SMEFT, the frog point of viewthoughts about what everybody has seen

Giampiero Passarino

Dipartimento di Fisica Teorica, Universita di Torino, Italy

INFN, Sezione di Torino, Italy

HEFT workshop, CP3, 16 April 2019

Paraphrasing Freeman Dyson:

The BSM picture is still incomplete. We can say that somephysicists are birds, others are frogs. Birds fly high in the air(top-down approach) and survey broad vistas out to the farhorizon. Frogs live in the mud below (bottom-up approach)and see only the flowers that grow nearby. They solveproblems one at a time. BSM physics needs both birds andfrogs. Physics is rich and beautiful because birds give it broadvisions and frogs give it intricate details.

The organizers for the invitation, A. David for a

continuous dialogue

2/39

Discussing several issues that arise in constructing an EFT, up toand including dim = 8 operators

¬ Considerations on validity

­ Local, non-local, hard, soft and all that or why you should not forget loopy

EFT and Landau singularities: an “improved” review

® Mixing or why SMEFT may not be as general as we think

¯ Fitting is not interpreting Fitters you can (should not) skip this part

° Linear vs. quadratic EFT representation . . . I hate the scent of blood

3/39

I do not make any warranties about the completeness of this information

¬ Solomon:2017nlh

­ Henning:2014wua, delAguila:2016zcb, Ellis:2016enq,Fuentes-Martin:2016uol, Ellis:2017jns, Donoghue:2017pgk,Henning:2016lyp

® Gorbahn:2015gxa, Wells:2017aoy, Gabelmann:2018axh

¯ Einhorn:2013kja, Gripaios:2015qya, Jiang:2016czg,Henning:2017fpj, Brivio:2017vri, Barzinji:2018xvu,Criado:2018sdb, Hays:2018zze, Helset:2018dht,Gripaios:2018zrz, Jiang:2018pbd, Quevillon:2018mfl,Bakshi:2018ics, Criado:2019ugp

° Brehmer:2015rna, Biekotter:2016ecg, Degrande:2016dqg,Boggia:2016asg, Boggia:2017hyq, Alte:2017pme,2018xgc

4/39

derivative-coupled field theories L = L0︸︷︷︸propagator

+ Li︸︷︷︸vertices

L0 =1

2φ(2−m2− a

Λ222)

φ Li =−1

4λ φ4

m Validity of Matthews’s theorem , i.e. the Feynman rules are just thoseobtained by using Li to determine the vertices and the covariant T∗

product to determine the propagators (in other words, one can readFeynman rules from the Lagrangian).

m The validity of the theorem has been proven long ago in Bernard:1974st wherean equivalent Lagrangian is obtained which contains only first derivativesbut yields the same results (original method due to Ostrogradsky)

5/39

see

BAKsli

des

Spectrum of the theory: there are two masses,

solutions of the equation

a

Λ2µ

4±−µ

2±+m2 = 0

¬ To exclude tachyons we must have

a> 0, am2

Λ2<

1

4

­ µ2− ∼m2 (1 +am2/Λ2); however, there is a negative metric for

the particle with the larger mass (µ+), i.e. there is a ghost

in the spectrum.

6/39

EFT Option (first order in 1/Λ2): the “dangerous” term (22) issubstituted by using EoMs where terms of O(Λ−2) are neglected.

(Of course one could work at second order in Λ−2, including dim = 8 operators).

EFT optionis an effective realization of the original L , where one

assumes that Leff will be replaced by a “well-behaved” Lnew at somelarger scale, therefore justifying a truncated perturbative expansion in1/Λ2, even in the quadratic part of L ∗.

m Nevertheless, the“Ostrogradsky” option

tells us something † about

the range of validity of Leff, i.e.

¬ 0 < m2

Λ2 a< 14 (no tachyons),

­ E� µ+, where µ+ ∼ a−1/2 Λ is the upper real (positive) rootand E is the scale at which we test the predictions of Leff, i.e.E must be well below the region where the (resummed) theorydevelops ghosts.

∗EFT does not have a ghost while remaining within its regime of validity.

†If we were to start probing energies high enough then we would worry about producing the ghost.

7/39

dim= 6dim= 8

8/39

The X tree...

X ′ etc are UV completions of X or the next theory in a tower of theories

RepG ⊃ heavy dof ⊃ X ′ or X ′ is F -invariant and G⊂F

XEFT

X

X ′ X ′...′

X ′′

· · ·

· · ·

G−invariant ← current theory E � Λ1c. 2019 X =SM

next theory Λ1 < E � Λ2 {X (′)} −→ Xfin ?

9/39

X ′→ XEFT← X up to one loop

m X ′ described by L (Φ), Φ = [{Φheavy} , {Φlight}]

¬ Expand Φ =source

Φc + φ (BFM)

­ Use (heat kernel, a very convenient tool for studying variousasymptotics of the effective action)

LBFM = Lc + 〈φ ,Dφ〉Z[Φc ] =

∫[Dφ]

both heavy and lightexp{i S}= exp{i Sc}det−1/2(D)

Make sure that D is self-adjoint w.r.t. 〈 . . .〉

10/39

Heat kernel for

m one-loop divergences and counterterms;

m 1/Λ expansion of the effective action.

m (in the “naive” version) does not capture the finite lnp2 partsand there are tricky details

LBFM = Lc +1

2φ†Qφ Q = 2−M+ Q(Φc)

m M is the squared mass matrix. Heat kernel expansion requirescomputing

Tr lnQ(x)δ4(x−y) =

∫d4x

d4q

(2π)4tr ln

[−q2−M +2+ 2 i q ·∂ + Q(x , ∂x )

]11/39

m When there is one field or Mij = M2 δij we write

ln[−(q2 +M2)(I+K)

]expand ln(I+K) in powers of K obtaining the largeM-expansion of Seff = 1/2 ln det(D) in terms of ( T ) tadpoleintegrals.

m Otherwise, with more heavy scales or mixed heavy-light scales,

. . . more math and ( Lt ) log-tadpole and non-tadpole(finite lnp2 parts) integrals are needed see BAK slides

Tj =∫

ddq

(q2 +M2)j, Ltj =

∫ddq

(q2)jln(q2 +M2)

12/39

Short tour in

LogMatLand...

tr ln(A+B) = tr lnA+tr lnB, if A,B are both positive−definite

ln(AB) = lnA+ lnB, if A,B commute.

To be more precise: let A,B ∈ Cn×n commute and have no eigenvalues on R−; if for every eigenvalue λi of A andthe corresponding eigenvalue µi of B, | argλi +argµi |< π, then lnAB = lnA+ lnB, the principal logarithm of

AB. Expanding the log of a matrix :

ln(A+B)− lnA =∫ ∞

0dµ

2[(A+ µ

2 I)−1− (A+B+ µ2 I)−1

]

The correct Taylor expansion isLashkari:2018tjh

ln(A+B)− lnA =∫ ∞

0dµ

2[A−1

+ BA−1+ −A−1

+ BA−1+ BA−1

+ + . . .],

where A+ = A+ µ2 I.

Ahem! For unbounded operators ∆ = A+B, the integral on the right-hand-side should be thought of as a limit ofRiemann sums in the strong operator topology induced by the domain of the logarithm of ∆. In principle ∆ shouldbe a positive operator.

13/39

Having said that you start computing . . .

m Example: Yukawa model + heavy scalar (SxYM). Local and non-local contributions to thehigher-dimensional Lagrangian are better understood in diagrammatic language: let M be the heavy massand m the light scalar mass (massless fermions).

m There will be terms like

non-tadpole︷︸︸︷L =

∫ddqddp

q2 (q2 +M2)(q+p)2exp{i p ·x}φc(p)ψc(x)ψ(x)

m giving local and non-local Yukawa couplings of O(1/M2

),

Lloc =i π2

M2ψc(x)ψ(x)φc(x),

Lnloc = − i π2

M2

∫ddp exp{i p ·x}ψc(x)ψ(x)φc(p) ln

p2− i 0

M2︸ ︷︷ ︸kinematic log

soft region(Λ� all scales) → i π2

M2

( 1

ε− ln

p2− i 0

µ2R

+ 2)

m with 1/ε = 2/(4−d)− γ− lnπ and where µR is the renormalization scale; as shown Lnloc has a branch

cut along the negative p2 -axis. BAK

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Having said that you start computing . . .

m Example: Yukawa model + heavy scalar (SxYM). Local and non-local contributions to thehigher-dimensional Lagrangian are better understood in diagrammatic language: let M be the heavy massand m the light scalar mass (massless fermions).

m There will be terms like

non-tadpole︷︸︸︷L =

∫ddqddp

q2 (q2 +M2)(q+p)2exp{i p ·x}φc(p)ψc(x)ψ(x)

m giving local and non-local Yukawa couplings of O(1/M2

),

Lloc =i π2

M2ψc(x)ψ(x)φc(x),

Lnloc = − i π2

M2

∫ddp exp{i p ·x} ψc(x)ψ(x)︸ ︷︷ ︸

massive

φc(p) β lnβ + 1

β −1︸ ︷︷ ︸kinematic log

soft region(Λ� all scales) → i π2

M2

( 1

ε− ln

p2− i 0

µ2R

+ 2)

m with 1/ε = 2/(4−d)− γ− lnπ and where µR is the renormalization scale; as shown Lnloc has a branchcut along the negative p2 -axis.

14/39

local non-local

Non-local EFT terms are present in loops with heavy and light(internal) lines and show the characteristic pattern of singularities (e.g.normal or anomalous thresholds) of 2(3 . . .)-point functions.

� Consider a one-loop diagram in X ′ with one heavy internal line andseveral light internal lines:

¬ heavy-light contribution ` shrink the heavy line to a point ≡ insertion of one dim = 6,

tree-generated, operator inside a one-loop diagram of the theory where the heavy fields have been removed

­ the latter is associated with nloopy XEFT ‡

® therefore, when going to nloopy (NLO) EFT one has to be careful in treating heavy-light contributionswhile matching. Non-local EFT goes beyond the heat kernel tadpoles.

‡The set of one-loop diagrams derivable from LEFT containing, at most, one dim = 6 vertex.

15/39

The upshot of this is that

m LEFT mimics the unknown UV by matching the hard-local part of theloops, i.e. the terms having a bounded number of derivatives.

soft-non local components in loops cancel on both sides of the (loopy)matching condition see BAK slides

but they are not a throwaway. N.B. we could

also introduce a non-local in space, one-loop effective, L =∫ddx ddy φ(x)L(x−y)φ(y),

L(z) = F{ln(p2)}

details

� Diagrams of X ′ with light external legs and heavy internal ones admit a local low-energy limit .

� Diagrams of X ′ with light external legs and mixed internal legs may show normal-threshold singularities

in the low-energy region and give inherently non-local parts. Beware, UV logs and kinematic logs are notthe same thing. Likewise non-local and “soft” are not synonyms.

kinematic ⇐ Λ2 � sij ...k =−(pi +pj + · · ·+pk )2 > (m1 +m2 + · · ·+mn)2 or Λ2 �| t |�m2

The key advantage of including the non-local behavior is the

appearance of some important kinematic dependence see BAK slides

m Important predictions of the EFT are often related to non-analyticcontributions which modify tails of distributions.

16/39

. . . . . .

m Non-local effects correspond to long distance propagationand hence to reliable predictions at low energy.

m Local terms by contrast summarize the unknown effectsfrom high energy.

m Having both local and non-local terms allows us to implementthe full (one-loop) EFT program.

m Heavy-light terms describe a multi scale scenario: the lightmasses, the Mandelstam invariants characterizing the processand the heavy scale; on the whole they are the leading term inthe Mellin-Barnes expansion of X ′ see BAK slides

.

17/39

X ′→ XEFT

tree → loop

loop / local

loop / non-localnormal threshold

(m 6= 0) ln p2

M2 → β ln β +1β−1 β 2 = 1 + 4m2

p2

X ′

↓XEFT TG loopy XEFT

XEFT LG

XEFT LG

=⇒=⇒=⇒ • =⇒=⇒=⇒ •

s →s →s →

normal threshold

s →s →s →

18/39

Proliferation of scalars and mixing.

m The lack of discovery of beyond-the-SM (BSM) physicssuggests that the SM is “isolated” (Wells:2017aoy), including smallmixing between light and heavy scalars. The small mixingscenario raises the following question:

if there are many scalars then we have to assume that there isat least the same small mixing for every one of them.

m This is no longer accidental but systematic, and so mustinvolve a principle; this principle is unknown

SMEFT assumes a Higgs doublet, so any mixing among scalars (in general among heavy and lightdegrees of freedom) in the high-energy theory brings us to the HEFT/SMEFT dichotomy. Although thereis a wide class of BSM models that support the linear SMEFT description, this realization does not alwaysprovide the appropriate framework.

19/39

Mixing, low-energy behavior of X ′ ⇐⇒ XEFT and

residual gauge invariance

m This question can be illustrated by starting with LSM

Aµ (Zµ ) → Aµ (Zµ ) + i g sW (cW )(Λ−W+

µ−Λ+W−

µ

)−∂µ ΛA(Z)

m and by integrating out the massive electroweak gauge bosons, the Higgsboson, and the top quark fields.

m The gauge group of the resulting low-energy effective field theory (LEFT)is QCD×QED (Jenkins:2017jig,Jenkins:2017dyc), Aµ →Aµ −∂µ Λ and the photonis not the U(1) field in SU(3) × SU(2) × U(1).

m Stated differently, W/Z are integrated out, not the SU(2) fields.

20/39

Mixing and low-energy

¬ The choice of Λ is crucial when going from X ′ to X ; Λ is (generally) aratio of masses and powers of couplings.

­ The low energy behavior of X ′ should be computed in the mass

eigenbasis, not in the weak eigenbasis mixing angles (Λ) 7→ Large number of1/Λ2 terms due to expansion of mixing angles, not to the integration ofheavy fields. see BAK slides

Consider the singlet extension of the SM (RxSM) where we haveone scalar doublet and one singlet; the SM scalar field Φ is

Φ =1√2

(h2 +

√2v + i φ0

√2 i φ−

)

while the singlet is χ = 1/√

2(h1 +vs ). There is a mixing angle, N.B. α(Λ) , such that

h = cosα h2− sinα h1 H = sinα h2 + cosα h1

are the mass eigenstates, one light Higgs (h) and one heavy Higgs (H).

21/39

m Gauge invariance of the low energy limit is complicated since we integratethe H field, and h does not transform as the SM Higgs boson.

SM decoupling limit can only be achieved by imposing furtherassumptions on the couplings.

� For instance L (h ,H = 0) alone is not invariant; the full L (h ,H) isSU(2) × U(1) invariant, but L (h , 0) is not.

m Working at O(1/Λ2

)we can split the total Lagrangian into

LH=0 = LSM(h) + ∑n=0,2 Λ2n−2 δL6−2n , LH →generates

L TG +L LG +L β

m The sum over n in LH=0 is due to the expansion of sinα(cosα) in terms of Λ. see BAK slidesAfter

integrating out H in LH we will have

¬ a tree generated Leff, i.e. L TG = L TG0 +O

(1/Λ2

),

­ a loop generated one, L LG, and the tadpole contributions

m The sum of LH=0 and of L TG0 restores invariance at O (1). The procedure can be iterated

order-by-order please read Boggia:2016asg

22/39

SMEFT, HEFT and mixing can we distinguish mixed from inert?

H → Vµ (p1)Vν (p2)︸ ︷︷ ︸simplest test

= FVV

D δµν +FVV

T

(p1 ·p2 δ

µν −pν1 p

µ

2

)︸ ︷︷ ︸Tµν

m SMEFT prediction isMW F

WWD = −g M2

W

[1 +

g6√2

(aφW +aφ2−1

4aφD )

],

MW FZZD = −g M2

Z ρ[1 +

g6√2

(aφW +aφ2 +1

4aφD )

],

MW FWWT =−2g

g6√2

aφW , MW FZZT =−2g

g6√2

aZZ ,

where aZZ = s2W

aφB +c2W

aφW −sW

cW

aφWB , ρ = M2W /(c2

WM2

Z ) and√

2g6 = 1/(GF Λ2).

� As a consequence, SMEFT predicts a change in the normalization of the SM -like term and the appearance

of the transverse term. OφD is Custodial Symmetry breaking .

23/39

Relevant quantities to be constrained, e.g. in

H→ 4 l

(FWWDMW− F

ZZD

MZ

)the forbidden even to speak of POs FVV

T

m a “measure” of aφD/Λ2 and a “measure” of a non-SM tensor structure

at O(1/Λ2

)(i.e. a “measure” of aφW and aZZ in SMEFT or of the

corresponding operators in HEFT).

m Rewrite

HW−µ (p1)W+

ν (p2) = κWWH (1 +

g6√2

δ κWWH )δµν −

√2

g

MWg6 aφW Tµν ,

HZµ (p1)Zν (p2) = κZZH (1 +

g6√2

δ κZZH )δµν −

√2

g

MWg6 aZZ Tµν ,

κWWH =−g MW , κZZ

H =−gM2

Z

MWρ, δκ

WW(ZZ)H = aφW +aφ2∓

1

4aφD,

24/39

Include HHVV and fit

(1 +cVV

1

Λ2H+

cVV

2

Λ2H2 + . . .)V

µV

µ+ . . . ∈ LHEFT

m

cVV

1,2 give some informationon the doublet structure of the

scalar field, e.g. SMEFT gives

cWW

2

cWW

1

=1

2

g

MW(1− g6√

2δ κWW

H )cZZ

2

cZZ

1

=1

2

g

MW

[1 +

g6√2

(aφD−δ κZZH )]

with δ κWWH = δ κZZ

H if aφD = 0 (custodial symm.), i.e. rZ = rW(1 +

g6√2

aφD)

the c coefficients can be computed in any X ′ theory.

(1 +d

VV1Λ2 H+

dVV2Λ2 H2)Fa

µν Faµν ⇒ consequences to the

kinematics BAKHH

25/39

Field transformations, aka EoM

m Consider a Lagrangian L (4) +L (6) containing one real scalar field,

L (4) =− 1

2∂µ φ∂µ φ− 1

2m2φ2− 1

4λ φ4 , L (6) = − 1

2a

Λ2 φ22 φ =− 1

Λ2O2φ

m If we perform the transformation

φ → φ +1

2

a

Λ22φ = φ +

1

Λ2O,

the Lagrangian transforms as L → L t = L + ∆L , with

∆L =[

δL (4)

δ φ− a

Λ222φ

] 1

Λ2O +

1

2

δ 2L

δ φ2

1

Λ4O2 + · · ·= δL (4)

δ φ1

Λ2O + higher orders

= 12

aΛ2 φ22 φ − 1

2

a

Λ2(m2 φ + λ φ3)2φ + higher orders︸ ︷︷ ︸

compensation

m The term of second order in the derivatives cancels in L t and the S -matrix remains unchanged.

m Note that we have neglected higher order terms since the goal was constructing the dim = 6 Lagrangian.Of course one could work at second order in Λ−2, including dim = 8 operators, etc.

26/39

m Usually we find statements like “by using EoM we can remove . . .”,meaning that many linear combinations of operators “vanish by theEoMs”.

If one is interested in the dim = 6 basis then the necessaryEoMs are going to be used at O (1), i.e. we can derive them from L (4)

alone

This last statement, taken out of context, creates the impression that thedim = 8 basis requires EoMs used naively at O

(1/Λ2

), which is

Scherer:1994wiu

No elimination without compensatione.g. in Warsaw basis

dim = 6 redundant operator field transformation

O(6)R = QL (Dµ Dµ Φ)uR +uR (Dµ Dµ Φ)† QL Φ→Φ−g

a6R

Λ2 uR QL

dim = 8 compensation→ (QL ΦuR)(QL Φc dR)

27/39

Higher order compensation of redundant operators.

m Removing a redundant O(6) with a Wilson coefficient a6R will propagate

a6R into the Wilson coefficients of dim = 8 operators.

m In the bottom-up approach it does not matter since we only “measure”combinations of Wilson coefficients, linear in the a8

i and quadratic ina6R

§. Thus, the shift due to the field redefinition can be absorbedinto the coefficients of operators that are already present in thetheory.

m However, the low energy limit of X ′ may contain some O(8)a as well as

some O(6)R whose dim = 8 compensations contain O

(8)a ; the Wilson

coefficient a8a is now computable in terms of the parameters of X ′ but

what we have “measured” at low energy is not a8a.

X ′ −→ O(8)a

←−−−− ←−−−−

O(6)R −→ O

(6)b , O

(6)c . . .

§Indeed, when constructing the original EFT, one must include all possible operators consistent with thesymmetries at every order in the 1/Λ expansion.

28/39

example

¬ Start by considering the Lagrangian

L (4) = −1

2∂µ φ∂µ φ− 1

2m2φ2− 1

4!gφ4

with a symmetry φ→−φ. How to construct L (6) and L (8)?

­ For dim = 6 we have 7 operators, reducible by IBP identities to

L (6)=

1

Λ2

[g4 a6

0 φ6 +a61 φ22φ + g2 a6

2 φ32φ]

® For dim = 8 we get

L (8)=

1

Λ4

{g6 a8

0 φ8 + g4 a81 φ52φ +a8

2 (2φ)22φ

+ g2[a8

3 φ322φ + a84 φ2 (∂µ ∂ν φ)(∂µ ∂ν φ) +a8

5 φ2 (2φ)2]}

29/39

example

¬ We eliminate all the operators containing 2nφ ; this can be achieved by

transforming φ.

­ After the transformation the Lagrangian becomes

L = . . . +g2

Λ4φ2 (∂µ ∂ν φ)2

[a8

4 + 6a83 +a8

2−9a61 a

62−2(a6

1)2]

m In fitting the data we constrain the combinations of coefficients appearing

in L ; after that : the Wilson coefficients are the pseudo-data.

m When interpreting the results we should remember that the coefficient of

φ2 (∂µ ∂ν φ)2 is not a84 , etc.

� ∴ caution should be used in constructing the coefficients in the dim = 8part of the basis if we want to extract the parameters of X ′ from thepseudo-data.

30/39

example

Perform canonical normalization (little more than LSZ, normalize sources,

make the propagators fully diagonal, with residue 1) and recombine Wilson coefficients

a60 = a6

0− 112 a

61− 1

2 a62, a6

1 = a61, a6

2 = 32 a

61 + 9a6

2

a80 = a8

0 + 1216 (2a8

5−6a83−a8

2 + 12a81), a8

1 = 136 (5a8

5−15a83−2a8

2 + 12a81),

a82 = 2a8

2 + 12a83, a8

3 = 14 (6a8

3−a82−6a8

5), a84 = a8

4 + 12 a

82

and obtain

L →L = . . . +g2

Λ4φ2

(∂µ ∂ν φ)(∂µ ∂ν φ)[a8

4− 12 (a6

1−12a62)a6

1

]= . . . +

g2

Λ4c8

4 φ2(∂µ ∂ν φ)(∂µ ∂ν φ)

31/39

. . . . . .

m Suppose that we use L XEFT

Abasis to fit the data and that c84 results to be

compatible with zero.

m Next, we consider X ′({p}); imagine that, after computing the low energy

limit, we obtain

¬ a set of dim = 8 operators, including φ2(∂µ ∂ν φ)(∂µ ∂ν φ) with

coefficient d({p}) ;

­ a set of dim = 6 operators, some of them redundant in A-basis

not the same basis

� Different extensions turn on different bases, to compare we need tochange basis, including the higher order compensations; therefore, wecannot conclude that d({p}) = 0.

32/39

Linear vs. quadratic representation (short version)

m Most of the SMEFT calculations include the extra term, i.e.

∣∣∣A(4) +1

Λ2A(6)

∣∣∣2 →∣∣∣A(4)

∣∣∣2 + 21

Λ2Re[A(4)

]∗A(6) + 1

Λ4

∣∣∣A(6)∣∣∣2

making positive definite (by construction) all the observables.

m dim = 8 operators are (yet) unavailable, but there is more than neglectingthe dim = 4/dim = 8 interfence:

we construct S -matrix elements at O(1/Λ4

)using a canonically

transformed L truncated at O(1/Λ2

)� What we have is

L = − 1

2(1 +

M2

Λ2δZ6

H + M4

Λ4 δZ8H )∂µ H ∂µ H + . . . +

1

Λ2

[pick at random︷ ︸︸ ︷aM3 HZµ Zµ + . . .

]+ 1

Λ4 ∑i a8i O

(8)i

where the frame box indicates that the terms are not available.

33/39

Linear vs. quadratic representation

m We should write

H = (1 +M2

Λ2η

6H +

M4

Λ4η

8H)H,

select

η6H =−1

2δZ6

H , η8H =

3

8

[δZ6

H

]2,

obtaining

L = −1

2∂µ H∂µ H+a

M3

Λ2(1− 1

2M2

Λ2 δZ6H )HZµ Zµ + . . .

where the round box gives terms that are neglected in the “naive”quadratic approach.

34/39

Linear vs. quadratic representation (longer version)

m To summarize, the proper definition of “quadratic” EFT is as follows: given a “truncated” Lagrangian

L = L (4) +1

Λ2L (6) +

1

Λ4L (8)

m we distinguish between redundant and non-redundant operators:

L (6,8) = L(6,8)NR + ∑

i∈RO

(6,8)i

δL (4)

δ φ

m redefine fields according to

φ → φ− ∑n=2,4

1

Λn ∑i∈R

O(n+4)i

m The corresponding shift in L will eliminate redundant operators leaving a (neglected) term

∆L = − 1

Λ4

[δL (4)

δ φ ∑i∈R

O(8)i +

δL (6)

δ φ ∑i∈R

O(6)i +

1

2

δ 2L (4)

δ φ2 ∑i ,j∈R

O(6)i O

(6)j

]

m Once again, ∆L will never generate terms that are not present in L (8) (symmetry)

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Linear vs. quadratic representation (longer version)

m however, we will see a difference when interpreting “fitted” Wilsoncoefficients in terms of the low-energy behavior of some X ′

L = −1

2Zij

φ ∂µ φi ∂µ φj −1

2Zijm φi φj +Lrest,

Zijφ = δ

ij +1

Λ2δZ

(6) ; ijφ +

1

Λ4δZ

(8) ; ijφ ,

Zijm = m2

i δij +

1

Λ2δZ

(6) ; ijm +

1

Λ4δZ

(8) ; ijm

m We rescale fields and masses (and possibly couplings) in order toreestablish canonical normalization.

� This additional transformation will affect Lrest

m Actually, this is not the end of the story since we have to link theLagrangian parameters to a given set of experimental data.

� These relations will, once again, change Lrest

36/39

Linear vs. quadratic representation (longer version)

m Once we have obtained the Lagrangian, up to O(1/Λ4

), we can obtain

Feynman rules and amplitudes. Furthermore, a given Atree containingterms up to O

(1/Λ4

)has single and double insertions of dim = 6

operators in the tree diagrams ∈ L (4) (plus set of diagrams having newstructures, 6∈L (4)). Given

A = A(4) +1

Λ2A(6) +

1

Λ4A(8)

mlinear means including the interference between A(4) and A(6),

mquadratic

“currently” means including the square of A(6) and Not

m thecomplete inclusion

of all terms giving 1/Λ4 (before consideringA(8)). N.B. heavy-light turn on Im A(6), i.e. π2 terms; without dim = 8 the 1/Λ4 terms are

basis-dependent

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A continuum EFT is not a model, but a sequence of low-energyeffective actions Seff(Λ), for all Λ < ∞ ¶.

EFT theories ‖ are being widely used in an effort to interpretexperimental measurements of SM processes. In this scenario,various consistency issues arise; one should critically examinethe issues and we argue for the necessity to learn more generallessons about new physics within the EFT approach;inconsistent results usually attributed to the EFTs are in fact

the consequence of unnecessary further approximations.

¶as highlighted in Costello, Renormalization and Effective Field Theory.

‖A theory is aimed at a generalized statement aimed at explaining a phenomenon. A model, on the other hand,is a purposeful representation of reality.

38/39

Thank you for your attention

39/39

40/39

dim = 8 a la Ostrogradsky

L8 =−1

2

a

Λ4φ23 φ ≡ 1√

aΛ2 ψ1 ψ2−ψ12ψ1 +

1√2

ψ22φ

m Mass spectrum: aΛ4 µ6−µ2 + m2 = 0aΛ4 µ6−µ2 + m2 = 0aΛ4 µ6−µ2 + m2 = 0

m All roots are real iff 0 < a< 4/27Λ4/m4

m however the product of the roots is −m2; therefore,

there is at least one tachyon in the spectrum .

41/39

Low-energy limit, Mellin-Barnes expansion

m Consider an light-heavy-light triangle

I =∫

ddq1

(q2 + m2)((q + p1)2 + M2)((q + p1 + p2)2 + m2)

where p2i =−m2 and s =−(p1 + p2)2. We introduce M2 = λ m2 and

s = 4m2 r.

m Mellin-Barnes representation:

I =1

2π i

∫ + i ∞

− i ∞dtλ

t−1B(t , 1− t)B(2−2t , t)∫ 1

0dy[1−4ry (1−y)

]−tm Limit λ → ∞: the t -integral will be closed over the left-hand complex

half-plane at infinity, with double poles at t = 0 ,−1 , . . . .

42/39

m The leading term in the expansion, O (1/λ ), gives

I =1

M2

{1 + lnλ −

∫ 1

0dy ln

[1−4ry (1−y)

]}+O

(1

M4

),

m For s < 4m2 (unphysical region) we can expand the second logarithm inpowers of r (Taylor expansion), as long as m 6= 0

m for s > 4m2 we are above the normal threshold and the y -integral is theUV finite part of a two-point functions, showing the non-local, kinematic,logarithm

−β lnβ + 1

β −1, β

2 = 1− 1

r

(ln−s− i 0

µ2R

when m = 0)

m To summarize, for√

s , m�M, we can Taylor expand only in the regions < 4m2

m the next term in the expansion is given by the residue of the pole at t =−1 and gives

1

M4

2

β 2−1

[1 + β

2−2(1−3β2) ln

M2

m2−4β

3 lnβ + 1

β −1

]

43/39

44/39

LSM−1

2M2

s S2 + µs Φ†ΦS expansions, loopy EFT

m p2i =−M2

H and s =−(p1 +p2)2.

I = µεR

∫ddq

1

(q2 +M2H )((q+p1)2 +M2

s )((q+p1 +p2)2 +M2H )

¬ → 1

(q+p1)2 +M2s

=1

M2s

(dim=4︷︸︸︷1 −

dim=6︷ ︸︸ ︷(q+p1)2

M2s

+ . . .)

I ∼ i π2

M2s

( 1

ε− ln

M2H

µ2R︸ ︷︷ ︸

+EFT C.T.

+ 2−β lnβ + 1

β −1︸ ︷︷ ︸soft

+ . . .)

M2s �| q2 |∼| p2

i |︸ ︷︷ ︸cancels out in the matching

­ → 1

(q+p1)2 +M2s

=1

q2 +M2s

(1− p2

1 + 2p1 ·qq2 +M2

s+ . . .

)respects UV at one loop

I ∼ i π2

M2s

(1 + ln

M2s

M2H

−β lnβ + 1

β −1︸ ︷︷ ︸soft + hard

+ . . .)

| q2 |∼M2s �| p2

i |

a la Mellin-Barnes → I∼ i π2

M2s

(1 + ln

M2s

M2H

−β ln β+1β−1

+ . . .)

Return44/39

A more difficult case: CxSM

bbb ttt

χχχ

a

b c

d e

gg→ t t reducible to 4 scalars

After 1/M2 Mellin-Barnes expansion

heavy line

a,c) generate Li2(β−1− ) +Li2(β

−1+ )

β± = 12 (1±β) β 2 = 1−

4m2b

s

b) generates lnm2b

µ2R

+ β ln β+1β−1

d,e) generate lnm2b

µ2R

gg→ bb, non-local only for s > 4m2t

45/39

X = sigma-modelfrom Donoghue:2017pgk

m Low-energy behavior of Afull(π+π0→ π+π0)

Afull 7→ t

v2+

1

v4Polynomial(s , t , u)

− 1

96π2 v4

[3t2 ln

−t

M2σ

+ s (s−u) ln−s

M2σ

+ u (u− s) ln−u

M2σ

]m AEFT computed using Σµν = ∂µU∂νU

† and

LEFT =v2

4TrΣ

µµ +a1

(TrΣ

µµ

)2+a2

(TrΣµµ

)2

m Match “full” and EFT, obtained by including one-loop bubbles (loopyEFT);

AEFT =t

v2+

1

v4Polynomial(s , t , u ; a1 , a2)

− 1

96π2 v4

[3t2 ln

−t

M2σ

+ s (s−u) ln−s

M2σ

+ u (u− s) ln−u

M2σ

]

46/39

sigma-model Cont’d

m derive (renormalized) Wilson coefficients

a1 =1

8

v2

M2σ

+1

384π2

(ln

M2σ

µ2R

− 35

6

)a2 =

1

192π2

(ln

M2σ

µ2R

− 11

6

)m Compare with the tree-level matching and conclude that we have taken

into account an important kinematic feature,

the logarithmic dependence upon the characteristic momentumtransfer in the problem

LEFT and AEFT have a different meaning; the Lagrangian is local (as itshould), the amplitude generates long-distance kinematic logarithms.

However, it is a nothing prevents us from introducing aone-loop, effective, non-local Lagrangian including all processes up to agiven order.

Return

47/39

Functional integration and non-tadpole integrals

LSxYM = − 1

2∂µ φ∂µ φ− 1

2m2φ2− 1

4gφ4−ψ (/∂ −λ1 φ)ψ

− 1

2∂µ S∂µ S− 1

2M2 S2− 1

4λ4 S4 + λ2 ψ ψS−λ3 φ2 S2

m Apply BFM, define η = (λ1 φ + λ2 S)ψc and generate non-tadpoles

Zf = exp{−Tr ln /∂

}[1− i λ1

∫d4z φc (z)

δ 2

δ η(z)δ η(z)+ . . .

]exp{i∫

d4xd4y η(x)SF(x−y)η(y)}

m Leff derived from (Φ contains both (light) φ and (heavy) S)

Z = exp{−Tr ln(/∂ )

} ∫ [D S

][D φ]

exp{i∫

d4xd4y LΦ(x ,y)}

×[1 + loops + i

∫d4x1d

4x2 η(x1)Γ(x1 ,x2)η(x2) + . . .]

LΦ =1

2Φ†(x)D(x)Φ(x)δ

4(x−y) + η(x)SF(x−y)η(y) Γ(x ,y) = ∑i ,j

λi1 λ

j2 Γij (x ,y)

m where Γij are open strings of γ -matrices and of propagators SF while “loops” indicates closed strings,generating loop diagrams with internal fermion lines

48/39

More on SxYM

m Non-local terms at the Lagrangian level (spectral decomposition)

∫d4xLnloc =− i π2

M2

∫d4xd4yψc (x)ψ(x) ΣM(x−y)φc (y) ΣM(z) =

∫∞0 dµ2

[ δ4(z)

µ2+M2 −∆F(z ; µ2)]

∆F(z ; µ2) = 14π2

µ

x K1(µ x)

m General form for non-local terms

non-local = (2π)4 N1

∫d4pd4q1d

4q2 δ(4)(p+q1 +q2)ψc (q1)ψc (q2)φc (p)

+ (2π)4 ∑i=1,2

N2i

∫d4p1d

4p2d4q1d

4q2 δ(4)(p1 +p2 +q1 +q2)

× ψc (q1)/pi ψc (q2)φc (p1)φc (p2)

N1 = λ1 λ22

p2

M2ln(p2) N2i = λ

21 λ

22

Fi (p1 , p2)

M2

m where the functions Fi are combinations of three-point tensor integrals and λ1,2 are SxYM couplings

λ1 ψ ψ φ λ2 ψ ψS

49/39

Canonical normalization is more than “normalization”

m field normalization, H→[1− 1

4M2

Λ2 (aφD−4aφ2)]H

m Process, u (x1 p1) +u(x2 p2)→H(−pH ) +Z(−pZ )

m Invariants, s =−2x1x2 p1 ·p2 and t = 2x1 p1 ·pZ +M2Z

m Look for aφ2 effects:

∑spin

|A |2 =3

4

g4 v2u

c4W

M2Z s |∆Z |2 (1 + 2

g6√2

aφ2)

+g4

c4W

g26 aφ2

[A2(s , t) |∆Z |2 +A1(s , t) Re ∆Z )

]

vu = 1− 83 s2

W∆Z = 1

s−M2Z

g6 = 1√2GF Λ2

m At O(1/Λ2

)the Wilson coefficient aφ2 modifies the normalization of the s distribution; at O

(1/Λ4

)the

shape of the t -distribution is modified.

50/39

hhh ZZZ

φ0φ0φ0HHH

+++ crossed

RxSM color map

sinα = λ2M

Ms

cosα = 1− 1

2

(λ2

M

Ms

)2M2

H = λ1(1 + λ 2

2

M2

M2s

)M2

s

Aµν =(1−2 λ 2

2M2

M2s

)AMSM

µν Tshµν = M2

Z δµν + p1µ p2ν

+ g2

c2W

λ 22

M2

M2s

[12 TG−1 δµν +

Tshµν

t−M2Z

+ +T

shνµ

u−M2Z

]

m SM, one-doublet , Mixing , TG , M2s = 1/4g2 (singlet VEV)2 , depending on RxSM parameters

51/39

ASMEFTµν =

[1 +

1

3√

2GF Λ2

(6aφW −aφD + 10aφ2

)]AMSM

µν

+1√

2GF Λ2

g2

c2W

[F1 δµν +F2 T

shµν +F3 T

shνµ + F4 T

sZµν +F5 Tt

µν +F6 Tuµν

]F1 = 12 M2

s−M2h

aφ + 14

ss−M2

h

(aφD −4aφ2)− 16 (7aφD −4aφ2)

F2 = 16

1t−M2

Z

(5aφD −8aφ2

)F3 = 1

61

u−M2Z

(5aφD −8aφ2

)F4 = 1

M2Z

(3

M2h

s−M2h

+ 1)aZZ F5 = 2

t−M2Z

aZZ F6 = 2u−M2

Z

aZZ

TsZµν = p3µ p4ν +

( 1

2s−M2

Z

)δµν

Ttµν =

(M2

h −M2Z −t

)δµν −p1µ p4ν −p2ν p3µ

Tuµν =

(M2

h −M2Z −u

)δµν −p1ν p3µ −p2µ p4ν

compare with SMEFT...

extra terms not reproduced even in higher orders in 1/Ms

Return 52/39

•••SM′ 7→SM′ 7→SM′ 7→ excluded

SM′′SM′′SM′′isolated7→ 7→ 7→

•••

Λ = ∞Λ = ∞Λ = ∞

SM →

Λ = 1 TeVΛ = 1 TeVΛ = 1 TeV

projection

allowed now

a(6)ia(6)ia(6)i -trajectory

•••

7→ 7→ 7→SM′′′SM′′′SM′′′

non-isolated

m SM extensions with heavy new particles that couple to any SM particle with O (< 1) couplings map intoSM(H)EFT(Λ)

53/39