eft approach to the electron edm at the two-loop level · eft approach to the electron edm at the...
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EFT approach to the electron EDM at the two-loop level
based on GP, A. Pomarol, M. Riembau 1810.09413 GP, M. Riembau, T. Vantalon 1712.06337
Giuliano PanicoUniversità di Firenze and INFN Firenze
DESY Zeuthen — 2/7/2020
The precision frontier
2
Precision measurements provide fundamental tests of the SM
… which means they can probe new physics
✦ They can significantly extend the reach of direct searches
‣ access hard-to-test parameter space points
‣ extend reach to new physics at higher energy scales
✦ Particularly relevant since no convincing direct signal of new physics has been seen at the LHC
The electron EDM
3
Excellent probes of new physics are providedby the Electric Dipole Moment (EDM) of the electron
1. predicted to be very small in the SM
2. usually enhanced in the presence of new physics
3. very well tested experimentally
The electron EDM in the SM
4
e eν
b s
t
c
W W
W
γ
CP-violating phases from CKM
EDM generated from radiative corrections with CP-violating interactions
In the SM the electron EDM is extremely small
• vanishing up to 3 loops
• severe cancellations due to GIM mechanism
de < 10�38 e cm
[ Khriplovich, Pospelov ’91 ]
BSM corrections much larger than SM prediction
The electron EDM beyond the SM
5
e e
e e
χ
γ
typical contribution to the electron EDM in the MSSM
BSM physics typically gives rise to additional contributions to EDMs
• additional sources of CP-violation
• cancellations are typically not present
• contributions can arise at low loop level(eg. 1-loop or 2-loop)
The experimental bounds
6
⇤ > 40TeV
⇤ > 3TeV
1-loop :
2-loop :
ACME II bounds relevant bound
even at 2-loop
dee
⇠ 1
16⇡2
me
⇤2
dee
⇠ 1
(16⇡2)2me
⇤2
The EFT approach
7
LHC results provide a strong hint that new physics scale should bewell above the EW scale
If new physics is heavy, we can adopt the Effective Field Theory (EFT) language
leading corrections from dimension-6 operators
L = Lsm +X
i
c(6)i
⇤2O(6)
i +X
i
c(8)i
⇤4O(8)
i + · · ·
O(6)i
‣ model independent
‣ bounds easy to be recast in explicit theories
New-physics effects encoded in deformations of the SM Lagrangian
EDMs in the EFT language
8
EDMs can be reinterpreted in terms of high-energy effective operators
H = �µ ~B ·~S
S� d ~E ·
~S
S
Ldipole
= �µ
2 �µ⌫F
µ⌫
� d
2 �µ⌫i�5F
µ⌫
L =ceW⇤2
�¯L�
µ⌫�aeR�HW a
µ⌫ +ceB⇤2
�¯L�
µ⌫eR�HBµ⌫
de(µ) =
p2v
⇤2Im [s✓WCeW (µ)� c✓WCeB(µ)]
relativistic limit
SM : SU(2)L×U(1)Y
EDMs in the EFT language
8
EDMs can be reinterpreted in terms of high-energy effective operators
H = �µ ~B ·~S
S� d ~E ·
~S
S
Ldipole
= �µ
2 �µ⌫F
µ⌫
� d
2 �µ⌫i�5F
µ⌫
L =ceW⇤2
�¯L�
µ⌫�aeR�HW a
µ⌫ +ceB⇤2
�¯L�
µ⌫eR�HBµ⌫
de(µ) =
p2v
⇤2Im [s✓WCeW (µ)� c✓WCeB(µ)]
relativistic limit
SM : SU(2)L×U(1)Y
Classifying EFT effects
9
A preliminary step to apply the EFT approach is to identify and organizethe most relevant new-physics effects
‣ loop order (we will consider affects up to 2 loops)
‣ additional enhancement from running (large log if there is a significant mass gap)
‣ power counting
Classification criteria:
H 02F
Selection rules for RGEs
10
F 3
H2F 2
4
2 2
H3 2
[ Elias-Miro, Espinosa, Pomarol ’14;Cheung, Shen ’15 ]
Running effects are controlled by several selection rules
Note: four-fermion operators in Weyl notation
e EDM(H 2F )
H 02F
Selection rules for RGEs
10
F 3
H2F 2
4
2 2
H3 2
[ Elias-Miro, Espinosa, Pomarol ’14;Cheung, Shen ’15 ]
Running effects are controlled by several selection rules
Note: four-fermion operators in Weyl notation
(finite)
1 loop
e EDM(H 2F )
log enhancement
H 02F
Selection rules for RGEs
10
F 3
H2F 2
4
2 2
H3 2
[ Elias-Miro, Espinosa, Pomarol ’14;Cheung, Shen ’15 ]
Running effects are controlled by several selection rules
Note: four-fermion operators in Weyl notation
(finite)
1 loop
H 02F
F 3
H2F 2
4
2 2
H3 2
2 loops
‘two-step’ RGEdouble log
‘one-step’ RGEsingle log
e EDM(H 2F )
Classifying RGE contribution
1-loop RGE
12
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
LLeR
QLuR
H W/B
Only one 4-fermion operator contributes at 1-loop
Oluqe = (LL uR)(QL eR)
d
d lnµ
CeB
CeW
!=
yug
16⇡2
� 1
2 t✓WNc(YQ + Yu)
14Nc
!Cluqe
• The structure of the other four-fermion operators does not allow for 1-loop diagrams
proportional to quark Yukawa
4
1-loop RGE
13
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
H W/B
LLeR
H W/B
d
d lnµIm
CeB
CeW
!= � yeg
16⇡2
0 2t✓W (YL + Ye)
32
1 0 t✓W (YL + Ye)
!0
B@CWfW
CB eB
CW eB
1
CA
proportional to electron Yukawa
OWfW = |H|2W aµ⌫fW aµ⌫ OB eB = |H|2Bµ⌫ eBµ⌫ OW eB = (H†�aH)W aµ⌫ eBµ⌫
Operators involving the Higgs and gauge bosons
2-loop double-log RGE
14
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
LLeR
QLuR
H W/B
LL
uR QL
eR
O(1)lequ = (LL eR)(QL uR)
Additional 4-fermion operator contributes at 2-loop double log 4
d
d lnµCluqe =
g2
16⇡2
⇥4(YL + Ye)(YQ + Yu)t
2✓W � 3
⇤C(1)
lequ
2-loop double-log RGE
15
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
Dipole operators contribute at 2-loop double logH 02F
Two RGE patterns:
• through the operatorsH2F 2
H W/B
LLeR
H W/B
W/B
W/BH
H
d
d lnµCW eB =� 2g
16⇡2Im
h2t✓W
�ye0(YL + Ye)Ce0W � yuNc(YQ + Yu)CuW + ydNc(YQ + Yd)CdW
�
+ ye0 Ce0B � yuNc CuB + ydNc CdB
i
d
d lnµCB eB = � 4g0
16⇡2Im
hye0(YL + Ye)Ce0B + yuNc(YQ + Yu)CuB + ydNc(YQ + Yd)CdB
i
d
d lnµCWfW = � 2g
16⇡2Im
hye0 Ce0W + yuNc CuW + ydNc CdW
iO W ,O B
2-loop double-log RGE
15
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
Dipole operators contribute at 2-loop double logH 02F
Two RGE patterns:
• through the operatorsH2F 2
• through the operator
LLeR
QLuR
H W/B
LLeR
QLuR
H W/B
OuW ,OuB
only
Oluqe
d
d lnµCluqe =
g ye16⇡2
h� 8t✓W (YL + Ye)CuB + 12CuW
i
2-loop double-log RGE
16
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
OfW = "abcfW a ⌫µ W b ⇢
⌫ W c µ⇢
operatorF 3
LLeR
H
W• finite 1-loop contributions
Im[CeW ] =3
64⇡2yeg
2CfW
2-loop double-log RGE
16
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
OfW = "abcfW a ⌫µ W b ⇢
⌫ W c µ⇢
operatorF 3
• finite 1-loop contributions
Im[CeW ] =3
64⇡2yeg
2CfW
H W/B
LLeR
H W/B
H H
W
W/B
• 2-loop double log contributionsd
d lnµC
WfW = � 1
16⇡215g3CfW ,
d
d lnµCW eB = +
1
16⇡26g0g2YHCfW
‣ 2-loop contributions dominant for ⇤ > 5TeV
2-loop single-log RGE
17
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
W/B
LLd
d lnµIm
CeB
CeW
!=
ydg3
(16⇡2)2Nc
4
3t✓W YQ + 4t3✓W (YL + Ye)(Y 2
Q + Y 2d )
12 + 2t2✓W (YL + Ye)YQ
!Cledq
d
d lnµIm
CeB
CeW
!=
ye0g3
(16⇡2)21
4
3t✓W YL + 4t3✓W (YL + Ye)(Y 2
L + Y 2e )
12 + 2t2✓W (YL + Ye)YL
!Clee0 l0
Oledq = (LL eR)(dR QL) Olee0 l0 = (LL eR)(e0R L0
L)
4-fermion operators contribute at 2-loop single log 2 2
cancellation suppresses leading contributions to electron EDM, only hypercharge terms survive
g2 ! g02/8
2-loop single-log RGE
18
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
W/B
LL
d
d lnµ
CeB
CeW
!=
g3
(16⇡2)23
4
t✓W YH + 4t3✓W Y 2
H(YL + Ye)12 + 2
3 t2✓W
YH(YL + Ye)
!Cye
Oye = |H|2LLeRH
Electron Yukawa corrections contribute at 2-loop single log
g2 ! g02
cancellation suppresses leading contributions to electron EDM, only hypercharge terms survive
EW scale threshold corrections
19
Additional contribution generated at the EW scale can be relevant
Example: threshold effects from Yukawa couplings (from finite Barr-Zee-type diagrams)
dee
' �16
3
e2
(16⇡2)2v
✓2 + ln
m2t
m2h
◆ImCye
⇤2
dee
' � e2
(16⇡2)24NcQ
2tme
mtv
✓2 + ln
m2t
m2h
◆ImCyt
⇤2
• Electron Yukawa
• Top Yukawa
larger than log-enhanced contributions for
no contribution from running
⇤ . 103 TeV
Implications for BSM Model-independent constraints
Constraints from ACME II
21
tree-level
CeW 5.5⇥ 10�5 yeg
CeB 5.5⇥ 10�5 yeg0
one-loop
Cluqe 1.0⇥ 10�3 yeyt
CWfW 4.7⇥ 10�3 g2
CB eB 5.2⇥ 10�3 g0 2
CW eB 2.4⇥ 10�3 gg0
CfW 6.4⇥ 10�2 g3
two-loops
Clequ 3.8⇥ 10�2 yeyt
C⌧W 260 y⌧g
C⌧B 380 y⌧g0
CtW 6.9⇥ 10�3 ytg
CtB 1.2⇥ 10�2 ytg0
CbW 64 ybg
CbB 47 ybg0
Cledq 10 yeyt(yt/yb)
Clee0 l0 0.63 yeyt(yt/y⌧ )
two-loops finite
Cye 14 ye�h
Cyt 14 yt�h
Cyb 2.9⇥ 103 yb�h
Cy⌧ 3.4⇥ 103 y⌧�h
Table 4: Bounds on the Wilson coe�cients coming from Eq. (1.1) taking ⇤ = 10 TeV. For a betterappreciation of the bound, we have extracted the Yukawa, gauge or Higgs coupling (�h = 0.1) thatwe naturally expect to carry these Wilson coe�cients. For Cledq and Clee0 l0 we have further extracteda factor (yt/yb) and (yt/y⌧ ) respectively to reflect the fact that these coe�cients can be potentiallylarger consistently with their natural sizes Eq. (3.2).
In fact, in most of the UV-complete BSM theories (e.g. supersymmetry, composite Higgs or theorieswith flavor symmetries only broken by Yukawas) we expect operators with chirality flips to carryyukawa couplings, i.e.,
CfV / yf , Cye / ye , Clequ / yeyu , Cluqe / yeyu , Cledq / yeyd , Clee0 l0 / yeye0 , (3.3)
implying that we only expect sizable contributions to the electron EDM from four-fermion operatorsinvolving the third family. All these considerations can be useful for a proper interpretation of therecent ACME bound.
In Table 4 we list the bounds on individual Wilson coe�cients that can be inferred from thenew electron EDM measurement Eq. (1.1). To derive the bounds we considered the various Wilsoncoe�cients one-by-one. Although typical BSM theories give rise to simultaneous contributions toseveral Wilson coe�cients, strong cancellations are typically not present. In such situation thebounds obtained on single Wilson coe�cients remain approximately valid.11
3.2 Leptoquarks and extra Higgs
As a first example of an application of the above EFT analysis, we focus here on new-physics modelscontaining states of Table 3. In particular, we focus on leptoquarks and heavy Higgs-like states.12
As can be seen from Table 3, four leptoquark multiplets can give rise to electron EDM contri-
11Bounds on e↵ective operators coming from measurements of the electron EDM were previously derived in theliterature in refs. [12, 18].
12Notice that, in addition to the electron EDM, leptoquarks and heavy Higgs-like states, as well as supersymmetricscenarios, can also be constrained by the EDM of 199Hg atom through the CP-odd electron-nucleon interaction [19].
17
Obtained by fixing and considering 3-rd generation fermions⇤ = 10TeV
ACME II results translate to very strong constraintson CP-violating effective operators
CfW ⇠ g2⇤g3
(16⇡2)2
Estimating BSM effects
22
Oye Fermion (1,2,�1/2)� (1,1(3),�1)
Fermion (1,2,�1/2)� (1,1(3),0)
Fermion (1,2,�3/2)� (1,1(3),0)
Scalar (1,2,1/2)
Oluqe Scalar (3,2,7/6)
Scalar (3,1,1/3)
O(1)lequ Scalar (1,2,1/2)
Scalar (3,1,1/3)
Oledq Vector (3,2,5/6)
Vector (3,1,2/3)
Scalar (1,2,1/2)
Olee0 l0 Vector (1,1,0)
Vector (1,2,1/2)
Scalar (1,2,1/2)
CfV ⇠ g3⇤g
16⇡2, CV eV ⇠ g2⇤g
2
16⇡2
Generated at tree-level:
Generated at loop:
Cye , Cluqe, Clequ, Cledq, Clee0 l0 ⇠ g2⇤
Classification in weakly-coupled renormalizable BSM theories
typical BSM coupling
2-loops1-loop
Constraints from electron mass
23
⇢CeV v
16⇡2,Cye v
3
⇤2,Clequ mu
16⇡2,Cluqe mu
16⇡2,Cledq md
16⇡2,Cleel me0
16⇡2
�. me
CfV / yf , Cye / ye , Clequ / yeyu , Cluqe / yeyu , Cledq / yeyd , Clee0 l0 / yeye0
To keep under control 1-loop corrections to the electron mass
Automatically satisfied in MFV-like theories
• chirality-flipping operators are proportional to Yukawa couplings
Contributions to the electron EDM
24
Hierarchy of effects taking into account power-counting
Oluqe1 loop log :
O(1)lequ2 loop double-log :
OV eV Oledq Olee0 l02 loop single-log :
Oyd2 loop : Oyu Oye0
Ol0VOuV OdV3 loop double-log :
OfW3 loop :
suppressed by Yukawa
suppressed by Yukawa
1 loop : OeV
Oye
Constraints from ACME II
25
������
���
���
��
������
��
��
�
���
�
�
Λ[���
]
CB eBCWfW CW eBCluqe ClequCeW CeB Cye Cyu CuW CuB CbW CbB
Constraints taking into account power-counting
( obtained by fixing )g⇤ = 1
1-loop log 1-loop 2-loop log2 2-loop log 2-loop 3-loop log2effective
loop order
Implications for BSM Constraints on specific BSM theories
Leptoquarks
mR2 & 420 TeV
s|Im(yLR
2 yRL2 )|
yeyt
mS1 & 400 TeV
s|Im(yLL
1 yRR⇤1 )|
yeyt
Scalar leptoquarks
28
• The R2 leptoquark (3, 2, 7/6)
L = �yRL2 tRR
a"abLbL1
+ yLR2 eRR
a⇤QaL3
+ h.c.yLR⇤2 yRL⇤
2
m2R2
Oluqe
• The S1 leptoquark (3, 1, 1/3)
L = yLL1 Q
C aL3
S1"abLb
L1+ yRR
1 tRS1eR + h.c.yLL⇤1 yRR
1
m2S1
hOluqe +O(1)
lequ
i
LL
uR eR
QL
R2, S1
Contribute to 4-fermion operators
mV2 & 5.5TeV
sIm(xLR
2 x
RL?2 )
yeyb
mU1 & 2.5TeV
sIm(xRR
2 x
LL?2 )
yeyb
Vector leptoquarks
29
• The U1 leptoquark (3, 1, 2/3)
• The V2 leptoquark (3, 2, 5/6)
L = x
RL2 b
CR�
µV
a2,µ"
abL
bL1
+ x
LR2 Q
C aL3
�
µ"
abV
b2,µeR + h.c.
L = x
LL1 Q
aL3�
µU1,µL
aL1
+ x
RR1 bR�
µU1,µeR + h.c.
Directly contribute to OeW , OeBe e
γ
b
V2, U1
SUSY
LL eR
!ℓL
M2µ
"W "Hu"Hd
HW/B
Constraints on electron partners
31
for
Large effects at 1-loop
Strong constraint on mass of electron superpartners
me& 25 (50) TeV
• Simple results in the limit of heavy partners
dee
⇠ g2
16⇡2
me
m2etan�
Im(M2µ)
m2e
log
|M2µ|m2
e
me= M2 = µ (me� M2 = µ)
Constraints on the stop
32
2-loop effects through Barr-Zee diagrams
A
eu
eqytµ
⇤
H
�
e e
• Can also be interpreted as running induced by AFF operator
met & 5 TeV
for
dee
⇠ e2
16⇡2
4
9
me
m2A
tan�|µAt|m2
etsin arg(µAt) log
m2et
m2A
Strong constraint on stop mass
tan� ⇠ sin arg(µAt) ⇠ 1 , mA ⇠ µ sinAt ⇠ 1TeV
W/B
M2
µ
!W
!Hu
!Hd
H
W/B
H
!W
for and CP-violating phases
Constraints from heavy EW-inos
33
CW
fW
= Cloop
�8 + 27⇢� 24⇢2 + 5⇢3 + 6⇢2 ln ⇢
16(⇢� 1)3
CB
eB
= t2✓W
Cloop
⇢(11� 16⇢+ 5⇢2 � 2(⇢� 4) ln ⇢)
16(⇢� 1)3
CW
eB
= t✓WC
loop
⇢(7� 8⇢+ ⇢2 + 2(⇢+ 2) ln ⇢)
8(⇢� 1)3
p|M2µ| & 4TeV
tan� ⇠ 1 O(1)
Cloop
⌘ g4 sin 2� sin'
16⇡2|M2µ|, ' = arg[m2
12µ⇤M⇤
2 ] , ⇢ ⌘ |M2/µ|2
Heavy electroweak-inos contribute to HHFF operators
ACME II bounds
for degenerate superpartner masses and
Squark-selectron-gauginos loop
34
Contribution to via squark-selectron-gauginos loopOluqe
ImCluqe = yeyu3g2 Im[µM2]
16⇡2 sin 2�
X
i
m2i lnm
2i
⇧i6=j(m2i �m2
j )
sum over Wino, Higgsinos and sfermion masses
mi & 7.5TeV
sin arg(µM2)/ sin 2� ⇠ 1
Strong bounds on superpartner mass scale
Composite Higgs
Anarchic composite Higgs
36
Anarchic flavor models generate lepton EDMs at 1-loop level(through the exchange of heavy vectors and electron partners)
dee
⇠ 1
8⇡2
me
f2f & 107TeV
e e
γ
E
ρ
electron partners
heavy vectors
compositeness scaleconnected to tuning!
Yukawa’s for each family generated at different energy scales
Llin = ✏fi fiOfi
Lbil =1
⇤dH�1i
(✏ifi)OH(✏jfj)
Multi-scale composite Higgs
37
CP-violating effects can be drastically reduced in multi-scale models[ G.P., Pomarol ’16 ]
⇤b
⇤s
⇤d
energy scale decouplingoperators
OdR , OQL1
⇤u OuR
OsR
OcR , OQL2
OtR , OQL3⇤t ⇠ ⇤ir
⇤c
ObR
High energy scale suppresses flavour effects
Main flavor effects from top
‣ negligible EDMs
Multi-scale composite Higgs
37
CP-violating effects can be drastically reduced in multi-scale models[ G.P., Pomarol ’16 ]
⇤b
⇤s
⇤d
energy scale decouplingoperators
OdR , OQL1
⇤u OuR
OsR
OcR , OQL2
OtR , OQL3⇤t ⇠ ⇤ir
⇤c
ObR
for and
Bounds on top partners
38
γ
γhe e
tL ⟨h⟩
Tct
Main contribution to the electron EDM from top parters
2-loop Barr-Zee diagrams
mT & 20 TeV
Im ct ⇠ 1
de
e⇠ e2
48⇡2ye
(Im ct
) sin ✓m
top
m2T
log
m2top
m2T
mixing betweentop and top partners
sin ✓ ⇠ O(1)
Strong bounds on top partners mass scale
Comparison with direct searches
39
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mixing parameter
tan ✓ =yL4
mT
Constraints from ACME II HL-LHC projections
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electron EDM bounds stronger than HL-LHC ones if Im ct & 0.01
Conclusions
Conclusions
41
OeW ,OeB
Oluqe
OW!W
,OB "B
,OW "B
1-loop
Oye
Oledq,Olel′e′2-loop
O(1)lequ
Oe′W ,OuW ,OdW
O!W
1-loop
Oe′B,OuB,OdB
1-loop (finite)
The electron EDM provides an excellent way to probe new physics
‣ clean observable, with strong sensitivity to new CP-violating sources
‣ can test new-physics at the 10 TeV scale even with 2-loop effects
New-physics effects can be cleanly classified within an EFT framework
Outlook
42
‣ Experimental bounds steadily improved in the past years
Outlook
42
Dates are indicative
‣ Experimental bounds steadily improved in the past years
‣ Significant boost is expected in the near future probe most BSM models well beyond the 102 TeV scale