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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
848137d670f8d176528dbd705f449acb2c7d6e87 | subsection | 255 | 285 | Behavior of | Since as a function of w_{11} and w_{22}, we have\tilde{\Phi }_\mathbf {m}^{(2)} (x_{12}w_{22}^*x_{21}w_{11}^* )\in \overline{\mathcal {P}_{\mathbf {m}}(\mathfrak {p}^+_{11})_{w_{11}}\boxtimes \mathcal {P}_{\mathbf {m}}(\mathfrak {p}^+_{22})_{w_{22}}},if f(x_{11},x_{22})\in \mathcal {P}_\mathbf {k}(\mathfrak {p}^+_{11}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.03589516505599022,
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... |
2760a167e27019d414015306a20591534b9d14d1 | subsection | 256 | 285 | Behavior of | Therefore\big (\tilde{\mathcal {F}}^i_{\lambda ,k,l}f\big )\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}
:=\lim _{(\nu _1,\nu _2)\rightarrow (\lambda _1,\lambda _2)}(\nu _1+\nu _2-\lambda _1-\lambda _2)^{i-\max \lbrace \mu ,0\rbrace }
(\mathcal {F}_{\nu ,k,l}f)\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.02... |
690085e330b928fe1318134776b0edc83b3cc23a | subsection | 257 | 285 | Behavior of | Similarly, \tilde{\mathcal {F}}^{i-1}_{\lambda ,k,l} is well-defined on M_{i}^{\fg _{11}}(\mu )\boxtimes \mathcal {O}_{\mu }(D_{22})_{\tilde{K}_{22}} +\mathcal {O}_{\mu }(D_{11})_{\tilde{K}_{11}}\boxtimes M_{i}^{\fg _{22}}(\mu ), and therefore \tilde{\mathcal {F}}^i_{\lambda ,k,l} is trivial on M_i^{\fg _{11}}(\mu )\bo... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0... |
4900f59a76dd7e5bbeab9b4521609001f6e83e71 | subsection | 258 | 285 | Behavior of | Moreover, if f\in M_{i+1}^{\fg _{11}}(\mu )\boxtimes M_{i+1}^{\fg _{22}}(\mu ), then for (g_{11},g_{22})\in \fg _{11}^\mathbb {C}\oplus \fg _{22}^\mathbb {C} we have{\rm d}\big (\rho _{(\nu _1+k)+(\nu _2+l)}^{\fg _{11}}\boxtimes \rho _{(\nu _1+l)+(\nu _2+k)}^{\fg _{22}}\big )(g_{11},g_{22})f \\
\qquad {} \in M_{i+1}^{\... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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-0.04220670461654663,
... |
b8b6c4e383d325771dd4e4822b30cb83b47c1968 | subsection | 259 | 285 | Behavior of | Therefore taking the limit (\nu _1,\nu _2)\rightarrow (\lambda _1,\lambda _2) in the both sides of{\rm d}\tau _{\nu _1+\nu _2}(g_{11},g_{22})(\nu _1+\nu _2-\lambda _1-\lambda _2)^{i-\max \lbrace \mu ,0\rbrace }\mathcal {F}_{\nu ,k,l}f \\
\qquad {} =(\nu _1+\nu _2-\lambda _1-\lambda _2)^{i-\max \lbrace \mu ,0\rbrace }\m... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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-0.044023964554071426,
0... |
08a7da0708ff3141c131ec2696e59169134dbcd3 | subsection | 260 | 285 | Behavior of | Therefore the restriction of \tilde{\mathcal {F}}_{\lambda ,k,l}^i,\tilde{\mathcal {F}}^i_{\lambda ,k,l}\colon \ \big (M_{i+1}^{\fg _{11}}(\mu )\boxtimes M_{i+1}^{\fg _{22}}(\mu )\big )
/\big (M_i^{\fg _{11}}(\mu )\boxtimes M_{i+1}^{\fg _{22}}(\mu )+M_{i+1}^{\fg _{11}}(\mu )\boxtimes M_i^{\fg _{22}}(\mu )\big ) \\
\hph... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.007622933480888605,
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-... |
67d075e375df34352c2cbca3e9264170b0a156e3 | subsection | 261 | 285 | Behavior of | Then for s^{\prime }\ge 1, for \mu \in \mathbb {Z}, \max \lbrace 0,\lfloor \mu \rfloor \rbrace \le i\le \big \lceil \frac{s^{\prime }}{2}\big \rceil ,
\tilde{\mathcal {F}}_{\lambda ,k}^i:=\lim _{\nu \rightarrow \lambda } (\nu -\lambda )^{i-\max \lbrace 0,\lfloor \mu \rfloor \rbrace }\mathcal {F}_{\nu ,k}\colon \\
\qqu... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.02897203341126442,
0.030787553638219833,
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-0.032099608331918716,
0... |
eb2ac7dfcca59c41fd556226b86751f7359d9a4f | subsection | 262 | 285 | Behavior of | Let (G,G_1)=(U(q,s),U(q^{\prime },s^{\prime })\times U(q^{\prime \prime },s^{\prime \prime })) with q=q^{\prime }+q^{\prime \prime }, s^{\prime }=s^{\prime }+s^{\prime \prime }, q^{\prime }\le q^{\prime \prime },s^{\prime },s^{\prime \prime }.
Let k,l\in \mathbb {Z}_{\ge 0} (k=0 if q^{\prime }\ne s^{\prime \prime }, l=... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.0031340180430561304,
0.029680656269192696,
-0.012169832363724709,
-0.03045891597867012,
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0.03506742790341377,
-0.011101634241640568,
-0.027864718809723854,
-0.0037921045441180468... |
a1670e493a5d0c67f892fd6e6783d985c485614c | subsection | 263 | 285 | Behavior of | Let (G,G_1)=(\operatorname{SO}^*(2s),\operatorname{SO}^*(2s^{\prime })\times \operatorname{SO}^*(2s^{\prime \prime })) with s=s^{\prime }+s^{\prime \prime }, 2\le s^{\prime }\le s^{\prime \prime }.
Let k\in \mathbb {Z}_{\ge 0} (k=0 if s^{\prime }\ne s^{\prime \prime }).
We assume \mu :=\lambda +2k\in \mathbb {Z}, \mu \... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.015437808819115162,
0.016925882548093796,
-0.01614750549197197,
-0.013575809076428413,
-0.0009624782833270729,
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0.02203875035047531,
0.010691235773265362,
0.03650740161538124,
-0.022130323573946953,
-0.010530982166528702,
-0.015224136412143707,... |
f418878c70f4253797ae357a04475ec1020f9f71 | subsection | 264 | 285 | Behavior of | Let (G,G_1)=(\operatorname{SO}_0(2,2+n^{\prime \prime }),\operatorname{SO}_0(2,2)\times \operatorname{SO}(n^{\prime \prime }))\simeq (\operatorname{SO}_0(2,2+n^{\prime \prime }),{\rm SL}(2,\mathbb {R})\times {\rm SL}(2,\mathbb {R})\times \operatorname{SO}(n^{\prime \prime })) (up to covering). Let (k_1,k_2)\in \mathbb ... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.033629097044467926,
0.023924874141812325,
-0.007228577509522438,
-0.021315719932317734,
0.011786967515945435,
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0.04553050175309181,
-0.024397877976298332,
-0.025298113003373146,
-0.02033919468522072,
... |
cca19c5a4fa1d2b515fcb59672c1978056a3f86b | subsection | 265 | 285 | Behavior of | Moreover,
\tilde{\mathcal {F}}_{\lambda ,k_1,k_2}^1\colon \ \big (\mathcal {O}_\mu (D_{11})_{\tilde{K}_{11}}/M_1^{\fg _{11}}(\mu )\big )
\boxtimes \big (\mathcal {O}_\mu (D_{22})_{\tilde{K}_{22}}/M_1^{\fg _{22}}(\mu )\big )\\
\hphantom{\tilde{\mathcal {F}}_{\lambda ,k_1,k_2}^1\colon }{} \ \boxtimes V_{(k_1-k_2,0,\ldot... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.018646083772182465,
0.039458900690078735,
-0.0047759609296917915,
-0.014007451012730598,
-0.009605327621102333,
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0.0013418085873126984,
0.03259250149130821,
0.02018721029162407,
0.04275476932525635,
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-0.03967252001166344,
-0.0073737483471632,
-0... |
d33ed4225bae9747f651445a83e066f59ebde42c | subsection | 266 | 285 | Behavior of | Then the intertwining operator\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{(\lambda +2k)+(\lambda +2l)}(D_1)\rightarrow \mathcal {O}_\lambda (D)is given by (REF ),(\mathcal {F}_{\lambda ,k,l}f)\begin{pmatrix}x_{11}&x_{12}\\{}^t\hspace{-1.0pt}x_{12}&x_{22}\end{pmatrix}=\det (x_{11})^k\det (x_{22})^l\\
\qquad {} \ti... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.015144324861466885,
-0.0030460311099886894,
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0.03051753155887127,
-0.03173823282122612,
0.0038776337169110775,
0.004837028682231903,
... |
1f30c9a94fbbb54f25b688ffe267f2209f2a7478 | subsection | 267 | 285 | Behavior of | Since as a function of w_{12}, \tilde{\Phi }_\mathbf {m}^{(1)} (x_{11}\overline{w_{12}}x_{22}w_{12}^* )\in \overline{\mathcal {P}_{2\mathbf {m}}(\mathfrak {p}^+_1)_{w_{12}}} holds, if f(x_{12})\in \mathcal {P}_\mathbf {k}(\mathfrak {p}^+_1) and \mathbf {m} satisfies 2m_j>k_j for some j, then we have\tilde{\Phi }_\mathb... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.055076658725738525,
0.0063353413715958595,
-0.031825460493564606,
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0.04088793322443962,
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0.036371953785419464,
0.02148142270743847,
-0.011305209249258041,
-0.008009763434529305,
-0.02572278305888176,... |
46d6a46d76994b0b8028b04b3856e369288c5f9f | subsection | 268 | 285 | Behavior of | Therefore if \mu \in \mathbb {Z}, then\big (\tilde{\mathcal {F}}^i_{\lambda ,k,l}f\big )\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}
:=\lim _{\nu \rightarrow \lambda }(\nu -\lambda )^{i-\max \lbrace 0,\lceil \mu \rceil \rbrace }(\mathcal {F}_{\nu ,k,l}f)\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatri... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.04452165961265564,
0.02348143793642521,
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0.0012244218960404396,
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-0.02686862275004387,
... |
8824ea7dfbeddf9bcedbf322f45fd6569759ef53 | subsection | 269 | 285 | Behavior of | Moreover, if i=\big \lfloor \frac{s^{\prime }}{2}\big \rfloor , then\tilde{\mathcal {F}}^{\lfloor s^{\prime }/2\rfloor }_{\lambda ,k,l}\colon \ \mathcal {O}_{2\mu }(D_1)_{\tilde{K}_1}
/M_{2\lfloor \frac{s^{\prime }}{2}\rfloor }^{\fg _1}(2\mu )\longrightarrow \mathcal {O}_\lambda (D)_{\tilde{K}}is clearly intertwining s... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.021713711321353912,
0.02568107843399048,
-0.018921295180916786,
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0.04382415860891342,
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-0.004787545185536146,
-0.024826569482684135... |
0cfec44fbbadbc3ee3527164226d26e37a0ea169 | subsection | 270 | 285 | Behavior of | Similarly, if \mu \in \mathbb {Z}+\frac{1}{2}, then\big (\tilde{\mathcal {F}}^i_{\lambda ,k,l}f\big )\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}
:=\lim _{\nu \rightarrow \lambda }(\nu -\lambda )^{i-\max \lbrace 0,\lceil \mu \rceil \rbrace }(\mathcal {F}_{\nu ,k,l}f)\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{2... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03582560643553734,
0.01660062186419964,
-0.020369330421090126,
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0.039090804755687714,
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-0.04366818070411682,
-0.014060177840292454,
-... |
c855e498c7fca75a42457671fc7e780b806fd3dc | subsection | 271 | 285 | Behavior of | On the other hand, for (G,G_1)=(\operatorname{SU}(s,s),\operatorname{Sp}(s,\mathbb {R})), (\operatorname{SU}(s,s),\operatorname{SO}^*(2s)), (\operatorname{SO}_0(2,n),\operatorname{SO}_0(2,n^{\prime })\times \operatorname{SO}(n^{\prime \prime })) (n^{\prime }: odd), a residue of \mathcal {F}_{\tau \rho } gives a well-de... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.01980981044471264,
0.03223291039466858,
-0.013140405528247356,
-0.01956562139093876,
0.001467040041461587,
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0.03519370034337044,
0.007600374054163694,
0.05460670590400696,
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-0.01980981044471264,
-0.002768108155578375,
0.... |
8f063c9f6dd4baeedb80c69e9d8e97fabf37dd4a | subsection | 272 | 285 | Behavior of | Then for s^{\prime }\ge 2, for \mu \in \mathbb {Z}, \max \lbrace 0,\lceil \mu \rceil \rbrace \le i\le \big \lfloor \frac{s^{\prime }}{2}\big \rfloor ,
\tilde{\mathcal {F}}^i_{\lambda ,k,l}=\lim _{\nu \rightarrow \lambda }(\nu -\lambda )^{i-\max \lbrace 0,\lceil \mu \rceil \rbrace }\mathcal {F}_{\nu ,k,l}
\colon \ M_{2... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03768083080649376,
0.028619125485420227,
-0.019542163237929344,
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0.04289817810058594,
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0.018489541485905647,
-0.03963352367281914,
... |
db0c0e0f94544ec20f11bcec10940957f37e2b66 | subsection | 273 | 285 | Behavior of | Moreover,
&\tilde{\mathcal {F}}^{\lfloor s^{\prime }/2\rfloor }_{\lambda ,k,l}\colon \ \mathcal {O}_{2\mu }(D_1)_{\tilde{K}_1}
/M_{2\lfloor \frac{s^{\prime }}{2}\rfloor }^{\fg _1}(2\mu )\longrightarrow \mathcal {O}_\lambda (D)_{\tilde{K}}\quad && (\mu \in \mathbb {Z}), &\\
&\tilde{\mathcal {F}}^{\lceil s^{\prime }/2\r... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0... |
7c5c0141f88859cf649e8dc0636893089bff2795 | subsection | 274 | 285 | Behavior of | Then
\tilde{\mathcal {F}}_{\lambda ,k,l}^i:=\lim _{\nu \rightarrow \lambda } (\nu -\lambda )^{i-\max \left\lbrace 0,\left\lfloor \frac{\mu }{2}\right\rfloor \right\rbrace }\mathcal {F}_{\nu ,k,l}
\colon \ M_{2i+2}^{\fg _1}(\mu )/M_{2i}^{\fg _1}(\mu )\longrightarrow \mathcal {O}_\lambda (D)_{\tilde{K}}
is well-defined... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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8c7b586df4c90d91cf8b4920fd8cf3aff767a8da | subsection | 275 | 285 | Behavior of | Let \max \big \lbrace 0,\mu -\frac{1}{2}\big \rbrace \le i\le \big \lfloor \frac{s}{2}\big \rfloor . Then
\tilde{\mathcal {F}}_{\lambda ,k}^i:=\lim _{\nu \rightarrow \lambda } (\nu -\lambda )^{i-\max \left\lbrace 0,\mu -\frac{1}{2}\right\rbrace }\mathcal {F}_{\nu ,k}
\colon \ M_{2i+2}^{\fg _1}(\mu )/M_{2i}^{\fg _1}(\m... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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9add8d8704353801a32839e8e94b9fc9c2a22412 | subsection | 276 | 285 | Behavior of | Moreover,
\tilde{\mathcal {F}}_{\lambda ,k,l}^{\lfloor s/2\rfloor }\colon \ \mathcal {O}_{2\mu }(D_1)_{\tilde{K}_1}/M_{\left\lfloor \frac{s}{2}\right\rfloor }^{\fg _1}(2\mu )\longrightarrow \mathcal {O}_\lambda (D)_{\tilde{K}}
intertwines the \big (\fg _1,\tilde{K}_1\big )-action.
Let (G,G_1)=(\operatorname{SO}_0(2,... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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9d721a6e55987caf6c3312cb854c007ffe66fa36 | subsection | 277 | 285 | Behavior of | Moreover,
\tilde{\mathcal {F}}_{\lambda ,k_1,k_2}^1\colon \ \big (\mathcal {O}_\mu (D_1)_{\tilde{K}_1}/M_2^{\mathfrak {so}(2,n^{\prime })}(\mu )\big )\boxtimes V_{(k_1-k_2,0,\ldots ,0)}^{[n^{\prime \prime }]\vee }
\longrightarrow \mathcal {O}_\lambda (D)_{\tilde{K}}
and the restriction
\tilde{\mathcal {F}}_{\lambda ... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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671a27b414deff2ff9d04f4c1860ccd589bb350d | subsection | 278 | 285 | Explicit calculation of intertwining operators: remaining case | In this section we again set (G,G_1)=(\operatorname{SU}(s,s),\operatorname{SO}^*(2s)) with s=2r+1\ge 2 odd,
and for k,l\in \mathbb {Z}_{\ge 0}, we determine the \tilde{G}_1-intertwining operator\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{2\lambda +4k}\big (D_1,V_{(2l,\dots ,2l,0)}^{(s)\vee }\big )\rightarrow \mat... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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ef6da538bab2a2ce61ff1e595f68440ac4c7e411 | subsection | 279 | 285 | Explicit calculation of intertwining operators: remaining case | Then we proved in Proposition REF that there exist monic polynomials \varphi _{\mathbf {m},-\mathbf {l}}(\mu )\in \mathbb {C}[\mu ] of degree
l-l_{r+1} such that the intertwining operator is given by(\mathcal {F}_{\lambda ,k,l}f)(x_1+x_2)=\det (x_2)^k\sum _{\mathbf {m}\in \mathbb {Z}_{++}^r}
\sum _{{\mathbf {l}\in (\ma... | {
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"doi": "10.48550/arxiv.1506.05919",
"end": 1520,
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"raw": "Nakahama R., Norm computation and analytic continuation of vector valued holomorphic discrete series representations, J. Lie Theory 26 (2016... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0970a9ff072cc41c5ff7b438842645a58dff84aa | subsection | 280 | 285 | Explicit calculation of intertwining operators: remaining case | Moreover, if \mu =2r-l-i-1 with i=0,1,\ldots ,l-1, thenM_i^l:=\bigoplus _{\mathbf {m}\in \mathbb {Z}_{++}^r}
\bigoplus _{{\mathbf {l}\in (\mathbb {Z}_{\ge 0})^{r+1},\; |\mathbf {l}|=l\\ 0\le l_j\le m_j-m_{j+1}\\ l-l_{r+1}\le i}}
V_{{(m_1+l,m_1+l-l_1,m_2+l,m_2+l-l_2,\ldots ,\;\\
\hspace{45.0pt}m_r+l,m_r+l-l_r,l-l_{r+1})... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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44a63daeaea0ca163a29e6061346a49c517161e6 | subsection | 281 | 285 | Explicit calculation of intertwining operators: remaining case | Then since
\mathcal {K}_{\mathbf {m},-\mathbf {l}}^{(1,4)}\left(x_2;\frac{1}{2}\overline{\frac{\partial }{\partial x_1}}\right)f(x_1)=0 holds if
f\in V_{(n_1+2l,n_1+2l-k_1,n_2+2l,n_2+2l-k_2,\ldots ,n_r+2l,n_r+2l-k_r,2l-k_{r+1})}^{(s)\vee }
with n_j<2m_j, n_j-k_j<2m_j-2l_j, or 2l-k_{r+1}<2l-2l_{r+1}, we have\operatornam... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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5f0f90b52609287a7b495ad18494e6783874555e | subsection | 282 | 285 | Explicit calculation of intertwining operators: remaining case | That is, the preimage \hat{M} contains a \tilde{K}_1-type V_M such thatV_M\lnot \subset M_{2i}^{2l}, \qquad {\rm d}\rho _{2r-2l-2i-1}(\mathfrak {p}^+_1)V_M\subset M_{2i}^{2l}.Then since the action {\rm d}\rho _{2r-2l-2i-1}(\mathfrak {p}^+_1) is given by 1st order differential operators of constant coefficients, in gene... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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027aa84013d1baa8bf9e02a5cf6bea586277f0e2 | subsection | 283 | 285 | Explicit calculation of intertwining operators: remaining case | Since \mathcal {O}_{2r-2l-2i-1}\big (D_1,V_{(2l,\dots ,2l,0)}^{(s)\vee }\big )_{\tilde{K}_1}/M_{2i}^{2l} is infinitesimally unitary,
this is completely reducible, and any \tilde{K}_1-type of this module is contained in some irreducible submodule. Therefore we have \operatorname{Ker}\tilde{\mathcal {F}}_{k,l}^i=\mathcal... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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1192d1ecdc312dce9698175f40afe442a44685d0 | subsection | 284 | 285 | Explicit calculation of intertwining operators: remaining case | Then the linear map\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{2\lambda +4k}\big (D_1,V_{(2l,\dots ,2l,0)}^{(s)\vee }\big )\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,k,l}f)(x_1+x_2)=\det (x_2)^k\sum _{\mathbf {m}\in \mathbb {Z}_{++}^{\lfloor s/2\rfloor }}
\sum _{{\mathbf {l}\in (\mathbb {Z}_... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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cb0304e2538e09cd01ccf78077e4330ae8936635 | abstract | 0 | 78 | Abstract | For a scalar theory whose classical scale invariance is broken by quantum
effects, we compute self-consistent bounce solutions and Green's functions.
Deriving analytic expressions, we find that the latter are similar to the
Green's functions in the archetypal thin-wall model for tunneling between
quasi-degenerate vacua... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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e4b4cdc6fad5b00d3bdbf83fdb11703dc6282690 | subsection | 1 | 78 | Introduction | A sufficiently large lifetime of metastable vacuum states , is an important criterion
for the viability of models of electroweak symmetry breaking , , , . For other sectors that are more or less closely tied to the electroweak one, false vacuum decay
can play an essential role in cosmology . Since tunneling events do n... | {
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"Bjorn Garbrecht",
"Peter Millington"
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f8de11ea416f2abbaf9b0ca9d98bb350b973ffcc | subsection | 2 | 78 | Introduction | This requires the knowledge of the eigenmodes, the complete set of which, however, does not appear
to be available in terms of analytic expressions for
the Fubini-Lipatov case. In Sec. , we therefore return
to the archetypal case of tunneling between quasi-degenerate vacua, originally considered
by Coleman and Callan ,... | {
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"arxiv_i... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
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"Bjorn Garbrecht",
"Peter Millington"
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9b81c719b7f4f1ee902c73abe45f63d775f685db | subsection | 3 | 78 | Setup | We work with the following Euclidean Lagrangian, comprising a real scalar field \Phi _x\equiv \Phi (x):\mathcal {L}\ =\ \frac{1}{2}\,(\partial _\mu \Phi )^2\:+\:U(\Phi )\:+\:\delta U(\Phi )\;,where the classical potential isU(\Phi )\ =\ \frac{1}{2}\,m^2\Phi ^2 \:+\:\frac{1}{3!}\,g\,\Phi ^3\:+\: \frac{1}{4!}\,\lambda \,... | {
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"Bjorn Garbrecht",
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95f5c4a3d3a8bf6966ae6b459d6a26e3530b2e6d | subsection | 4 | 78 | Setup | These results should give a first approximation to the tunneling rate per unit volume/V\ \sim \ e^{-B/\hbar }\;,whenever m\ll R^{-1}, which is what we will also find explicitly when including radiative corrections, cf. the results for the bounce solutions shown in Sec. .Now, radiative corrections will, in general, brea... | {
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d36d680fc86c787328f1ce2c14da51375925b703 | subsection | 5 | 78 | Setup | The effective potential resulting from our choice of renormalization conditions is given by (see App. )U^{\rm ren}_{\rm eff}(\varphi )\ &=\ \frac{1}{2}\,m^2\varphi ^2\:+\:\frac{\lambda }{4!}\,\varphi ^4\\
&\quad +\:\frac{1}{256\pi ^2}\Big \lbrace \big (2m^2\:+\:\lambda \varphi ^2\big )^2\ln \frac{2m^2\:+\:\lambda \varp... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
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92bcf416444dd5ac5cda07ac40ff64bb7df29d5a | subsection | 6 | 78 | Spectrum of fluctuations | We aim to compute radiative corrections to tunneling transitions using Green's functions.
While these can be obtained as direct solutions to their defining
equations, additional insights can be gained through consideration of the particular contributions
from the fluctuation spectrum.
Unfortunately, in the Fubini-Lipat... | {
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in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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dfb6982dfb44f9f13d9a55b2b7c75a90501e8a6f | subsection | 7 | 78 | Spectrum of fluctuations | The radial eigenvalue equation is&\bigg [
-\:\frac{{\rm d}^2}{{\rm d}r^2}\:-\:\frac{3}{r}\,\frac{\rm d}{{\rm d} r}\:+\:\frac{j(j+2)}{r^2}\\&\qquad \qquad +\:U^{\prime \prime }(\varphi )
\bigg ]\phi _{\lambda ^X j}\ =\ \lambda ^X \phi _{\lambda ^X j}\;.This applies to both the Fubini-Lipatov and the thin-wall cases, whi... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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a720a5c6829e3db7a11ea019cc706ab344b01e32 | subsection | 8 | 78 | Spectrum of fluctuations | Therefore, and in order to gain further insight into
the nature of the radiative effects, it is useful to represent the Green's function as a
spectral sum, which can be written in the following form:&G(x,x^{\prime })\ =\ \frac{1}{2\pi ^2}\sum _{j\,=\,0}^{\infty }(j+1)U_j(\cos \theta )\\&\qquad \times \left[\sum _{\lamb... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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5df531e26f006b1afb9db10f31ef70f9c62764b1 | subsection | 9 | 78 | Zero and negative modes | We first consider the eigenmodes with zero and negative eigenvalues. The zero modes are
associated with the spontaneous breakdown of symmetries: translations in both the Fubini-Lipatov
and thin-wall cases, and, in addition, dilatations around the tree-level Fubini-Lipatov instantons.
A negative eigenmode is a hallmark ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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bd9fd3b4299fba0c2095cf0968779322e53c4db1 | subsection | 10 | 78 | Thin wall | In the thin-wall regime, the gradients of the bounce are negligible everywhere except in the vicinity of the bubble wall. We can therefore make the following series of approximations for the damping term in the equation of motion (REF ):-\:\frac{3}{r}\,\frac{{\rm d}}{{\rm d}r}\,\varphi (r)\ \approx \ -\:\frac{3}{R}\,\f... | {
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]... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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01fa827e734868b1c400cd718d3b79e7743c4821 | subsection | 11 | 78 | Thin wall | Acting on the equation of motion for the bounce with the operator \partial _r, we find\partial _r\bigg [-\:\frac{{\rm d}^2}{{\rm d}r^2}\,\varphi \:-\:\frac{3}{r}\,\frac{{\rm d}}{{\rm d}r}\,\varphi \:+\:U^{\prime }(\varphi )\bigg ]\ &=\ 0\\
\Leftrightarrow \ \bigg [-\:\frac{{\rm d}^2}{{\rm d}r^2}\:-\:\frac{3}{r}\,\frac{... | {
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"start": 113... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
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b99e079172c2011c2cf26783483b308e7d26490f | subsection | 12 | 78 | Thin wall | However, acting instead on the equation of motion for the bounce, we find\bar{D}\bigg [-\:\frac{{\rm d}^2}{{\rm d}r^2}\,\varphi \:-\:\frac{3}{r}\,\frac{{\rm d}}{{\rm d}r}\,\varphi \:+\:U^{\prime }(\varphi )\bigg ]\ &=\ 0\\
\Leftrightarrow \ \bigg [-\:\frac{{\rm d}^2}{{\rm d}r^2}\:-\:\frac{1}{r}\,\frac{{\rm d}}{{\rm d}r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
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bb16c8b806a2bbcc6f0b03d5c2520e2eda79d94f | subsection | 13 | 78 | Thin wall | Nevertheless,
for each j, there are two discrete eigenvalues and a continuum starting for positive energies
in the corresponding quantum mechanical problem .In the thin-wall regime, we can make the approximations (see Ref. )\frac{j(j+2)}{r^2}\ \longrightarrow \ \frac{j(j+2)}{R^2}\;,\\
-\:\frac{3}{r}\,\frac{\rm d}{{\rm ... | {
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... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
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] | [
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e13af12071a11d2ec80b1caa10760917a4170882 | subsection | 14 | 78 | Thin wall | 1 - 3 \,
\frac{(1 - u^{\prime })(1 - \omega + u^{\prime })}{(1 - \omega )(2 - \omega )} \bigg )
+ (u \leftrightarrow u^{\prime }) \bigg ] \; ,where \vartheta (z) is the generalized unit-step function.It is helpful, however, to proceed slightly differently, by substituting for the function \tilde{G}_j(r,r^{\prime })\equ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
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b3f0ed6cf39a80f2bf98a592557952395766da74 | subsection | 15 | 78 | Thin wall | Specifically, the coincident limit of the Green's function isG(r, r) \ & = \ \frac{\gamma }{4 \pi ^2 R^3}
\sum _{j\, =\, 0}^{\infty } \frac{(j + 1)^2}{\omega }\bigg [ \frac{1}{\gamma ^2} \\&\qquad + 3
\big ( 1 - u^2 \big ) \sum _{n\, =\, 1}^2
\frac{(-1)^n (n - 1 - u^2) }{\gamma ^2(\omega ^2 - n^2)} \bigg ] \;,where the... | {
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... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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9251d68c925da33caf28f0bc253d1adc7279e401 | subsection | 16 | 78 | Fubini-Lipatov instanton | We now turn our attention to the spectrum of fluctuations over the Fubini-Lipatov instanton in Eq. (REF ). For this given background, the eigenvalue equation (REF ) has the form&\bigg [-\:\frac{{\rm d}^2}{{\rm d} r^2}\:-\:\frac{3}{r}\,\frac{{\rm d}}{{\rm d}r}\:+\:\frac{j(j+2)}{r^2}\\&\qquad -\:\frac{24R^2}{(r^2+R^2)^2}... | {
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} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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f73a1a00db6cc7b10ab36dbbe1117c19df4c26cd | subsection | 17 | 78 | Fubini-Lipatov instanton | (REF )
for j=0 and \lambda ^{\rm FL}=0, we find&D\left[\left(-\:\frac{\rm d^2}{{\rm d}r^2}\:-\:\frac{3}{r}\,\frac{\rm d}{{\rm d}r}\right)\varphi \:+\:U^\prime (\varphi )\right]
\\&
\quad -\:\left(-\:\frac{\rm d^2}{{\rm d}r^2}\:-\:\frac{3}{r}\frac{\rm d}{{\rm d}r}\:+\:U^{\prime \prime }(\varphi )\right)D\varphi \\
&\qua... | {
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... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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677d3a2964c40760c11f71145a50de71834dcfe3 | subsection | 18 | 78 | Fubini-Lipatov instanton | Introducing the dimensionless variable z=r/R and defining f(z)\equiv (1+z^2)^2\phi _0(r), we are looking for the solution to&-\,z\,(1+z^2)f^{\prime \prime }(z)\:+\:(5z^2-3)f^{\prime }(z)\\&\qquad \qquad +\:z(|\lambda ^{\rm FL}_{20}|-8+|\lambda ^{\rm FL}_{20}|\,z^2)f(z)\ =\ 0\;.By the Frobenius method, we obtain the fol... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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0e122c9e9a53b33359a466d0034d0933fec83f31 | subsection | 19 | 78 | Fubini-Lipatov instanton | (REF ) takes the form:L^{\rm FL}_{{\rm d}0}\ &=\ \left\lbrace \lambda _{20}^{\rm FL}\right\rbrace \;,\\
L^{\rm FL}_{{\rm d}j}\ &=\ \emptyset \;\;\textnormal {for}\ j\: >\: 0\;,\\
L^{\rm FL}_{{\rm c}j}\ &=\ \left\lbrace \lambda ^{\rm FL}\;|\;\lambda ^{\rm FL}\: >\: 0\right\rbrace \;,where the superscript {\rm FL} indica... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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7a4009e952e1f64a7659ad5dc1512665ba7f9317 | subsection | 20 | 78 | Fubini-Lipatov instanton | In that situation, the above spectrum is modified
toL^{\rm FL}_{{\rm d}0}\ &=\ \left\lbrace \lambda _{20}^{\rm FL}\right\rbrace \;,\\
L^{\rm FL}_{{\rm d}1}\ &=\ \left\lbrace 0\right\rbrace \;,\\
L^{\rm FL}_{{\rm d}j}\ &=\ \emptyset \;\;\textnormal {for}\ j\:>\:1\;,\\
L^{\rm FL}_{{\rm c}j}\ &=\ \left\lbrace \lambda ^{\r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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2b78b1f25c4a020215e54fab9a29b3d7c9d621e8 | subsection | 21 | 78 | Fubini-Lipatov instanton | (REF ), we haver^{(\prime )}\ =\ R\bigg (\frac{1-u^{(\prime )}}{1+u^{(\prime )}}\bigg )^{1/2}\;.This change of variables leads to&\Bigg [\frac{\rm d}{{\rm d}u}\,(1-u^2)\,\frac{\rm d}{{\rm d}u}\:-\:\frac{\omega ^2}{1-u^2}\:+\:6\Bigg ]F_j(u,u^{\prime })\\&\qquad =\ -\,\bigg (\frac{1-u}{1+u}\bigg )\bigg (\frac{1+u^{\prime... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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9b362011da2288eda85e6bfc9196d671007cf4a3 | subsection | 22 | 78 | Fubini-Lipatov instanton | REF .
We may recognize zero modes for j=0, n=1 and j=1, n=2, which correspond to the five Goldstone modes that arise from the spontaneous
breakdown of symmetries in presence of the bounce solutions: one (j=0) arises from broken dilatational and four (j=1) from broken translational invariance. In addition,
there is one ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
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994050ba7d1c12b1bae79b126bd70dea3a580582 | subsection | 23 | 78 | Spectral sum | In order to understand the apparent mismatch between the eigenspectra and the form
of the coincident Green's function in the Fubini-Lipatov background (REF ), it would be interesting to compare with an explicit construction via a spectral sum in the form of Eq. (REF ). However, this cannot be carried out analytically f... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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071f41f16bb206054cfca36232d3891e89bfa28f | subsection | 24 | 78 | Spectral sum | (REF ) and changing coordinates via Eq. (REF ), the eigenproblem becomes\bigg [\frac{\rm d}{{\rm d}u}\,(1-u^2)\,\frac{\rm d}{{\rm d}u}\:-\:\frac{\varpi ^2}{1-u^2}\:+\:6\bigg ]\tilde{\phi }_{\lambda ^{\rm TW} j}^{\pm }(u)\ =\ 0\;,where\varpi ^2\ \equiv \ \omega ^2\:-\:\bar{\lambda }^{\rm TW}\;,\bar{\lambda }^{\rm TW}\eq... | {
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"raw": "B. Garbrecht and P. Millington, Phys. Rev. D 91 (2015) no. 10, 105021 [arXiv:1501.07466 [hep-th]].",
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... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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079043bcd2cbb76c6124383f74e5dd97e726e050 | subsection | 25 | 78 | Spectral sum | However, rather than dealing with the associated Legendre functions of imaginary order, we can definef^{\pm }_{\lambda ^{\rm TW} j}(u)\ =\ \bigg (\frac{1-u}{1+u}\bigg )^{\!\pm \varpi /2}\tilde{\phi }^{\pm }_{\lambda ^{\rm TW} j}(u)and recast the eigenvalue problem in terms of the Jacobi differential equation\bigg [(1-u... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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bfca4e1fb99d6efe2f4e9e9bdb8dbac5005ed518 | subsection | 26 | 78 | Spectral sum | These \bar{\lambda }^{\rm TW}=\omega ^2-\varpi ^2 of the thin-wall basis should not be confused with those of the thin-wall case, which differ in the value of \omega . We emphasize that the Fubini-Lipatov value of \omega =j+1 [cf. Eq. (REF ) for the thin-wall case] does not change under the basis transformation.In orde... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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4bb6df6c5a6cb4015bc4a2ab6b471d944e242a50 | subsection | 27 | 78 | Spectral sum | Moreover, it indicates that the transformation, described here and in App. for illustrative purposes, is not a basis transformation in the proper sense, since the two bases span different Hilbert spaces.Returning to the problem of finding the spectral sum representations in the thin-wall basis, the solutions to Eq. (R... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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e26da3aeaf354bea6b2af586901e6ca31c8d70f4 | subsection | 28 | 78 | Spectral sum | For \varpi =n\in \lbrace 1,2\rbrace , we can show that these are normalizable (see App. )\int _{-1}^{+1}\frac{{\rm d}u}{1-u^2}\;\bigg (\frac{u+1}{u-1}&\bigg )^{\!+\frac{n}{2}}\bigg (\frac{u+1}{u-1}\bigg )^{\!-\frac{n^{\prime }}{2}}P_2^{(-n,+n)}(u)\\\times \:P_2^{(+n^{\prime },-n^{\prime })}(u)\ &=\ \frac{(-1)^n\pi }{4\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.023... |
734d1856c1a6cdcdb5d7e8d42d4f0a9270264d9a | subsection | 29 | 78 | Spectral sum | The contributions of the apparent continuum modes to the Green's functions areG_{\rm c}(r,r)\ &=\ \frac{1}{2\pi ^2R^2}\begin{Bmatrix} \frac{1}{\gamma R}\\ \Big (\frac{1+u}{1-u}\Big )\end{Bmatrix}\\&\qquad \times \sum _{j\,=\,0}^{\infty }(j+1)^2\int _{-\infty }^{+\infty }\frac{{\rm d}\xi }{2\pi }\;\frac{4}{(4+\xi ^2)(1+... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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17e20ea87f7e9bb3abf7100be87cfbe200e0dffc | subsection | 30 | 78 | Spectral sum | The eigenspectra for both cases are summarized in Table REF .We remark that, while one might be tempted to remove the apparent discrete zero modes in the thin-wall basis, this will not eliminate the infrared divergences, since these reside also in the continuum, as is clear from the interplay of the apparent discrete a... | {
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{
"arxiv_id": "",
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"raw": "A. Andreassen, W. Frost and M. D. Schwartz, Phys. Rev. D 97 (2018) no. 5, 056006 [arXiv:1707.08124 [hep-ph]].",
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"start": 356
... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.018503611907362938,
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... |
15ae079196284f3797b1d46be1b37ee5ae8f36d0 | subsection | 31 | 78 | Zero modes | In order to deal with the translational zero modes, we first decompose the field as\Phi (x)\ =\ \varphi (x-y)\:+\:\sum _{\mu \,=\,1}^{4}a_{\mu }\phi _{\mu }(x-y)\:+\:\Phi ^{\prime }(x-y)\;,where \varphi (x-y) is the bounce, y is its coordinate centre, \Phi ^{\prime }(x-y) contains the contributions of the negative- and... | {
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{
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"raw": "J. L. Gervais and B. Sakita, Phys. Rev. D 11 (1975) 2943.",
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"start": 1457
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"arxiv_id": "",
"doi": ""... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
-0.011841812171041965,
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... |
1e3c450e84cb2b87326ccd47d300a6f7d8286138 | subsection | 32 | 78 | Zero modes | We takef_{\mu }(y)\ =\ \int \!{\rm d}^4x\;\Phi (x)\partial _{\mu }^{(x)}\varphi (x-y)\;,from which it follows that\partial _{\mu }^{(y)}f_{\mu }(y)\ =\ -\:\int \!{\rm d}^4x\;\Phi (x)\partial _{\mu }^{(x)}\partial _{\mu }^{(x)}\varphi (x-y)\;.By virtue of the orthogonality of the eigenmodes, we can quickly show thatf_{\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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6c87333e51d52d8f6ac58c2dc772ee27685acce5 | subsection | 33 | 78 | Zero modes | (REF ) gives&\int \mathcal {D}\Phi \ =\ \bigg (\frac{1}{2\pi \hbar {\cal N}^2}\bigg )^{\!2}\int \mathcal {D}\Phi ^{\prime }\int \!{\rm d}^4y\\&\quad \times \:\prod _{\mu \,=\,1}^4\bigg (1-{\cal N}^2 \int \!{\rm d}^4x\;\Phi ^{\prime }(x-y)\partial _{\mu }^{(x)}\partial _{\mu }^{(x)}\varphi (x-y)\bigg )\;.The integral wi... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
-0.039032477885484695,
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0.0006756868096999824,
0.035309284925460815,
... |
90c6f52cf36ca9056d8f39a1b35b9f4d84bdeadf | subsection | 34 | 78 | Zero modes | We have also made explicit the bookkeeping factors of \hbar ^{1/2}.The Faddeev-Popov procedure described above then leads to
(We use the more compact index notation where, e.g., f(x)\equiv f_x for functions
and \int \!{\rm d}^4 x\equiv \int _x for integrals.)\mathcal {Z}[J]\ &=\ VT\,\frac{B^2}{(2\pi \hbar )^2}
\\&\quad... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.03250572085380554,
0.01406... |
ca47508cf83d2aaab3c2c3ca4050a526d3e1af96 | subsection | 35 | 78 | Zero modes | Nevertheless, the inversion is well defined in the subspace perpendicular to the
zero modes, where the subtracted two-point function G^\perp is the solution to &\int _z\;G^{-1}_{xz}\;G^{\perp }_{zy}\ =\ \delta ^4_{xy}-\:\,\sum _{\mu \,=\,1}^4\big (\phi ^{(0)}_{\mu }\big )_x\,\big (\phi ^{(0)}_{\mu }\big )_y\;,with the ... | {
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{
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"raw": "A. A. Aleinikov and E. V. Shuryak, Sov. J. Nucl. Phys. 46 (1987) 76 [Yad. Fiz. 46 (1987) 122].",
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} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.027160000056028366,
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0.011527741327881813,
0.007... |
ac45ffdde5d1b703692f21bae13fb71aa1cc252a | subsection | 36 | 78 | Zero modes | We then define\varphi ^\perp _x\ &=\ \hbar \,\frac{1}{\mathcal {Z}^{\perp }[0]}\,\frac{\delta }{\delta J_x}\,\mathcal {Z}^\perp [J]\bigg |_{J\,=\,0}\,,\\
\hbar \,{\cal G}^\perp _{xy}\ &=\ \hbar ^2\,\frac{1}{\mathcal {Z}^{\perp }[0]}\,\frac{\delta ^2}{\delta J_x\delta J_y}\,\mathcal {Z}^\perp [J]\bigg |_{J\,=\,0}\;,\\
... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
-0.06420818716287613,
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0.00794973038136959,
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-0.000367160071618855,
0.0003817034012172371,
-0.0112... |
64cbac00732f5ac3ce92a842443e316d6ffb914f | subsection | 37 | 78 | Zero modes | Note that Goldstone's theorem implies that not only are \partial _\mu \varphi ^{(0)}
zero modes of the inverse Green's function G^{-1} at tree-level but that
the same holds true at each order in perturbation theory
up to the exact solutions \partial _\mu \varphi ^{\mathrel {\hbox{\rule []{3pt}{.4pt}}\hspace{-2.22214pt}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
-0.032777346670627594,
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0.024964498355984688,
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0.010933412238955498,
... |
a25928f932e2070d355c935ca694e01f47b520ca | subsection | 38 | 78 | Zero modes | (REF ),
\Pi and \Sigma , as well as the solution \varphi and
{\cal G}, are given to a certain order in this expansion
and we consider an infinitesimal translation
\varphi \rightarrow \varphi +\varepsilon \partial _\varrho \varphi of Eq. (REF ) in the
\varrho direction. In this way, we obtain&\int _z
\bigg [\delta _{xz}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.0026901152450591326,
0.0015625563682988286,
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0.013820705935359001,
0.036173466593027115,
0.008280212990939617,
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0.02852666936814785,
-0.017964627593755722,
0.0... |
b17baa442f7c57f91451230cec8db42e1cdfb5ab | subsection | 39 | 78 | Zero modes | (REF ) in the
form of
\frac{\delta }{\delta \varphi _z} \left(
\int _{w} G^{-1}_{xw}G^\perp _{wy}\:+\:\sum \limits _{\mu \,=\,1}^4 \big (\phi _\mu \big )_{x}\big (\phi _\mu \big )_{y}
\right)
\ =\ \frac{\delta }{\delta \varphi _z}\,\delta _{xy}\ =\ 0\;,
we obtain
&\frac{\delta G^\perp _{xy}}{\delta \varphi _z}\ =\ ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.02905752882361412,
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0.011102600954473019,
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-0.023365061730146408,
-0.0... |
68cefc112a04b2dd736b158fe737534d6ff5b648 | subsection | 40 | 78 | Zero modes | The remaining
second term in curly brackets leads to the contribution
&-\:\sum _{\nu \,=\,1}^4\int _{ywz}
\bigg [\left(
(\partial _\varrho \varphi )_{z}
\frac{\delta }{\delta \varphi _z}
(\phi _{\nu }\phi _{\nu })_{xw}\right)G^\perp _{wy}
\\&\qquad \qquad +\:G^\perp _{xw}\left((\partial _\varrho \varphi )_{z}\frac{\de... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.022465655580163002,
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0.033973198384046555,
0.02216041460633278,
-0.053935885429382324,
0.006612258963286877,
0.0357435904443264,
0.015... |
df47660b50868925796cb1c4a1418d34f4b00777 | subsection | 41 | 78 | Zero modes | The functions
\varphi ^{\mathrel {\hbox{\rule []{3pt}{.4pt}}\hspace{-2.22214pt}\hbox{\usefont {U}{lasy}{m}{n})}}} or {\cal G}^{\mathrel {\hbox{\rule []{3pt}{.4pt}}\hspace{-2.22214pt}\hbox{\usefont {U}{lasy}{m}{n})}}} can be obtained to order \hbar as\varphi ^{\mathrel {\hbox{\rule []{3pt}{.4pt}}\hspace{-2.22214pt}\hbox... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
0.0038874659221619368,
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0.0003712453763000667,
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0.032503642141819,
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0.0324120819568634,
0.0157... |
1d77b3dc61a9299bf231903de4d1053b5fa1d989 | subsection | 42 | 78 | Zero modes | (REF ). It would be desirable to
derive a formally exact expression for {\cal G}^{\mathrel {\hbox{\rule []{3pt}{.4pt}}\hspace{-2.22214pt}\hbox{\usefont {U}{lasy}{m}{n})}}-1} that also
points to systematic approximations for this quantity to all orders.An expression for the proper self-energy that should appear in
{\cal... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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... |
baed6baee6ef14450e06f70486ebac72cdd8618b | subsection | 43 | 78 | Self-consistent solutions in the classically scale-invariant model | We now apply the more general considerations of the previous sections
in order to obtain solutions in a classically scale-invariant setup,
i.e. self-consistent radiative corrections of the Fubini-Lipatov instanton.
For the contributions j\ge 2, the Green's functions can be
computed straightforwardly in the background o... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.028... |
7491f32bde881edb259b3b15291b7bb2257356ef | subsection | 44 | 78 | Modes | For j\ge 2, we can proceed straightforwardly, i.e. we can substitute the Fubini-Lipatov instanton,
which is the tree-level solution for \varphi from Eq. (REF ), in Eq. (REF ) for the Green's functions. The solutions G_j, to leading accuracy in the gradient corrections, are presented in Sec. REF and App. . | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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7857b53dc9a68e299640aba591e01f7d3c3c536b | subsection | 45 | 78 | Spectator fields | In order to force the action to have an extremum at the bounce, we add to the model in Eq. (REF ) N_\chi spectator fields \chi , as in Eq. (REF ). At leading order, the Green's functions for each of the fields \chi are determined as the solution to&\bigg [
-\:\frac{{\rm d}^2}{{\rm d}r^2}\:-\:\frac{3}{r}\frac{\rm d}{{\r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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... |
8a7f53a51339390d708abd27d4ed1613995489f8 | subsection | 46 | 78 | Resummation of loop corrections, renormalization and the local approximation | For the modes j=0 and j=1, it is necessary to account for infrared effects.
The required resummation is carried out through
the inclusion of self-energy terms in the equation of motion (REF )
for the bounce and in Eq. (REF ) for the Green's function,
which acquire loop corrections as-\:\frac{{\rm d}^2}{{\rm d}r^2}\,\va... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.019... |
c98e3b6a26598081efdbcaf206e72d11bc79e8bd | subsection | 47 | 78 | Resummation of loop corrections, renormalization and the local approximation | \Pi ^{\rm ren} and \Pi ^{\rm ren}_{\alpha }.
The former is given by\Pi ^{\rm ren}(r)\ =\ \frac{\lambda }{2}\,{\cal G}^\perp (r,r)\:+\:\delta m^2\:+\:\frac{\delta \lambda }{6}\,\varphi ^2(r)\;,where we have introduced counterterms \delta m^2 and \delta \lambda . While we
have expressed this in terms of the exact Green's... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.0330... |
49d201a07a5eb45e7ffc9523fc2e7de63ffb8ebf | subsection | 48 | 78 | Resummation of loop corrections, renormalization and the local approximation | This leads to\delta m^2\ =\ -\:\frac{\lambda }{2}\,\frac{\partial }{\partial \varphi }\,\varphi \,G^{\rm hom}\big (\sqrt{m^2+\lambda \,\varphi ^2/2};r,r\big )\Big |_{\varphi \,=\,0}\ \\
=\ -\:\frac{\lambda }{2}\,G^{\rm hom}(m;r,r)\;,\\
\delta \lambda \ =\ -\:\frac{\lambda }{2}\,\frac{\partial ^3}{\partial \varphi ^3}\,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2164,
"openalex_id": "",
"raw": "J. Berges, S. Borsanyi, U. Reinosa and J. Serreau, Phys. Rev. D 71 (2005) no. 10, 105004 [hep-ph/0409123].",
"source_ref_id": "438f16372b7c846fff7534b479dde9045075f9ca",
"start": 1994
... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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... |
cfc5aa2ec12f02151212afaf09cf434d3a7e28cc | subsection | 49 | 78 | Resummation of loop corrections, renormalization and the local approximation | Instead of explicitly specifying local counterterms for j=0 and j=1, we therefore make the replacement{\cal G}_{j}(r,r)
\ &\longrightarrow \ {\cal G}_{j}(r,r)\:-\:{\rm Re}\big [G^{\rm hom}_{j}\big (M_\varphi (\varphi );r,r\big )\\&\qquad \qquad -\:G^{\rm hom}_{j}\big (\sqrt{m^2+\lambda \,\varphi ^2/{2}};r,r\big )\big ]... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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... |
21a01200f4d55147b1ba4c12d1035f068b9baa68 | subsection | 50 | 78 | Resummation of loop corrections, renormalization and the local approximation | Taking for the external momentum
(\partial _\mu \varphi )/\varphi and comparing the square of this with the squared mass in the loop,
we arrive at the same estimate as in Eq. (REF ).At the centre of the bubble, for small r, the local approximation should therefore be accurate, and, for
j\ne 0, we should replace m_{\rm ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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-0.025609850883483887,... |
a1a71489575c033567836c08ea8dd365c4460a40 | subsection | 51 | 78 | Mode | The case j=1 requires special treatment because of the apparent singularity in the tree-level Green's function (REF ) and because of the presence of the translational zero modes. Besides the divergence in the denominator, we also note that P_2^{-j-1}(u) is proportional to P_2^{j+1}(u) in the limit j\rightarrow 1, such ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.03... |
e3e20a278040ac3aa3ea43d6c841315bb0b36275 | subsection | 52 | 78 | Mode | The solutions f_\pm can be obtained numerically and are, in general, not orthogonal to \tilde{\phi }^{\rm tr}. The coefficient a can be determined by imposing orthogonality to the zero mode\int \limits _0^\infty {\rm d}r\;r \tilde{\phi }^{\rm tr}(r) \tilde{\cal G}_1(r,r^\prime )\ =\ 0\;.This condition can be solved for... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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-0... |
5ccf481518d522c9ec02d6115916513d30ffa1d5 | subsection | 53 | 78 | Mode | This problem does not occur when we account for the deviation
from the Fubini-Lipatov form. In particular, the mass term leads to an exponential decay of
the modes for r\rightarrow \infty , such that the integral Eq. (REF ) is convergent.Nevertheless, we can compare the result (REF ) for \tilde{{\cal G}}_ 1, based on t... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.0241... |
f1abfa6b3d89a2c7139c0895e9e26c0c3c4d8a40 | subsection | 54 | 78 | Mode | (REF ) is solved by&\tilde{G}_0(u,u^\prime )\ =\ \vartheta (u-u^\prime )\frac{1}{6}\, P_2^1(u)Q_2^1(u^\prime )
\\&\qquad +\:\vartheta (u^\prime -u)\frac{1}{6}\, P_2^1(u^\prime )Q_2^1(u)
\:+\: b\,P_2^1(u) P_2^1(u^\prime )
\;.Note that rr^\prime \tilde{G}_0(r,r^\prime ) is regular everywhere, but the parameter b remains ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.014346604235470295,
0.01... |
d9cc9bb989aa51de4d45ef917368bd46ce90b73d | subsection | 55 | 78 | The negative eigenmode in the loop expansion | The functional integral over the negative eigenmode can be defined only by analytic continuation via the method of steepest descent. One might be concerned that this will lead to subtleties in the diagrammatic expansion with respect to the contributions from the negative eigenmode in the j=0 mode of the (subtracted) Gr... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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-0.0015605016378685832,
0.... |
39be3d5467caef555ff6f9819cc5ce3342069d17 | subsection | 56 | 78 | The negative eigenmode in the loop expansion | (REF )] can now be obtained straightforwardly by functional differentiation:\hbar \,(G^{\perp }_0)_{xy}\: =\: \frac{\hbar ^2}{\mathcal {Z}^{\prime (0)}[0,0]}\,\frac{\delta }{\delta (J_{0})_{x}}\,\frac{\delta }{\delta (J_{0})_{y}}\,\mathcal {Z}^{\prime (0)}[J,J_0]\bigg |_{J,J_0\,=\,0}\,.The superscript (0) on \mathcal {... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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-0.007171371020376682,
0.008239447139203548,
... |
61ab8b36f355ea48a637ebf8e77eb67d328cb716 | subsection | 57 | 78 | The negative eigenmode in the loop expansion | Applying the method of steepest descent, we obtain 1/2 of the integral over a_0^{\prime }=-ia_0\in (-\infty ,\infty ):&\hbar \,(G^{\perp }_0)_{xy}\ =\ \hbar \,(\phi _{0})_{x}(\phi _{0})_{y}\\&\ \times \:\frac{1}{\mathcal {Z}^{\prime (0)}[0,0]}\,\bigg \lbrace \frac{i}{2}\,\exp \bigg [-\:\frac{1}{\hbar }\bigg (S[\varphi ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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... |
05526de9768f2704348628ef5550ec4b45b13c07 | subsection | 58 | 78 | Parametric example | Now, we numerically solve Eq. (REF ) for the bounce and Eqs. (REF ) and () for the Green's functions self-consistently by running several iterations over these equations. This procedure can be initialized by calculating the bounce in the Coleman-Weinberg effective potential. Since the iterations are repeated until the ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.01... |
673f4c0b054d15bac5613ed8d0d5e7edb60c7709 | subsection | 59 | 78 | Parametric example | The dotted line corresponds to the analytic solution based on Eq. () with b=0.][Figure: Plots of the j=0 contribution to the renormalized self-energy \Pi ^{\rm ren}_{j\,=\,0}(\varphi ) with (solid) and without (dashed) gradients. The dotted line corresponds to the analytic solution based on Eq. () with a fitted value f... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.030434... |
a38d80bfb5746e2587be2e5a9464f7a20decb71e | subsection | 60 | 78 | Conclusions | In this paper, we have presented a Green's function method for calculating loop-improved bounce solutions in
classically scale-invariant models.While the problem of tunneling in classically
scale-invariant scalar theory has been addressed in a number of earlier articles , , , , , the present method is
complementary in ... | {
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"raw": "G. Isidori, G. Ridolfi and A. Strumia, Nucl. Phys. B 609 (2001) 387 [hep-ph/0104016].",
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"arxi... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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d1e32b7b02d8658f2e5b9bb7483e7bfc8c8c6205 | subsection | 61 | 78 | Conclusions | The solutions in the full Hilbert space
can be obtained from these solutions by applying the corrections from the Jacobian. These should be included
in the future when, e.g., aiming to compute the decay rate to leading-loop order.
Analytic form of the Green's functions and fluctuation spectra.
We have found an intrigu... | {
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"raw": "S. R. Coleman, Phys. Rev. D 15 (1977) 2929 [Erratum: Phys. Rev. D 16 (1977) 1248].",
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in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
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9274f1262851e35c6d62e3747e83c567829e9c58 | subsection | 62 | 78 | Coleman-Weinberg effective potential | The renormalized one-loop Coleman-Weinberg effective potential for the model in Eq. (REF ) with m^2=0 and g=0 takes the formU^{\rm ren}_{\rm eff}\ =\ U\:+\:\delta U\:+\:\frac{1}{2}\int \!\frac{{\rm d}^4 k}{(2\pi )^4}\,\big [\ln \big (k^2\:+\:\lambda \,\varphi ^2/2\big )\:-\:\ln \,k^2\big ]\;,where the one-loop correcti... | {
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]
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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b154ef1c84ef2e5c8b6cb85c3ca9da6ab2dead13 | subsection | 63 | 78 | Coleman-Weinberg effective potential | (REF ), we take m^2\ne 0 and make the choice\frac{\partial ^2 U_{\rm eff}^{\rm ren}(\varphi )}{\partial \varphi ^2}\Bigg |_{\varphi \,=\,0}\ =\ m^2\;,\qquad \frac{\partial ^4 U^{\rm ren}_{\rm eff}(\varphi )}{\partial \varphi ^4}\Bigg |_{\varphi \,=\,0}\ =\ \lambda \;,yielding the counterterms\delta m^2\ =&\ -\:\frac{\l... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
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40731ca7136421a706619118320c2ee72efddf38 | subsection | 64 | 78 | Fubini-Lipatov Green's function | In this appendix, we outline the calculation of the Green's function in the Fubini-Lipatov background. Beginning from the transformed problem in Eq. (REF ) with \omega =j+1, we recognize the homogeneous equation\Bigg [\frac{\rm d}{{\rm d}u}\,(1-u^2)\,\frac{\rm d}{{\rm d}u}\:-\:\frac{\omega ^2}{1-u^2}\:+\:6\Bigg ]F_j(u,... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
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f539ad0f056d29619c56b2eab0fc9709086967a0 | subsection | 65 | 78 | Fubini-Lipatov Green's function | For the time being, however, it is technically simpler to deal with the associated Legendre polynomials.Matching around the discontinuity, we requireF_j^>(u^{\prime },u^{\prime })\ =\ F_j^<(u^{\prime },u^{\prime })\;,\\
\lim _{u\,\rightarrow \,u^{\prime }}\Bigg [\frac{{\rm d}}{{\rm d} u}\,F_j^>(u,u^{\prime })-\frac{{\r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
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80bd658f7b9514aee1fc0dc76bd0aa018727b543 | subsection | 66 | 78 | Fubini-Lipatov Green's function | For \nu =2, the polynomial expansion terminates, and we have&P_2^{(\pm \mu ,\mp \mu )}(u)\ =\ \frac{1}{2}\Big [(1\pm \mu )(2\pm \mu )\\&\qquad -\:3(2\pm \mu )(1-u)\:+\:3(1-u)^2\Big ]for all \mu . Substituting this expansion into Eq. (REF ) with \mu =\omega =j+1 and after some algebra, we arrive at the expression for th... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
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9666db244763dbdaedf10bda674adb982aa460ca | subsection | 67 | 78 | Orthonormality of the Jacobi polynomials | The associated Legendre polynomials satisfy the familiar orthonormality condition\int _{-1}^{+1}\frac{{\rm d}u}{1-u^2}\;P_{\nu }^{\mu }(u)P_{\nu }^{\mu ^{\prime }}(u)\ =\ \frac{(\nu +\mu )!}{\mu (\nu -\mu )!}\,\delta _{\mu \mu ^{\prime }}\;.Using the identityP_{\nu }^{\mu }(u)\ =\ (-1)^{\mu }\frac{(\nu +\mu )!}{(\nu -\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
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672c802c3b7e6e4dda82e15fab5d5785689da42d | subsection | 68 | 78 | Orthonormality of the Jacobi polynomials | (REF ), giving&\int _{-1}^{+1}\frac{{\rm d}u}{1-u^2}\;\bigg (\frac{u+1}{u-1}\bigg )^{+\frac{\mu }{2}}\bigg (\frac{u+1}{u-1}\bigg )^{-\frac{\mu ^{\prime }}{2}}\\&\quad \times P_{\nu }^{(-\mu ,+\mu )}(u)P_{\nu }^{(+\mu ^{\prime },-\mu ^{\prime })}(u)\\&\quad =\ \frac{(-1)^{\mu }}{\mu }\,\frac{\delta _{\mu \mu ^{\prime }}... | {
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"doi... | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
"hep-th",
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4b6ca7dc2d1e45330d783840ca32d532c1300988 | subsection | 69 | 78 | Orthonormality of the Jacobi polynomials | Namely,&\int _{-1}^{+1}\frac{{\rm d}u}{1-u^2}\;\bigg (\frac{u+1}{u-1}\bigg )^{+\frac{i\xi }{2}}\bigg (\frac{u+1}{u-1}\bigg )^{-\frac{i\xi ^{\prime }}{2}}\\&\quad \times P_{\nu }^{(-i\xi ,+i\xi )}(u)P_{\nu }^{(+i\xi ^{\prime },-i\xi ^{\prime })}(u)\\&\quad =\ \frac{2\sinh (\pi \xi )}{\xi }\,\frac{\delta (\xi -\xi ^{\pri... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.016001 | 1804.04944 | Fluctuations about the Fubini-Lipatov instanton for false vacuum decay
in classically scale invariant models | [
"Bjorn Garbrecht",
"Peter Millington"
] | [
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