Pseudo-observables in Higgs physics · 105 width taking into account the ﬁnal state QED and QCD...

LHCHXSWG-INT-2016-xxx1

May 1, 20162

LHC Higgs Cross Section Working Group 2 (Higgs Properties)3

Pseudo-observables in Higgs physics4

Main authors of the note (at present. . . ):5

Admir Greljo, Gino Isidori, Jonas M. Lindert, David Marzocca, Giampiero Passarino6

We acknowledge contributions and feedback from (at present. . . ):7

Marzia Bordone, André David, Michael Duehrssen-Debling, Adam Falkowski, Martin Gonzales-8

Alonso, Sabine Kraml, Andrea Pattori, Kerstin Tackmann.9

Contents10

1 Introduction 211

2 Two-body decay modes 412

2.1 h→ f f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

2.2 h→ γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

3 Three-body decay modes 715

3.1 h→ f f γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

4 Four-fermion decay modes 917

4.1 h→ 4 f neutral currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918

4.2 h→ 4 f charged currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119

4.3 h→ 4 f complete decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 1120

4.4 Physical PO for h→ 4` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221

4.5 Physical PO for h→ 2`2ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422

5 PO in Higgs electroweak production: generalities 1523

5.1 Amplitude decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724

5.1.1 Vector boson fusion Higgs production . . . . . . . . . . . . . . . . . . 1725

5.1.2 Associated vector boson plus Higgs production . . . . . . . . . . . . . 1926

1

6 PO in Higgs electroweak production: phenomenology 2027

6.1 Vector Boson Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2028

6.2 Associated vector boson plus Higgs production . . . . . . . . . . . . . . . . . 2429

6.3 Validity of the momentum expansion . . . . . . . . . . . . . . . . . . . . . . . 2530

6.4 Illustration of NLO QCD effects . . . . . . . . . . . . . . . . . . . . . . . . . 2631

7 Parameter counting and symmetry limits 2832

7.1 Yukawa modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2833

7.2 Higgs EW decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2834

7.3 EW production processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3135

7.4 Additional PO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3236

8 PO meet SMEFT 3337

8.1 SMEFT summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3438

8.2 Theoretical uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3639

8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3640

8.4 SMEFT and physical PO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4641

8.5 Summary on the PO-SMEFT matching . . . . . . . . . . . . . . . . . . . . . . 4842

9 Conclusions 4943

1 Introduction44

The idea of PO has been formalized the first time in the context of electroweak observables45

around the Z pole [1]. A generalization of this concept to describe possible deformations from46

the SM in Higgs production and decay processes has been discussed in Refs. [2,3,4,5,6,7]. The47

basic idea is to identify a set of quantities that are48

I. experimentally accessible,49

II. well-defined from the point of view of QFT,50

and capture all relevant New Physics (NP) effects (or all relevant deformations from the SM)51

without losing information and with minimum theoretical bias. The last point implies that52

changes in the underlying NP model should not require any new processing of raw experimental53

data. In the same spirit, the PO should be independent from the theoretical precision (e.g. LO,54

NLO, ...) at which NP effects are computed. Finally, the PO are obtained after removing (via55

a proper deconvolution) the effect of the soft SM radiation (both QED and QCD radiation),56

that is assumed to be free from NP effects. In the case of observables around the Z pole, the57

Γ(Z→ f f ) partial decay rates provide good examples of PO.58

2

The independence from NP models can not be fulfilled in complete generality. However, it can59

be fulfilled under very general assumptions. In particular, we require the PO to60

III. capture all relevant NP effects in the limit of no new (non-SM) particles propagating on-61

shell (in the amplitudes considered) in the kinematical range where the decomposition is62

assumed to be valid.63

Under this additional hypothesis, the PO provide a bridge between the fiducial cross-section64

measurements and the determination of NP couplings in explicit NP frameworks.65

On a more theoretical footing, the Higgs PO are defined from a general decomposition of66

on-shell amplitudes involving the Higgs boson – based on analyticity, unitarity, and crossing67

symmetry – and a momentum expansion following from the dynamical assumption of no new68

light particles (hence no unknown physical poles in the amplitudes) in the kinematical regime69

where the decomposition is assumed to be valid. These conditions ensure the generality of this70

approach and the possibility to match it to a wide class of explicit NP model, including the71

determination of Wilson coefficients in the context of Effective Field Theories.72

The old κ framework [8,9] satisfied the conditions I and II, but not the condition III, since the73

framework was not general enough to describe modifications in (n > 2)-body Higgs decays74

resulting in non-SM kinematics. Similarly, the old κ framework could not describe modifica-75

tions of the Higgs-cross sections that cannot be reabsorbed into a simple overall re-scaling with76

respect to the SM.77

Similarly to the case of electroweak observables, it is convenient to introduce two complemen-78

tary sets of Higgs PO:79

• a set of physical PO, namely a set of (idealized) partial decay rates and asymmetries;80

• a set of effective-couplings PO, parameterizing the on-shell production and decay ampli-81

tudes.82

The two sets are in one-to-one correspondence: by construction, the effective-couplings PO83

are directly related to the physical PO after properly working out the decay kinematics. The84

effective-couplings PO are particularly useful to build tools to simulate data, taking into account85

the effect of soft QCD and QED radiation.1 This is why, from the practical point of view, the86

effective-couplings PO are first extracted from data in the LHC Higgs analysis, and from these87

the physical PO are indirectly derived. As we discuss below, the latter provide a more intuitive88

and effective presentation of the measurements performed.89

The note is organized as follows: the PO for Higgs decays are discussed in Section 2–4, sepa-90

rating two, three, and four-body decay modes. General aspects of PO in electroweak production91

processes are discussed in Section 5, whereas the specific implementation for VH and VBF is92

presented in Section 6. The total number of PO to discuss both production and decay processes93

1A first public tool for Higgs PO is available in Ref. [10].

3

is summarized in Section 7, where we also address the reduction of the number of independent94

terms under specify symmetry assumptions (in particular CP conservation and flavor universal-95

ity). Finally, a discussion about the matching between the PO approach and the SM Effective96

Field Theory (SMEFT) is presented in Sections 8. The latter section is not needed to discuss97

the PO implementation in data analyses, but it provides a bridge between this chapter of the YR98

(Measurements and Observables) and the one devoted to the EFT approaches.99

2 Two-body decay modes100

In the case of two-body Higgs decays into on-shell SM particles, namely h→ f f and h→ γγ ,101

the natural physical PO for each mode are the partial decay widths, and possibly the polariza-102

tion asymmetry if the spin of the final state is accessible.103

In the h→ f f case the main issue to be addressed is the optimal definition of the partial decay104

width taking into account the final state QED and QCD radiation.105

In the h→ γγ case the point to be addressed is the extrapolation to real photons of electromag-106

netic showers with non-vanishing invariant mass.107

2.1 h→ f f108

For each fermion species we can decompose the on-shell h→ f f amplitude in terms of two109

effective couplings (y fS,P), defined by110

A (h→ f f ) =− i√2

(y f

S f f + iy fP f γ5 f

), (1)

where f , f in the right hand side are spinor wave functions. These couplings are real in the limit111

where we neglect re-scattering effects, that is an excellent approximation (also beyond the SM112

if we assume no new light states), for all the accessible h→ f f channels. If h is a CP-even state113

(as in the SM), then y fP is a CP-violating coupling.114

In order to match our notation with the κ framework [8], we define the two effective couplings115

PO of the h→ f f decays as follows:116

κ f =y f

S

y f ,SMS

, δCPf =

y fP

y f ,SMS

. (2)

Here y f ,SMS is the SM effective coupling that provides the best SM prediction in the κ f → 1 and117

δ CPf → 0 limit.118

4

The measurement of Γ(h → f f )(incl) determines the combination |κ f |2 + |δ CPf |2, while the119

δ CPf /κ f ratio can be determined only if the fermion polarization is experimentally accessi-120

ble. With this notation, the inclusive decay rates, computed assuming a pure bremsstrahlung121

spectrum can be written as122

Γ(h→ f f )(incl) =[|κ f |2 + |δ CP

f |2]

Γ(h→ f f )(SM)(incl) , (3)

where fermion-mass effects, of per-mil level even for the b quark, have been neglected. In ex-123

periments Γ(h→ f f )(incl) cannot be directly accessed, given tight cuts on the f f invariant mass124

to suppress the background: Γ(h→ f f )(incl) is extrapolated from the experimentally accessible125

Γ(h→ f f )(cut) assuming a pure bremsstrahlung spectrum, both as far as QED and as far as126

QCD (for the qq channels only) radiation is concerned.127

The SM decay width is given by128

Γ(h→ f f )(SM)(incl) = N f

c|y f ,SM

eff |216π

m2H , (4)

where the color factor N fc is 3 for quarks and 1 for leptons. Using the best SM prediction of the

branching ratios in these channels [8], for mH = 125.0 GeV and ΓtotH = 4.07× 10−3 GeV, we

extract the values of the |y f ,SMeff | couplings in Eq. (4):

bb ττ

B(h→ f f ) 5.77×10−1 6.32×10−2

|y f ,SMeff | 1.77×10−2 1.02×10−2

,

cc µµ

B(h→ f f ) 2.91×10−2 2.19×10−4

|y f ,SMeff | 3.98×10−3 5.99×10−4

,

As anticipated, the physical PO sensitive to δ CPf /κ f necessarily involve a determination (direct129

or indirect) of the fermion spins. Denoting by~k f the 3-momentum of the fermion f in the Higgs130

center of mass frame, and with {~s f ,~s f } the two fermion spins, we can define the following CP-131

odd asymmetry [11]132

A CPf =

1

|~k f |〈~k f · (~s f ×~s f )〉=−

δ CPf κ f

κ2f +(δ CP

f )2(5)

As pointed out in Ref. [12], in the h→ τ+τ−→ Xτ+Xτ− decay chains asymmetries proportional133

to A CPf are accessible through the measurement of the angular distribution of the τ± decay134

products.135

5

Note that, by construction, the effective couplings PO depend on the SM normalization. This136

imply an intrinsic theoretical uncertainty in their determination related to the theory error on137

the SM reference value. On the other hand, the physical PO are independent of any reference138

to the SM. Indeed the (conventional) SM normalization of κ f cancels in Eq. (3).139

2.2 h→ γγ140

The general decomposition for the h→ γγ amplitude is141

A[h→ γ(q,ε)γ(q′,ε ′)

]= i

2vF

ε′µεν

[εγγ(gµν q·q′−qµq′ν)+ ε

CPγγ ε

µνρσ qρq′σ], (6)

where εµνρσ is the fully antisymmetric tensor and ε0123 = 1 in our convention. From this we142

identify the two effective couplings εγγ and εCPγγ that, similarly to y f

S,P, can be assumed to be real143

in the limit where we assume no new light states and small deviations from the SM limit. Here144

vF = (√

2GF)−1/2, and GF is the Fermi constant extracted from the muon decay. We define the145

effective couplings PO for this channels as146

κγγ =Re(εγγ)

Re(εSMγγ )

, δCPγγ =

Re(εCPγγ )

Re(εSMγγ )

, (7)

where εγγ

SM is the value of the PO which reproduces the best SM prediction of the decay width.147

By construction, the SM expectation for the two PO is κSMγγ = 1 and (δ CP

γγ )SM = 0.148

If the photon polarization is not accessible, the only physical PO for this channel is Γ(h→149

γγ). Starting from realistic observables, where the electromagnetic showers have non-vanishing150

invariant mass, Γ(h→ γγ) is defined as the extrapolation to the limit of zero invariant mass for151

the electromagnetic showers. The relation between Γ(h→ γγ) and the two effective couplings152

PO is153

Γ(h→ γγ) =[κ

2γγ +(δ CP

γγ )2]

Γ(h→ γγ)(SM) , (8)

where154

Γ(h→ γγ)(SM) =|εSM,eff

γγ |216π

m3H

v2F. (9)

Using the SM prediction for the branching ratios in two photons [8], for vF = 246.22 GeV,155

mH = 125.0 GeV and ΓtotH = 4.07×10−3 GeV, we obtain156

B(h→ γγ)SM = 2.28×10−3 → εγγ

SM = 3.8×10−3 . (10)

This value corresponds to the 1-loop contribution in the SM, which also fixes the relative sign.157

Similarly to the f f case, the SM normalization cancels in the definition of the physical PO.158

6

The physical PO linear in the CP-violating coupling δ CPγγ necessarily involves the measurement159

of the photon polarization and is therefore hardly accessible at the LHC (at least in a direct way,160

see for example [13]). Denoting by~q1,2 the 3-momenta of the two photons in the centre of mass161

frame, and with~ε1,2 the corresponding polarization vectors, we can define:162

A CPγγ =

1mh〈(~q1−~q2) · (~ε1×~ε2)〉 ∝

δ CPγγ κγγ

κ2γγ +(δ CP

γγ )2. (11)

3 Three-body decay modes163

The guiding principle for the definition of PO in multi-body channels is the decomposition164

of the decay amplitudes in terms of contributions associated to a specific single-particle pole165

structure. In the absence of new light states, such poles are generated only by the exchange166

of the SM electroweak bosons (γ , Z, and W ) or by hadronic resonances (whose contribution167

appears only beyond the tree level and is largely suppressed). Since positions and residues on168

the poles are gauge-invariant quantities, this decomposition satisfies the general requirements169

for the definitions of PO.170

3.1 h→ f f γ171

The general form factor decomposition for these channels is172

A[h→ f (p1) f (p2)γ(q,ε)

]= i

2vF

∑f= fL, fR

( f γµ f )εν ×

×[F f γ

T (p2)(p·q gµν −qµ pν)+F f γ

CP(p2)εµνρσ qρ pσ

], (12)

where p = p1 + p2. The form factors can be further decomposed as173

F f γ

T (p2) = εZγ

g fZ

PZ(p2)+ εγγ

eQ f

p2 +∆SMf γ (p2) , (13)

F f γ

CP(p2) = εCPZγ

g fZ

PZ(p2)+ ε

CPγγ

eQ f

p2 . (14)

Here g fZ are the effective PO describing on-shell Z→ f f decays2 and PZ(q2)= q2−m2

Z+ imZΓZ .174

In other words, we decompose the form factors identifying the physical poles associated to the175

Z and γ propagators.176

2We have absorbed a factor g/cos(θW ) with respect to the definition of the effective Z couplings adopted atLEP-1, see Eq. (24).

7

The term ∆SMf γ

(p2) denotes the remnant of the SM h→ f f γ loop function that is regular both177

in the limit p2→ 0 and in the limit p2→m2Z . This part of the amplitude is largely subdominant178

(being not enhanced by a physical single-particle pole) and cannot receive non-standard con-179

tributions from operators of dimension up to 6 in the EFT approach to Higgs physics. For this180

reason it is fixed to its SM value.181

In this channel we thus have four effective couplings PO, related to the four εX terms in Eqs. (13)182

and (14), two of which are accessible also in h→ 2γ .3183

Similarly to the h→ 2γ case, it is convenient to define the PO normalizing them the correspond-184

ing reference SM values of the amplitudes. We thus define185

κZγ =Re(εZγ)

Re(εSMZγ

), δ

CPZγ =

Re(εCPZγ

)

Re(εSMZγ

), (15)

where the numerical value of the SM contribution εSMZγ

is obtained from the best SM prediction186

for the h→ Zγ decay width.187

The simplest physical PO that can be extracted from this channel is Γ(h→ Zγ), where both the188

Z boson and the photon are on-shell. By construction, this can be written as189

Γ(h→ Zγ) =[κ

2Zγ +(δ CP

Zγ )2]

Γ(h→ Zγ)(SM) , (16)

where190

Γ(h→ Zγ)(SM) =|εSM,eff

Zγ|2

8π

m3H

v2

(1− m2

Z

m2H

)3

. (17)

The SM prediction for this decay rate [8] provides the value of εZγ

SM:191

B(h→ Zγ)(SM) = 1.54×10−3 → εSMZγ = 6.9×10−3 . (18)

The independent physical PO linear in the coupling δ CPZγ

is the following CP-odd asymmetry at192

the Z peak:193

A CPZγ =

1|~p||~q|〈~p · (~q×~εγ)〉

∣∣∣∣(p2=m2

Z)

∝δ CP

ZγκZγ

κ2Zγ

+(δ CPZγ

)2, (19)

where all 3-momenta are defined in the Higgs center of mass frame.194

3 In the decomposition (12) we have also neglected possible dipole-type (helicity-suppressed) amplitudes. Thelatter necessarily give a strongly suppressed contribution to the decay rate since the interference with the leadingh→ f f γ amplitude vanishes in the limit m f → 0. There is no obstacle, in principle, to add a corresponding setof PO for these suppressed amplitudes. However, for all light fermions they will be un-measurable in realisticscenarios even in the high-luminosity phase of the LHC (see e.g. [14] for a numerical discussion in the h→ 2µγ

case).

8

This channel is also sensitive to Γ(h→ γγ) and A CPγγ via the effective couplings κγγ (or εγγ ) and195

δ CPγγ (or εCP

γγ ). Determining such couplings from a fit to the from factors in the low p2 region, one196

can indirectly determine Γ(h→ γγ) and A CPγγ by means of Eq. (8) and Eq. (11), respectively.197

4 Four-fermion decay modes198

Similarly to the three-body modes, also in this case the guiding principle for the definition of PO199

is the decomposition of the decay amplitudes in terms of contributions associated to a specific200

pole structure. Such decomposition for the h→ 4 f channels has been presented in Ref. [2]. The201

effective coupling PO that appear in these channels consist of four sets:202

• 3 flavor-universal charged-current PO: {κWW ,εWW ,εCPWW};203

• 7 flavor-universal neutral-current PO, 4 of which are appearing already in h→ γγ and h→204

f f γ : {κγγ ,δCPγγ ,κZγ ,δ

CPZγ}, and another 3 which are specific for h→ 4 f : {κZZ,εZZ,ε

CPZZ };205

• the set of flavor non-universal charged-current PO: {εW f };206

• the set of flavor non-universal neutral-current PO: {εZ f }.207

While the number of flavor-universal PO is fixed, the number of flavor non-universal PO depend208

on the fermion species we are interested in. For instance, looking only at light leptons (`= e,µ),209

we have 4 flavor non-universal PO contributing to h→ 4` modes (εZ f , with f = eL,eR,µL,µR)210

and 4 PO contributing to h→ 2`2ν modes (εWeL ,εW µL ,εZνe ,εZνµ). The definition of these PO is211

done at the amplitude level, separating neutral-current and charged-current contributions to the212

h→ 4 f processes, as discussed below.213

Starting from each of the effective couplings PO we can define a corresponding physical PO. In214

particular, Γ(h→ ZZ) is defined as the (ideal) rate extracted from the full Γ(h→ 4 f ), extrapo-215

lating the result in the limit κZZ 6= 0 and all the other effective couplings set to zero. Similarly216

Γ(h→ Z f f ) is defined from the extrapolation in the limit εZ f 6= 0 and all the other effective217

couplings set to zero (see extended discussion below).218

4.1 h→ 4 f neutral currents219

Let us consider the case of two different (light) fermion species: h→ f f + f ′ f ′. Neglecting220

helicity-violating terms (yielding contributions suppressed by light fermion masses in the rates),221

we can decompose the neutral-current contribution to the amplitude in the following way222

An.c.[h→ f (p1) f (p2) f ′(p3) f ′(p4)

]= i

2m2Z

vF∑

f= fL, fR∑

f ′= f ′L, f′R

( f γµ f )( f ′γν f ′)T µνn.c. (q1,q2)

9

T µνn.c. (q1,q2) =

[F f f ′

L (q21,q

22)g

µν +F f f ′T (q2

1,q22)

q1·q2 gµν −q2µq1

ν

m2Z

+F f f ′CP (q2

1,q22)

εµνρσ q2ρq1σ

m2Z

], (20)

where q1 = p1 + p2 and q2 = p3 + p4. The form factor FL describes the interaction with the223

longitudinal part of the current, as in the SM, the FT term describes the interaction with the224

transverse part, while FCP describes the CP-violating part of the interaction (if the Higgs is225

assumed to be a CP-even state).226

We can further expand the form factors in full generality around the poles, providing the defini-227

tion of the neutral-current PO [2]:228

F f f ′L (q2

1,q22) = κZZ

g fZg f ′

Z

PZ(q21)PZ(q2

2)+

εZ f

m2Z

g f ′Z

PZ(q22)

+εZ f ′

m2Z

g fZ

PZ(q21)

+∆SML (q2

1,q22) , (21)

F f f ′T (q2

1,q22) = εZZ

g fZg f ′

Z

PZ(q21)PZ(q2

2)+ εZγ

(eQ f ′g

fZ

q22PZ(q2

1)+

eQ f g f ′Z

q21PZ(q2

2)

)+ εγγ

e2Q f Q f ′

q21q2

2

+∆SMT (q2

1,q22), (22)

F f f ′CP (q2

1,q22) = ε

CPZZ

g fZg f ′

Z

PZ(q21)PZ(q2

2)+ ε

CPZγ

(eQ f ′g

fZ

q22PZ(q2

1)+

eQ f g f ′Z

q21PZ(q2

2)

)+ ε

CPγγ

e2Q f Q f ′

q21q2

2. (23)

Here g fZ are Z-pole PO extracted from Z decays at LEP-I, the translation to the notation used at229

LEP being very simple230

g fZ =

2mZ

vFgLEP

f , and (g fZ)SM = 2mZ

vF(T f

3 −Q f s2θW

) . (24)

As anticipated, all the parameters but εZ f and g fZ are flavor universal, i.e. they do not depend on231

the fermion species. In fact, flavor non-universal effects in g fZ have been very tightly constrained232

at LEP-I [15], however, sizeable effects in εZ f are possible and should be tested at the LHC.233

In the limit where we neglect re-scattering effects, both κZZ and εX are real. The functions234

∆SML,T (q

21,q

22) denote subleading non-local contributions that are regular both in the limit q2

1,2→ 0235

and in the limit q21,2→ m2

Z . As in the 3-body decay case, this part of the amplitude is largely236

subdominant and not affected by operators with dimension up to 6, therefore it is fixed it to its237

SM value.238

10

4.2 h→ 4 f charged currents239

Let us consider the h→ `ν` ¯′ν`′ process.4 Employing the same assumptions used in the neutral240

current case, we can decompose the amplitude in the following way:241

Ac.c.[h→ `(p1)ν`(p2)ν`′(p3) ¯′(p4)

]= i

2m2W

vF( ¯Lγµν`L)(ν`′Lγν`

′L)T

µνc.c. (q1,q2)

T µνc.c. (q1,q2) =

[G``′

L (q21,q

22)g

µν +G``′T (q2

1,q22)


ν

m2W

+G``′CP(q

21,q

22)

εµνρσ q2ρq1σ

m2W

], (25)

where q1 = p1 + p2 and q2 = p3 + p4. The decomposition of the form factors, that allows us to242

define the charged-current PO, is [2]243

G``′L (q2

1,q22) = κWW

(g`W )∗g`′

W

PW (q21)PW (q2

2)+

(εW`)∗

m2W

g`′

W

PW (q22)

+εW`′

m2W

(g`W )∗

PW (q21)

, (26)

G``′T (q2

1,q22) = εWW

(g`W )∗g`′

W

PW (q21)PW (q2

2), (27)

G``′CP(q

21,q

22) = ε

CPWW

(g`W )∗g`′

W

PW (q21)PW (q2

2), (28)

where PW (q2) is the W propagator defined analogously to PZ(q2) and g fW are the effective cou-244

plings describing on-shell W decays (we have absorbed a factor of g compared to standard245

notations). In the SM,246

(gikW )SM =

g√2

Vik , (29)

where V is the CKM mixing matrix.5 In absence of rescattering effects, the Hermiticity of the247

underlying effective Lagrangian implies that κWW , εWW and εCPWW are real couplings, while εW`248

can be complex.249

4.3 h→ 4 f complete decomposition250

The complete decomposition of a generic h→ 4 f amplitude is obtained combining neutral- and251

charged-current contributions depending on the nature of the fermions involved. For instance252

h→ 2e2µ and h→ ` ¯qq decays are determined by a single neutral current amplitude, while253

4 The analysis of a process involving quarks is equivalent, with the only difference that the εW f coefficients arein this case non-diagonal matrices in flavor space, as the gW

ud effective couplings.5More precisely, (gik

W )SM = g√2Vik if i and k refers to left-handed quarks, otherwise (gik

W )SM = 0.

11

the case of two identical lepton pairs is obtained from Eq. (20) taking into account the proper254

(anti-)symmetrization of the amplitude:255

A[h→ `(p1) ¯(p2)`(p3) ¯(p4)

]= An.c.

[h→ f (p1) f (p2) f ′(p3) f ′(p4)

]f= f ′=`

− An.c.[h→ f (p1) f (p4) f ′(p3) f ′(p2)

]f= f ′=`

. (30)

The h→ e±µ∓νν decays receive contributions from a single charged-current amplitude, while256

in the h→ ` ¯νν case we have to sum charged and neutral-current contributions:257

A[h→ `(p1) ¯(p2)ν(p3)ν(p4)

]= An.c.

[h→ `(p1) ¯(p2)ν(p3)ν(p4)

]

− Ac.c.[h→ `(p1)ν(p4)ν(p3) ¯(p2)

]. (31)

4.4 Physical PO for h→ 4`258

To define the idealised physical PO we start with the quadratic terms for each of the form259

factors in Eqs. (21-23), and compute their contribution to the double differential decay rate for260

h→ e+e−µ+µ− (for κZZ , εZZ and εCPZZ ) and for h→ Z`+`− (for the contact terms εZ`). It is261

important to stress that physical PO as defined here, are extracted from the effective-coupling262

PO, and represent a more intuitive presentation of the experimental measurements.263

Decay channel h→ e+e−µ+µ−264

We choose this particular decay channel for the (conventional) definition of the physical PO265

because it depends on all the PO relevant for h→ 4` and because it does not contain interference266

between the two fermion currents as in Eq. (30). The independent contributions of the three267

form factors to the decay rate are:268

dΓLL

dm1dm2=

λpβ10

2304π5m4

Zm3h

v2F

m1m2 ∑f , f ′

∣∣∣F f f ′L

∣∣∣2,

dΓTT

dm1dm2=

λpβ4

1152π5m3

h

v2F

m31m3

2 ∑f , f ′

∣∣∣F f f ′T

∣∣∣2,

dΓCP

dm1dm2=

λpβ2

1152π5m3

h

v2F

m31m3

2 ∑f , f ′

∣∣∣F f f ′CP

∣∣∣2,

(32)

where f = eL,eR, f ′ = µL,µR, m1(2) ≡√

q21(2) and269

λp =

√√√√1+(

m21−m2

2m2

h

)2

−2m2

1 +m22

m2h

, βN = 1+m4

1 +Nm21m2

2 +m42

m4h

−2m2

1 +m22

m2h

. (33)

12

By integrating over m1 and m2 we obtain the partial decay rate as270

Γ(h→ 2e2µ) = Γ(h→ 2e2µ)SM×∑j≥i

X2e2µ

i j κiκ j , (34)

where X2e2µ

i j are the numerical coefficients reported in Ref. [3], while κi are the corresponding271

effective-coupling PO. We define the physical PO as the specific contribution to the partial decay272

rate. Namely, physical PO is the partial decay rate due to a single PO, while setting other PO to273

zero. In particular,274

Γ(h→ 2e2µ)[κZZ] = 4.929×10−2(|gZeL |2 + |gZeR|2)(|gZµL |2 + |gZµR |2) |κZZ|2 MeV

Γ(h→ 2e2µ)[εZZ] = 4.458×10−3(|gZeL |2 + |gZeR|2)(|gZµL |2 + |gZµR |2) |εZZ|2 MeV

Γ(h→ 2e2µ)[εCPZZ ] = 1.884×10−3(|gZeL |2 + |gZeR|2)(|gZµL |2 + |gZµR |2) |εCP

ZZ |2 MeV(35)

The numerical coefficients in Eq. (35) have been obtained neglecting QED corrections. The275

latter must be included at the simulation level by appropriate QED showering programs, such276

as PHOTOS [16]. As shown in Ref. [17]: the impact of such corrections is negligible after277

integrating over the full phase space, hence in the overall normalization of the partial rates278

in Eq. (35), while they can provide sizable distortions of the spectra in specific phase-space279

regions.280

Since each effective coupling PO correspond to a well-defined pole contribution to the ampli-281

tude (with one or two poles of the Z boson), and a well-defined Lorentz and flavor structure, we282

can associate to those contributions to partial rates an intutive physical meaning. In particular,283

we define the following physical PO for the h→ 4` decays:284

Γ(h→ ZLZL)≡Γ(h→ 2e2µ)[κZZ]

B(Z→ 2e)B(Z→ 2µ)= 0.209 |κZZ|2 MeV

Γ(h→ ZT ZT )≡Γ(h→ 2e2µ)[εZZ]

B(Z→ 2e)B(Z→ 2µ)= 0.0189 |εZZ|2 MeV

ΓCPV(h→ ZT ZT )≡

Γ(h→ 2e2µ)[εCPZZ ]

B(Z→ 2e)B(Z→ 2µ)= 0.00799 |εCP

ZZ |2 MeV

(36)

where, due to the double pole structure of the amplitude, we have removed the (physical)285

branching ratios of the Z→ e+e− and Z→ µ+µ− decays. Here286

B(Z→ 2`) =Γ0

ΓZR`((g`L

Z )2 +(g`RZ )2)≈ 0.4856

((g`L

Z )2 +(g`RZ )2), (37)

where Γ0 =mZ24π

, ΓZ is the total decay width and R` =(1+ 3

4πα(mZ)

)describes final state QED287

radiation. The relative uncertainty in the numerical coefficients in Eqs. (36,37), due to the288

experimental error on ΓZ and mZ [18], is at the ∼ 10−3 level, thus negligible even given the289

expected long-term sensitivity in these Higgs measurements.290

13

Decay channel h→ Z`+`−291

The idealised physical PO related to the contact terms can be defined directly from the on-292

shell decay h→ Z`+`−, where ` = eL,eR,µL,µR and the Z boson is assumed to be on-shell293

(narrow width approximation). We compute this decay rate, neglecting QED corrections and294

light lepton masses, in presence of the contact terms εZ` only. The Dalitz double differential295

rate in s12 ≡ (p`+ + p`−)2 and s23 ≡ (p`−+ pZ)2 is296

dΓ

ds12ds23=

1(2π)3

132m2

h

4|εZ`|2v2

(s12 +

(s23−m2Z)(m

2h− s12− s23)

m2Z

), (38)

The allowed kinematical region is 0 < s12 < (mh−mZ)2 and, for any given value of s12, smin

23 <297

s23 < sMax23 with298

smin(Max)23 = (E∗2 +E∗Z)

2−(

E∗2 ±√(E∗Z)2−m2

Z

)2

, (39)

where E∗2 =√

s12/2 and E∗Z =m2

h−s12−m2Z

2√

s12. The decay rate defines the relation between the299

physical PO and the effective couplings PO as:300

Γ(h→ Z`+`−) = 0.0366|εZ`|2 MeV . (40)

As before, the physical PO is defined as the contribution to partial decay rate from the relevant301

PO, assuming others are set to zero. The only inputs in the numerical coefficient in Eq. (40)302

are the Z and Higgs masses, as well as the Higgs vev, therefore the relative uncertainty is at the303

10−3 level. Together with the physical PO already defined for h→ γγ and h→ Zγ , we have304

thus established a complete mapping between the effective couplings PO and the physical PO305

appearing in h→ 4` decays.306

4.5 Physical PO for h→ 2`2ν307

Physical PO for charged-current processes can be defined in a very similar way as the neutral-308

current ones. In particular, we use the h→ e+νeµ−νµ process for the physical PO correspond-309

ing to kWW , εWW , and εCPWW , and h→W+`ν` for the contact terms.310

Decay channel h→ e+νeµ−νµ311

Integrating the differential distributions analogous to Eq. (32) we obtain the expression of the312

partial decay rate in this channel, in the limit where only one PO is turned on:313

Γ(h→ eµ2ν)[κWW ] = 2.20×10−4|gWeL |2|gW µL |2 |κWW |2 MeV

14

Γ(h→ eµ2ν)[εWW ] = 4.27×10−5|gWeL |2|gW µL |2 |εWW |2 MeV

Γ(h→ eµ2ν)[εCPWW ] = 1.77×10−5|gWeL |2|gW µL |2 |εCP

WW |2 MeV(41)

As in the neutral channel, the physical PO are defined from these quantities by factorizing the314

W branching ratios:315

Γ(h→WLWL)≡Γ(h→ eµ2ν)[κWW ]

B(W → eνe)B(W → µνµ)= (0.841±0.016) |κWW |2 MeV

Γ(h→WTWT )≡Γ(h→ eµ2ν)[εWW ]

B(W → eνe)B(W → µνµ)= (0.1634±0.0030) |εWW |2 MeV

ΓCPV(h→WTWT )≡

Γ(h→ eµ2ν)[εCPWW ]

B(W → eνe)B(W → µνµ)= (0.0677±0.0012) |εCP

WW |2 MeV ,

(42)

where the uncertainty has been obtained from the experimental error on ΓW [18] that, as recently316

pointed out in [19], it has a non-neglible impact in the prediction of h→ 2`2ν branching ratios.317

The W branching ratios are given by318

B(W → `ν`) =Γ0

ΓW(gW`L)

2 ≈ 0.511(gW`L)2 , (43)

where Γ0 =mW24π

, ΓW is the total decay width.319

Decay channel h→W+`ν`320

Also in this case the physical PO corresponding to the charged-current contact terms are defined321

in complete analogy to the neutral-current case, starting from the 3-body decay h→W+`ν`. The322

partial decay width computed in the limit where only the contact term PO is switched on defines323

the relation between the physical PO and the effective couplings PO as:324

Γ(h→W+`ν`) = 0.143|εW`|2 MeV , (44)

where the relative uncertainty in the coefficient due to the experimental error on mW [18] is325

below ∼ 2×10−3.326

5 PO in Higgs electroweak production: generalities327

The PO decomposition of h→ 4 f amplitude discussed above can naturally be generalized to328

describe electroweak Higgs-production processes, namely Higgs-production via vector-boson329

fusion (VBF) and Higgs-production in association with a massive SM gauge boson (VH).330

15

PO Physical PO Relation to the eff. coupl.κ f , δ CP

f Γ(h→ f f ) = Γ(h→ f f )(SM)[(κ f )2 +(δ CP

f )2]

κγγ , δ CPγγ Γ(h→ γγ) = Γ(h→ γγ)(SM)[(κγγ)

2 +(δ CPγγ )2]

κZγ , δ CPZγ

Γ(h→ Zγ) = Γ(h→ Zγ)(SM)[(κZγ)2 +(δ CP

Zγ)2]

κZZ Γ(h→ ZLZL) = (0.209 MeV)×|κZZ|2εZZ Γ(h→ ZT ZT ) = (1.9×10−2 MeV)×|εZZ|2εCP

ZZ ΓCPV(h→ ZT ZT ) = (8.0×10−3 MeV)×|εCPZZ |2

εZ f Γ(h→ Z f f ) = (3.7×10−2 MeV)×N fc |εZ f |2

κWW Γ(h→WLWL) = (0.84 MeV)×|κWW |2εWW Γ(h→WTWT ) = (0.16 MeV)×|εWW |2εCP

WW ΓCPV(h→WTWT ) = (6.8×10−2 MeV)×|εCPWW |2

εW f Γ(h→W f f ′) = (0.14 MeV)×N fc |εW f |2

κg σ(pp→ h)gg−fusion = σ(pp→ h)SMgg−fusionκ2

g

κt σ(pp→ tth)Yukawa = σ(pp→ tth)SMYukawaκ2

t

κH Γtot(h) = ΓSMtot (h)κ

2H

Table 1: Summary of the effective coupling PO and the corresponding physical PO. The parameter N fc

is 1 for leptons and 3 for quarks. In the case of the charged-current contact term, f ′ is the SU(2)L partnerof the fermion f . See the main text for a discussion about the errors on the numerical coefficient in thetable and the reference values of {κX ,δX ,εX} within the SM.

The interest of such production processes is twofold. On the one hand, they are closely con-331

nected to the h→ 4`,2`2ν decay processes by crossing symmetry, and by the exchange of332

lepton currents into quark currents. As a result, some of the Higgs PO necessary to describe the333

h→ 4`,2`2ν decay kinematics appear also in the description of the VBF and VH cross sections334

(independently of the Higgs decay mode). This facts opens the possibility of combined analyses335

of production cross sections and differential decay distributions, with a significant reduction on336

the experimental error on the extraction of the PO. On the other hand, the production cross sec-337

tions allow to explore different kinematical regimes compared to the decays. By construction,338

the momentum transfer appearing in the Higgs decay amplitudes is limited by the Higgs mass,339

while such limitation is not present in the production amplitudes. The higher energies probed in340

the production processes provide an increased sensitivity to new physics effects. This fact also341

allows to test the momentum expansion that is intrinsic in the PO decomposition, as well as in342

any effective field theory approach to physics beyond the SM.343

Despite the similarities at the fundamental level, the phenomenological description of VBF344

and VH in terms of PO is significantly more challenging compared to that of Higgs decays.345

16

On the one hand, QCD corrections plays a non-negligible role in the production processes.346

Although technically challenging, this fact does not represent a conceptual problem for the PO347

approach: the leading QCD corrections factorize in VBF and VH, similarly to the factorization348

of QED corrections in h→ 4`. This implies that NLO QCD corrections can be incorporated in349

general terms with suitable modifications of the existing Montecarlo tools. On the other hand,350

the relation between the kinematical variables at the basis of the PO decomposition (i.e. the351

momentum transfer of the partonic currents, q2) and the kinematical variables accessible in352

pp collisions is not straightforward, especially in the VBF case. This problem finds a natural353

solution in the VBF case due to strong correlation between q2 and the pT of the VBF tagged354

jets, while in the VH case invariant mass of the VH system is correlated to the vector pT .355

5.1 Amplitude decomposition356

Neglecting the light fermion masses, the electroweak production processes VH and VBF or,357

more precisely, the electroweak partonic amplitudes f1 f2 → h+ f3 f4, can be completely de-358

scribed by the three-point correlation function of the Higgs boson and two (color-less) fermion359

currents360

〈0|T{

Jµ

f (x),Jν

f ′(y),h(0)}|0〉 , (45)

where all the states involved are on-shell. The same correlation function controls also the four-361

fermion Higgs decays discussed above. In the h→ 4`,2`2ν case both currents are leptonic362

and all fermions are in the final state. In case of VH associate production one of the currents363

describes the initial state quarks, while the other describes the decay products of the (nearly364

on-shell) vector boson. Finally, in VBF production the currents are not in the s-channel as in365

the previous cases, but in the t-channel. Strictly speaking, in VH and VBF the quark states are366

not on-shell; however, their off-shellness can be neglected compared to the electroweak scale367

characterizing the process (both within and beyond the SM).368

As in the h→ 4 f case, we can expand the correlation function in Eq. (45) around the known369

physical poles due to the propagation of intermediate SM electroweak gauge bosons. The PO370

are then defined by the residues on the poles and by the non-resonant terms in this expansion. By371

construction, terms corresponding to a double pole structure are independent from the nature372

of the fermion current involved. As a result, the corresponding PO are universal and can be373

extracted from any of the above mention processes, both in production and in decays [7].374

5.1.1 Vector boson fusion Higgs production375

Higgs production via vector boson fusion (VBF) receives contribution both from neutral- and376

charged-current channels. Also, depending on the specific partonic process, there could be two377

different ways to construct the two currents, and these two terms interfere with each other. For378

17

example, for uu→ uuh one has the interference between two neutral-current processes, while379

in ud → udh the interference is between neutral and charged currents. In this case it is clear380

that one should sum the two amplitudes with the proper symmetrization, as done in the case of381

h→ 4e.382

We now proceed describing how each of these amplitudes can be parametrized in terms of PO.383

Let us start with the neutral-current one. The amplitude for the on-shell process qi(p1)q j(p2)→384

qi(p3)q j(p4)h(k) can be parametrized by385

An.c(qi(p1)q j(p2)→ qi(p3)q j(p4)h(k)) = i2m2

Zv

qi(p3)γµqi(p1)q j(p4)γνq j(p2)Tµν

n.c. (q1,q2),

(46)where q1 = p1− p3, q2 = p2− p4 and T µν

n.c. (q1,q2) is the same tensor structure appearing in386

h→ 4 f decays. Indeed, proceeding as in Eq. (20), using Lorentz invariance we decompose this387

tensor structure in term of three from factors:388

T µνn.c. (q1,q2) =

[Fqiq j

L (q21,q

22)g

µν +Fqiq jT (q2

1,q22)


ν

m2Z

+Fqiq jCP (q2

1,q22)

εµνρσ q2ρq1σ

m2Z

]. (47)

Similarly, the charged-current contribution to the amplitude for the on-shell process ui(p1)d j(p2)→389

dk(p3)ul(p4)h(k) can be parametrized by390

Ac.c(ui(p1)d j(p2)→ dk(p3)ul(p4)h(k)) = i2m2

Wv

dk(p3)γµui(p1)ul(p4)γνd j(p2)Tµν

c.c. (q1,q2),

(48)where, again, T µν

c.c. (q1,q2) is the same tensor structure appearing in the charged-current h→ 4 f391

decays:392

T µνc.c. (q1,q2) =

[Gi jkl

L (q21,q

22)g

µν +Gi jklT (q2

1,q22)


ν

m2W

+Gi jklCP (q2

1,q22)

εµνρσ q2ρq1σ

m2W

](49)

The amplitudes for the processes with initial anti-quarks can easily be obtained from the above393

ones.394

The next step is to perform a momentum expansion of the form factors around the physical poles395

due to the propagation of SM electroweak gauge bosons (γ , Z and W±), and to define the PO396

(i.e. the set {κi,εi}) from the residues of such poles. We stop this expansion neglecting terms397

which can be generated only by local operators with dimension higher than six. A discussion398

about limitations and consistency checks of this procedure will be presented later on. The399

decomposition of the form factors closely follows the procedure already introduced for the400

18

decay amplitudes and will not be repeated here. We report explicitly only expression of the401

longitudinal form factors, where the contact terms not accessible in the leptonic decays appear:402

403

Fqiq jL (q2

1,q22) = κZZ

gqiZ gq j

Z

PZ(q21)PZ(q2

2)+

εZqi

m2Z

gq jZ

PZ(q22)

+εZq j

m2Z

gqiZ

PZ(q21)

+∆SML,n.c.(q

21,q

22) ,

Gi jklL (q2

1,q22) = κWW

gikW g jl

W

PW (q21)PW (q2

2)+

εWik

m2W

g jlW

PW (q22)

+εW jl

m2W

gikW

PW (q21)

+∆SML,c.c(q

21,q

22) .

(50)

Here PV (q2) = q2 −m2V + imV ΓV , while g f

Z and gikW are the PO characterizing the on-shell404

couplings of Z and W boson to a pair of fermions, see Eqs. (24) and (29). The functions405

∆SML,n.c.(c.c.)(q

21,q

22) denote non-local contributions generated at the one-loop level (and encoding406

multi-particle cuts) that cannot be re-absorbed in the definition of κi and εi. At the level of407

precision we are working, taking into account also the high-luminosity phase of the LHC, these408

contributions can be safely fixed to their SM values.409

As anticipated, the crossing symmetry between h→ 4 f and 2 f → h2 f amplitudes ensures that410

the PO are the same in production and decay (if the same fermions species are involved). The411

amplitudes are explored in different kinematical regimes in the two type of processes (in partic-412

ular the momentum-transfers, q21,2, are space-like in VBF and time-like in h→ 4 f ). However,413

this does not affect the definition of the PO. This implies that the fermion-independent PO asso-414

ciated to a double pole structure, such as κZZ and κWW in Eq. (50), are expected to be measured415

with higher accuracy in h→ 4` and h→ 2`2ν rather than in VBF. On the contrary, VBF is416

particularly useful to constrain the fermion-dependent contact terms εZqi and εWuid j , that appear417

only in the longitudinal form factors.418

5.1.2 Associated vector boson plus Higgs production419

The VH production process denote the production of a Higgs boson with a nearly on-shell420

massive vector boson (W or Z). For simplicity, in the following we will assume that the vector421

boson is on-shell and that the interference with the VBF amplitude can be neglected. However,422

we stress that the PO formalism clearly allow to describe both these effects (off-shell V and423

interference with VBF in case of V → qq decay) simply applying the general decomposition of424

neutral- and charged-current amplitudes as outlined above.425

Similarly to VBF, Lorentz invariance allows us to decompose the amplitudes for the on-shell426

processes qi(p1)qi(p2)→ h(p)Z(k) and ui(p1)d j(p2)→ h(p)W+(k) in three possible tensor427

19

structures: a longitudinal one, a transverse one, and a CP-odd one,428

A (qi(p1)qi(p2)→ h(p)Z(k)) = i2m2

Zv

qi(p2)γνqi(p1)εZ∗µ (k)×

×[

FqiZL (q2)gµν +FqiZ

T (q2)−(q · k)gµν +qµkν

m2Z

+FqiZCP (q2)

εµναβ qαkβ

m2Z

],

(51)

429

A (ui(p1)d j(p2)→ h(p)W+(k)) = i2m2

Wv

d j(p2)γνui(p1)εW∗µ (k)×

×[

Gqi jWL (q2)gµν +Gqi jW

T (q2)−(q · k)gµν +qµkν

m2W

+Gqi jWCP (q2)

εµναβ qαkβ

m2W

],

(52)

where q = p1 + p2 = k+ p. In the limit where we neglect the off-shellness of the final-state V ,430

the form factors depend only on q2. Already from this decomposition of the amplitude it is clear431

the importance of providing measurements of the differential cross-section as a function of q2,432

as well as differential measurements in terms of the angular variables that allow to disentangle433

the different tensor structures.434

Performing the momentum expansion of the form factors around the physical poles, and defin-435

ing the PO as in Higgs decays and VBF, we find436

FqiZL (q2) = κZZ

gZqiPZ(q2)

+εZqim2

ZGqi jW

L (q2) = κWW(g

uid jW )∗

PW (q2)+

ε∗Wuid j

m2W

FqiZT (q2) = εZZ

gZqiPZ(q2)

+ εZγeQqq2 Gqi jW

T (q2) = εWW(g

uid jW )∗

PW (q2)

FqiZCP (q2) = εCP

ZZgZqi

PZ(q2)− εCP

Zγ

eQqq2 Gqi jW

CP (q2) = εCPWW

(guid jW )∗

PW (q2)

(53)

where we have omitted the indication of the (tiny) non-local terms, fixed to their corresponding437

SM values. As in the VBF case, only the longitudinal form factors FL and GL contain PO not438

accessible in the leptonic decays, namely the quark contact terms εZqi and εWuid j .439

6 PO in Higgs electroweak production: phenomenology440

6.1 Vector Boson Fusion441

At the parton level (i.e. in the qq→ hqq hard scattering) the ideal observable relevant to extract442

the momentum dependence of the factor factors would be the double differential cross section443

d2σ/dq21dq2

2, where q1 = p1− p3 and q2 = p2− p4 are the momenta of the two fermion currents444

20

entering the process (here p1, p2 (p3, p4) are the momenta of the initial (final) state quarks).445

The q2i are also the key variables to test and control the momentum expansion at the basis of the446

PO decomposition.447

A first nontrivial task is to choose the proper pairing of the incoming and outgoing quarks, givenwe are experimentally blind to their flavor. For partonic processes receiving two interferingcontributions when the final-state quarks are exchanged, such as uu→ huu or ud → hud, thedefinition of q1,2 is even less transparent since a univocal pairing of the momenta can not beassigned, in general, even if one knew the flavor of all partons. This problem can be simplyovercome at a practical level by making use of the VBF kinematics, in particular the fact thatthe two jets are always very forward. This implies one can always pair the momenta of thejet going, for example, on the +z direction with the initial parton going in the same direction,and viceversa. The same argument can be used to argue that the interference between differentamplitudes (e.g. neutral current and charged current) is negligible in VBF. In order to checkthis, we have performed a leading order parton level simulation of the VBF Higgs production(pp→ h j j) using MADGRAPH5_AMC@NLO [20] (version 2.2.3) at 13 TeV c.m. energy. Wehave imposed the basic set of cuts,

pT,j1,2 > 30 GeV, |ηj1,2|< 4.5, and mj1j2 > 500 GeV. (54)

In Fig. 1, we show the distribution in the opening angle of the incoming and outgoing quark448

momenta for the two different pairings. The left plot is for the SM, while the right plot is449

for a specific NP benchmark point. Shown in blue is the pairing based on the leading color450

connection using the color flow variable while in red is the opposite pairing. The plot shows that451

the momenta of the color connected quarks tend to form a small opening angle and the overlap452

between the two curves, i.e. where the interference effects might be sizable, is negligible. This453

implies that in the experimental analysis the pairing should be done based on this variable.454

Importantly, the same conclusions can be drawn in the presence of new physics contributions to455

the contact terms.456

There is a potential caveat to the above argument: the color flow approximation ignores the457

interference terms that are higher order in 1/NC, where NC is number of colors. Let us con-458

sider a process with two interfering amplitudes with the final state quarks exchanged, for ex-459

ample in uu→ uuh. The differential cross section receives three contributions proportional460

to |F f f ′L (t13, t24)|2, |F f f ′

L (t13, t24)Ff f ′

L (t14, t23)| and |F f f ′L (t14, t23)|2, where ti j = (pi − p j)

2 =461

−2EiE j(1− cosθi j). For the validity of the momentum expansion it is important that the mo-462

mentum transfers (ti j) remain smaller than the hypothesized scale of new physics. On the other463

hand, imposing the VBF cuts, the interference terms turns out to depend on one small and one464

large momentum transfer. However, thanks to the pole structure of the form factors, these inter-465

ference effects turns out to give a very small contribution. Therefore, we can safely state that466

the momentum transfers marked with the leading color flow are reliable control variables of the467

momentum expansion validity.468

21

Correct color flow Wrong color flow

κZZ=1, κWW=1, ϵZuL=0, ϵZuR=0,ϵZdL=0, ϵZdR=0, ϵWuL=0

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.05

0.10

0.15

0.20

Δθ jet-parton

Higgs VBF @ 13 TeV LHC

Correct color flow Wrong color flow

κZZ=1, κWW=1, ϵZuL=0, ϵZuR=0,ϵZdL=0, ϵZdR=0, ϵWuL=0.05

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.05

0.10

0.15

Δθ jet-parton


Figure 1: Leading order parton level simulation of the Higgs VBF production at 13 TeV pp c.m. energy(from Ref. [7]). Show in blue is the distribution in the opening angle of the color connected incomingand outgoing quarks ](~p3,~p1), while in red is the distribution for the opposite pairing, ∠(~p3,~p2). Theleft plot is for the SM, while the plot on the right is for a specific NP benchmark.


0 100 200 300 400 500 600 7000

100

200

300

400

500

600

-q2 (GeV )

p T(jet)(GeV

)


1.×10-5

1.3×10-4

1.6×10-3

2.×10-2κZZ=1, κWW=1, ϵZuL=0, ϵZuR=0,ϵZdL=0, ϵZdR=0, ϵWuL=0.05

0 500 1000 15000

500

1000

1500

-q2 (GeV )

p T(jet)(GeV

)


1.×10-5

1.6×10-4

2.5×10-3

4.×10-2

Figure 2: Leading order parton level simulation of the Higgs VBF production at 13 TeV pp c.m. en-ergy [7]. Shown here is the density histogram in two variables; the outgoing quark pT and the momentumtransfer

√−q2 with the initial “color-connected” quark. The left plot is for the SM, while the plot on the

right is for a specific NP benchmark.

22

In some realistic experimental analyses, after reconstructing the momenta of the two VBF469

tagged jets and the Higgs boson, one can compute the relevant momentum transfers q1 and q2,470

adopting the pairing based on the opening angle. However, for some interesting Higgs decays471

modes, such as h→ 2`2ν , it is not possible to reconstruct the Higgs boson momentum. In this472

case, a good approximation of the momentum transfer is the jet pT . This can be understood by473

explicitly computing the momentum transfer q21,2 in the limit |pT | � E jet and for a Higgs pro-474

duced close to threshold. Let us consider the partonic momenta in c.o.m. frame for the process:475

p1 = (E,~0,E), p2 = (E,~0,−E), p3 = (E ′1,~pT 1,√

E ′21 − p2T 1) and p4 = (E ′2,~pT 2,

√E ′22 − p2

T 2).476

Conservation of energy for the whole process dictates 2E = E ′1+E ′2+Eh, where E2h is the Higgs477

energy, usually of order mh if the Higgs is not strongly boosted. In this case E−E ′i = ∆Ei� E478

since the process is symmetric for 1↔ 2. For each leg, energy and momentum conservation479

(along the z axis) give480

{qz

i = E−√

E ′2i − p2Ti

q0i = E−E ′i

→

q0i −qz

i =√

E ′2i − p2Ti−E ′i ≈−

p2Ti

2E ′i

q0i +qz

i ≈ 2∆Ei +p2

Ti2E ′i

. (55)

Putting together these two relations one gets481

q2i = (q0

i )2− p2

Ti− (qzi )

2 =−p2Ti +(q0

i −qzi )(q

0i +qz

i )≈−p2Ti−

p2Ti∆Ei

2E ′i+O(p4

Ti/E ′2) . (56)

We can thus conclude that, for a Higgs produced near threshold (∆Ei << E ′), q2 ≈−p2T .482

To illustrate the above conclusion, in Fig. 2 we show a density histogram in two variables: the483

outgoing quark pT and the momentum transfer√−q2 obtained from the correct color flow pair-484

ing (the left and the right plots are for the SM and for a specific NP benchmark, respectively).485

The plots indicate the strong correlation of the jet pT with the momentum transfer√−q2 as-486

sociated with the correct color pairing. We stress that this conclusion holds both within and487

beyond the SM.488

Given the strong q2↔ p2T correlation, we strongly encourage the experimental collaborations to489

report the unfolded measurement of the double differential distributions in the two VBF tagged490

jet pT ’s: F(pT j1, pT j2). This measurable distribution is closely related to the form factor en-491

tering the amplitude decomposition, FL(q21,q

22), and encode (in a model-independent way) the492

dynamical information about the high-energy behavior of the process. Moreover, the extraction493

of the PO in VBF must be done preserving the validity of the momentum expansion: the latter494

can be checked and enforced setting appropriate upper cuts on the pT distribution. As an ex-495

ample, in Fig. 3, we show the prediction in the SM (left plot) and in the specific NP benchmark496

(right plot) of the normalized pT -ordered double differential distribution.497

23


100 200 300 400 500

100

200

300

400

500

pT( j1) (GeV)

p T(j2)(GeV

)Higgs VBF @ 13 TeV LHC

5.×10-7

5.×10-6

5.×10-5

5.×10-4

5.×10-3

5.×10-2

5.×10-1κZZ=1, κWW=1, ϵZuL=0, ϵZuR=0,ϵZdL=0, ϵZdR=0, ϵWuL=0.05

100 200 300 400 500

100

200

300

400

500

pT( j1) (GeV)

p T(j2)(GeV

)


5.×10-7

5.×10-6

5.×10-5

5.×10-4

5.×10-3

5.×10-2

5.×10-1

Figure 3: Double differential distribution in the two VBF-tagged jet pT for VBF Higgs production at13 TeV LHC [7]. The distribution is normalized such that the total sum of events in all bins is 1. (Left)Prediction in the SM. (Right) Prediction for NP in εWuL = 0.05.

6.2 Associated vector boson plus Higgs production498

Higgs production in association with a W or Z boson are respectively the third and fourth Higgs499

production processes in the SM, by total cross section. Combined with VBF studies, they offer500

other important handles to disentangle the various Higgs PO. Due to the lower cross section,501

this process is mainly studied in the highest-rate Higgs decay channels, such as h→ bb and502

WW ∗. The drawback of these channels is the background, which is overwhelming in the bb503

case and of the same order as the signal in the WW ∗ channels. Nonetheless, kinematical cuts,504

such as the Higgs pT in the bb case, and the use of multivariate analysis allow the experiments505

to precisely extract the the signal rates from these measurements.506

An important improvement for future studies of these channels with the much higher luminos-507

ity which will be available, is to study differential distributions in some specific kinematical508

variables. In Section 5.1.2 we showed that the invariant mass of the V h system is the most509

important observable in this process, since the form factors directly depend on it. In those510

channels where the V h invariant mass can not be reconstructed due to the presence of neu-511

trinos, another observable which shows some correlation with the q2 is the pT of the vector512

boson, or equivalently of the Higgs, as can be seen in the Fig. 4. Even though this corre-513

lation is not as good as the one between the jet pT and the momentum transfer in the VBF514

channel, a measurement of the vector boson (or Higgs) pT spectrum, i.e. of some form factor515

FV h(pTV ) would still offer important information on the underlying structure of the form fac-516

tors appearing in Eq. (53), FqiZL (q2) or Gqi jW

L (q2). The invariant mass of the V h system is given517

by m2V h = q2 = (pV + ph)

2 = m2V +m2

h +2pV · ph. Going in the center of mass frame, we have518

24

kZZ=1, ϵZuL=0, ϵZuR=0,ϵZdL=0, ϵZdR=0

200 400 600 800 10001200140016000

200

400

600

800

mZh [GeV ]

p TZ[GeV

]Zh production @ 13 TeV LHC in the SM

1.× 10-5

1.3× 10-4

1.6× 10-3

2.× 10-2

kZZ=1, ϵZuL=0.1, ϵZuR=0,ϵZdL=0, ϵZdR=0

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

mZh [GeV ]

p TZ[GeV

]

Zh production @ 13 TeV LHC

1.× 10-5

1.× 10-4

1.× 10-3

1.× 10-2

Figure 4: The correlation between the Zh invariant mass and the pT of the Z boson in Zh associateproduction at the 13TeV LHC in the SM (left plot) and for a BSM point κZZ = 1, εZuL = 0.1 (rightplot) [7]. A very similar correlation is present in the Wh channel.

pV = (EV ,~pT , pz) and ph = (Eh,−~pT ,−pz), where Ei =√

m2i + p2

T + p2z (i =V,h). Computing519

m2V h explicitly:520

m2V h = m2

V +m2h +2p2

T +2p2z +2

√m2

V + p2T + p2

z

√m2

h + p2T + p2

z|pT |→∞−→ 4p2

T . (57)

For pz = 0 this equation gives the minimum q2 for a given pT , which can be seen as the left edge521

of the distributions in the Fig. 4. This is already a valuable information, for example the boosted522

Higgs regime used in some bb analysis implies a lower cut on the q2: a bin with pT > 300 GeV523

implies√

q2& 630 GeV, which could be a problem for the validity of the momentum expansion.524

In the Wh process, if the W decays leptonically its pT can not be reconstructed independently of525

the Higgs decay channel. One could think that the pT of the charged lepton from the W decay526

would be correlated with the Wh invariant mass, but we checked that there is no significant527

correlation between the two observables.528

6.3 Validity of the momentum expansion529

In order to control the momentum expansion at the basis of the PO composition, it is necessary530

to set an upper cut on appropriate kinematical variables. These are the pT of the leading VBF-531

tagged jet in VBF, and the V h invariant mass (or the pT of the massive gauge boson) in VH.532

The momentum expansion of the form factors in Eq. (50) makes sense only if the higher order533

terms in q21,2 are suppressed. Comparing the first two terms in this expansion, this leads to the534

consistency condition,535

εX f q2max . m2

Z g fX , (58)

25

where q2max is the largest momentum transfer in the process. A priori we don’t know which is536

the size of the εX f or, equivalently, the effective scale of new physics. However, a posteriori537

we can verify by means of Eq. (58) if we are allowed to truncate the momentum expansion538

to the first non-trivial terms. In VBF, setting a cut-off on pT we implicitly define a value of539

qmax. Extracting the εX f for p jT < (p j

T )max ≈ qmax we can check if Eq. (58) is satisfied. Ideally,540

the experimental collaborations should perform the extraction of the εX f for different values of541

(p jT )

max optimizing the range according to the results obtained. The issue is completely analog542

in VH, where the q2max is controlled by m2

V h.543

It must be stressed that if data indicate non-vanishing values for the εX f , and the condition (58) is544

falsified, this does not necessarily imply a break down of the momentum expansion.6 However,545

in such case it is important to check with data the size of additional terms in this expansion.546

From this point of view, it is very important to complement the PO approach with the differential547

measurements of the cross-section as function of p jT that could be achieved via the so-called548

template-cross-section method. In such distributions a possible break-down of the momentum549

expansion at the basis of the PO decomposition could indeed be seen (or excluded) directly by550

data.551

A further check to assess the validity of the momentum expansion is obtained comparing the552

fit performed including the full quadratic dependence of the distributions, as function on the553

PO, with the fit in which such distributions are linearized in δκX ≡ κX −κSMX and εX . The idea554

behind this procedure is that the quadratic corrections to physical observable in δκX and εX555

are formally of the same order as the interference of the first neglected term in Eq. (50) with556

the leading SM contribution. If the two fits yields significantly different results, the difference557

can be used as an estimate of the uncertainty due to the neglected higher-order terms in the558

momentum expansion. However, as discussed in detail in Ref. [7], such procedure naturally559

leads to a large overestimate of the uncertainty. This is because in the linearized fit only a560

few linear combinations of the PO enter the observables, and thus the number of independent561

constraints derived from data is effectively reduced. On general grounds, the fit obtained with562

the full quadratic dependence should be considered as the most reliable result, provided that the563

obtained PO satisfy the consistency condition in Eq. (58).564

6.4 Illustration of NLO QCD effects565

Higgs production via Higgsstrahlung and VBF are very stable with respect to NLO QCD566

corrections. At the inclusive level NLO scale uncertainties are as small as a few percent567

[22,23,24,25,8] and further reduced at NNLO [26,27,28,29,30,31]. Given an appropriate scale568

6 The break-down of the momentum expansion occurs only when we approach, with a given kinematical vari-able, a new pole in the amplitude due to the exchange of a NP particle. On the other hand, the dominance of thecontact terms leading to a violation of the condition (58) could also occur far from the NP poles in case of stronglyinteracting theories (see e.g. Ref. [21]).

26

SM: κZZ = 1

µ0 = HT/2

LONLO

10−5

10−4

10−3

10−2pp → ZH @ 13 TeV

dσ

/dp

T,Z

[pb/

GeV

]

0 100 200 300 400 5000.40.60.8

11.21.41.6

pT,Z [GeV]

dσ

NL

O/d

σL

OPO: κZZ = 1, ǫZuL = 0.0195

µ0 = HT/2

LONLO

10−3

10−2

pp → ZH @ 13 TeV

dσ

/dp

T,Z

[pb/

GeV

]

0 100 200 300 400 5000.40.60.8

11.21.41.6

pT,Z [GeV]

dσ

NL

O/d

σL

O

Figure 5: Differential pT,Z distributions in the process pp → ZH in the SM (left) and including anexample of NP within the PO framework with κZZ = 1,εZuL = 0.0195 (right). Shown are LO (red) andNLO (blue) predictions and corresponding (7-pt) scale variations employing a central scale µ0 = HT/2.

choice similar conclusion also hold at the differential level with residual NLO scale uncertain-569

ties at the 10% level.570

As already discussed above, similar to the factorization in QED, the dominant QCD corrections571

are universal and factorize from any new physics effects in EW Higgs production. Conse-572

quently, with respect to possible small deformations from the SM, parametrized via effective573

form factor contributions in the PO framework, we expect a very limited sensitivity to QCD574

effects assuming a similar stabilization of higher order corrections as observed for the SM. In575

order to verify this assumption (and to make a corresponding tool available), the PO frame-576

work has been implemented in the Sherpa + OpenLoops [32,33,34,35] framework, where the577

implementation within Sherpa is based on a model independent UFO interface [36]. As in illus-578

tration, in Fig. 5 we compare the NLO QCD corrections to the pT distribution of the Z-boson579

in pp→ ZH between the SM and a NP point with κZZ = 1,εZuL = 0.0195. Despite the very580

different shape of the two distributions, higher order QCD corrections are very similar and do581

only show a very mild shape dependence. Detailed studies for VBF and VH including parton582

shower matching are under way.583

After the inclusion of NLO QCD corrections the dominant theoretical uncertainties to Higgs584

observables in VBF and VH are of EW type and dominated by large EW Sudakov logarithms585

at large energies [37,38,39]. The dominant NLO EW effects are factorizable corrections which586

can be reabsorbed into a future redefinition of the PO.587

27

7 Parameter counting and symmetry limits588

We are now ready to identify the number of independent pseudo-observables necessary to de-589

scribe various sets of Higgs decay amplitudes and productions cross sections. We list them590

below separating four set of observables:591

i) the Yukawa decay modes (h→ f f );592

ii) the EW decays (h→ γγ, f f γ,4 f );593

iii) the EW production production cross sections (VBF and VH);594

iv) the non-EW production cross sections (gluon fusion and ttH) and the total Higgs decay595

width.596

We list the PO needed for a completely general analysis, and the reduction of the number of597

independent PO obtained under well-defined symmetry hypotheses, such as CP invariance or598

flavor universality. The latter can be more efficiently tested considering specific sub-sets of599

observables.600

7.1 Yukawa modes601

As discussed in Sec. 2.1 the h→ f f amplitudes are characterized by two independent PO (κ f602

and δ CPf ) for each fermion species. Considering only the decay channels relevant for LHC, the603

full set of 8 parameters is:604

κb,κc,κτ ,κµ ,δCPb ,δ CP

c ,δ CPτ ,δ CP

µ . (59)

Assuming CP conservation (that implies δ CPf = 0 for each f ) the number of PO is reduced to 4.605

This is also the number of independent PO effectively measurable if the spin polarization of the606

final-state fermions is not accessible. The corresponding physical PO are the Γ(h→ f f ) partial607

widths (see Table 1).608

7.2 Higgs EW decays609

The category of EW decays includes a long list of channels; however, not all of them are ac-610

cessible at the LHC. The clean neutral current processes h→ e+e−µ+µ−, h→ e+e−e+e− and611

h→ µ+µ−µ+µ−, together with the photon channels h→ γγ and h→ `+`−γ , can be described612

in terms of 11 real parameters:613

κZZ,κZγ ,κγγ ,εZZ,εCPZZ ,δ

CPZγ ,δ

CPγγ ,εZeL ,εZeR,εZµL ,εZµR (60)

28

(of which only the subset {κγγ ,κZγ ,δCPγγ ,δCP

Zγ} is necessary to describe h→ γγ and h→ `+`−γ).614

The charged-current process h→ νeeµνµ needs 7 further independent real parameters to be615

completely specified:616

κWW ,εWW ,εCPWW (real) + εWeL ,εW µL (complex) . (61)

Finally, the mixed processes h→ e+e−νν and h→ µ+µ−νν can be described by a subset of617

the coefficients already introduced plus 2 further real contact interactions coefficients:618

εZνe ,εZνµ. (62)

This brings the total number of (real) parameters to 20 for all the (EW) decays involving muons,619

electrons, and photons.620

The extension to discuss h→ 4 f or h→ f f γ decays with one or two pairs of tau leptons is621

straightforward: it requires the introduction of the corresponding set of contact terms (εZτL ,εZτR ,622

εWτL ,εZντ). Similarly, quark contact terms need to be introduced if one or two lepton pairs are623

replaced by a quark pair.624

A first simple restriction in the number of parameters is obtained by assuming flavor universal-625

ity. This hypothesis imply that the contact terms are the same for all flavors. In particular, for626

muon and electron modes, this implies627

εZeL = εZµL , εZeR = εZµR , εZνe = εZνµ, εWeL = εW µL . (63)

Technically, this correspond to assume an underlying U(N`)2 flavor symmetry, for the N` gen-628

erations of leptons considered (namely the maximal flavor symmetry compatible with the SM629

gauge group).630

Since the εW`L parameters are complex in general, the relations (63) allow to reduce the total631

number of parameters to 15. This assumption can be tested directly from data by comparing the632

extraction of the contact terms from h→ 2e2µ , h→ 4e and h→ 4µ modes.633

The assumption that CP is a good approximate symmetry of the BSM sector and that the Higgs634

is a CP-even state, allows us to set to zero six independent (real) coefficients:635

εCPZZ = δ

CPZγ = δ

CPγγ = ε

CPWW = ImεWeL = ImεW µL = 0 . (64)

Assuming, at the same time, flavor universality, the number of free real parameters reduces to636

10.637

The various cases are prorated in the upper panel of Table 2: in the second column we list the638

10 PO needed assuming both CP invariance and flavor universality, while in the third and fourth639

column we list the additional PO needed if these hypotheses are relaxed (for the clean modes640

involving only muons and electrons). The corresponding physical PO are the partial widths641

reported in Table 1.642

29

Higgs (EW) decay amplitudes

Amplitudes Flavor + CP Flavor Non Univ. CPVh→ γγ,2eγ,2µγ κZZ,κZγ ,κγγ ,εZZ εZµL ,εZµR εCP

ZZ ,δCPZγ

,δCPγγ4e,4µ,2e2µ εZeL ,εZeR

h→ 2e2ν ,2µ2ν ,eνµνκWW ,εWW εZνµ

, Re(εW µL) εCPWW , Im(εWeL)

εZνe , Re(εWeL) Im(εW µL)

Higgs (EW) production amplitudes

Amplitudes Flavor + CP Flavor Non Univ. CPV

VBF neutral curr.[

κZZ,κZγ ,κγγ ,εZZ]

εZcL ,εZcR

[εCP

ZZ ,δCPZγ

,δCPγγ

]

and Zh εZuL ,εZuR,εZdL ,εZdR εZsL ,εZsR

VBF charged curr. [ κWW ,εWW ] Re(εWcL) [εCPWW ], Im(εWuL)

and Wh Re(εWuL) Im(εWcL)

EW production and decay modes, with custodial symmetry

Amplitudes Flavor + CP Flavor Non Univ. CPV

production & decays κZZ,κZγ ,κγγ ,εZZ εCPZZ ,δ

CPZγ

,δCPγγ

VBF and VH only εZuL ,εZuR ,εZdL ,εZdR

εZcL ,εZcR

εZsL ,εZsR

decays only εZeL ,εZeR , Re(εWeL) εZµL ,εZµR

Table 2: Summary of the effective couplings PO appearing in EW Higgs decays and in the VBF andVH production cross-sections (see main text). The terms between square brakes in the middle table arethe PO present both in production and decays. The last table denote the PO needed to describe bothproduction and decays under the assumption of custodial symmetry.

30

7.3 EW production processes643

The fermion-independent PO present in Higgs decays appear also in EW production processes.644

The additional PO appearing only in production (assuming Higgs decays to quark are not de-645

tected) are the contact terms for the light quarks. In a four-flavor scheme, in absence of any646

symmetry assumption, the number of independent parameters for the neutral currents contact647

terms is 16 (εZqi j , where q = uL,uR,dL,dR, and i, j = 1,2): 8 real parameters for flavor diagonal648

terms and 4 complex flavour-violating parameters. Similarly, there are 16 independent parame-649

ters in charged currents, namely the 8 complex terms εWuiLd j

Land εWui

Rd jR. However, we can safely650

reduce the number of independent PO under neglecting the terms that violates the U(1) f flavour651

symmetry acting on each of the light fermion species, uR, dR, sR, cR, q(d)L , and q(s)L , where q(d,s)L652

denotes the two quark doublets in the basis where down quarks are diagonal. This symmetry is653

an exact symmetry of the SM in the limit where we neglect light quark masses. Enforcing it at654

the PO level is equivalent to neglecting terms that do not interfere with SM amplitudes in the655

limit of vanishing light quark masses. Under this (rather conservative) assumption, the number656

of independent neutral currents contact terms reduces to 8 real parameters,657

εZuR, εZcR , εZdR, εZsR, εZdL , εZsL , εZuL , εZcL , (65)

and only 2 complex parameters appear in the charged-current case:658

εWuiLd j

L≡Vi jεWu j

L, εWui

Rd jR= 0 . (66)

Similarly to the decays, a further interesting reduction of the number of parameters is obtained659

assuming flavor universality or, more precisely, under the assumption of an U(2)3 symmetry660

acting on the first two generations of quarks. The latter is the the maximal flavor symmetry for661

the light quarks compatible with the SM gauge group. In this case the independent parameters662

in this case reduces to six:663

εZuL , εZuR, εZdL , εZdR , εWuL , (67)

where εWuL is complex, or five if we further neglect CP-violating contributions (in such case664

εWuL is real). This case is listed in the second column of Table 2 (middle panel), where the665

terms between brackets denote the PO appearing also in decays.666

Custodial symmetry and the combination of EW production and decay modes667

Assuming flavor universality and CP conservation, the number of independent PO necessary to668

describe all EW decays and production cross sections is 15. These are the terms listed in the669

second column of the first two panels of Table 2.670

A further reduction of the number of independent PO is obtained under the hypothesis of cus-671

todial symmetry, that relates charged and neutral current modes. The complete list of custodial672

31

symmetry relations can be found in Refs. [2,7]. Here we only mention the one between κWW673

and κZZ , noting that the presence of contact terms modify it with respect to the one known in674

the context of the kappa-framework:675

κWW −κZZ =−2g

(√2εW`L +2

mW

mZεZ`L

), (68)

where ` = e,µ . After imposing flavor and CP conservation, custodial symmetry allow a re-676

duction of the number of independent PO from 15 down to 11, as shown in the lower panel of677

Table 2.678

7.4 Additional PO679

The remaining PO needed for a complete description of Higgs physics at the LHC are those680

related to the non-EW production processes (gluon fusion and ttH) and to the total Higgs decay681

width (i.e. NP effects in invisible or undetected decay modes).682

A detailed formalism, similar to the one developed for EW production and decay process, has683

not been developed yet for gluon fusion and ttH production process. However, it should be684

stressed that the latter are on a very different footing compared to EW processes since they685

involve a significantly smaller number of observables. Moreover, a smaller degrees of mod-686

elization is required in order to analyze the corresponding data in generic NP frameworks. As687

a result, the combination of PO for the total cross sections, and template-cross-section analyses688

of the kinematical distributions, provide an efficient way to report data in a sufficiently general689

and unbiased way.690

More precisely, for the time being we suggest to introduce the following two PO691

κ2g =

σ(pp→ h)σSM(pp→ h)

∣∣∣∣gg−fusion

, κ2t =

σ(pp→ tth)σSM(pp→ tth)

∣∣∣∣Yukawa

, (69)

in close analogy to what it is presently done within the κ formalism. As far as the gluon692

fusion is concerned, it is well known that the Higgs pT distribution carries additional dynamical693

information about the underlying process. However, such distribution can be efficiently reported694

via the template-cross-section method. Moreover, the steep fall of the pT spectrum (that is a695

general consequence of the infrared structure of QCD) implies that the determination of κgg is696

practically unaffected by possible NP effects in this distribution.697

Finally, as far as the Higgs width is concerned, we need to introduce a single effective physical698

PO to account for all the invisible or undetected Higgs decay modes. This additional partial699

width must be added to the various visible partial widths in order to determine the total Higgs700

width. Alternatively, it is possible to define an effective coupling PO as the ratio701

κ2H =

Γtot(h)ΓSM

tot (h). (70)

32

8 PO meet SMEFT702

One of the main goals of the LHC is to perform high-precision studies of possible deviations703

from the SM. Ideally, this would require the following four steps: i) for each process write704

down some (QFT-compatible) amplitude allowing for SM-deviations, both for the main signal705

analyzed (e.g. a given Higgs cross-section, close to the resonance) as well as for the background706

(non-resonant signal); ii) compute fiducial observables; iii) fit the signal (SM+NP) via an appro-707

priate set of conventionally-defined PO, without subtracting the SM background; iv) using the708

PO thus obtained to derive information on the Wilson coefficients of an appropriate Lagrangian709

allowing for deviations from the SM.710

In the previous sections we have discussed a convenient choice for the definition of the PO711

relevant to resonant Higgs physics (steps i and iii). In this Section we outline how to address712

the last step in the case of the so-called SM Effective Field Theory (SMEFT), i.e. how to extract713

the Wilson coefficients of the SMEFT from the measured PO.714

Before starting, it is worth stressing that PO are not Wilson coefficients, despite one can derive715

a linear relation between the two sets of parameters when working at the lowest-order (LO) in a716

given Lagrangian framework. The distinction between PO and Wilson coefficients is quite clear717

from their different “status” in QFT: the PO provide a general parameterization a given set of718

on-shell scattering amplitudes and are not Lagrangian parameters. Once a PO is observed to de-719

viate from its SM value we cannot, without further theoretical assumptions, predict deviations720

in other amplitudes. The latter can be obtained only using a given Lagrangian and after ex-721

tracting from data (or better from PO) the corresponding set of Wilson coefficients. Conversely,722

Wilson coefficients are scale and scheme dependent parameters that require specific theoretical723

prescriptions to be extracted from physical observables. This is why the PO can be measured724

including only the SM THU7, while the extraction of SMEFT Wilson coefficients require also725

an estimate of the corresponding SMEFT THU8.726

There is a line of thought where the Wilson coefficients in any LO EFT approach to physics727

beyond the SM are not actual Wilson coefficients, but parameters encoding deformation possi-728

bilities. According to this line of though, PO and and Wilson coefficients are somehow the same729

object. But this way of proceeding has a limited applicability, especially if a deviations from730

the SM is found. Proceeding along this line one could write an ad-hoc effective Lagrangian,731

do some calculations at LO (deviation parameters at tree-level), interpret the data, and limit the732

considerations to answer the question “are there deviations from the SM?". If we want to go733

a step further, viz. answering the question “What do the deviations from the SM mean?" then734

it is important to separate the role of PO and Wilson coefficients. Indeed after extracting the735

7By THU we mean theoretical uncertainty which has two components, parametric (PU) and missing higherorder uncertainties (MHOU)

8Although SMEFT converges to SM in the limit of zero Wilson coefficients, SMEFT and SM are differenttheories in the UV.

33

PO, two possibilities appear: i) top-down, namely employ a specific UV model, compute the736

PO and try to figure out if it matches or not with the observed deviation; in such case there will737

be an uncertainty in projecting down the UV model to the parameters and in the choice of the738

input parameter set (IPS); ii) bottom-up, namely do a SMEFT analysis to extract from the PO739

conclusions on the actual Wilson coefficients; here there will be an uncertainty from the order at740

which the calculation is done, as well as a parametric uncertainty. In the following we illustrate741

the basic strategy of for the latter (bottom-up) approach.742

8.1 SMEFT summary743

To establish our notations we observe that in the SMEFT a (lepton number preserving) ampli-744

tude can be written as745

A =∞

∑n=N

gn A(4)

n +∞

∑n=N6

n

∑l=1

∞

∑k=1

gn[

1(√

2GF Λ2)k

]l

A(4+2k)

nl k , (71)

where g is a SM coupling. GF is the Fermi coupling constant and Λ is the cut off scale. l746

is an index that indicates the number of SMEFT operator insertions leading to the amplitude,747

and k indicates the inverse mass dimension of the Lagrangian terms inserted. N is a label for748

each individual process, that indicates the order of the coupling dependence for the leading non749

vanishing term in the SM (e.g. N = 1 for h→VV etc. but N = 3 for h→ γγ). N6 = N for tree750

initiated processes in the SM. For processes that first occur at loop level in the SM, N6 = N−2751

when operators in the SMEFT can mediate such decays directly thought a contact operator, for752

example, through a dim = 6 operator for h→ γγ . For instance, the hγγ (tree) vertex is generated753

by OHB = Φ† ΦBµν Bµν , by O8HW = Φ† Bµν Bµρ Dρ Dν Φ etc. Therefore, SMEFT is a double754

expansion: in g and g6 = v2F/Λ2 for pole observables and in g,g6 E2/v2

F for off-shell ones;755

furthermore, the combination of parameters gg6 A(6)

1,1,1 defines the LO SMEFT expression for756

the process while g3 g6 A(6)

3,1,1 defines the NLO SMEFT amplitude in the perturbative expansion.757

To summarize, LO SMEFT refers to dim = 6 operators in tree diagrams, sometimes called758

“contact terms” while NLO SMEFT refers to one loop diagrams with a single insertion of759

dim = 6 operators. One can make additional assumptions by introducing classification schemes760

in SMEFT. One example of a classification scheme is the Artz-Einhorn-Wudka “potentially-761

tree-generated” (PTG) scenario [40,41]. In this scheme, it is argued that classes of Wilson762

coefficients for operators of dim = 6 can be argued to be tree level, or loop level (suppressed by763

g2/16π2)9. In these cases the expansion in Eq.(71) is reorganized in terms of TG (we assume a764

BSM model where PTG is actually TG) and LG insertions, i.e. LG contact terms and one loop765

TG insertions, one loop LG insertions and two loop SM etc. It is clear that LG contact terms766

alone do not suffice.767

9This classification scheme corresponds only to a subset of weakly coupled and renormalizable UV physicscases.

34

Strictly speaking we are considering here the virtual part of SMEFT, under the assumptions768

that LHC PO are defined à la LEP, i.e. when QED and QCD corrections are deconvoluted.769

Otherwise, the real (emission) part of SMEFT should be included and it can be shown that the770

infrared/collinear part of the one-loop virtual corrections and of the real ones respect factoriza-771

tion: the total = virtual + real is IR/collinear finite at O(g4 g6).772

It is worth nothing that SMEFT has limitations, obviously the scale should be such that E� Λ.773

Understanding SM deviations in tails of distributions requires using SMEFT, but only up to the774

point where it stops to be valid, or using the kappa–BSM-parameters connection, i.e. replace775

SMEFT with BSM models, optimally matching to SMEFT at lower scales.776

In any process, the residues of the poles corresponding to unstable particles (starting from max-777

imal degree) are numbers while the non-resonant part is a multivariate function that requires778

some basis, i.e. a less model independent, underlying, theory of SM deviations. That is to say,779

residue of the poles can be PO by themselves, expressing them in terms of Wilson coefficients780

is an operation the can be eventually postponed. The very end of the chain, the non-resonant781

part, may require model dependent BSM interpretation. Numerically speaking, it depends on782

the impact of the non-resonant part which is small in gluon-fusion but not in Vector Boson Scat-783

tering. Therefore, the focus for data reporting should always be on real observables, fiducial784

cross sections and pseudo-observables.785

To explain SMEFT in a nutshell consider a process described by some SM amplitude786

ASM = ∑i=1,n

A(i)

SM , (72)

where i labels gauge-invariant sub-amplitudes. In general, the same process is given by a contact787

term or a collection of contact terms of dim = 6; for instance, direct coupling of h to VV (V =788

γ,Z,W ). In order to construct the theory one has to select a set of higher-dimensional operators789

and to start the complete procedure of renormalization. Of course, different sets of operators790

can be interchangeable as long as they are closed under renormalization. It is a matter of fact791

that renormalization is best performed when using the so-called Warsaw basis, see Ref. [42].792

Moving from SM to SMEFT we obtain793

A LOSMEFT = ∑

i=1,nA

(i)SM + ig6 κc , A NLO

SMEFT = ∑i=1,n

κi A(i)

SM + ig6 κc +g6 ∑i=1,N

ai A(i)

nfc , (73)

where g−16

=√

2GF Λ2. The last term in Eq.(73) collects all loop contributions that do not fac-794

torize and the coefficients ai are Wilson coefficients. The κi are linear combinations of Wilson795

coefficients.10 We conclude that Eq.(73) gives a consistent and convenient generalization of796

the original κ-framework at the price of introducing additional, non-factorizable, terms in the797

amplitude.798

10We denote these combinations of Wilson coefficients κi, rather than κi, in order to distinguish them from thePO defined in the previous sections.

35

There are several reasons why loops should not be neglected in SMEFT, one is as follows:799

consider the “off-shell”gg→ h fusion [43,44,45,46], the “contact” term is real while the SM800

amplitude crosses normal thresholds, e.g. at s = 4m2t , where s is the Higgs virtuality. Therefore,801

in the interference one misses the large effect induced by the SM imaginary part while this effect802

(of the order of 5% above the tt -threshold) is properly taken into account by the inclusion of803

SMEFT loops, also developing an imaginary part after crossing the same normal threshold. To804

summarize, the LO part (contact term) alone shows large deviations from the SM around the tt -805

threshold while the one-loop part reproduces, with the due rescaling, the SM lineshape in a case806

where there is no reason to neglect the insertion of PTG operators in loops. Only the formulation807

including loops gives an accurate result, with deviations of O(5%) wrt tree (uncritical as long808

as experimental precision is� 10% but experiments are getting close).809

8.2 Theoretical uncertainty810

A theoretical uncertainty arises when the value of the Wilson coefficients in the PO scenario811

is inferred. A fit defined in a perturbative expansion must always include an estimate of the812

missing higher order terms (MHOU) [1]. Various ways exist to estimate this uncertainty, at any813

order in perturbation theory. One can compute the same observable with different “options”,814

e.g. linearization or quadratization of the squared matrix element, resummation or expansion815

of the (gauge invariant) fermion part in the wave function factor for the external legs (does816

not apply at tree level), variation of the renormalization scale, GF renormalization scheme or817

α -scheme, etc.818

A conservative estimate of the associated theoretical uncertainty is obtained by taking the enve-819

lope over all “options”; the interpretation of the envelope is a log-normal distribution (this is the820

solution preferred in the experimental community) or a flat Bayesian prior [47,48] (a solution821

preferred in a large part of the theoretical community). It is clear that MHOU for the SM should822

always be included.823

The notion of MHOU has a long history but it is worth noting that there is no statistical foun-824

dation and that it cannot be derived from a set of consistent (incomplete) principles. Ideally,825

calculations should be repeated using a well defined (and definable) set of options, results from826

different calculations should be compared and their MHOU assumptions subjectable to falsifi-827

cation. Therefore, no estimate of the theoretical errors is general enough and it is clear that there828

are several ways to approach the problem with conceptual differences between the bottom-up829

and the top-down scenarios.830

8.3 Examples831

In this Section we provide a number of examples connecting PO to Wilson coefficients; results832

are based on the work of Refs. [4,49], and of Refs. [50,51,52,53]. For simplicity we will confine833

36

the presentation to CP-even couplings.834

• h→ bb At tree level this amplitude is given by835

AHbb =−12

gmb

mW

[ASM

Hbb +g6

(aφ W−

14

aφ D +aφ�−ab φ

)], (74)

giving the following connection with Eq. (1) (note that all deviations are real):836

ybS =− i√

2g

mb

mW

[ASM

Hbb +g6

(aφ W−

14

aφ D +aφ�−ab φ

)]. (75)

• h→ γγ The amplitude for the process h(P)→ γµ(p1)γν(p2) can be written as837

Aµν

HAA = iTHAA Tµν , m2h Tµν = pµ

2 pν1 − p1 · p2 gµν . (76)

The S -matrix element follows from Eq.(76) when we multiply the amplitude by the photon838

polarizations eµ(p1)eν(p2); in writing Eq.(76) we have used p · e(p) = 0. A convenient way839

for writing the amplitude is the following: after renormalization we neglect all fermion masses840

but mt,mb and write841

THAA =g3

F s2W

8π2 ∑I=W,t,b

ρHAAI T I

HAA ; LO +gF g6

m2h

mWaAA +

g3F g6

π2 T nfcHAA , (77)

where g2F = 4

√2GF m2

W and cW = mW/mZ . Note that, at this point we have selected the842

{GF , mZ , mW} IPS, alternatively one could use the {α , GF , mZ} IPS where843

g2A =

4π α

s2θ

s2θ=

12

[1−√

1−4π α√

2GF m2Z

], (78)

with a numerical difference that enters the MHOU. Referring to Eq.(73) we have844

κHAAI =

g3F s2

W

8π2 ρHAAI , κ

HAAc = gF

m2h

mWaAA . (79)

In writing deviations in terms of Wilson coefficients we introduce the following combinations:845

aZZ = s2W

aφ B + c2W

aφ W− sW cW aφ WB , aAA = c2W

aφ B + s2W

aφ W + sW cW aφ WB ,

aAZ = 2cW sW (aφ W−aφ B)+(

2c2W−1)

aφ WB . (80)

The process dependent ρ -factors are given by846

ρprocI = 1+g6 ∆ρ

procI , (81)

37

and there are additional, non-factorizable, contributions. For h → γγ the ∆ρ factors are as847

follows:848

∆ρHAAt =

316

m2h

sW m2W

at WB +(2− s2W)

cW

sW

aAZ +(6− s2W)aAA

− 12

[aφ D +2s2

W(c2

WaZZ−at φ−2aφ�)

] 1s2

W

,

∆ρHAAb = −3

8m2

h

sW m2W

ab WB +(2− s2W)

cW

sW

aAZ +(6− s2W)aAA

− 12

[aφ D +2s2

W(c2

WaZZ +ab φ−2aφ�)

] 1s2

W

,

∆ρHAAW = (2+ s2

W)

cW

sW

aAZ +(6+ s2W)aAA−

12

[aφ D−2s2

W(2aφ�+ c2

WaZZ)

] 1s2

W

. (82)

In the PTG scenario we only keep at φ,ab φ,aφ D and aφ� in Eq.(82). The advantage of Eq.(77) is849

to establish a link between the EFT and the κ-framework, which has a validity restricted to LO.850

As a matter of fact Eq.(77) tells us that appropriate κ -factors can be introduced also at the loop851

level; they are combinations of Wilson coefficients but we have to extend the scheme with the852

inclusion of process dependent, non-factorizable, contributions.853

We also derive the following result for the non-factorizable part of the amplitude:854

T nfcHAA = mW ∑

a∈{A}T nfc

HAA(a)a , {A}= {at WB,ab WB,aAA,aAZ,aZZ} . (83)

In the PTG scenario all non-factorizable amplitudes for h→ γγ vanish. Comparing with Eq. (6)855

we obtain856

εγγ =−12

vF

m2hTHAA , T LO

HAA = T SMHAA +gF g6

m2h

mWaAA . (84)

• h→ 4 f857

Few additional definitions are needed: by on-shell S-matrix for an arbitrary process (involving858

external unstable particles) we mean the corresponding (amputated) Green’s function supplied859

with LSZ factors and sources, computed at the (complex) poles of the external lines [54,55]. For860

processes that involve stable particles this can be straightforwardly transformed into a physical861

PO.862

The connection of the hVV,V = Z,W (on-shell) S-matrix with the off shell vertex h→ VV863

and the full process pp→ 4ψ is more complicated and is discussed in some detail in Sect. 3 of864

Ref. [6]. The “on-shell” S-matrix for hVV , being built with the the residue of the h−V−V poles865

in pp→ 4ψ is gauge invariant by construction (it can be proved by using Nielsen identities) and866

represents one of the building blocks for the full process: in other words, it is a PO. Technically867

38

speaking the “on-shell” limit for external legs should be understood “to the complex poles”868

(for a modification of the LSZ reduction formulas for unstable particles see Ref. [56]) but, as869

well known, at one loop we can use on-shell masses (for unstable particles) without breaking870

the gauge parameter independence of the result. In order to understand the connection with871

Eqs. (21)–(23), defining neutral current PO we consider the process872

h(P)→ e−(p1)+ e+(p2)+ f (p3)+ f (p4) (85)

where f 6= e,νe, and introduce the following invariants: sH = P2, s1 = q21 = (p1 + p2)

2 and s2 =873

q22 = (p3 + p4)

2, while si, i = 3, . . . ,5 denote the remaining invariants describing the process.874

We also introduce sZ, the Z complex pole. Propagators are875

∆A(i) =1si, ∆Z(i) = P−1

Z (si) =1

si− sZ. (86)

The total amplitude for process Eq.(85) is given by the sum of different contributions, doubly Z876

resonant etc.877

A(h→ e−e+ f f

)= AZZ (sH , s1 , s2) ∆Z(s1)∆Z(s2)

+ AAA (sH , s1 , s2) ∆A(s1)∆A(s2)+AAZ (sH , s1 , s2) ∆A(s1)∆Z(s2)

+ AAZ (sH , s2 , s1) ∆Z(s1)∆A(s2)+AZ (sH , s1 , s2) ∆Z(s1)

+ AZ (sH , s2 , s1) ∆Z(s2)+AA (sH , s1 , s2) ∆A(s1)

+ AA (sH , s2 , s1) ∆A(s2)+ANR . (87)

To describe in details the various terms in Eq.(87) we introduce fermion currents defined by878

Jµ

Z f (p ; q,k) = u f (q)γµ

[V f (p2)+A f (p2)γ

5]

v f (k)

= V +f (p2) u f L(q)γ

µ v f L(k)+V −f (p2) u f R(q)γµ v f R(k) ,

Jµ

A f (p ; q,k) = Q f (p2) u f (q)γµ v f (k) , (88)

where p = q+ k. At tree level we have879

V +f =

gcW

(I(3)f −Q f s2

W

), V −f =−gQ f

s2W

cW

, Q f = gQ f sW . (89)

The amplitude for h(P)→ γ(q1)+ γ(q2) is880

AAA (sH , s1 , s2) = THAA (sH , s1 , s2) Tµν (q1 , q2) Jµ

Ae (q1 ; p1, p2) JνA f (q2 ; p3, p4) , (90)

with q21 = s1 and q2

2 = s2 . Similarly, the amplitude for h(P)→ Z(q1)+Z(q2) is881

AZZ (sH , s1 , s2) =[PHZZ (sH , s1 , s2) q2µ q1ν −DHZZ (sH , s1 , s2) gµν

]

39

s2s2s2

s1s1s1

Z , AZ , AZ , A

Z , AZ , AZ , A

p1p1p1

p2p2p2

p3p3p3

p4p4p4

Figure 6: Doubly-resonant (ZZ, AA or AZ) part of the amplitude for the process of Eq.(85).

Z , AZ , AZ , A

s1(s2)s1(s2)s1(s2)

p1(p3)p1(p3)p1(p3)

p2(p4)p2(p4)p2(p4)

p3(p1)p3(p1)p3(p1)

p4(p2)p4(p2)p4(p2)

Figure 7: Singly-resonant (Z or A) part of the amplitude for the process of Eq.(85).

× Jµ

Z e (q1 ; p1, p2) JνZ f (q2 ; p3, p4) . (91)

The ZZ, AA or AZ, doubly-resonant parts of the amplitude are shown in Fig. 6 while the singly,882

Z or A, parts are shown in Fig. 7. For the singly-resonant amplitudes we write883

AZ (sH , s1 , s2) = u f (p3)FHZ µ (sH , s1 , s2)v f (p4)Jµ

Z,e (q1 ; p1, p2) , (92)

where the form factor F is again decomposed as follows:884

u f (p3)FHZ µ v f (p4) = ∑i F iHZ Ci µ ,

Ci µ = { u f (p3)γµ v f (p4) , u f (p3)γµ γ5 v f (p4) , . . .} . (93)

Having the full amplitude we start expanding, e.g.885

AZZ (sH , s1 , s2) = AZZ (sH , sZ , s2)+(s1− sZ) A(1)ZZ (sH , s1 , s2) ,

A(1)ZZ (sH , s1 , s2) = A(1)

ZZ (sH , s1 , sZ)+(s2− sZ) A(12)ZZ (sH , s1 , s2) , (94)

etc. The total amplitude of Eq.(87) can be split into several components, Z doubly resonant (DR)886

. . . Z singly resonant (SR) . . . non resonant (NR). Note that NR includes multi-leg functions,887

up to pentagons:888

ADR; ZZ = AZZ (sH , sZ , sZ) ∆Z(s1)∆Z(s2) ,

40

ADR; AA = AAA (sH , 0 , 0) ∆A(s1)∆A(s2) ,

ASR; Z =[AZ (sH , sZ , s2)+A(2)

AZ (sH , s2 , sZ)+A(2)ZZ (sH , sZ , s2)

]∆Z(s1) ,

+[AZ (sH , sZ , s1)+A(1)

AZ (sH , s1 , sZ)+A(1)ZZ (sH , s1 , sZ)

]∆Z(s2) ,

ASR; A =[AA (sH , 0 , s2)+A(2)

AA (sH , 0 , s2)+A(2)AZ (sH , 0 , s2)]∆A(s1)

+[AA (sH , 0 , s1)+A(1)

AA (sH , s1 , 0)+A(1)AZ (sH , 0 , s1)]∆A(s2)

ADR; AZ = AAZ (sH , sZ , s1)[∆Z(s1)∆A(s2)+∆A(s1)∆Z(s2)

]

ANR = A(1,2)DR; ZZ (sH , s1 , s2)+A(2,1)

DR; ZZ (sH , s1 , s2)+A(1,2)DR; AZ (sH , s2 , s1)

+ A(2,1)DR; AZ (sH , s1 , s2)+A(1,2)

DR; AA (sH , s1 , s2)+A(2,1)DR; AA (sH , s1 , s2)

+ A(1)SR; Z (sH , s1 , s2)+A(2)

SR; Z (sH , s2 , s1)+A(1)SR; A (sH , s1 , s2)

+ A(2)SR; A (sH , s2 , s1)+ANR . (95)

Each A amplitude is gauge parameter independent. Let us consider SMEFT at tree level, so889

that890

AAA (sH , s1 , s2) = ASMAA (sH , s1 , s2)+∆AAA (sH , s1 , s2) , (96)

etc.. In ∆A we do not include loops with dim = 6 insertions. Taking f = µ− and neglecting891

fermion masses we obtain the following result892

∆A(h→ e−e+µ

−µ+)=− ig3 g6 Jµ

L (q1 ; p1, p2) Jµ L (q2 ; p3, p4) ∆AL

− ig3 g6

mW

(q2µ q1ν −q1 ·q2 gµν

)Jµ

L (q1 ; p1, p2) JνL (q2 ; p3, p4)

]∆AT , (97)

where Jµ

L is the left-handed fermion current (fermion masses are neglected) and ∆AL ,T are the893

longitudinal and transverse parts of the LO SMEFT deviations. We obtain894

∆AT = 2s2W

aAA ∆A(s1)∆A(s2)+12

vlsW

cW

aAZ

[∆A(s1)∆Z(s2)+∆A(s2)∆Z(s1)

]

− 12

v2l

aZZ

c2W

∆Z(s1)∆Z(s2) , (98)

895

∆AL =14

vl

c2W

1mW

(aφ l V +aφ l A

) [∆Z(s1)+∆Z(s2)

]

− 116

[(7−20s2

W+12s4

W

) aφ D

c4W

+4v2l

aφ�

c4W

+8vl

c4W

(aφ l V +aφ l A

)

+ 4(

3−7s2W+4s6

W

) aZZ

c4W

−4(

5−12s2W+4s4

W

) sW

c4W

(sW aAA + cW aAZ

)]

41

× mW ∆Z(s1)∆Z(s2) , (99)

where vl = 1−2s2W

. With the help of Eqs.(98)–(99) it is straightforward to establish the relation896

between the PO of Sect. 4 and the SMEFT Wilson coefficients (when the complex poles are897

identified with on-shell masses). It is worth noting that we have not included dipole operators.898

It is worth noting that there are subtleties when the h is off-shell, they are described in Ap-899

pendix C.1 of Ref. [55]. Briefly, there is a difference between performing an analytical con-900

tinuation (h virtuality → h on-shell mass) in the off-shell decay width and using leading-pole901

approximation (LPA) of Ref. [57], i.e. the DR part, where the matrix element (squared) is pro-902

jected but not the phase-space. Analytical continuation is a unique, gauge invariant procedure,903

the advantage of LPA is that it allows for a straightforward implementation of cuts.904

In order to extend the SMEFT-PO connection to loop-level SMEFT we have to consider various905

ingredients separately.906

• h→ ZZ The amplitude for h(P)→ Zµ(p1)Zν(p2) can be written as907

Aµν

HZZ = i(P11

HZZ pµ

1 pν1 +P12

HZZ pµ

1 pν2 +P21

HZZ pµ

2 pν1 +P22

HZZ pµ

2 pν2 −DHZZ gµν

). (100)

The result in Eq.(100) is fully general and can be used to prove Ward-Slavnov-Taylor identities908

(WSTI). As far as the partial decay width is concerned only P21HZZ ≡PHZZ will be relevant, due909

to p · e(p) = 0 where e is the polarization vector. Note that computing WSTI requires additional910

amplitudes, i.e. h→ φ0γ and h→ φ

0φ

0. The result can be written as follows:911

DHZZ = −gFmW

c2W

ρHZZD; LO +

g3F

π2

[∑

I=t,b,Wρ

HZZI ;D;NLO D

IHZZ ; NLO +D

(4) ;nfcHZZ +g6 ∑

{a}D

(6) ;nfcHZZ (a)

],

PHZZ = 2gF g6

aZZ

mW+

g3F

π2

[∑

I=t,b,Wρ

HZZI ;P ;NLO P

IHZZ ; NLO +g6 ∑

{a}P

(6) ;nfcHZZ (a)

]. (101)

912

∆ρHZZD; LO = s2

WaAA +

[4+ c2

W

(1− m2

h

m2W

)]aZZ + cW sW aAZ +2aφ� , (102)

913

∆ρHZZq ;D;NLO = ∆ρ

HZZq ;P;NLO = 2I(3)q aq φ +2aφ�−

12

aφ D +2aZZ + s2W

aAA ,

∆ρHZZW ;D; NLO =

112

(4+

1c2

W

)aφ D +2aφ�+ s2

WaAA

+ s2W

(3cW +

53

1cW

)aAZ +

(4+ c2

W

)aZZ ,

∆ρHZZW ;P ; NLO = 4aφ�+

52

aφ D +12aZZ +3s2W

aAA (103)

42

It is convenient to define sub-amplitudes; however, to respect a factorization into t,b and bosonic914

components, we have to introduce the following quantities:915

Wh = Wh W +Wh t +Wh b WZ = WZ W +WZ t +WZ b +∑gen WZ ; fdZ g = dZ g ;W +∑gen dZ g ; f dZ cW

= dZ cW ;W +dZ cW ; t +dZ cW ;b +∑gen dZ cW ; f

dZ mW = dZ mW ;W +∑gen dZ mW ; f(104)

where WΦ ;φ denotes the φ component of the Φ (LSZ) wave-function factor etc. By dZ par we916

denote the (UV finite) counterterm that is needed in connecting the renormalized parameters to917

an input parameter set (IPS). In the actual calculation we use IPS = {GF , mZ , mW}. Further-918

more, ∑gen implies summing over all fermions and all generations, while ∑gen excludes t and b919

from the sum.920

D(4) ;nfcHZZ =

132

mW

c2W

(2∑gen

WZ ; f −∑gen

dZmW ; f +4∑gendZcW ; f −2 ∑

gendZg ; f

). (105)

The connection to Eqs. (20)–(24) is given by921

vF g fZ g f ′

Z PHZZ =−2εZZ , vF g fZ g f ′

Z (p1 · p2 PHZZ−DHZZ) = 2m2Z κZZ . (106)

• h→WW The derivation of the amplitude for h→WW follows closely the one for h→922

ZZ.923

DHWW = −gF mW ρHWWD; LO +

g3F

π2

[∑

I=q,Wρ

HWWI ;D; NLO D

IHWW ; NLO +D

(4) ;nfcHWW +g6 ∑

{a}D

(6) ;nfcHWW (a)

],

PHWW = 2gF g6

1mW

aφ W +g3

Fπ2

[∑I=q

ρHWWI ;P ;NLO P

IHWW ; NLO +g6 ∑

{a}P

(6) ;nfcHWW (a)

], (107)

where we have introduced924

D(4) ;nfcHWW =

132

mW ∑gen

(2WW ; f −dZ mW ; f −2dZ g ; f

). (108)

925

∆ρHWWD; LO = s2

W

(m2

hmW−5mW

)(aAA +aAZ +aZZ)+

12

mW aφ D−2mW aφ� , (109)926

∆ρHWWq ;D;NLO = ∆ρ

HWWq ;P;NLO = aφ t V +aφ t A +aφb V +aφb A−ab φ +2aφ�−

12

aφ D

+ sW ab WB + cW ab BW +5s2W

aAA +5cW sW aAZ +5c2W

aZZ ,

∆ρHWWW ;D; NLO =

196

s2W

c2W

aφ D +2312

aφ�−3596

aφ D

43

+ 4 s2W

aAA +1

12sW

(3

1cW

+49cW

)aAZ +

12

(9c2

W+ s2

W

)aZZ ,

∆ρHWWW ;P ; NLO = 7aφ�−2aφ D +5s2

WaAA +5cW sW aAZ +5c2

WaZZ . (110)

These results allow us to write εWW and κWW of Eqs. (25)–(27) in terms of Wilson coefficients.927

• Z→ f f Let us consider the Z f F vertex, entering the process h→ 4 f :928

Jµ

Z f (p ; q,k) = u f (q){

γµ

[V f (p2)+A f (p2)γ

5]+T f (p2)σ

µν pν v f (k)}. (111)

At lowest order we have deviations defined by929

V f = i I3)f

gcW

(v f +g6 ∆V f

), A f = i I(3)f

gcW

(12+g6 ∆A f

), T f =−

g2

g6

m f

mWaf WB ,

(112)930

∆V f = aφ f V +[v f + c2

W

(v f −1

)](

aAA +cW

sW

aAZ−1

4s2W

aφ D

)+(1− v f

)c2

WaZZ ,

∆A f = aφ f A +12

(aφ W−

14

aφ D

). (113)

where the vector couplings are vu = 1/2−2Qu s2W

, vd = 1/2+2Qd s2W

and u,d are generic up,931

down fermions. When loops are included the decomposition in gauge invariant sub-amplitudes932

is not as simple as in the previous case, fermion loops and boson loops. Here the decomposition933

is given in terms of abelian and non-abelian (Z and W ) parts, Q-components (those proportional934

to γµ ) and L-parts (those proportional to γµ γ+). Details can be found in Sect. 6.15 of Ref. [58].935

The general expression in SMEFT will not be reported here. It is worth noting that936

AZ f f (P2) = AinvZ f f (m

2Z)+

(P2−m2

Z)

Aξ

Z f f (P2) , (114)

where ξ denotes the collection of gauge parameters.937

• W → ud Similarly to the Z vertex we obtain938

ig

2√

2γ

µ

[V(+)

F γ++V(−)

F γ−]+

g4

g6

(m2

U

mWaU W−

m2D

mWaD W

)σ

νµ pν , (115)

939

V(+)F = 1+

√2g6

(a(3)

φF +12

aφ W

), V(−)

F =√

2g6 aφU D . (116)

Here F is a generic doublet of components U = u or νl and D = d or l. Note that aφU D = 0 for940

leptons. The general expression in SMEFT will not be reported here.941

44

• Z→WW Triple gauge boson couplings are dscribed by the following deviations (all mo-942

menta flowing inwards):943

Vµνρ

ZWW (p1 , p2 , p3) = gcW Fµνρ (p1 , p2 , p3)+32

gg6 Hµνρ (p1 , p2 , p3)aQW

m2W

+ gg6cW Fµνρ (p1 , p2 , p3)[(

1−2s2W

) ( sW

cW

aφ WB−aφ W

)+2s2

Waφ B +

14

aφ D

]

+ gg6cW sW Gµνρ (p1 , p2 , p3)[(

1−2s2W

) sW

cW

+2cW

]aφ WB , (117)

944

Fµνρ (p1 , p2 , p3) = pρ

1 pµ

2 pν3 − pν

1 pρ

2 pµ

3 +gνρ(

pµ

3 p1 · p2− pµ

2 p1 · p3)

+ gνµ(

pρ

2 p1 · p3− pρ

1 p2 · p3)+gρµ (pν

1 p2 · p3− pν3 p1 · p2) ,

Gµνρ (p1 , p2 , p3) = gνρ(

pµ

2 − pµ

3)+gνµ

(pρ

1 − pρ

2)+gρµ (pν

3 − pν1 ) ,

Hµνρ (p1 , p2 , p3) = gνρ(

pµ

3 − pµ

2)+gνµ

(pρ

2 − pν3). (118)

• VBF The process that we want to consider is945

u(p1)+u(p2)→ u(p3)+ e−(p4)+ e+(p5)+µ−(p6)+µ

+(p7)+u(p8) . (119)

At LO SMEFT we introduce the triply-resonant (TR) part of the amplitude (t -channel propaga-946

tors are never resonant):947

Jµ

±(pi , p j) = upi γµ

γ± up j , ∆−1Φ(p) = s−M2

Φ , s = p2 , (120)948

A TRLO =

[Jµ

−(p4 , p5)(1− vl)+ Jµ

+(p4 , p5)(1+ vl)][

J−µ (p6 , p7)(1− vl)+ J+µ (p6 , p7)(1+ vl)]

×[Jν−(p3 , p2)(1− vu)+ Jν

+(p3 , p2)(1+ vu)][

J−ν (p8 , p1)(1− vu)+ J+ν (p8 , p1)(1+ vu)],

(121)949

A TRSMEFT = − g6

4096∆h(q1 +q2) ∏

i=1,4∆Z(qi)

m2W

c8θ

ρLO A TRLO +g6 g6 A TR;nfc

SMEFT

∆ρLO = 2aφ�−2m2

Z−2m2h +q1 ·q2 +q2 ·q2

m2W

c2W

aZZ , (122)

where q1 = p8− p1, q2 = p3− p2 are the incoming momenta in VBF and q3 = p4 + p5, q4 =950

p6 + p7 are the outgoing ones. Furthermore, γ± = 1± γ5.951

45

8.4 SMEFT and physical PO952

In this Section we describe the connection between a possible realization of physical PO and953

SMEFT.954

Multi pole expansion (MPE) has a dual role: as we mentioned, poles and their residues are955

intimately related to the gauge invariant splitting of the amplitude (Nielsen identities); residues956

of poles (after squaring the amplitude and after integration over residual variables) can be inter-957

preted as physical PO, which requires factorization into subprocesses. However, gauge invariant958

splitting is not the same as “factorization” of the process into sub-processes, indeed phase space959

factorization requires the pole to be inside the physical region. For all technical details we refer960

to the work in Sect. 3 of Ref. [6] which is based on the following decomposition of the square961

of a propagator962

∆ =1

(s−M2)2+Γ2 M2

=π

M Γδ(s−M2)+PV

[1

(s−M2)2

]. (123)

and on the n-body decay phase space963

dΦn (P, p1 . . . pn) =1

2πdQ2 dΦn− j+1

(P,Q, p j+1 . . . pn

)dΦ j

(Q, p1 . . . p j

). (124)

To “complete” the decay (dΦ j) we need the δ -function in Eq.(124). We can say that the δ -part964

of the resonant (squared) propagator opens the corresponding line allowing us to define physical965

PO (t -channel propagators cannot be cut). Consider the process qq→ f1 f1 f2 f2 j j, according to966

the structure of the resonant poles we have different options in extracting physical PO, e.g.967

σ(qq→ f1 f1 f2 f2 j j) PO7−→ σ(qq→ h j j)Br(h→ Z f1 f1)Br(Z→ f2 f2) ,

σ(qq→ f1 f1 f2 f2 j j) PO7−→ σ(qq→ ZZ j j)Br(Z→ f1 f1)Br(Z→ f2 f2) . (125)

There are fine points when factorizing a process into “physical” sub-processes (PO): extracting968

the δ from the (squared) propagator, Eq.(123), does not necessarily factorize the phase space; if969

cuts are not introduced, the interference terms among different helicities oscillate over the phase970

space and drop out, i.e. we achieve factorization, see Refs. [59]. Furthermore, MPE should be971

understood as “asymptotic expansion”, see Refs. [60,61], not as Narrow-Width-Approximation972

(NWA). The phase space decomposition obtains by using the two parts in the propagator expan-973

sion of Eq.(124): the δ -term is what we need to reconstruct PO, the PV-term (understood as a974

distribution [60]) gives the remainder and PO are extracted without making any approximation.975

It is worth noting that, in extracting PO, analytic continuation (on-shell masses into complex976

poles) is performed only after integrating over residual variables [55].977

We can illustrate the SMEFT - MPE - PO connection by using a simple but non-trivial example:978

Dalitz decay of the Higgs boson, see Refs. [62,6]. Consider the process979

h(P)→ f (p1)+ f (p2)+ γ(p3) , (126)

46

∑spin

∑spin

∑spin

∫cut

dΦ1→3

∫cut

dΦ1→3

∫cut

dΦ1→3

222

CCC

222

CCC

222

µµµνννppp

δµν →∑

λ

[eλµ(p)

]∗eλν(p)δµν →

∑λ

[eλµ(p)

]∗eλν(p)δµν →

∑λ

[eλµ(p)

]∗eλν(p)

conserved current

CCC

2 =2 =2 = polarization

222 222

222

λλλ λλλ1s−sZ1

s−sZ1

s−sZ

∑λ

∑λ

∑λ

| ∑λ f(λ) |2= ∑λ | f(λ) |2 + rest| ∑λ f(λ) |2= ∑λ | f(λ) |2 + rest| ∑λ f(λ) |2= ∑λ | f(λ) |2 + rest

¬

®

Figure 8: Building a physical PO.

and introduce invariants sH = −P2, s = −(p1 + p2)2 and propagators, ∆A(i) = 1/si, ∆Z(i) =980

1/(si− sZ). With sH = µ2H− i µH γH we denote the h complex pole etc. In the limit m f → 0 the981

total amplitude for process Eq.(126) is given by the sum of three contributions, Z,A -resonant982

and non-resonant:983

A (h→ f f γ) =[Aµ

Z (sH , s) ∆Z(s)+Aµ

A (sH , s) ∆A(s)]

eµ (p3, l)+ANR , (127)

where eµ is the photon polarization vector. The two resonant components are given by984

Aµ

V (sH , s) = THAV (sH , s) Tµ

ν (q , p3) JνV f (q ; p1, p2) , (128)

where Jµ

V f is the V fermion ( f ) current, V = A,Z, q = p1 + p2 and Tµν (k1 , k2) = k1 · k2 gµν −985

kν1 kµ

2 . Having the full amplitude we start expanding (MPE) according to986

THAZ (sH , s) = THAZ (sH , sZ)+(s− sZ) T(1)

HAZ (sH , s) etc. (129)

Derivation continues till we define physical PO:987

ΓPO (h→ Zγ) =1

16π

1MH

(1−

µ2Z

m2h

)Fh→Zγ

(sZ , µ

2Z

),

ΓPO (Z→ f f ) =1

48π

1µZ

FZ→ f f(sZ , µ

2Z

).

47

ΓSR (h→ f f γ) =12

ΓPO (h→ Zγ)1γZ

ΓPO (Z→ f f )+ remainder . (130)

In the NWA the remainder is neglected while we keep it in our formulation where the goal is988

extracting PO without making approximations. The interpretation in terms of SMEFT is based989

on THAZ (sH , sZ). A convenient way for writing THAZ is the following:990

THAZ =g3

Fπ2mZ

∑I=W,t,b

ρHAZI T I

HAZ ;LO +gF g6

m2h

MaAZ +

g3F g6

π2 T nfcHAZ . (131)

The factorizable part is defined in terms of ρ -factors991

∆ρHAZq =

(2I(3)q aq φ +2aφ�−

12

aφ D +3aAA +2aZZ

),

∆ρHAZW =

1+6c2W

c2W

aφ�−14

1+4c2W

c2W

aφ D−12

1+ c2W−24c4

W

c2W

aAA ,

+14

(1+12c2

W−48c4

W

) sW

c3W

aAZ +12

1+15c2W−24c4

W

c2W

aZZ . (132)

In the PTG scenario we only keep at φ,ab φ,aφ D and aφ� in Eq.(132). We also derive the following992

result for the non-factorizable part of the amplitude:993

T nfcHAZ = ∑

a∈{A}T nfc

HAZ(a)a , (133)

where {A} = {aφ t V,at BW,at WB,aφb V,ab WB,ab BW,aφ D,aAZ,aAA,aZZ}. In the PTG scenario there994

are only 3 non-factorizable amplitudes for h→ γZ, those proportional to aφ t V,aφb V and aφ D.995

8.5 Summary on the PO-SMEFT matching996

As we have shown, there are different layers of measurable parameters that can be extracted997

from LHC data. An external layer, where the kinematics is kept exact, is represented by phys-998

ical PO such as ΓSR (h→ f f γ) of Eq.(130): these are similar to the σpeakf measured at LEP999

and, similarly to the LEP case, can be extracted from data via a non-trivial NWA. A first in-1000

termediate inner layer is represented by the effective-couplings PO introduced in Section 2-61001

(and summarized in Section 7): these are similar to effective Z-boson couplings (geV A) mea-1002

sured at LEP and control the parameterization of on-shell amplitudes. A further internal layer1003

is represented by the κi introduced in this Section, that are appropriate combinations of Wilson1004

coefficients in the SMEFT. Finally, the innermost layer is represented by the Wilson coefficients1005

(or the Lagrangian couplings) of the specific EFT (or explicit NP model) employed to analyze1006

the data. When moving to the innermost layer in the SMEFT context we still have the option1007

of performing the tree-level translation, which is well defined and should be integrated with the1008

corresponding estimate of MHOU, or we can go to SMEFT at the loop level, again with its own1009

MHOU.1010

48

9 Conclusions1011

The experimental precision on the kinematical distributions of Higgs decays and production1012

cross sections is expected to significantly improve in the next few years. This will allow us1013

to investigate in depth a wide class of possible extensions of the SM. To reach this goal, an1014

accurate and sufficiently general parameterization of possible NP effects in such distributions is1015

needed.1016

The Higgs PO presented in this note are conceived exactly to fulfill this goal: they provide a1017

general decomposition of on-shell amplitudes involving the Higgs boson, based on analyticity,1018

unitarity, and crossing symmetry. A further key assumption is the absence of new light particles1019

in the kinematical regime of interest, or better no unknown physical poles in theses amplitudes.1020

These conditions ensure the generality of this approach and the possibility to match it to a wide1021

class of explicit NP models, including the determination of Wilson coefficients in the context of1022

Effective Field Theories.1023

As we have shown, the PO can be organized in two complementary sets: the so-called physi-1024

cal PO, that are nothing but a series of idealized Higgs partial decay widths, and the effective-1025

couplings PO, that are particularly useful for the developments of simulation tools. The two sets1026

are in one-to-one correspondence, and their relation is summarized in Table 1. The complete set1027

of effective-couplings PO that can be realistically accessed in Higgs-related measurements at the1028

LHC, both in production and in decays, is summarized in Section 7. The reduction of indepen-1029

dent PO obtained under specific symmetry assumptions (in particular flavor universality and CP1030

invariance) is also discussed in Section 7. In two-body processes the effective-couplings PO are1031

in one-to-one correspondence with the parameters of the original κ framework. A substantial1032

difference arises in more complicated processes, such as h→ 4 f or VBF and VH production.1033

Here, in order to take into account possible kinematical distortions in the decay distributions1034

and/or in the production cross-sections, the PO framework requires the introduction of a series1035

of additional terms. These terms encode generic NP effects in the hV f f amplitudes and their1036

complete list is summarized in Table 2.1037

The PO framework can be systematically improved to include the effect of higher-order QCD1038

and QED corrections, recovering the best up-to-date SM predictions in absence of new physics.1039

The effective-couplings PO should not be confused with EFT Wilson coefficients. However,1040

their measurement can facilitate the extraction of Wilson coefficients in any EFT approach to1041

Higgs physics, as briefly illustrated in Section 8 in the context of the so-called SMEFT. The1042

physical and the effective-couplings PO can be considered as the most general and external1043

layers in the characterization of physics beyond the SM, whose innermost layer is represented1044

by the couplings of some explicit NP model.1045

49

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http://arxiv.org/abs/1412.6038

http://dx.doi.org/10.1140/epjc/s10052-015-3345-5






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Pseudo-observables in Higgs physics · 105 width taking into account the ﬁnal state QED and QCD...

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