexit> <latexit sha1_base64="97jrfsmbfjoqm782w/ls2m/jskc=">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</latexit> Gauge fixing Bounce solution NG boson L GF =(@ µ A µ g ) 2 L GF = 1 (@ µa µ ) 2 Numerical Evaluation Treatment of gauge zero modes Semi-analytical Evaluation Phys. Lett. B771(2017)281 JHEP11(2017)074

15 <latexit sha1_base64="bs8qllljyyde7wbnfuyzjjm+zmk=">aaacvxichvg7sgnbfd1zx3f9jmfgsamgifw4a6nyctawezghxbb2n6mu2ewuu5tadp5awgslkwur8tns/agl9dzigchgwrubgkiod5izm2fm3dlzr3nmw/ojbhfpynjqeiy6k8/nywulsfh8ybpbri6kum3abkvtpwealij6hm+kiumktawzoqw1d4p9cke4nmfb+37xebwwemqzh4au+kxl6/euzsim5e+gjeek47djnzhaazz0tngcgawfsqkvhrcqfbac5mromecymsj9gvpirg3zkcenvgabpb7xqjpmlv4hob1qrfmtjnexlumk6zfuaugpdefp9p5rrl6yi/ds5vkbayvtj/wxc2//qlo8+zj+vp3p2cchtkkvbnt3qiz4ht7sd07oh4xtflq3rlf0wv4vaud3/akr86pf50t+ajlxx/le7z+gtjfrkkpkcfgsybxrxozn7gapwrt5ugb6ois8sbkugp2tfbl/wajfqlr6aayjh7c=</latexit> <latexit sha1_base64="rgphyuqiltd7uoxfxlzxrmvgyvs=">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</latexit> <latexit sha1_base64="rgphyuqiltd7uoxfxlzxrmvgyvs=">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</latexit> <latexit sha1_base64="97jrfsmbfjoqm782w/ls2m/jskc=">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</latexit> Gauge fixing Bounce solution NG boson L GF =(@ µ A µ g ) 2 L GF = 1 (@ µa µ ) 2 Numerical Evaluation Treatment of gauge zero modes Standard Model A. Andreassen, W. Frost, M. D. Schwartz; 17 S. Chigusa, T. Moroi, YS; 17 Semi-analytical Evaluation Phys. Lett. B771(2017)281 JHEP11(2017)074

16 Standard Model!&# S. Chigusa, T. Moroi, YS; 18 [ log 10 γpl Gyr Gpc 3] [ ]!! /!"#!(&!(%!($!(" -!(#!"#!""!"$!"%!"&!'#!! / )*+ cf.) A. Andreassen, W. Frost, M. D. Schwartz; 17 S. Chigusa, T. Moroi, YS; 17

17 How about other models?

18 <latexit sha1_base64="xrjroppnvfd49eucc0a5/n2xopg=">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</latexit> <latexit sha1_base64="xrjroppnvfd49eucc0a5/n2xopg=">aaacghichvfnlwnbgh6s7/oqlhkxrkoc6l0krcirlo5tkrkl2v3tmtjubna3tvg9upoddk4kiulcb3dxbxz8bhekcxhwdrujihgnm/pmm+/zzjmzumnkzyd6bfkaw1rb2js6y13dpb198f6bnc+uuibigbzpuxu65gltwilns98ug44rtljuinv9b6m+v14vridta9xfd8rwwstzsignzweqeb9enm8xxc0izvoo3j6sbvohezplo9phrrbpuorcspweagssicjtxy+rxw5sgkigdaelpmmtgjxum1bbcjjbqsccy0ig+wi1xfhb4szbgrqzezyweluzsrav6zw9ug3wksz3l5ujjnidxdel3dm1pdh7r7wcsebdyz7pekmrnelf8ddk27+qms8+dj9vf3r2ucrs6fwydydk6rcwgvrqwcnlylx2nbijc3pm/2f0shd8a6v6alxkrpyumf4a9ftz/wrrkymvumpmormwgh1fb4yxgnf+7xksyblp5pjci1zibrekoowre4rasfwais0gvoqy9wfpajqc</latexit> <latexit sha1_base64="xrjroppnvfd49eucc0a5/n2xopg=">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</latexit> <latexit sha1_base64="xrjroppnvfd49eucc0a5/n2xopg=">aaacghichvfnlwnbgh6s7/oqlhkxrkoc6l0krcirlo5tkrkl2v3tmtjubna3tvg9upoddk4kiulcb3dxbxz8bhekcxhwdrujihgnm/pmm+/zzjmzumnkzyd6bfkaw1rb2js6y13dpb198f6bnc+uuibigbzpuxu65gltwilns98ug44rtljuinv9b6m+v14vridta9xfd8rwwstzsignzweqeb9enm8xxc0izvoo3j6sbvohezplo9phrrbpuorcspweagssicjtxy+rxw5sgkigdaelpmmtgjxum1bbcjjbqsccy0ig+wi1xfhb4szbgrqzezyweluzsrav6zw9ug3wksz3l5ujjnidxdel3dm1pdh7r7wcsebdyz7pekmrnelf8ddk27+qms8+dj9vf3r2ucrs6fwydydk6rcwgvrqwcnlylx2nbijc3pm/2f0shd8a6v6alxkrpyumf4a9ftz/wrrkymvumpmormwgh1fb4yxgnf+7xksyblp5pjci1zibrekoowre4rasfwais0gvoqy9wfpajqc</latexit> Manifestly gauge invariant analytic formulas γ = Bubble nucleation rate (decay rate of a vacuum) d ln R 1 R 4 [ A (h) A (σ) A (ψ) A (A µ,ϕ) e B] MS, [ ] 1-loop [ ] ln A (h) = 3 MS 4 6 ln A G ln π 3 5 λ ln ln µr 2, [ ] ln A (σ) = 1 MS 2 S σ(z κ )+ κ ( λ + κ2 1+γ 3 λ 2 E + ln µr ), 2 [ ] ( ln A (ψ) = y4 1+γ MS 3 λ 2 E + ln µr ) 2y2 2 3 λ [ ] [ ]... ( γ E + ln µr 2 ) + S ψ (z y ), B = Tree [A. Andreassen, W. Frost, M. D. Schwartz; 17, S. Chigusa, T. Moroi, YS; 18]

19 ELVAS c++ package for ELectroweak VAcuum Stability

20 Fermions make our vacuum unstable!#$ S. Chigusa, T. Moroi, YS; 18 $#) $#(!! $#' #'$ $#" $'% $'" $#$!"#$!"#%!&#$!&#%!'#$!'#%!%#$ *+,!" [!! / -./ ] %'# $'! %'"!!!! $'& %'! $'% %'& $'$!! " #$ #% #& ()*!" [!! / +,- ] %'%! " # $% $& $! ()*!" [!! / +,- ]

21 BSM scalars Scalars are more interesting! working with S. Oda, D. Takahashi at OIST * Bottom and Tau Yukawa couplings can become large Large bottom and tau couplings with a large tanβ * Additional quartic couplings Vacuum structure becomes more non-trivial * Charge breaking vacuum may appear We need to include quantum corrections from photon as well * Bounce is multi-field First example of one-loop calculation for multi-field bounce solution (but it can be reduced to single-field)

22 Summary We found a way to treat the gauge zero modes gauge invariantly and the final result is manifestly gauge invariant. We applied it to the standard model and confirmed that the lifetime of the EW vacuum is much longer than the age of the Universe. We provide a c++ package that can calculate decay rates for generic models with approximate scale invariance.