EFT approach to the electron EDM at the two-loop level · EFT approach to the electron EDM at the two-loop level based on GP, A. Pomarol, M. Riembau 1810.09413 GP, M. Riembau, T.

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


10

F 3

H2F 2

4

2 2

H3 2




(finite)

1 loop

e EDM(H 2F )

log enhancement

H 02F


10

F 3

H2F 2

4

2 2

H3 2




(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


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


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


15

OeW ,OeB

Oluqe

OW!W

,OB "B

,OW "B

1-loop

Oye


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


15

OeW ,OeB

Oluqe

OW!W

,OB "B

,OW "B

1-loop

Oye


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


16

OeW ,OeB

Oluqe

OW!W

,OB "B

,OW "B

1-loop

Oye


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


16

OeW ,OeB

Oluqe

OW!W

,OB "B

,OW "B

1-loop

Oye


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


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


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

�

�

��

��

��

��

��

��

��

��

��

��

��[��]

�� (��)

mixing parameter

tan ✓ =yL4

mT

Constraints from ACME II HL-LHC projections

��

��

��(�� ) = �

��Γ/� = ��

��

��/��

��/��

�/��

��

��

��

��

��

��/� [��]

� �

_�-�� +� �� = � ξ = ��_

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


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

EFT approach to the electron EDM at the two-loop level · EFT approach to the electron EDM at the two-loop level based on GP, A. Pomarol, M. Riembau 1810.09413 GP, M. Riembau, T.

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