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Hadronic decays of a light Higgs-like scalar

Alexander Monin,

1,*

Alexey Boyarsky,

and Oleg Ruchayskiy

Department of Theoretical Physics, University of Geneva 24 quai Ernest-Ansermet, 1211 Geneva

Lorentz Institute, Leiden University, Niels Bohrweg 2, Leiden, NL-2333 CA, Netherlands

Discovery Center, Niels Boh r Institute, Copenhagen University,

Blegdamsvej 17, DK-2100 Copenhagen, Denmark

(Received 27 August 2018; published 10 January 2019)

We revisit the question of hadronic decays of a giga-electron-volt-mass Higgs-like scalar. A number of

extensions of the Standard Model predict the Higgs sector with additional light scalars. Currently operating

and planned Intensity Frontier experiments will probe for the existence of such particles, while theoretical

computations are plagued by uncertainties. The goal of this paper is to bring the results in a consolidated

form that can be readily used by experimental groups. To this end, we provide a physically motivated fitting

ansatz for the decay width that reproduces the previous nonperturbative numerical analysis. We describe

systematic uncertainties of the nonperturbative method and provide explicit examples of the influence of

extra resonances above 1.4 GeV onto the total decay width.

DOI: 10.1103/PhysRevD.99.015019

I. MOTIVATION

The Standard Model of particle physics provides a

closed and self-consistent description of known elementary

particles interacting via strong, weak, and electromagnetic

forces. The Standard Model coupled with general relativity

has also been very successful in describing the evolution

of the Universe as a whole. However, the impressive

success of the Standard Model at accelerator and cosmic

frontiers has also revealed with certainty that the Standard

Model fails to explain a number of observed phenomena in

particle physics, astrophysics, and cosmology. These major

unsolved challenges are commonly known as “beyond the

Standard Model” (BSM) problems. They include neutrino

masses and oscillations, dark matter, baryon asymmetry of

the Universe, etc.

A range of possible scenarios capable of resolving the

BSM puzzles is extremely wide. At the one end, there

are models such as the neutrino minimal Standard Model

(νMSM) [1,2] that postulate only three extra particles

lighter than electroweak scale, providing resolutions of

major BSM puzzles and leading to a Standard Model–like

quantum field theory up to very high scales [3–5]. At the

other end, there are models in which one is completely

agnostic about the structure of the hidden (“dark”) sectors

and explores portals—mediator particles that both couple

to states in the “hidden sectors” and interact with the

Standard Model. Such portals can be renormalizable (mass

dimension ≤ 4) or be realized as higher-dimensional oper-

ators suppressed by the dimensionful couplings Λ

−n

, with

Λ being the new energy scale of the hidden sector. Mediator

couplings to the Standard Model sector can be sufficiently

small to allow for the portal particles to be (much) lighter

than the electroweak scale. Such models can be explored

with Intensity (rather than Energy) frontier experiments.

In this paper, we focus on scalar (or “Higgs”) portal

[6]—gauge singlet scalar S interacting with the Higgs

doublet H via the SH

†

H term. Such particles “inherit”

their interactions from the Higgs boson (albeit suppressed

by a small dimensionless parameter θ). New generation of

Intensity Frontier experiments, such as NA62 [7–9],

SHiP [10,11], MATHUSLA [12,13], FASER [14]),

CODEX-b [15], and SeaQuest [16] will probe for the

existence of such scalars with the masses ∼GeV. The

lifetime of such scalars is dominated by the decay into

light mesons (S → ππ, S →

KK, etc.) The question of

computation of the decay width of such particles was

studied in the 1980s [17–23] in the context of hadronic

decays of the light Higgs boson. Based on the data for

→ ψππ and ϒ

→ ϒππ decays, Ref. [18] argued in

favor of extrapolating the results obtained with the help of

chiral perturbation theory (ChPT) up to 1.5 GeV. At the

same time, the nonperturbative analysis of Ref. [23]

produced results differing from Ref. [18] by as much as

an order of magnitude.

This discrepancy, crucial for the new generation of

experiments, warrants the current work. We critically

alexander.monin@unige.ch

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribut ion to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW D 99, 015019 (2019)

2470-0010=2019=99(1)=015019(19) 015019-1 Published by the American Physical Society

review existing methods of computation of the scalar’s

hadronic width and assess the uncertainties. We mainly

reconfirm findings of Ref. [23] but provide a way to

assess its uncertainties and speculate up to what scales

the nonperturbative approach should be used (read:

trusted).

The paper is organized as follows. In Sec. II, we discuss

briefly the properties of the scalar portal and define the

form factors through which the hadronic decay width Γ

ππ

expressed. We review computation of the hadronic decay in

the chiral perturbation theory, reproducing the results of

Ref. [18] in Sec. III. The unitarity arguments that allow for

nonperturbative treatment of the relevant form factors are

summarized in Sec. IV. The review of dispersive methods is

given in Sec. V. Section VI summarizes our results and

compares with previous works. Section VII provides the

error estimate and domain of validity. We conclude in

Sec. VIII and provide supplementary material in the

Appendices.

II. SETUP

We start by laying down the ground rules for com-

puting the desired decay rate. We consider a scalar field

S weakly coupled to the Standard Model Higgs field H;

see Ref. [10] for details. For masses below 1 GeV, the

relevant UV couplings of the scalar are only those to quarks

and leptons

int

¼ −

qq −

ll; ð1Þ

since the only interesting decay channels are ππ, μ

−

, and

possibly

KK. In Eq. (1), v

≡ v cot θ, where v ¼ 246 GeV

is the Higgs vacuum expectation value (vev) and θ ≪ 1

parametrizes the interaction of the scalar S with the

Standard Model particles.

From the Lagrangian (1), the decay rate S → μ

−

can

be immediately found:

−

8π



1 −



3=2

: ð2Þ

Computing the width due to hadronic decays is somewhat

more involved. The difficulty stems from the strong

coupling of QCD in the regime of interest. Therefore,

considering only the tree-level process—which in the case

of the leptonic decay leads to (2)—is not enough. Quarks

and gluons are not adequate degrees of freedom for

describing the low-energy physics. Instead, to compute

hadronic decay rates, matrix elements of the Lagrangian (1)

between low-energy hadronic states should be computed

directly. For instance, in the case of S → π

, where a and

b are isospin indices, the amplitude is defined as

ðm

Þδ

≡ hπ

ðp

Þπ

ðp

ÞjiL

int

jSi

¼ −

hπ

ðp

Þπ

ðp

Þj

qqj0i: ð3Þ

Integrating it over the phase space gives the decay width

ππ

32π

ðm

Þj

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 −

; ð4Þ

where we summed over all species and took into account

that the particles in the final state are identical.

The sum in (3) contains contributions from light (u, d, s)

and heavy (c, b, t) quarks. The latter can be expressed in

terms of the former and the energy-momentum tensor

by using a clever trick based on the knowledge of the

trace anomaly and the renormalization group invariance

of the energy-momentum tensor (for more details, see

Refs. [18,24,25] and [20,21,26]). It uses two different

representations of the energy-momentum tensor at energies

immediately above and below the c-quark mass. On the one

hand, using the UV description (all quarks), the trace of the

energy-momentum tensor, due to the anomaly, is given by

βðα

4α

all

qq; α

4π

¼ G

μν

;

ð5Þ

with the one-loop beta function for the strong coupling α

defined as

βðα

Þ¼−

bα

2π

;b¼ 9 −

; ð6Þ

where N

is the number of heavy quarks (in our case 3). On

the other hand, from the IR perspective (after integrating out

heavy quarks), the energy-momentum tensor becomes

βðα

4α

light

qq þ Oð1=m

Þ; ð7Þ

where the reduced beta function corresponds to only light

quarks (u, d, s),

βðα

Þ¼−

9α

2π

: ð8Þ

As a result, one concludes that

heavy

qq ¼−

8π



−

light



; ð9Þ

and the interaction Lagrangian can therefore be rewritten as

MONIN, BOYARSKY, and RUCHAYSKIY PHYS. REV. D 99, 015019 (2019)

015019-2

int

¼ −





1 −



light



; ð10Þ

which immediately leads to the expression for the amplitude

(3) in the leading order in α

ðm

Þ¼



ðm



1 −



½Γ

ðm

ÞþΔ

ðm

Þ



; ð11Þ

where the following notations for the form factors were

introduced:

ðsÞδ

¼hπ

uu þ m

ddj0i; ð12aÞ

ðsÞδ

¼hπ

ssj0i; ð12bÞ

ðsÞδ

¼hπ

jθ

j0i: ð12cÞ

The problem of computing the width (4) thus boils down

to computing these form factors. In the next sections, we

present several approximations when it can be done using

different techniques such as the ChPT and unitarity. It is

also important to note that the expression (11) does not

capture effects suppressed by heavy quark masses 1=m

III. ChPT

For very small energies, the form factors (12a)–(12c)

can be easily computed using the chiral perturbation

theory [18,27]. We will not go to great lengths to introduce

the ChPT and refer the reader to numerous sources (for

instance, Refs. [28–30]). Instead, we give just the key

results allowing us to demonstrate how the computation

is done.

The low-energy dynamics of QCD can be described

[in the case of the SUð2Þ × SUð2Þ chiral symmetry] by

introducing an SUð2Þ matrix,

Σ ¼ e

iσ

; ð13Þ

parametrized by pion fields π

, with f

¼ 93 MeV being

the pion decay constant. The action of the chiral symmetry

group on the space of these matrices is realized by the right

and left multiplications,

¼ U

ΣU

†

: ð14Þ

In building the Lagrangian, one has to make sure that the

symmetry (14) is preserved. It is straightforward to show

that the leading (derivative expansion) order Lagrangian is

given by

L ¼

Tr∂

Σ∂

†

TrðM

†

Σ þ Σ

†

MÞ; ð15Þ

where M is the quark mass matrix

M ¼



0 m



ð16Þ

and B is a constant. Expanding the Lagrangian up to

quadratic order shows that the pion mass is given by

¼ Bðm

þ m

Þ. At this order, the trace of the energy-

momentum tensor is given by

¼ 2m

− ð∂πÞ

: ð17Þ

Therefore, the corresponding form factor (12c) becomes

ðsÞ¼s þ 2m

: ð18Þ

The other two form factors (12a) and (12b) can be

computed in a similar way:

ðsÞ¼m

; Δ

ðsÞ¼0: ð19Þ

As a result, the amplitude (11) becomes [18]





; ð20Þ

which—upon using (4)—leads to the following expression

for the S → ππ decay rate:

ππ

216π

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 −





: ð21Þ

By construction, the ChPT is only reliable for suffi-

ciently small energies. It is thus clear that the result (20) is

valid at energies low compared to QCD scale, i.e.,

QCD

≡ 4πf

≈ 1 GeV. At the same time, we are inter-

ested in finding the decay rate for scalars with masses

comparable to 1 GeV. To do that, it is not enough to

compute the next-to-leading-order correction in ChPT. In

the next section, we will present a nonperturbative

approach based on dispersion relations.

IV. BEYOND ChPT AND UNITARITY

In the previous section, we showed how the decay rate

of a light scalar into pions can be computed using the

power of the effective field theory. ChPT corrections to the

leading-order result are suppressed by powers of s=Λ

QCD

Therefore, perturbative computations become unreliable

at energies close to the cutoff. However, the precise point

where corrections become comparable with the leading-

order computation depends on the specifics of an observable.

HADRONIC DECAYS OF A LIGHT HIGGS-LIKE SCALAR PHYS. REV. D 99, 015019 (2019)

015019-3

There are indications that for the form factor Γ

ðsÞit happens

for energies s much smaller

than Λ

QCD

Using the definition of the quadratic scalar radius of the

pion

S;π

¼ 6

∂ log Γ

ðsÞ

∂s



s¼0

; ð22Þ

the form factor Γ

ðsÞ around s ¼ 0 can be written as

ðsÞ¼Γ

ð0Þ



1 þ

shr

S;π

þ …



; ð23Þ

where Γ

ð0Þis given by Eq. (19). The quadratic scalar radius

of the pion was first computed in Ref. [32] using ChPT at

one loop, with the result hr

S;π

¼ 0.55  0.15 fm

.The

method of Ref. [23] (to be discussed shortly) produced

S;π

¼ 0.600  0.052 fm

. Later, using a better input for

the pi-pi phases, the analysis was repeated in Ref. [33],

producing

S;π

¼ 0.61  0.04 fm

: ð24Þ

Recently, this computation was corroborated by lattice

computations in Ref. [34].

Such a value of the quadratic

scalar radius implies that ChPT cannot be trusted when

shr

S;π

∼ 1 or, equivalently, at

ﬃﬃﬃ

∼ 600–700 MeV

(see also Refs. [20,21]). Therefore, for masses of a scalar

≲ 1 GeV, a nonperturbative approach should be used.

A. Analyticity and unitarity

Such a method by definition should use only the most

general constraints on form factors without alluding—if

possible—to any specific perturbative computation. The

first constraint comes from analyticity. It can be proven (see

Refs. [35,36]) that form factors are analytic functions in the

complex plane of the variable s with the cut s>4m

, and

it can be established (using the high-energy behavior of

QCD [37]) that their behavior at infinity should be ∼1=s.

The second constraint is due to unitarity. To discuss it,

we have to introduce the notion of the scattering matrix for

s waves with isospin zero. For two-to-two (π

→ π

)

scattering, the S matrix, defined as

abcd

ðs; t; uÞ¼

out

hπ

ðp

Þπ

ðp

Þjπ

ðp

Þπ

ðp

Þi

; ð25Þ

depends on all Mandelstam variables (s, t, and u) and has

an arbitrary tensor structure in the space of isospin indices

a; …;d. However, it can be expanded in partial waves with

fixed angular momentum J and isospin I. We are interested

in scalar (isoscalar) form factors; therefore, we consider

only s-wave isospin-0 (J ¼ I ¼ 0) scattering, by projecting

(25) on the corresponding subspace (see Chap. 19 in

Ref. [28]). It is this component that we refer to as an S

matrix in what follows. For energies below the inelastic

threshold (4m

), the S matrix is completely determined by

the pion phase shift

SðsÞ¼e

2iδ

ðsÞ

; 4m

<s<16m

: ð26Þ

As energy grows, channels 2π → 4π, 2π → 6π, and then

2π →

KK open up. Correspondingly, the S matrix can be

represented by a finite-dimensional matrix, S

, with i and j

running in the space of channels [38]. It is observed

experimentally that the mixing with multiparticle (four

and more) states for energies below Λ

dat

¼ 1.4–1.6 GeV is

small [39,40]. Therefore, in this region, there are effectively

only two relevant channels ðππ → ππ; ππ →

KKÞ.

In the case of two channels—the generalization to an

arbitrary number of channels seems straightforward—

unitarity constraints for form factors, similar to the optical

theorem (see, e.g., the textbook [41], Sec. 6-3-4) for

amplitudes, can be derived in the following way. We define

ðsÞδ

≡ hπ

jXj0i;

ðsÞδ

αβ

≡

ﬃﬃﬃ

jXj0i; ð27Þ

where X is any of the operators appearing in ((12a)–(12c).

The relative factor 2=

ﬃﬃﬃ

is due to the normalization of the

isospin-0 eigenstates

jππi¼

ﬃﬃﬃ

a¼1

jπ

i; and jKKi¼

a¼1

ð28Þ

Also, we introduce the T matrix via

ðsÞ≡ δ

þ2iT

ðsÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðsÞσ

ðsÞ

Θðs − 4m

ÞΘðs − 4m

Þ;

ð29Þ

where ΘðsÞ is the Heaviside theta function (representing

the opening of the corresponding channel), and the factors

ðsÞ

ðsÞ ≡

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 −

ð30Þ

are responsible for the phase space volume. One can be

straightforwardly convinced that the unitarity of the S

matrix [which is represented schematically in Fig. 1(a)]

translates into the following constraint on T:

It is known that final-state interaction effects can be rather

strong [31].

It is precisely the result of Ref. [33] and lattice computations

that will be used in the following sections to fix unknown

coefficients in several form factors.

MONIN, BOYARSKY, and RUCHAYSKIY PHYS. REV. D 99, 015019 (2019)

015019-4

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