R/32

Kimiko YAMASHITASMEFT~lo"4 "MOVS
EXTREMAL RAY~4 "MO
2

1

60

:
Cen Zhang (Shuang-Yong Zhou (

),

(

)

1

)

JHEP 01 (2021) 095 (arXiv: 2009.04490 [hep-ph])

51

Kimiko YAMASHITA


S/32

Table of contents
1. Effective Field Theory (EFT)and Standard Model Effective Field Theory (SMEFT)
2. Positivity Bounds
3. Positivity Bounds on F4 operators
1. Elastic Positivity Bounds
2. Extremal Positivity Bounds
4. Comparison
5. SummaryT/32

Kimiko YAMASHITA

Effective Field Theory (EFT)and Standard Model Effective Field Theory (SMEFT)
• EFT• heavy degrees of freedom decouple• for large-distance phenomena

or small momentum scale• [\T interaction terms:

L<latexit sha1_base64="m2J283pKGoE8rog77GU7cUpNFHs=">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</latexit>

(5)

+L

(6)

+L

Wilson coefficients

nd 5
X
c

(7)

+L

(8)

+ ···

6
7
8
c
c
c
(5)
(6)
(7)
(8)
i
i
i
i
L=
Oi + 2 O i + 3 Oi + 4 Oi + · · ·
⇤
⇤
⇤
⇤
i=1
<latexit sha1_base64="GddOH25d+YtM3ZjBYlFnu0f0lGE=">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</latexit>

Kimiko YAMASHITA

U/32

Effective Field Theory (EFT)and Standard Model Effective Field Theory (SMEFT)



SMEFT
• Extending the SM by adding massive particles heavier

than the energy scale _f scatterings• SUC(3)g︎SUL(2)g︎UY(1) invariant

higher-dimensional operator built by SM fields• Model independent way to search for new physics

V/32

Kimiko YAMASHITA

Positivity Bounds• EFT is for the energy scale of

E << Λ (typical energy scale of the UV physics)• Many UV models correspond with EFT• From the general feature of UV theory,
Next next page

can we bound on Wilson coefficients of EFT?
general feature
restrictcorrespondenceW/32

Kimiko YAMASHITA

Positivity Bounds• One of the way to do this is Positivity bounds
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, R.Rattazzi, JHEP 0610, 014(2006)

• Positivity bounds: the signs of certain combinations of

Wilson coefficients in EFT have to be positive,e.g. W4 operators: One of the positivity bounds:

2FT,0 + 2FT,1 + FT,2<latexit sha1_base64="hU6kRYtvshXBW6LVJ1eDwFv2ULc=">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</latexit>

0Kimiko YAMASHITA

X/32

Positivity Bounds
If we base on the local Quantum Field Theory(QFT) for thegeneral feature of UV theory,
1. Special relativity
2. Conservation of probability
3. Causality

Kimiko YAMASHITA

Y/32

Positivity Bounds

If we base on the local Quantum Field Theory(QFT) for thegeneral feature of UV theory,
1. Special relativity

Lorentz invariance2. Conservation of probability
Unitarity
3. Causality
Analyticity

Z/32

Kimiko YAMASHITA

Positivity Bounds
Forward limit positivity bounds are from:
1. Lorentz Invariance
2. Unitarity c Optical theorem:

e.g., elastic case,

ImM(k1 , k2 ! k1 , k2 ) = s tot (k1 , k2 ! anything)<latexit sha1_base64="sBmNCBYTdTmaLfZjIwiLFCEVQbE=">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</latexit>

3. Analyticity* c Froissart Bound:

Positive|M(s, cos ✓ = 1)| < Const. s(ln s)2<latexit sha1_base64="Z2aQOglPA3GkOBQZsyXrif/IMVo=">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</latexit>

forward, elasticFroissart, Martin 1960’s *Analyticity of the amplitude besides poles and branch cuts on real axis

Kimiko YAMASHITA

RQ/32

Positivity Bounds
Even though the EFT as IR theory seems totally okay byseeing the Lagrangian and check it with
ü Lorentz invarianceü Perturbative Unitarity(Unitary S-matrix at energies far lower than the cutoff), etc.,
It can be excluded out when we suppose the UV completed
theory!
This “wrong” EFT leads to the superluminal fluctuations aroundnon-trivial backgrounds, and making it impossible to define local and causal
evolution, and implying an IR breakdown of the EFT.
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, R.Rattazzi, JHEP 0610, 014(2006)

RR/32

Kimiko YAMASHITA

Positivity Bounds• Effective Theory Forward Amplitude (IR):23*Ref: Slides by Francesco Riva
https://indico.ph.tum.de/event/
4408/contributions/3825/
attachments/3292/3974/
Berlin-2.pdf

s
s
s
s4
M = C0 + C1 2 + C2 4 + C3 6 + C4 8 + · · ·MMMM<latexit sha1_base64="QlHhWZkLeNMVfxH7w179ytKP6eY=">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</latexit>

M

UVcompletion:

IRE behaviors<latexit sha1_base64="tm7wEyV81tpuj/gQcbEvUS8vqc0=">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</latexit>

Lorentz invariance
Unitarity
Causality

C2 > 0,
C4 > 0, · · ·<latexit sha1_base64="gmSuuOUXkxZxmxv/HKzbVqZ3Gso=">AAACa3ichVFNLwNRFD0d3/XRYiNYTDQVC2nulFAWImwsabWaIM3MeJiYzkxmpk1o/AFLGws2JCLiZ9j4AxZ+gohVJTYW7kwrYoE7mXnnnXvPnfPu0xzT8Hyip4jU0trW3tHZFe3u6e2LxfsHCp5dcXWR123Tdoua6gnTsETeN3xTFB1XqGXNFBvawXKQ36gK1zNsa90/dMR2Wd2zjF1DV32misultLwg02QpnqAUESmKIgdAmZ0hBnNzmbSSkZUgxZFAM1bt+A22sAMbOiooQ8CCz9iECo+fTSggOMxto8acy8gI8wLHiLK2wlWCK1RmD/i7x7vNJmvxPujphWqd/2Ly67JSRpIe6Zbq9EB39Ewfv/aqhT0CL4e8ag2tcEqxk6Hc+7+qMq8+9r9Vf3r2sYtM6NVg707IBKfQG/rq0Vk9N59N1sbpil7Y/yU90T2fwKq+6ddrInuOKF/A15Tl30EhnVKmUum16cTiUvMqOjGCMUzwvGexiBWsIh/O+RTnuIi8SoPSsDTaKJUiTc0gfoSU/ATXcYuu</latexit><latexit sha1_base64="pfR99i5cN/NdWcCVGy+4jsLkD5o=">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</latexit>

Positivity

M<latexit sha1_base64="hQSCc10sN5b4XNdxDS5RhQmwIjM=">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</latexit>E<latexit sha1_base64="E3TzOztZQp9S7UkvqH5Sb+t0qtg=">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</latexit>RS/32

Kimiko YAMASHITA

Positivity Bounds
massless scalar 2-2 forward elastic scattering:O
forward: t=0<latexit sha1_base64="t622VdjxjRkXAy7kaKGgKM/qGEU=">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</latexit><latexit sha1_base64="BjH2E51gHZOhLRB0m/0wzBBs34Q=">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</latexit><latexit sha1_base64="BjH2E51gHZOhLRB0m/0wzBBs34Q=">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</latexit><latexit sha1_base64="t622VdjxjRkXAy7kaKGgKM/qGEU=">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</latexit>

I+II →I+II
elastic
Let us consider the amplitude (divided by sn+1) of this:OM(s, 0)n = 2, 4, ...
sn+1<latexit sha1_base64="7wGbv7c2He3G7qK5j0S8io3poFo=">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</latexit><latexit sha1_base64="9qq7/0t/wKG1GEcLSnmIpxwyo+k=">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</latexit>RT/32

Kimiko YAMASHITA

Positivity Bounds
massless scalar 2-2 forward elastic scattering amplitude:
FroissartOM(s, 0)Im(s)
Bound
0
n+1
s
<latexit sha1_base64="9qq7/0t/wKG1GEcLSnmIpxwyo+k=">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</latexit>n = 2, 4, ...
<latexit sha1_base64="7wGbv7c2He3G7qK5j0S8io3poFo=">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</latexit>

Analytical
→0

s=-u
crossing sym.
→ A(s,0)=A(-u,0)

UV

IR

Re(s)

M >0<latexit sha1_base64="hQSCc10sN5b4XNdxDS5RhQmwIjM=">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</latexit>

Unitarity(Optical theorem)
This is a circle that EFT can apply.

RU/32

Kimiko YAMASHITA

Positivity Bounds<latexit sha1_base64="WPo2ltFXNaAJSKQBkuYjzQxQ8Jg=">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</latexit>Z

1
M(s, 0)M(s, 0)ds n+1 = 0 =ds n+1=0s⇡isAnalytical
Froissart Bound
Z
Z 1
2
1
M(s, 0)ImM(s)
ds n+1 =ds n+1 >0
⇡i
s
⇡ sIR
s

1
⇡i

<latexit sha1_base64="HNiqJ54o5K5bKdXunD7sN26GSfM=">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</latexit>

Z

Res[M(s, 0)]s=0 = Cn<latexit sha1_base64="uhx1JuHrEexpROFUE2nuoG3D2Tw=">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</latexit>

Positivity

sIR ⌧ M2<latexit sha1_base64="6Ql1a2/AwVTOfXwuedl1HF5D3WE=">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</latexit>

Unitarity(Optical theorem)

Cn > 0(n = 2, 4, · · · ) from UV general features
<latexit sha1_base64="m5moTWMgmx1Mm8Ecz9NHHCfjGBk=">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</latexit>

s
s2
s3
s4
M = C0 + C1 2 + C2 4 + C3 6 + C4 8 + · · ·MMMM<latexit sha1_base64="QlHhWZkLeNMVfxH7w179ytKP6eY=">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</latexit>

e.g., From Dim.8 operators

RV/32

Kimiko YAMASHITA

Positivity Bounds: Example of Positivity for Dim.8• EFT Lagrangian →Amplitudes

1• SU(2)L Higgs doublet: =p2
O = [(D )† D ][(Dµ )† D⌫ ]<latexit sha1_base64="xkf6Oo+64r+zPfblUsb95glUIIc=">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</latexit>

S,0

µ

⌫
µ

<latexit sha1_base64="04GLp4har258Kpjk2wED4qziGfc=">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</latexit>

OS,1 = [(Dµ )† D ][(D⌫ )† D⌫ ]OS,2 = [(Dµ )† D⌫ ][(D⌫ )† Dµ ]<latexit sha1_base64="nY8ITvL2ddvcsJEBt5l3cafGCaU=">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</latexit><latexit sha1_base64="IPA9nZKSi5q07k2sefqZWcMfjuM=">AAACaXichVFNLwNRFD0d3/VVbBo2jYZYNfeVUFbCxrI+WhKkmRkPj+nMZGbahMYfsLITrEhExM+w8Qcs/ASxJLGxcGdaEQvcybx33rn33Hfee4ZrKT8geoppTc0trW3tHfHOru6e3kRff9F3Kp4pC6ZjOd6aofvSUrYsBCqw5JrrSb1sWHLV2J8P86tV6fnKsVeCA1dulvUdW20rUw+YKm64u6okSok0ZYhICJEKgZiaJAbT07msyKVEmOJIoxF5J3GDDWzBgYkKypCwETC2oMPnbx0CBJe5TdSY8xipKC9xhDhrK1wluUJndp/HHV6tN1ib12FPP1KbvIvFv8fKFEbokW7plR7ojp7p49detahH6OWAZ6OulW6p9zi5/P6vqsxzgN1v1Z+eA2wjF3lV7N2NmPAUZl1fPTx9XZ5ZGqmN0hW9sP9LeqJ7PoFdfTOvF+XSBeL8AF+3nPodFLMZMZ7JLk6kZ+caT9GOIQxjjO97CrNYQB4F3ncPJzjDeexF69OS2mC9VIs1NAP4EVr6E4nsjBk=</latexit><latexit sha1_base64="vA4N4LuhXKa/9Mi46XqmJ6/0oc4=">AAADD3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQusUvGPrw7WMaxVsFWI1nCJj8ktVYgJyMjUjItJSUxPTy1ScImDicWCFeRhUQAVi1WJieFShZhohMdEuCEgE+OwmgizUkVBTSEGCBS4uOIFlA30DMBAAZNhCGUoM0BBQL7AcoYYhhSGfIZkhlKGXIZUhjyGEiA7hyGRoRgIoxkMGQwYCoBisQzVQLEiICsTLJ/KUMvABdRbClSVClSRCBTNBpLpQF40VDQPyAeZWQzWnQy0JQeIi4A6FRhUDa4arDT4bHDCYLXBS4M/OM2qBpsBckslkE6C6E0tiOfvkgj+TlBXLpAuYchA6MLr5hKGNAYLsFszgW4vAIuAfJEM0V9WNf1zsFWQarWawSKD10D3LzS4aXAY6IO8si/JSwNTg2YzgCLAED24MRlhRnqGxnpGgUbKDk7QqOBgkGZQYtAAhrc5gwODB0MAQyhDMqMAoxmjPaMD00Sm7Uz7mA5AlDIxQvUIM6AAppMAooTBng==</latexit>

✓

1+i 2
3+i 4

1

◆

forward: t=01<latexit sha1_base64="k+6GjT1gB3Y7YrljhoCs9akYxjc=">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</latexit><latexit sha1_base64="ocbGbUgEO23M8NFhl3LZm64LFz4=">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</latexit><latexit sha1_base64="k+6GjT1gB3Y7YrljhoCs9akYxjc=">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</latexit>L<latexit sha1_base64="xuy1lhlx7leElocqga4AHUfz6E8=">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</latexit><latexit sha1_base64="bR9bUwjgLgKL+A1sZnGvxqedFoc=">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</latexit>(8)1<latexit sha1_base64="IPA9nZKSi5q07k2sefqZWcMfjuM=">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</latexit>

cS,0
cS,1
cS,2
I+II →I+II, elastic
= 4 OS,0 + 4 OS,1 + 4 OS,2
⇤
⇤
⇤

1 1 !

1 1 scattering2

M = (cS,0 + cS,1 + cS,2 )s /⇤<latexit sha1_base64="WvcylL+WagGuP9xpyihrTlsQXVw=">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</latexit>

1

4

RW/32

Kimiko YAMASHITA

Positivity Bounds: Example of Positivity for Dim.8• EFT Lagrangian →Amplitudes

1• SU(2)L Higgs doublet: =p2
O = [(D )† D ][(Dµ )† D⌫ ]<latexit sha1_base64="xkf6Oo+64r+zPfblUsb95glUIIc=">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</latexit>

S,0

µ

⌫
µ

<latexit sha1_base64="04GLp4har258Kpjk2wED4qziGfc=">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</latexit>

OS,1 = [(Dµ )† D ][(D⌫ )† D⌫ ]OS,2 = [(Dµ )† D⌫ ][(D⌫ )† Dµ ]<latexit sha1_base64="nY8ITvL2ddvcsJEBt5l3cafGCaU=">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</latexit><latexit sha1_base64="IPA9nZKSi5q07k2sefqZWcMfjuM=">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</latexit><latexit sha1_base64="vA4N4LuhXKa/9Mi46XqmJ6/0oc4=">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</latexit>

✓

1+i 2
3+i 4

1

◆

forward: t=01<latexit sha1_base64="k+6GjT1gB3Y7YrljhoCs9akYxjc=">AAACaXichVFNLwNRFD0dX1VfxaZh02iI1eROUa2VsLGsj5YEaWbGw2M6M5mZNqHxB6zsBCsSEfEzbPwBCz9BLElsLNxOK2KB+/Leve+8e+497z3DtaQfED1FlJbWtvaOaGesq7unty/eP1D0nYpnioLpWI63Zui+sKQtCoEMLLHmekIvG5ZYNfbn6+erVeH50rFXggNXbJb1HVtuS1MPGCpuuLuypJXiKVJzOZrUMklSp4jSmRwHNJHOZjJJTaXQUmha3onfYANbcGCigjIEbAQcW9Dh81iHBoLL2CZqjHkcyfBc4Agx5lY4S3CGzug+rzu8W2+iNu/rNf2QbXIXi6fHzCRG6ZFu6ZUe6I6e6ePXWrWwRl3LAXujwRVuqe84sfz+L6vMPsDuN+tPzQG2kQ21Stbuhkj9FmaDXz08fV2eWRqtjdEVvbD+S3qie76BXX0zrxfF0gVi/AFfr5z8PSimVW1CTS9Opmbnml8RxTBGMM7vPY1ZLCCPAvfdwwnOcB55UfqVhDLUSFUiTc4gfpiS+gS1p4wv</latexit><latexit sha1_base64="ocbGbUgEO23M8NFhl3LZm64LFz4=">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</latexit><latexit sha1_base64="k+6GjT1gB3Y7YrljhoCs9akYxjc=">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</latexit>L<latexit sha1_base64="xuy1lhlx7leElocqga4AHUfz6E8=">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</latexit><latexit sha1_base64="bR9bUwjgLgKL+A1sZnGvxqedFoc=">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</latexit>(8)1<latexit sha1_base64="IPA9nZKSi5q07k2sefqZWcMfjuM=">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</latexit>

cS,0
cS,1
cS,2
I+II →I+II, elastic
= 4 OS,0 + 4 OS,1 + 4 OS,2
⇤
⇤
⇤

1 1 !

1 1 scattering2

M = (cS,0 + cS,1 + cS,2 )s /⇤<latexit sha1_base64="WvcylL+WagGuP9xpyihrTlsQXVw=">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</latexit>

1

4

>0

Positivity: the signs of certain linear combinations ofeffective operators have to be positiveRX/32

Kimiko YAMASHITA

Positivity Bounds●• Positivity bounds can apply for dim-8 operatorsin tree-level SMEFT← Froissart Bound (bAnalyticity)• Dim-8 operators are more suppressed by Λ than lower
dimensional ones, however, for dim-8 aQGC operators,
LHC experimentalists have been and currently working on
constraining them
CMS-PAS-SMP-18-001
20

Expected 68% CL
Expected 95% CL
Expected 99% CL
Observed 95% CL

Preliminary

35.9 fb-1 (13 TeV)

fS,1/ Λ4 (TeV-4)

fM,1/ Λ4 (TeV-4)

35.9 fb-1 (13 TeV)

CMS

100

10

CMS

Expected 68% CL
Expected 95% CL
Expected 99% CL
Observed 95% CL

Preliminary

50

0
0

−50
−10

−100
−10

0

10
4

-4

fM,0/ Λ (TeV )

−50

0

50

fS,0/ Λ4 (TeV-4)• In the future, more dim-8 effects may become accessible
(e.g. new observable proposed for DY process[Alioli, Boughezal, Mereghetti, Petriello, 2003.11615])

RY/32

Kimiko YAMASHITA

Positivity Bounds
Positivity bounds are important as they offer complementarybounds to the experiments Q. Bi, C. Zhang, S.-Y. Zhou JHEP 1906 (2019) 137OS,1 = [(Dµ )† Dµ ][(D⌫ )† D⌫ ]<latexit sha1_base64="vA4N4LuhXKa/9Mi46XqmJ6/0oc4=">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</latexit>

OS,0 = [(Dµ )† D⌫ ][(Dµ )† D⌫ ]<latexit sha1_base64="nY8ITvL2ddvcsJEBt5l3cafGCaU=">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</latexit>

Kimiko YAMASHITA

RZ/32

Positivity Bounds
• Positivity bounds can be obtained traditionally by considering

eelastic forward scattering by using EFT operators• Recently new approach for positivity bounds is established:

fextremal positivity bounds
C. Zhang, S.-Y. ]hou, arXiv: 2005.03047 [hep-ph]

• Comparing f with e should be useful• To see this, we use F4 operators (next page) as

a concrete example• Deriving bounds on F4 operators itself is also useful as

offering complementary bounds to the experiments

Kimiko YAMASHITA

Positivity Bounds• \4 operators (this work)• WWWW• WWBB• BBBB

SQ/32

SR/32

Kimiko YAMASHITADispersion Relation (for Positivity Bounds)
Forward scattering amp,
at low energy (EFT)

Mijkl =
<latexit sha1_base64="6sm+f7YldGOoyA34SEnTnVQfY5Q=">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</latexit>

Amplitude of SM →X✏1<latexit sha1_base64="/WmeCJViMdv31Cd9AfCrjckTXpY=">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</latexit>

ΣX: BSM states, X summation&
LIPS integration

sau crossingSS/32

Kimiko YAMASHITADispersion Relation
• Useful to rewrite Dispersion Relation for Positivity Bounds
(Amp by Dim.8)
d (F/Λ4) s2
ijkl
F ↵ M↵
Mijkl = ⇤4<latexit sha1_base64="4iGmz3SIwzhijZ2NTkBUGJsdT+E=">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</latexit>

where

• When i=k, j=l, RHS complete squares >=0ijijM
0 because<latexit sha1_base64="aDRcPGKa+hkdT+hl5+AvWMOUFK4=">AAACdHichVFNLwNRFD0d3/XRYiNh0WgqVs0tEmIlbGwk1SoSH83MeOrVdGbMTJtU0z/gD1h0RYiIn2HjD1j4CWJJ2Fi4nU4iCO7kzTvvvHvuO+9dzTak6xE9hJS29o7Oru6ecG9f/0AkOji07lplRxc53TIsZ1NTXWFIU+Q86Rli03aEWtIMsaEdLjX3NyrCcaVlrnlVW+yU1IIp96Wuekzlo5GV3ZosymI9tl0QRzHKR+OUJD9iP0EqAHEEkbaiV9jGHizoKKMEARMeYwMqXP62kALBZm4HNeYcRtLfF6gjzNoyZwnOUJk95H+BV1sBa/K6WdP11TqfYvBwWBlDgu7pmp7pjm7okd5/rVXzazS9VHnWWlph5yMnI9m3f1Ulnj0cfKr+9OxhH3O+V8nebZ9p3kJv6SvHp8/Z+UyiNkHn9MT+z+iBbvkGZuVFv1wVmQbC3IDU9+f+Cdankqnp5NTqTHxhMWhFN0Yxjkl+71ksYBlp5PyeNHCBy9CrMqbElUQrVQkFmmF8CSX5AWc4j5M=</latexit>• More generally,
Elastic Forward Scattering between Superposed States :
with(generalized) Elastic Positivity BoundsST/32

Kimiko YAMASHITAElastic Positivity Bounds on F4 operators (Result)
• All possible superposed states for the elastic forward
scattering are considered:
• Linear Positivity Bounds

other superposed
states don’t givenew bounds

• Some Quadratic and Cubic Bounds• Parameter space is constrained to ~0.7% of totalSU/32

Kimiko YAMASHITA

Extremal Positivity Bounds on F4 operators
C. Zhang, S-Y. ]hou, arXiv: 2005.03047 [hep-ph]

• Dispersion Relation

Dynamics

SymmetryPositive

i(j|k|l): j, l indices are
symmetrized

• r runs all irreps of the SO(2) rotation
around the forward scattering axis
and SM gauge symmetries• M must stay inside the cone
A convex cone C:
if x, yC then αx+βy
for α,β>=02<latexit sha1_base64="zxo/ZDpXw8OeiI7GZ8paEjUvjGI=">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</latexit>

2<latexit sha1_base64="zxo/ZDpXw8OeiI7GZ8paEjUvjGI=">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</latexit>

C

Edges of this coneare extremal raysSV/32

Kimiko YAMASHITA

Extremal Positivity Bounds on F4 operators
C. Zhang, S-Y. ]hou, arXiv: 2005.03047 [hep-ph]

• Dispersion Relation

Dynamics

SymmetryPositive

i(j|k|l): j, l indices are
symmetrized• Dynamics by the SMEFT determine:
S of {Mijkl}

• The positivity bounds are obtained byKimiko YAMASHITA

SW/32

Extremal Positivity Bounds on F4 operators
• WWWW case
Find projection operators:
1.
(N=3)
2.
the in SO(2) rotation around the forward direction (N=2)

Combining both 1) and 2) projectors withβ, σ indices symmetrized→ 9 potential ERs
à 8 ERs for C ER: cannot be a sum of two other elements in the cone
Calculate the EFT amplitudesàS and find the intersections:Kimiko YAMASHITASX/32

Extremal Positivity Bounds on F4 operators (Result)
• WWWW case

Better than Elastic Positivity ApproachSY/32

Kimiko YAMASHITA

Extremal Positivity Bounds on F4 operators (Result)
• Full F4 operators
Linear

WWWW

WWBB

BBBB

Kimiko YAMASHITA

SZ/32

Extremal Positivity Bounds on F4 operators (Result)
• Full F4 operators
QuadraticTQ/32

Kimiko YAMASHITA

Comparison• Elastic Positivity Approach• Extremal Positivity Approach

Allowed Parameter Space✅

Differences mainly arise in W sector

Potential Ability to Apply for More Complicated CaseNot yet algorithmized
It should become
more complicated✅algorithmized

we directly wrote down usinggroup theory as a guideline toknow the edge of the cone

Numerical Calculation Speed✅ faster← the error is negligible

Kimiko YAMASHITA

TR/32

Summary• If the UV theory possesses Unitarity and Analyticity of

the S-matrix→ the signs of certain linear combinationsof Wilson coefficients in EFT have to be positive• This positivity can be applied for many kind of EFTs
• Here we applied the positivity to SMEFT, especially,

F4 operators by using elastic & new extremal positivity
bounds→ bounds are very strong (99.3% excluded) and give a
useful guide to experimental searches• Extremal positivity approach has a potential ability to apply

for more complicated case

Kimiko YAMASHITA

TS/32

Other progresses
(Almost) after our work, there are progresses on improving
positivity bounds, e.g.,• N. Arkani-Hamed, T. C. Huang, Y.T. Huang
The EFT-Hedron
arXiv: 2012.15849 [hep-ph]• B. Bellazzini, J. Elias Miro, R. Rattazzi, M. Riembau, F. Riva
Positive Moments for Scattering Amplitudes
arXiv: 2011.00037 [hep-ph]• X. Li, C. Yang, H. Xu, C. Zhang, S. Y. Zhou
Positivity in Multi-Field EFTs
arXiv: 2101.01191 [hep-ph]Positivity is one of the active areas!