Implications of a matter-antimatter mass asymmetry in Penning-trap experiments Ting Cheng1Manfred Lindner1and Manibrata Sen1 1Max-Planck-Institut für Kernphysik Saupfercheckweg 1 69117 Heidelberg Germany

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Implications of a matter-antimatter mass asymmetry in Penning-trap experiments

Ting Cheng,1, ∗Manfred Lindner,1, †and Manibrata Sen1, ‡

1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

The Standard Model (SM) of particle physics, being a local, unitary and Lorentz-invariant quan-

tum ﬁeld theory, remains symmetric under the combined action of Charge, Parity, and Time Re-

versal (CPT) symmetry. This automatically implies that fundamental properties of particles and

antiparticles should be equal in magnitude. These fundamental tenets of the CPT principle have

been put to stringent tests in recent Penning-trap experiments, where the matter-antimatter mass

asymmetry has been measured. In light of these recent advances, we compare the bounds arising

on CPT invariance from kaon systems with those from Penning-trap experiments. Using a simple

yet powerful argument of mass decomposition of hadrons, we show that bounds on quark-antiquark

mass diﬀerences from kaon oscillations are way beyond the reach of Penning-trap experiments. We

lay out a roadmap to discuss possible reformulations of our understanding of the SM in the case of

a discovery of CPT violation by these precision experiments.

Keywords: CPT violation, precision experiments

I. INTRODUCTION

The Standard Model (SM) of Particle Physics is a lo-

cal, Lorentz invariant, Hermitian quantum ﬁeld theory

(QFT). As explored in a series of celebrated papers [1–3],

one of the fundamental tenets of such a local, Lorentz in-

variant theory is the conservation of CPT symmetry, that

is, invariance under the combined operations of charge

conjugation, parity inversion, and time reversal. The

conservation of CPT guarantees that physical proper-

ties of matter and antimatter are related, for example,

their masses should be identical, their charges, if any,

should be equal and opposite. In fact, the requirement

that a Hermitian QFT is causal automatically warrants

the existence of antimatter which should have the ex-

act same mass of the corresponding matter ﬁeld. There-

fore, a test of whether there exists a mass asymmetry

between matter and antimatter, aptly dubbed here as

the matter-antimatter mass asymmetry (MAMA), auto-

matically translates to a test of the sacred principle of

CPT invariance, and in turn, the foundations of the SM.

Theoretically, a number of motivations exist for CPT

symmetry to be exact, relating the properties of matter

and antimatter. However, the baryon asymmetry of the

Universe implies a matter dominated Universe. This in-

dicates that some form of asymmetry between matter and

antimatter must have been introduced through a new yet-

unknown mechanism in the early Universe. While models

of successful baryogenesis usually follow a CPT symmet-

ric approach, focusing on the Sakharov conditions [4], a

baryon asymmetry could also arise in thermal equilib-

rium in the presence of CPT violation, and baryon num-

ber violation [5]. Additionally, extensions of the SM to in-

corporate a quantum theory of gravity often induces CPT

violation [6]. Phenomenological motivations include the

∗ting.cheng@mpi-hd.mpg.de

†manfred.lindner@mpi-hd.mpg.de

‡manibrata.sen@mpi-hd.mpg.de

search for violation of Lorentz invariance (LI), or viola-

tion of locality (L), leading to CPT violation [7–15].

Experimental tests of CPT invariance can be twofold:

testing the properties of particles and antiparticles di-

rectly, or probing the indirect impact of CPT violation

on other processes. In the absence of a speciﬁc model,

it is diﬃcult to compare diﬀerent experimental results at

the same footing. Therefore, in this letter, we aim to

bridge diﬀerent mass measuring experiments in a model

independent manner, such that any new physics resulting

in CPT breaking can be investigated through a bottom-

up approach. By treating the constraints from each ex-

perimental result using dimensionless parameters, the

strongest constraint on CPT symmetry itself is currently

from the kaon oscillation experiments. The MAMA pa-

rameter is the diﬀerence between the two diagonal terms

of the Hamiltonian of (K0,¯

K0)in ﬂavor space, and is

tested using the Bell-Steinberger relation constructed un-

der assumptions of unitarity [16,17]. Note that neu-

tral kaon-antikaon oscillations involve a process where

strangeness is violated by 2 units at one loop, through

a box-diagram (see [18], and references therein). In such

a scenario, the test of MAMA could be more sensitive

to violations of the principle of locality, the underlying

process being a loop process. On the other hand, neu-

trino oscillation experiments also provide another inter-

esting test of CPT conservation. Here, for a given neu-

trino energy, and baseline of the experiment, oscillation

parameters (mass-squared diﬀerence, mixing angles) are

ﬁtted separately for the neutrino and antineutrino spec-

tra [7,8,19–30]. In this case, MAMA is measured with

respect to the dispersion relation of the propagating neu-

trino, and therefore, provides a more sensitive probe of

Lorentz-invariance violation [31]. For a discussion re-

garding tests of non-locality using neutrino oscillation

experiments, see [32] and references therein. In all these

cases, tests of CPT conservation usually quote the re-

sults in terms of the mass diﬀerence between the parti-

cles and the antiparticles. It is important to emphasize

that while the deﬁnition of the measured “mass” may be

arXiv:2210.10819v2 [hep-ph] 17 Jul 2023

diﬀerent from experiment to experiment, these systems

can eventually be sensitive to multiple underlying prin-

ciples behind the breaking of CPT: non-locality, and/or

Lorentz invariance violation (LI-V). On the other hand,

not all CPT breaking eﬀects are covered in our analysis,

since we have narrowed down the arbitrariness of the ori-

gin of CPT violation by considering only MAMA tests.

Signiﬁcant progress has also been achieved by precision

experiments in testing the tenets of the CPT principle.

The ALPHA experiment at CERN uses trapped antihy-

drogen to study its charge-neutrality [33,34], the ratio

of gravitational mass to inertial mass [35], as well as a

measurement for the hyperﬁne splitting in neutral an-

tihydrogen [36]. The experiment has, hence far, shown

that the SM is consistent with CPT conservation. Similar

tests have been performed using other species of antimat-

ter such as antiprotons, which can be trapped for longer

times in Penning-traps [37]. The BASE collaboration [38]

at CERN measures the charge-to-mass ratio (q/m) of the

proton/antiproton by comparing the cyclotron frequency

(νc) of a single antiproton ¯pto those of a single negatively

charged hydrogen H−(to avoid systemic uncertainties by

having the charge diﬀerent from the antiproton). This

is done using the Brown-Gabrielse invariance theorem:

ν2

c=ν2

++ν2

z+ν2

−, where ν+, νz, ν−are three eigen-

frequencies, namely, the modiﬁed cyclotron frequency,

the axial frequency and the magnetron frequency, and

νc= 1/(2π)(q/m)B0. The Penning-trap captures a H−

and a ¯pproduced at CERN; then by measuring the three

eigenfrequencies under a homogeneous magnetic ﬁeld of

B0= 1.945 T, one can obtain the ratio of the proton to

antiproton’s inertial mass. Since the inertial mass is di-

rectly measured in this setup, measurement of the annual

modulation of the MAMA would relate directly to tests

of extension of the weak equivalence principle, such as

having scalar-tensor theories of modiﬁed gravity [11,39].

With technology advancing in leaps and bounds, the

sensitivities of the Penning-trap experiments are ex-

pected to get even better with time. In light of these

facts, it is important to compare the bounds on CPT

arising from kaon systems, and neutrino systems, with

those from current and upcoming precision experiments.

In this letter, we present a simple yet crucial argument

relying on the mass decomposition of hadrons using the

energy momentum tensor in Quantum Chromodynam-

ics (QCD), which allows a hadron mass to be separable

into individual quark contributions, and those coming

from gluons, kinetic terms, as well as anomaly terms. To

zeroth order, this allows the MAMA in the hadron sys-

tem to be written as the mass diﬀerence between quark

and antiquarks. Using this parameterization, the existing

bounds from kaon systems translates to bounds on the

mass diﬀerence between quarks and antiquarks, which

are well beyond the sensitivity of Penning-trap experi-

ments to such quark-antiquark mass diﬀerences. Similar

outcomes with regards to the diﬀerences in gravitational

forces exerted on matter and antimatter have been pre-

sented in [40]. This emphasizes that any discovery by

these precision experiments would warrant a serious re-

formulation of our understanding of QFTs in order to be

compatible with the results from the kaon systems.

Our discussion is organized as follows. In the next sec-

tion, we outline the basic framework of decomposition of

a hadron mass in QCD, and how it relates to the con-

stituent quark masses. Then, we apply it to diﬀerent

systems: protons, kaons, and discuss the bounds aris-

ing on MAMA. We also compare bounds on CPT arising

from neutrino oscillations. We ﬁnally discuss the theo-

retical implications of a positive measurement of MAMA

from the Penning-trap experiments, and/or other exper-

iments, and conclude.

II. HADRON MASS DECOMPOSITION IN QCD

The masses of hadrons in QCD are governed primarily

by the interaction amongst the constituent quarks and

gluons. In fact, while hadron masses have been quite

well measured experimentally, disentangling the mass

contributions of the individual quarks from the quark-

gluon strong interaction dynamics is still a matter of ac-

tive research. The incredible complexity of the problem

arises from the fact that at low energies, QCD is non-

perturbative, and hence one needs to resort to lattice

computations to get a clearer understanding.

However, it is possible to obtain a phenomenological

decomposition of the mass of a hadron using the energy-

momentum tensor (Tµν )in QCD [41,42]. The symmet-

ric, conserved energy-momentum tensor in QCD can be

formally written in Euclidean space as

Tµν =1

4¯

ψγ(µ

←→

Dν)ψ+FµαFνα −1

4δµν F2,(1)

where −→

D=−→

∂µ+igAµand ←−

D=←−

∂µ−igAµ, for a

gluon ﬁeld Aµand coupling g, while the () in the kinetic

term denotes symmetrization over the indices. The QCD

Hamiltonian operator can be deﬁned through Tµν as

HQCD =−Zd3x T44(x).(2)

This Hamiltonian can be decomposed into four contribu-

tions, coming from the kinetic energy of the quarks, the

gluon ﬁeld energy, the bare quark masses, and the QCD

anomaly respectively, given by

HQCD =HE+Hg+Hm+Ha,(3)

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Implicationsofamatter-antimattermassasymmetryinPenning-trapexperimentsTingCheng,1,∗ManfredLindner,1,†andManibrataSen1,‡1Max-Planck-InstitutfürKernphysik,Saupfercheckweg1,69117Heidelberg,GermanyTheStandardModel(SM)ofparticlephysics,beingalocal,unitaryandLorentz-invariantquan-tumfieldtheory,remainssym...

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Implications of a matter-antimatter mass asymmetry in Penning-trap experiments Ting Cheng1Manfred Lindner1and Manibrata Sen1 1Max-Planck-Institut für Kernphysik Saupfercheckweg 1 69117 Heidelberg Germany

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