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Symmetry and Geometry in generalized HEFT
Yoshiki Uchida
In collaboration with Ryo Nagai (ICRR)
Masaharu Tanabashi (Nagoya U.) Koji Tsumura (Kyoto U.)
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 1
Nagoya University
arXiv : 1904.07618
Symmetry and Geometry in generalized HEFT
Yoshiki Uchida
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 2
Nagoya University
arXiv : 1904.07618
The sequel to the presentation by Nagai-san in PPP2018
2019/7/30
・Composite Higgs ・SUSY ・Extra dim.
SM is not complete
・Hierarchy problem ・Dark Matter …
Beyond the standard model (BSM) is needed !
In many BSM, the scalar sector is extended from SM
…
CMS-PAS-HIG-17-031
Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 3
2019/7/30
・Composite Higgs ・SUSY ・Extra dim.
SM is not complete
・Hierarchy problem ・Dark Matter …
Beyond the standard model (BSM) is needed !
In many BSM, the scalar sector is extended from SM
…
CMS-PAS-HIG-17-031
Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 4
Explore the scalar sector in model-independent way
Our Goal
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 5
II. Higgs cancels the divergence in oblique corrections
I. Higgs unitalize W_L scattering amplitude at tree level (tree level unitarity)
Peskin Takeuchi Phys. Rev. Lett. 65 (1990) 964
( in SM )
<Standard Model>
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 6
II. Higgs cancels the divergence in oblique corrections
I. Higgs unitalize W_L scattering amplitude at tree level (tree level unitarity)
Peskin Takeuchi Phys. Rev. Lett. 65 (1990) 964
( in SM )
<Standard Model>
What will happen if deviate from 1 ?
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 7
II. Higgs cancels the divergence in oblique corrections
I. Higgs unitalize W_L scattering amplitude at tree level (tree level unitarity)
unitarity sum rules
finiteness conditions
<Singlet extension>
J F Gunion, H E Haber, J Wudka Phys. Rev. D 43 904 (1991)
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 8
We impose the two conditions on the extended scalar sector
I. tree level unitarity
II. finiteness of oblique parameter ( 1-loop )
Explore the scalar sector in model-independent way Our Goal
V V S, T, U
= 0
= 0
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 9
We impose the two conditions on the extended scalar sector
I. tree level unitarity
II. finiteness of oblique parameter ( 1-loop )
Explore the scalar sector in model-independent way Our Goal
V V S, T, U
= 0
= 0
Independent condition ?
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 10
We impose the two conditions on the extended scalar sector
I. tree level unitarity
II. finiteness of oblique parameter ( 1-loop )
Explore the scalar sector in model-independent way Our Goal
V V S, T, U
φ φ
φ φ
= 0
= 0
Independent condition ?
・singlet extension w custodial sym. unitarity sum rules finiteness conditions
Exactly same !
・singlet extension w/ custodial sym.
unitarity sum rules finiteness conditions R Nagai, M Tanabashi, K Tsumura Phys. Rev. D 91, 034030 (2015)
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 11
We impose the two conditions on the extended scalar sector
I. tree level unitarity
II. finiteness of oblique parameter ( 1-loop )
Explore the scalar sector in model-independent way Our Goal
V V S, T, U
φ φ
φ φ
= 0
= 0
Independent condition ? unitarity sum rules finiteness conditions
in arbitrary scalar sector ?
• Introduction
• unitarity vs oblique corrections
• T parameter
• Summary
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 12
• Introduction
• unitarity vs oblique corrections
• T parameter
• Summary
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 13
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 14
Approach based on Symmetry and Geometry
unitarity sum rules finiteness conditions in arbitrary scalar sector ?
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Approach based on Symmetry and Geometry
unitarity sum rules finiteness conditions in arbitrary scalar sector ?
R. Alonso et al. JHEP08(2016)101
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 16
SM + Singlet scalar
Integrate out
Effective field theory approach
New Physics is encoded in the , …
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 17
SM + Singlet scalar
Integrate out
Approach based on Symmetry and Geometry
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 18
SM + Singlet scalar Approach based on Symmetry and Geometry
New Physics is encoded in and
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 19
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 20
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
SU(2)L & U(1)Y Killing vectors, respectively
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 21
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 22
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
( ) ( )
( ) ( )
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 23
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 24
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 25
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 26
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq.
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 27
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq. Nontrivial relation ?
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 28
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq. Nontrivial relation ?
①
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 29
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq. Nontrivial relation ?
①
②
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 30
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq. Nontrivial relation ?
①
②
③ ③
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 31
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq. Nontrivial relation ?
①
②
③ ③
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 32
4-point perturbative unitarity condition :
①
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 33
4-point perturbative unitarity condition :
①
unitarity is ensured by
N-point φ
φ
φ
φ
φ
‘s
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 34
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq. Nontrivial relation ?
①
②
③ ③
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 35
Approach based on Symmetry and Geometry
Geometry
( Riemann tensor )
Killing eq. Nontrivial relation ?
① ③ ③
Symmetry
( Killing vector )
, V V S, T, U ②
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is invariant under
②
SU(2)L
U(1)Y
( a = 1,2,3 )
, V V
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 37
②
Killing vector conserved current
, V V
,
W^3 B
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 38
②
Killing vector conserved current
, V V
,
W^3 B
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 39
②
Killing vector conserved current
, V V
,
W^3 B
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 40
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
Killing eq. Nontrivial relation ?
①
②
③ ③
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 41
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
①
②
Killing eq. Nontrivial relation ?
③ ③
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 42
Geometry
Symmetry
…
③
pertubative unitary if
S is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 43
Geometry
Symmetry
…
③
Perturbative unitarity ensures the finiteness of S ?
pertubative unitary if
S is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 44
Geometry
Symmetry
…
③
pertubative unitary if
+ ,
S is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 45
Geometry
Symmetry
…
③
pertubative unitary if
S is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 46
Geometry
Symmetry
…
③
pertubative unitary if
If perturbarive unitarity is ensured,
S is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 47
Geometry
Symmetry
…
③
pertubative unitary if
If perturbarive unitarity is ensured,
Then 1-loop finiteness of S is guaranteed
S is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 48
Geometry
Symmetry
…
③
pertubative unitary if
If perturbarive unitarity is ensured,
U is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 49
Geometry
Symmetry
…
③
pertubative unitary if
If perturbarive unitarity is ensured,
U is 1-loop finite if
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 50
Geometry
Symmetry
…
③
pertubative unitary if
If perturbarive unitarity is ensured,
Then 1-loop finiteness of U is guaranteed
U is 1-loop finite if
• Introduction
• unitarity vs oblique corrections
• T parameter
• Summary
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 51
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 52
T parameter
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 53
T parameter
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 54
T parameter
if
= 0 ( SM ) = 0 ( 2HDM ) = 0 ( Georgi Machacek Model)
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New !!
unitarity is ensured by
N-point φ
φ
φ
φ
φ
‘s
New !!
Summary
BACK UP
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2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 57
zz
Perturbative unitary
zRenormalizable
S & U 1-loop finite
Oblique correction finiteness v.s. Unitarity v.s. Renoralizability
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 58
We neglect the effects of operators
Ex.) aQGC violate tree level unitarity,
But we neglect the tree level effects
from the operators
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 59
HEFT
SMEFT : is written in terms of
: is written in terms of (NGBs) and
・ HEFT is more general than SMEFT
・ New Physics effect is encoded in or
SMEFT v.s. HEFT
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We know that the Killing eq. connect the geometry to the symmetry
Using commutation relation of the Killing vectors
We get the following expression
This expression is “ universal ” in any new physics model
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 61
Suppose that is invariant under the following transformation
: Killing vector
If transform linearly, and symmetric transformation consists of
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 62
EWPT
and parameter can be written in terms of EW current,
They can be expressed in terms of Killing vectors of scalar manifold
This expression is “ universal ” in any new physics model
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 63
・# of indices = # of scalar fields
・form = expression of the scalars
SM SM + Singlet
2HDM
…
Nagai-san’s slide at PPP2018
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 64
connect the Geometry and the Symmetry
in MCHM should satisfy Killing eq.
in SM should satisfy Killing eq.
in SM + Singlet Model should satisfy Killing eq.
and
and
and
Killing eq. ③
① ②
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 65
HEFT
SMEFT : is written in terms of
: is written in terms of (NGBs) and
・ HEFT is more general than SMEFT
・ New Physics effect is encoded in or
Effective field theory approach
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 66
II. Higgs cancels the divergence in oblique corrections
I. Higgs unitalize W_L scattering amplitude at tree level (tree level unitarity)
Peskin Takeuchi Phys. Rev. Lett. 65 (1990) 964
< Singlet extension >
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 67
Approach based on Symmetry and Geometry
Geometry
Symmetry
( Killing vector )
,
( Riemann tensor )
V V S, T, U
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 68
N point amplitude
φ
φ
φ
φ
φ
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 69
N point amplitude
φ
φ
φ
φ
φ
We divide the verification prosedue into two steps
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 70
N point amplitude
φ
φ
φ
φ
φ
…
Part 1
Part 2
We divide the verification prosedue into two steps
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 71
N point amplitude
φ
φ
φ
φ
φ
…
Part 1
Part 2
We divide the verification prosedue into two steps
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 72
N point amplitude
φ
φ
φ
φ
φ
…
denote the absence of and
i.e.
Part 1
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 73
In order to verify N point amplitude
φ
φ
φ
φ
φ
We must confirm that
the relation above hold for arbitrary n
U. Muller et al Gen. Rel. Grav. 31 (1999) 1759
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 74
We can confirm that
the relation above hold for arbitrary n
Actually,
once we admit , , … are proportional to
( the derivative of ) Riemann tensor.
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 75
We can confirm that
the relation above hold for arbitrary n
Actually,
once we admit , , … are proportional to
( the derivative of ) Riemann tensor.
For example
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 76
For example
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 77
For example
Conclusion :
Verification :
① Write down in terms of
symmetrize the indices of
② Write down the relation above in terms of (*)
… (*)
(*) trivial
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 78
N point amplitude
φ
φ
φ
φ
φ
…
Part 1
Part 2
We divide the verification prosedue into two steps
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 79
N point amplitude
φ
φ
φ
φ
φ
…
Part 1
We divide the verification prosedue into two steps
Part 2
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In , we verify
N point amplitude φ
φ
φ
φ
φ
Part 1
φ φ
φ φ
4 point
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 81
In , we verify
N point amplitude φ
φ
φ
φ
φ
Part 1
φ φ
φ φ
In Part 1 , we also found
4 point
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 82
In , we verify
N point amplitude φ
φ
φ
φ
φ
Part 1
φ φ
φ φ
In Part 1 , we also found
Using
( Bianchi identity ) &
we can verify
4 point
φ φ
φ φ
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 83
φ
φ
φ
φ
φ
=
Using
we can verify
φ
φ
φ
φ
φ
5 point
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 84
φ
φ
φ
φ
φ
=
Using
we can verify
φ
φ
φ
φ
φ
5 point
Difficult to write down the general formula for N point amp. ( we must verify case by case )
Gram determinant constraints ( N 6 )
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 85
φ
φ
φ
φ
φ
=
Using
we can verify
φ
φ
φ
φ
φ
5 point
Difficult to write down the general formula for N point amp. ( we must verify case by case )
Gram determinant constraints ( N 6 )
We come up with a good idea solving problems above !!
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 86
Compute 4-point amp. in following limit
φ φ
φ φ
We get, however, the most stringent condition :
We expect to get the necessary conditions for the perturbative unitarity
is the necessary and sufficient conditions for the perturbative unitarity
other
2019/7/30 Symmetry and geometry in generalized HEFT [ Yoshiki Uchida ] 87
φ φ
φ φ
We get, however, the most stringent condition :
We expect to get the necessary conditions for the perturbative unitarity
is the necessary and sufficient conditions for the perturbative unitarity
In 6-point amp., Gram det. constraint is
Compute 4-point amp. in following limit
we must chose appropriate limit consistent with Gram det. constraint
other