Hidden Symmetries and HEFTAndrea Quadri

Dip. di Fisica, Università di Milano & INFN, Sez. di Milano

Bad Honnef, February 2017

based on arXiv:1610.00150

OutlookProbing the Higgs potentialThe modelFunctional Symmetries and RenormalizationExpanding around the p.c. renormalizable theory: the privileged operator Applications: anomalous trilinear Higgs couplingConclusions

@µ(�†�)@µ(�†�)

Probing the Higgs potentialThe discovery at LHC of a particle compatible with the Standard Model Higgs boson in 2012

has been an unprecedented breakthrough in the study of the electroweak symmetry breaking mechanism.

Parameterization of Beyond-the-Standard-Model (BSM) physicsat the LHC is usually carried out

within the Effective Field Theory approach.

Recent summary: Handbook of LHC Higgs cross sections:

4. Deciphering the nature of the Higgs sectorarXiv:1610.07922

Higgs Effective Field TheoriesAdd operators of higher dimension to the SM Lagrangian

compatible with the symmetries of the theory

c are the Wilson coefficients, is some large energy scale ⇤

Dimension-six bosonic operators

From arXiv:1610.07922

UV properties of the HEFTPower-counting renormalizability is lost

Physical Unitarity (cancellation of ghost states) guaranteed by BRST symmetry & Slavnov-Taylor identities

Froissart bound usually not respected

In general all possible terms allowed by symmetrymust be included in an EFT approach

J.Gomis, S.Weinberg, Are nonrenormalizable gauge theories renormalizable?Nucl.Phys. B469 (1996) 473-487

One-loop anomalous dimensions in the HEFT

However a tour de force computation of one-loop anomalous dimensions in the HEFT

has revealed surprising cancellations.

R.Alonso, E.Jenkins, A.Manohar, M.TrottarXiv:1308.2627 , arXiv:1310.4838 , arXiv:1312.2014 , arXiv:1409.0868

Not all mixings in principle allowed by the symmetriesdo indeed arise at one loop level

HolomorphyHolomorphic part of the Lagrangian: built out of

X+, R, L̄

Anti-holomorphic part of the Lagrangian: built out of

X�, R̄, LThe remnant is non-holomorphic

Basic idea: holomorphic operators do not mix with anti-holomorphic and non-holomorphic operators

True at the one-loop level (up to some breaking proportional to Yukawa couplings)on the S-matrix elements (not off-shell)

One-loop Anomalous Dimensions for a gauge theory

From C.Cheung and C.Shen, arXiv: 1505.01844

UV properties of HEFT dimension-six operators from derivative interactions

We restrict ourselves to the Higgs potential sector.In the first approximation we do not couple the scalars

with the gauge bosons and the fermions

S =

Zd

4x

h12@

µ�@µ� +

1

2@

µ�a@µ�a �

M

2

2X

22

+1

v

(X1 +X2)⇤⇣12�

2 + v� +1

2�

2a � vX2

⌘i.

SU(2) doublet SU(2) singlet� =1p2

�i�1 + �2

� + v � i�3

�X2

Equations of motion for the auxiliary field�S

�X1=

1

v⇤⇣12�2 + v� +

1

2�2a � vX2

⌘

Going on-shell with the auxiliary field and neglecting zero modes of the Laplacian

we obtain

X2 =1

2v�2 + � +

1

2v�2a

= �†�� v2

2

On-shell equivalence with the standard formulation

By substituting the e.o.m. solution for the singlet fieldin the classical action one gets back the ordinary quartic potential

S|on�shell

=

Zd

4

x

h12@

µ�@µ� +

1

2@

µ�a@µ�a �

1

2

M

2

v

2

⇣12�

2 + v� +1

2�

2

a

⌘2

i

with coupling constant

� =1

2

M2

v2right sign of the quartic

potential needed to ensure stability from the sign of the mass term,in turn fixed by the requirement

of the absence of tachyons

Dangereous interactions for renormalizability

The model contains derivative interactions of the schematic form

�⇤�2

i.e. an operator of dimension 5.

Renormalizability?

More symmetries needed

BRST implementation of the on-shell constraint

BRST symmetry (it does not originate from gauge invariance)

sX1 = vc , sc = 0 , s� = s�a = sX2 = 0 ,

sc̄ =1

2�2 + v� +

1

2�2a � vX2 .

Ghost action

S

ghost

= �Z

d

4x c̄⇤c .

Invariance under the nilpotent BRST symmetry

formally associated with a U(1)constr group

A.Q., Phys.Rev. D73 (2006) 065024

Off-shell there is one more scalar field X1.What about this field? Physical or unphysical?

Full tree-level vertex functional�(0) =

Zd

4x

h12@

µ�@µ� +

1

2@

µ�a@µ�a �

M

2

2X

22

� c̄⇤c+1

v

(X1 +X2)⇤⇣12�

2 + v� +1

2�

2a � vX2

⌘

+ c̄

⇤⇣12�

2 + v� +1

2�

2a � vX2

⌘i.

Slavnov-Taylor identityZ

d

4x

⇣vc

��

�X1+

��

�c̄

⇤��

�c̄

⌘= 0

Antifield

Propagators

��0�0 =i

p2, ��a�b =

i�abp2

, �c̄c =i

p2

�X1X1 = � i

p2, �X2X2 =

i

p2 �M2.

The quadratic part is diagonalized by � = �0 +X1 +X2

�XX =iM2

p2(p2 �M2)

The derivative interaction only depends on ,whose propagator has an improved UV behaviour

X = X1 +X2

Thus the derivative interactionis harmless and p.c. renormalizability still holds

Physical statesThey are described by the standard BRST construction

of the physical Hilbert space in terms of the asymptotic charge QHphys = ker Q/Im Q

X1,�0 62 Hphys

X2 2 Hphys

c = {Q,X1} , [Q, c̄] = �0 +X1

The scalar singlet is physical:

The only physical degrees of freedom are the scalar X2 and the components of the

SU(2) doublet �a

Functional identities & RenormalizationGhost and antighost equation

X1 equation

X2 equation & shift symmetry

Global SU(2) invariance

��

�c̄= �⇤c ,

��

�c= ⇤c̄

��

�X1=

1

v⇤ ��

�c̄⇤

�X1(x) = ↵(x) , �X2(x) = �↵(x)��

�X2(x)=

1

v

⇤ ��

�c̄

⇤ +⇤(X1 +X2)�M

2X2 � vc̄

⇤

Zd

4x

h� 1

2↵a�a

��

��

+⇣12(� + v)↵a +

1

2✏abc�b↵c

⌘��

��a

i= 0 .

Stability of the theory

The ghosts remain free fields also at the quantum level.

The X1 and X2 equations fix the amplitudes involving at least one X1 and X2 fields in terms of amplitudes without.

The latter have a good UV behaviour

Global SU(2) invariance is a symmetry of the theory

UV indices

Field/ext. source UV index

c̄⇤ 2

�01

�a 1

There is no power-counting with positive UV index on the physical singlet and the auxiliary field X1,

however they are controlled by the functional identitiesin terms of ancestor amplitudes with external legs

with positive UV dimensions

Counterterms for ancestor amplitudesLct = Lct,1 + Lct,2

Lct,1 =� Z⇣12@µ�@µ� +

1

2@µ�a@µ�a

⌘�M

⇣12�2 + v� +

1

2�2a

⌘

� G⇣12�2 + v� +

1

2�2a

⌘2�R1c̄

⇤⇣12�2 + v� +

1

2�2a

⌘.

Lct,2 = �R2c̄⇤ � 1

2R3(c̄

⇤)2

SM with the Derivative Representationof the Higgs potential

The coupling to the gauge fields and the fermions happens through the doublet

@µ�†@µ� ! Dµ�

†Dµ�

�GY ̄L� R � G̃Y ̄L R�C + h.c.

The mass term for X2, the (X1+X2)-dependent sectoras well as the ghosts and antighosts associatedwith the SU(2) constraint are left unchanged.

Custodial SymmetryStill true in the derivative formulation

of the Higgs potential

⌦ = (�C ,�)

1

2�2 + v� +

1

2�2a =

1

2Tr (⌦†⌦)

By requiring X1, X2 to be invariant under global SU(2)L x SU(2)R we recover the usual custodial symmetry

⌦ ! L⌦R†

in the limit where the hypercharge coupling vanishes.

SU(2) x U(1) x U(1)constr BRST symmetryThe full BRST symmetry of the theory is

s = s0 + s1

s0 is the SU(2)xU(1) BRST symmetry of the SMs1 is the U(1)constr BRST symmetry

s0� = 0 for � 2 {X2, X1, c̄, c}s1� = 0 on gauge fields, fermions and the � doublet

s2 = 0 , s20 = s21 = 0 , {s0, s1} = 0

The z-modelThe X2 equation is not the most general functional symmetry

holding true for the vertex functional.The breaking term on the R.H.S. of the shift symmetry

stays linear in the quantum fields even if one adds a kinetic termfor the scalar singletZ

d

4x

z

2@

µX2@µX2

Upon integration over the auxiliary field this is equivalentto the addition of the dimension-six operator

Zd

4x

z

v

2@µ�

†�@µ�†�

Modified identities and propagatorsThe X2 equation changes

��

�X2(x)=

1

v

⇤ ��

�c̄

⇤ +⇤(X1 + (1� z)X2)�M

2X2 � vc̄

⇤

�X2X2 =i

(1 + z)p2 �M2

The X2 propagator becomes

and p.c. renormalizability is lost, as a consequenceof the worsened UV behaviour of the X field

�XX =i(�zp2 +M2)

p2[(1 + z)p2 �M2]

z-dependence of the vertex functional@�

@z

=

Zd

4x

��

�R(x)

R(x) coupled to �1

2X2⇤X2

Thus we can expand the non p.c.-renormalizable theory around z=0in terms of insertion of the kinetic operator for X2 at zero momentum

z-dependence from tree-level Feynman rulesRescale the X2 lines by a factor 1/(1+z)

and redefine the mass throughM2 ! M2/(1 + z)

Side remark:No power-counting renormalizability,

however no freedom to add finite countertermsvanishing at z=0

(otherwise further sources of z-dependence would be added)

Application: anomalous trilinear Higgs coupling

G.Degrassi, P.Giardino, F.Maltoni, D.PaganiarXiv:1607.04251

VH3 = �3vH3

induced e.g. by an anomalous dim. 6 operator (�†�)3

Mixing with @µ(�†�)@µ(�†�)?Gauge invariance?

NLO correctionsAccording to arXiv:1607.04251 there are two types of corrections:

⌃NLO = ZH⌃LO(1 + �C1)

ZH is related to the wave function correction associated with the diagram

is a process-dependent coefficient arising from the interferencefrom one-loop virtual EW corrections

C1

Examples of EW vertex corrections

Processes involving massive vector bosons

Contributions to tt̄H

The X2-equation in the presence of a cubic interaction

One needs one more external source L to define the X2-equation in the presence of a cubic interaction

g6X32

Zd

4xLX

22

��

�X2=

1

v⇤ ��

�c̄⇤+ 3g6

��

�L+⇤(X2 + 1(1� z)X2)�M2X2 � vc̄⇤ + 2LX2

The X2-equation becomes

valid to all orders in the loop expansion

The two-point function�(j)X2(�p)X2(p)

=1

v2p4�(j)

c̄⇤(�p)c̄⇤(p)

� 3g6v

p2�(j)L(�p)c̄⇤(p) �

3g6v

p2�(j)L(p)c̄⇤(�p)

+ 9g26�(j)L(�p)L(p)

ext. source (L, c̄⇤)

�

�

At one loop level (j=1) the amplitudesinvolving the external sources on the r.h.s.

are logarithmically divergentOne generates divergences formonomials involving X2 and

up to four derivatives

⇠j=1 (@µX2)2

⇠j=1 (@µX2@µX2)

2

⇠j=1 X22

Basis dependenceThe form of the X2-equation depends on the basis.

It simplifies if one uses X = X1 +X2, X2,�

��

�X2= 3g6

��

�L�⇤X �M2X2 � z⇤X2 � vc̄⇤

so that one has for the two-point function�(j)X2(�p)X2(p)

= 9g26�(j)L(�p)L(p) ⇠j=1 X2

2

However one needs to consider mixed propagators �XX2 ,��X2

and mixed 1-PI Green’s functions �(j)�X2

,�(j)XX2

yielding back the divergences described before

Vertex functionsAnomalous coupling associated with g6X3

2

SM contribution controlled by the derivative interactionand the sigma-dependent part of the action

At the one-loop level diagrammatic(gauge-invariant) separation of the SM and BSM part

in each process

Sample SM and BSM diagrams

X2

X2

X2

UV divergences in vertex functionsAgain controlled by the X2-equation

More complicated relations involving external sources insertions

The latter have better UV propertiesthan the field X2

Valid off-shell, to any order in the loop expansion

Outlook

• A particular dimension-six operator has special UV properties in the HEFT extensions of the SM

• The latter properties are transparent in the derivative Higgs interaction formalism (application to the study of anomalous interactions)

• The non-power-counting renormalizable theory is constructed as an expansion around the p.c. renormalizable theory at z=0