Searching for CPT Violation with Neutral-Meson Oscillations

TL;DR

This paper introduces a comprehensive method to detect CPT violation in neutral-meson oscillations using an effective field theory approach, analyzing effects of CPT-violating operators of various dimensions and extracting experimental bounds.

Contribution

It develops a general formalism incorporating CPT-violating operators of arbitrary dimension into neutral-meson oscillation analysis, and provides first experimental constraints on dimension-five operators.

Findings

Established a formalism for CPT violation in meson oscillations

Extracted experimental bounds on dimension-five CPT-violating operators

Discussed observable effects of CPT violation in meson systems

Abstract

A general technique is presented for treating CPT violation in neutral-meson oscillations. The effective field theory for a complex scalar with CPT-violating operators of arbitrary mass dimension is incorporated in the formalism for the propagation and mixing of neutral mesons. Observable effects are discussed, and first measurements of CPT-violating operators of dimension five are extracted from existing experimental results.

Tables

Table 1. Table 1: Constraints on ( k a , P ( 5 ) ) T ​ T ​ μ (k_{a,P}^{(5)}){}_{TT\mu} for K 0 superscript 𝐾 0 K^{0} , D 0 superscript 𝐷 0 D^{0} , B d 0 superscript subscript 𝐵 𝑑 0 B_{d}^{0} , and B s 0 superscript subscript 𝐵 𝑠 0 B_{s}^{0} mesons.

Coefficient	Constraint (GeV-1)	Ref.
$|\left(k_{a,K}^{\left(5\right)}\right){}_{TTT}{}^{}-0.97\left(k_{a,K}^{\left(5\right)}\right){}_{TTZ}{}^{}|$		[13]
$|\left(k_{a,K}^{\left(5\right)}\right){}_{TTX}{}^{}|,|\left(k_{a,K}^{\left(5\right)}\right){}_{TTY}{}^{}|$		[7]
$\left(k_{a,K}^{\left(5\right)}\right){}_{TTT}$	$\left(-2.4\pm 4.4\right)\times {10}^{-17}$	[10]
$\left(k_{a,K}^{\left(5\right)}\right){}_{TTX}$	$\left(1.2\pm 2.8\right)\times {10}^{-18}$	[10]
$\left(k_{a,K}^{\left(5\right)}\right){}_{TTY}$	$\left(2.7\pm 2.7\right)\times {10}^{-18}$	[10]
$\left(k_{a,K}^{\left(5\right)}\right){}_{TTZ}$	$\left(4.2\pm 3.0\right)\times {10}^{-18}$	[10]
$N^D[\left(k_{a,D}^{\left(5\right)}\right){}_{TTT}{}^{}+0.97\left(k_{a,D}^{\left(5\right)}\right){}_{TTZ}{}^{}]$	$\left(-5.1\mathrm{to}8.8\right)\times {10}^{-20}$	[8]
$N^D\left(k_{a,D}^{\left(5\right)}\right){}_{TTX}{}^{},N^D\left(k_{a,D}^{\left(5\right)}\right){}_{TTY}$	$\left(-6.5\mathrm{to}3.5\right)\times {10}^{-20}$	[8]
$N^{B_d}[\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTT}{}^{}-0.7\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTZ}{}^{}]$	$\left(-1.3\pm 1.0\right)\times {10}^{-16}$	[9]
$N^{B_d}\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTX}$	$\left(-3.6\pm 1.1\right)\times {10}^{-16}$	[9]
$N^{B_d}\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTY}$	$\left(-4.5\mathrm{to}-0.7\right)\times {10}^{-16}$	[9]
$\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTT}{}^{}-0.8\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTZ}$	$\left(-1.3\pm 17\right)\times {10}^{-20}$	[12]
$\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTX}$	$\left(1.0\pm 0.8\right)\times {10}^{-19}$	[12]
$\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTY}$	$\left(2.3\pm 8.1\right)\times {10}^{-20}$	[12]
$\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTT}{}^{}-0.7\left(k_{a,B_d}^{\left(5\right)}\right){}_{TTZ}$	$\left(-8.0\pm 5.1\right)\times {10}^{-16}$	[18]
$\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTT}$	$\left(1.0\pm 1.1\right)\times {10}^{-14}$	[16]
$|\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTX}{}^{}|,|\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTY}{}^{}|$		[11]
$\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTT}{}^{}-0.4\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTZ}$	$\left(-1.6\mathrm{to}8.3\right)\times {10}^{-16}$	[11]
$\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTT}{}^{}-0.8\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTZ}$	$\left(-1.1\pm 2.1\right)\times {10}^{-18}$	[12]
$\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTX}$	$\left(0.5\pm 1.4\right)\times {10}^{-18}$	[12]
$\left(k_{a,B_s}^{\left(5\right)}\right){}_{TTY}$	$\left(-1.9\pm 1.4\right)\times {10}^{-18}$	[12]

Equations

$$\Lambda={\textstyle{1\over 2}}\Delta\lambda\left(\begin{array}[]{cc}U+\xi&VW^{-1}\\ &\\ VW&U-\xi\end{array}\right),$$

$${\mathcal{L}}_{\mathrm{CPT}}\supset -\frac{1}{2}i{\varphi }^{†}({\hat{k}}_a){}^{\mu }{\partial }_{\mu }\varphi +h.c.,$$

$$({\hat{k}}_a){}^{\mu }$$

$$\xi =\sum _{d=3}^{\infty }{\xi }^{(d)},$$

$${\xi }^{(d)}=\frac{m^{d-3}}{\Delta \lambda }(k_a^{(d)}{)}^{TT\cdots T},(\mathrm{rest},\mathrm{Sun}-\mathrm{centered}\mathrm{frame}).$$

$${\xi }^{(d)}$$

$${\xi }^{(d)}=\frac{m^{d-3}{\gamma }^{d-2}}{\Delta \lambda }\sum _k{}^{(d-2)}{C}_k{\beta }^k{\hat{\beta }}^{j_1}\cdots {\hat{\beta }}^{j_k}(k_a^{(d)}{)}_{t\cdots t{j}_1\cdots j_k}^{\mathrm{lab}},$$

$${\xi }^{(d)}=\frac{m^{d-3}{\gamma }^{d-2}}{\Delta \lambda }\sum _k{}^{(d-2)}{C}_k{\beta }^k{\hat{\beta }}^{\prime J_1}\cdots {\hat{\beta }}^{\prime J_k}(k_a^{(d)}{)}_{T\cdots T{J}_1\cdots J_k},$$

$${\hat{\beta }}^{\prime }{}^X$$

$${\hat{\beta }}^{\prime }{}^Y$$

$${\hat{\beta }}^{\prime }{}^Z$$

$$(k_a^{(5)}){}_{\mu ZZ}\equiv (k_a^{(5)}){}_{\mu TT}-(k_a^{(5)}){}_{\mu XX}-(k_a^{(5)}){}_{\mu YY},$$

$${\xi }^{(5)}$$

$${\mathcal{A}}_0$$

$${\mathcal{A}}_1$$

$${\mathcal{B}}_1$$

$${\mathcal{A}}_2$$

$${\mathcal{B}}_2$$

$${\mathcal{A}}_3$$

$${\mathcal{B}}_3$$

$$(k_{a,P}^{(3)}){}_T$$

$$(k_{a,P}^{(3)}){}_X$$

$$(k_{a,P}^{(3)}){}_Y$$

$$(k_{a,P}^{(3)}){}_Z$$

Full text

The four neutral mesons $K^{0}$, $D^{0}$, $B_{d}^{0}$, $B_{s}^{0}$ and their antiparticles play a central role in investigations of fundamental symmetries. Although electrically neutral, each of these mesons carries nonzero flavor. The weak interactions mix each meson with its antiparticle, thereby inducing flavor oscillations that can serve as a sensitive interferometer in searches for physics beyond the Standard Model (SM) and General Relativity (GR). The existence of CP violation in nature was uncovered using this technique [1].

Among the fundamental symmetries that are accessible to testing with neutral-meson oscillations is CPT invariance. This symmetry is guaranteed to hold in any relativistic quantum field theory, including the SM [2]. Interest in hypothetical violations of CPT invariance has increased in recent years, stimulated by the realization that they may arise in an underlying theory unifying quantum physics and gravity such as strings [3]. The advent of a comprehensive framework for describing CPT violation at the level of effective field theory, the Standard-Model Extension (SME) [4, 5, 6], has led to numerous experimental investigations [7, 8, 9, 10, 11, 12] based on specific and detailed SME predictions for experimental observables for CPT violation in neutral-meson systems [13, 14, 15, 16, 17, 18, 19].

The propagation and oscillation of a neutral meson can be described using a 2$\times$2 effective hamiltonian $\Lambda$ acting in the space of meson and antimeson quantum states. In this formalism, CPT violation is controlled by a complex parameter, denoted here by $\xi$, that is proportional to the difference between the diagonal elements of $\Lambda$. Using this approach, early analyses studying CPT violation in neutral-meson oscillations began over 50 years ago [20]. These efforts preceded the establishment of the SM and relied primarily on generic arguments at the level of quantum mechanics. In this context, it seemed natural to assume that the parameter $\xi$ controlling CPT violation is a single complex number that is a universal constant quantity for all meson species. However, the widespread acceptance of quantum field theory and the advent of the SM has radically changed this understanding. We now know that the SM provides an excellent description of nature and that small deviations from it can be described using the tools of effective field theory [21]. For CPT violation, adopting this methodology yields the SME. In this comprehensive framework, it turns out that the parameter $\xi$ can depend on the meson flavor and, crucially, must depend on the direction and magnitude of the meson 3-momentum and on the meson energy [13]. Moreover, all laboratories represent noninertial frames due to the rotation of the Earth, which implies that experimental observables involving $\xi$ also depend on the location of the laboratory on the Earth’s surface and on the sidereal time [13]. The phenomenological description of CPT violation is therefore considerably more interesting and involved than the historical approach would suggest. Indeed, experiments that traditionally were regarded as measuring the same quantity turn out in fact to be measuring distinct phenomena.

The essential idea underlying the SME is to extend the SM coupled with GR to an effective field theory that includes all operators violating Lorentz symmetry [22]. In realistic effective field theory, any operators violating CPT violate Lorentz symmetry as well [5, 23], so the SME also provides a general framework for studying CPT violation. Terms in the SME beyond the SM and GR can be organized as a series of operators of increasing mass dimension $d$, each controlled by a coefficient that determines the magnitude of effects and that is the target of experimental measurements [24]. The minimal SME is the restriction of the full SME to operators of $d\leq 4$, which in Minkowski spacetime produces a renormalizable theory. Nonminimal operators with $d\geq 5$ also play a central role in formal contexts including studies of string theory [3, 25], causality and stability [26], and Riemann-Finsler geometry [27, 28, 29], in searches for Lorentz-invariant geometric extensions of GR including torsion [30] and nonmetricity [31], and in phenomenological investigations including supersymmetric models [32] and noncommutative quantum field theories [33]. The key notion in the present context is that the SME can be used to predict specific observable signals for CPT violation in the neutral-meson systems.

To date, all SME analyses of CPT violation in neutral-meson oscillations have been performed using the minimal SME. In this work, we develop techniques to handle also CPT-violating operators of arbitrary nonminimal dimension $d$. This permits a comprehensive treatment of CPT violation in meson propagation and mixing, which can be used to analyze experimental data in any of the neutral-meson systems. The primary idea is to treat a meson-antimeson system using CPT-violating scalar effective field theory [29], which bypasses many of the complications associated with the nonminimal sector of the SME. In what follows, we present the theory and then discuss some applications to experiments.

The Schrödinger time evolution of a linear combination of the wave functions for a generic neutral meson $P^{0}$ and its antiparticle $\overline{P^{0}}$ is governed by a $2\times 2$ effective hamiltonian $\Lambda$. The physical propagating mesons are the eigenstates of $\Lambda$, which play a role paralleling that of the normal modes in a two-dimensional mechanical oscillator [34]. The eigenvalues of $\Lambda$ are complex numbers $\lambda_{a}\equiv m_{a}-{\textstyle{1\over 2}}i\gamma_{a}$ and $\lambda_{b}\equiv m_{b}-{\textstyle{1\over 2}}i\gamma_{b}$, where $m_{a}$, $m_{b}$ are the physical masses and $\gamma_{a}$, $\gamma_{b}$ are the corresponding decay rates. It is convenient to define also the quantities $\lambda=\lambda_{a}+\lambda_{b}$ and $\Delta\lambda=\lambda_{a}-\lambda_{b}$.

Independent of phase conventions and of the size of CPT violation, the effective hamiltonian $\Lambda$ can be expressed as [15]

[TABLE]

where $U\equiv\lambda/\Delta\lambda$, $V\equiv\sqrt{1-\xi^{2}}$. The complex parameter $\xi$ controls CPT violation and is determined by the difference $\Delta\Lambda$ of diagonal elements of $\Lambda$, while the modulus $w$ of the complex parameter $W=w\exp(i\omega)$ controls T violation. The phase $\omega$ is unobservable. Relationships between these quantities and various other parametrizations found in the literature are given in Ref. [15].

The explicit form of $\xi$ is determined by the underlying theory, and it can be found by direct calculation in the context of effective field theory. In the minimal SME, the expression for $\xi$ is known at leading order in the coefficients for CPT violation [13]. However, incorporating the nonminimal sector of the SME is more challenging. In the strong and electromagnetic sectors, a technique has recently been devised for constructing all nonminimal terms including both propagation and interaction [35], but the full SME is yet to appear in the literature. In the present work we introduce a different approach that avoids the complexities of the nonminimal SME, instead using scalar effective field theory to determine key features of $\xi$. The idea is to treat the corrections to the neutral-meson propagator as arising from the CPT-violating operators of arbitrary $d$ that have recently been constructed in scalar effective field theory [29]. This has the advantage of permitting the immediate calculation of the explicit form of $\xi$ while isolating for independent derivation [36] the specific connection between the coefficients for CPT violation in the scalar effective field theory and the standard SME coefficients.

To implement this idea, we can follow the general perturbative approach previously adopted for the minimal SME but now applied instead to scalar effective field theory. In the SM limit, the effective hamiltonian $\Lambda$ is CPT invariant and the meson wave functions can be taken as unperturbed states. In the minimal SME, the contribution to $\Lambda$ at leading order in coefficients for CPT violation can be found by evaluating the expectation values of the CPT-violating operators in the unperturbed states [4]. The dominant effects turn out to arise from CPT-violating corrections to the valence-quark propagators, while the CPT-violating effects from the sea are suppressed. This procedure yields the explicit form of $\xi$ at leading order in minimal-SME coefficients [13]. An analogous methodology can be applied in the present context of scalar effective field theory, with the CPT-violating operators being evaluated in the unperturbed meson states. Note that this approach avoids the complexities of determining the valence-quark and sea contributions in the presence of nonminimal SME operators, instead treating the meson as a point particle in scalar effective field theory. In what follows, we assume for definiteness that the dominant effects at arbitrary $d$ arise from operators correcting the meson propagation rather than from meson interactions. This assumption is known to hold in the minimal SME [13] and appears reasonable for nonminimal interactions, which either appear suppressed by other couplings or are controlled by independent SME coefficients. A detailed investigation of contributions from nonminimal interactions is an interesting open topic for future investigation.

Following this approach, we model the field operator for the $P^{0}$ and $\overline{P^{0}}$ mesons using a complex scalar $\phi$. The required modifications to the meson propagation can then readily be handled using effective field theory, even though a complete description of the meson behavior is challenging to model using only a single field in a hermitian Lagrange density because the finite meson lifetime implies nonhermitian contributions to the effective hamiltonian $\Lambda$. Explicitly, the CPT-violating contributions ${\cal L}_{\rm CPT}$ to the Lagrange density describing the propagation of $\phi$ in scalar effective field theory can be written as [29]

[TABLE]

where $(\widehat{k}_{a}){}^{\mu}$ is constructed as a series containing arbitrary even powers of derivatives $\partial_{\alpha}$. We assume here that the CPT violation is perturbative and maintains energy-momentum conservation, which implies $(\widehat{k}_{a}){}^{\mu}$ is independent of spacetime position. The constant coefficients appearing in the series $(\widehat{k}_{a}){}^{\mu}$ can then be taken hermitian without loss of generality. Note that the expression ${\cal L}_{\rm CPT}$ preserves the U(1) symmetry associated with the meson flavor, as expected for diagonal contributions to the effective hamiltonian $\Lambda$. In contrast, hermitian quadratic contributions to the Lagrange density simultaneously violating the flavor U(1) and CPT symmetries reduce to total-derivative terms leaving unaffected the physics. Hermitian quadratic terms violating the U(1) symmetry while preserving CPT also exist and can affect oscillations, but they are irrelevant in determining $\xi$ because they contribute only to off-diagonal components of $\Lambda$. The terms (2) are therefore the only ones of relevance for the calculation of $\xi$, and taking appropriate expectation values in unperturbed meson states can be expected to yield hermitian corrections to the effective hamiltonian and hence real contributions to $\Delta\Lambda$.

In momentum space with the identification $i\partial_{\mu}\leftrightarrow p_{\mu}$, we can write

[TABLE]

where the sum is over odd values of $d$. To avoid possible issues with an infinite range for this sum, we can take the number of Lorentz-violating terms to be arbitrary but finite [37]. The quantities $(k_{a}^{(d)})^{\mu\alpha_{1}\alpha_{2}\ldots\alpha_{d-3}}$ are the coefficients for CPT violation in the scalar effective field theory. The label $d$ specifies the mass dimension of the corresponding operator in ${\cal L}_{\rm CPT}$, and the coefficients can be taken as real spacetime constants with dimension $4-d$. Each coefficient has $d-2$ spacetime indices and is totally symmetric, so it is convenient to use a symmetric notation for the indices, writing the coefficient as $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$. Field redefinitions permit the traces of a coefficient at fixed $d$ to be absorbed into coefficients at lower $d$ [29], so all coefficients can be taken as traceless. The number of independent traceless components of $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ at fixed $d$ is $(d-1)^{2}$. We remark in passing that the properties of $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ imply that the mesons propagate along geodesics of a Riemann-Finsler spacetime [27], which for the case $d=3$ is a Randers spacetime [38] and for $d\geq 5$ is a $k$ spacetime [29]. In this picture, the flavor conversion from $P^{0}$ to $\overline{P^{0}}$ corresponds to a transition between two partial geodesics with opposite 4-velocities in Randers or $k$ space, matching the usual notion in relativistic quantum mechanics in which an antiparticle is interpreted as a particle moving backwards in time with opposite 3-velocity.

Measuring the coefficients $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ is the goal of experiments searching for CPT violation. The components of the coefficients are frame dependent, so reporting a measurement requires specifying a chosen inertial frame. Due to the rotation of the Earth and its revolution about the Sun, all laboratories on the surface of the Earth are noninertial. Instead, the canonical frame used in the literature to report results is the Sun-centered frame [39], for which the time coordinate $T$ has origin at the vernal equinox 2000, and the spatial coordinates $X^{J}=(X,Y,Z)$ are defined so that $Z$ parallels the Earth’s rotation axis, $X$ is directed from the Earth to the Sun at $T=0$, and $Y$ is the orthogonal axis in a right-handed system. The coefficients of experimental interest in the present context are therefore the components of $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ evaluated in the Sun-centered frame.

The general formalism (1) is valid for nonrelativistic mesons, and the quantities $\xi$ and $w$ governing CPT and T violation are associated with the meson and defined in a comoving frame. The frame dependence of the coefficients $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ means that some care is required in deriving an explicit expression for $\xi$. Several approaches are possible, all yielding the same results. We follow here a stepwise derivation inspired by the method introduced in Ref. [13] in the context of the minimal SME. Since the Sun-centered frame is the canonical choice for reporting results on $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$, we first consider a meson at rest in this frame. Results for a boosted meson and in other frames can then be derived by judicious use of particle and observer Lorentz transformations. Also, although any experimental measurement can involve coefficients with multiple values of $d$, for practical purposes it is often useful and conventional to restrict attention to only one value of $d$ at a time [24]. We therefore write

[TABLE]

where the sum is over odd values of $d$, and we present a derivation of $\xi^{\hskip-1.0pt(d)}{}$ at any fixed $d$.

According to the formalism (1), the value of $\xi^{\hskip-1.0pt(d)}{}$ is related to the difference of energies of the $P^{0}$ and $\overline{P^{0}}$ mesons. The leading-order shift of the energy of a meson at rest in the Sun-centered frame is given by the expectation value of the perturbation to the nonrelativistic hamiltonian. The scalar in the effective field theory has a Klein-Gordon kinetic term, so the relativistic Lagrange density contains the combination $\phi^{\dagger}p^{2}\phi+{\cal L}_{\rm CPT}(p)$. The CPT-violating correction to the unperturbed hamiltonian for a meson at rest in the Sun-centered frame can therefore be identified as $-{\cal L}_{\rm CPT}(m)/2m$, where $m$ is the $P^{0}$ mass. This constant shift gives equal and opposite contributions to the effective energies for $P^{0}$ and $\overline{P^{0}}$, which implies that the explicit form of $\xi^{\hskip-1.0pt(d)}{}$ for mesons at rest in the Sun-centered frame is

[TABLE]

To obtain a result valid for mesons moving in the Sun-centered frame, one must perform a particle boost while leaving unchanged the coefficients. Noting that the meson 4-velocity $\beta^{\mu}$ takes the form $\beta^{\mu}=(1,0,0,0)$ for mesons at rest, we can write $(k_{a}^{(d)})_{TT\cdots T}=\beta^{\mu_{1}}\beta^{\mu_{2}}\cdots\beta^{\mu_{d-2}}(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$. Performing the particle boost changes the meson 4-velocity to $\beta^{\mu}=\gamma(1,\vec{\beta})$ but leaves the coefficients $(k_{a}^{(d)})^{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ unchanged. This shows that the expression for $\xi^{\hskip-1.0pt(d)}{}$ for moving mesons in the Sun-centered frame is

[TABLE]

Note that this result reveals that a moving meson in the Sun-centered frame is affected by more components of coefficients for CPT violation than a meson at rest. Contributions from coefficients with different $d$ come with different powers of $\beta^{\mu}$ and so are physically distinct. This also confirms that the coefficients can be taken traceless, as any trace contribution is proportional to at least one power of the Minkowski metric and hence eliminates two or more factors of $\beta^{\mu}$ via the identity $\eta_{\mu\nu}\beta^{\mu}\beta^{\nu}=1$.

The expression (6) is invariant under changes of observer frame, which transform both the meson velocities and the coefficients for CPT violation. In particular, this means that the explicit expression for $\xi^{\hskip-1.0pt(d)}{}$ in an inertial frame instantaneously comoving with a laboratory on the Earth takes the form

[TABLE]

where $\beta^{\mu}$ is now the meson 4-velocity in the laboratory and the coefficients $(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}^{\rm lab}$ are measured in the laboratory frame. The equation relating $(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}^{\rm lab}$ in the laboratory to $(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ in the Sun-centered frame involves an instantaneous observer Lorentz transformation, so the rotation and revolution of the Earth imply that the coefficients $(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}^{\rm lab}$ depend on the time $T$ and on the location of the laboratory.

In practical applications, it is convenient to perform a time-space decomposition of the expression (7) and then express the results in terms of coefficients in the Sun-centered frame. To achieve this, first write $\beta^{\mu}=\gamma(1,\beta\hat{\beta})$ where $\beta$ is the magnitude of the meson three-velocity in the laboratory frame, $\hat{\beta}$ is a unit vector along its spatial direction, and $\gamma=1/\sqrt{(1-\beta^{2})}$ is the meson boost factor. Denoting the time index by $t$ and the spatial ones by $j$ with $j=x,y,z$, we can then decompose $\xi^{\hskip-1.0pt(d)}{}$ in the laboratory frame as

[TABLE]

where the sum over $k$ spans $0\leq k\leq d-2$ and ${}^{(d-2)}C_{k}$ is the usual binomial coefficient.

Next, note that the laboratory boost due to the rotation and revolution of the Earth is of order $10^{-4}$, so at zeroth order in this small quantity the transformation between $(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}^{\rm lab}$ in the laboratory and $(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ in the Sun-centered frame is a rotation. Implementing the rotation yields the desired final result,

[TABLE]

where $\hat{\beta}^{\prime J}$ is a unit vector with components in the Sun-centered frame given by

[TABLE]

Here, $\cos\chi\equiv\hat{z}\cdot\hat{Z}$ is the projection of the unit vector in the $z$ direction onto the unit vector in the $Z$ direction. This projection can be expressed as a trigonometric combination of the colatitude and orientation of the laboratory. The frequency $\omega_{\oplus}\simeq 2\pi/(23{\rm~{}h}~{}56{\rm~{}min})$ is the Earth’s sidereal rotation frequency. The time $T_{\oplus}$ is any convenient local sidereal time differing from the canonical time $T$ in the Sun-centered frame by an appropriate adjustment of the time zero. For example, one choice for $T_{\oplus}$ is associated with the standard laboratory frame defined in Ref. [39] and is shifted relative to $T$ by an integer number of sidereal days and an additional amount that depends on the laboratory longitude according to Eq. (43) of Ref. [40].

The result (9) exhibits several interesting features. As anticipated, the parameter $\xi^{\hskip-1.0pt(d)}{}=\xi^{\hskip-1.0pt(d)}{}(\beta,\hat{\beta},T_{\oplus},\chi)$ for CPT violation is found to depend explicitly on the magnitude and direction of the meson velocity, on the sidereal time, and on geometric factors associated with the location of the laboratory. The oscillatory dependence on sidereal time includes components ranging from the zeroth to the $(d-2)$th harmonic in the Earth’s sidereal frequency $\omega_{\oplus}$. For the special case of $d=3$ only the zeroth and first harmonics appear, in agreement with known results in the context of the minimal SME [13]. Expanding the expression (9) and comparing to the analogous SME result reveals that for $d=3$ the connection between the coefficient $(k_{a}^{(3)})_{\mu}$ in the scalar effective field theory and the SME coefficient combination $\Delta a_{\mu}$ is comparatively simple: $(k_{a}^{(3)})_{\mu}\approx 2\Delta a_{\mu}$. However, the corresponding matches for higher $d$ are more involved and lie outside our present scope [36].

In the nonminimal sector, the CPT-violating operator of lowest dimension has $d=5$. The corresponding coefficient is $(k_{a}^{(5)})_{\lambda\mu\nu}$, and its 20 symmetric components can conventionally be chosen as $(k_{a}^{(5)}){}_{TTT}$, $(k_{a}^{(5)}){}_{TTX}$, $(k_{a}^{(5)}){}_{TTY}$, $(k_{a}^{(5)}){}_{TTZ}$, $(k_{a}^{(5)}){}_{TXX}$, $(k_{a}^{(5)}){}_{TXY}$, $(k_{a}^{(5)}){}_{TXZ}$, $(k_{a}^{(5)}){}_{TYY}$, $(k_{a}^{(5)}){}_{TYZ}$, $(k_{a}^{(5)}){}_{TZZ}$, $(k_{a}^{(5)}){}_{XXX}$, $(k_{a}^{(5)}){}_{XXY}$, $(k_{a}^{(5)}){}_{XXZ}$, $(k_{a}^{(5)}){}_{XYY}$, $(k_{a}^{(5)}){}_{XYZ}$, $(k_{a}^{(5)}){}_{XZZ}$, $(k_{a}^{(5)}){}_{YYY}$, $(k_{a}^{(5)}){}_{YYZ}$, $(k_{a}^{(5)}){}_{YZZ}$, and $(k_{a}^{(5)}){}_{ZZZ}$. The requirement of tracelessness implies that $(k_{a}^{(5)})_{\mu\alpha}{}^{\alpha}=0$. This can conveniently be used to eliminate the four components $(k_{a}^{(5)}){}_{\mu ZZ}$, $\mu=T,X,Y,Z$ via

[TABLE]

thereby leaving 16 independent observable components.

With the conditions (11) understood, the expression (9) for $\xi^{(5)}$ takes the form

[TABLE]

The seven amplitudes ${\cal A}_{0}$, ${\cal A}_{1}$, ${\cal B}_{1}$, ${\cal A}_{2}$, ${\cal B}_{2}$, ${\cal A}_{3}$, ${\cal B}_{3}$ of the harmonic oscillations are functions of the coefficients $(k_{a}^{(5)})_{\lambda\mu\nu}$ for CPT violation in the Sun-centered frame, the meson velocity $\beta^{j}=(\beta^{x},\beta^{y},\beta^{z})$ in the laboratory frame, and the geometric factors $c\equiv\cos\chi$ and $s\equiv\sin\chi$. The zeroth harmonic is independent of sidereal time and has amplitude ${\cal A}_{0}$ given by

[TABLE]

The amplitudes ${\cal A}_{1}$ and ${\cal B}_{1}$ for the first harmonics are

[TABLE]

and

[TABLE]

The amplitudes ${\cal A}_{2}$ and ${\cal B}_{2}$ for the second harmonics are

[TABLE]

and

[TABLE]

Finally, the amplitudes ${\cal A}_{3}$ and ${\cal B}_{3}$ for the third harmonics are

[TABLE]

and

[TABLE]

If sufficient data are taken in a given experiment, then the expressions (13)-(19) reveal that binning in sidereal time permits the measurement of seven independent linear combinations of the 16 observable components of $(k_{a}^{(5)}){}_{\lambda\mu\nu}$. Since the specific linear combinations depend on the meson boost, an experiment with a sufficiently broad meson spectrum can obtain more independent measurements by binning in momentum as well. The dependence on geometric factors implies that distinct experiments with the same meson spectra can also have different sensitivities. For any specific meson species, all 16 components of $(k_{a}^{(5)}){}_{\lambda\mu\nu}$ appear in the above amplitudes with distinct multiplicative factors, so each component is therefore independently measurable in principle. However, if this separation is infeasible in a given experiment, then insight about the comparative sensitivities achieved for different components can nonetheless be obtained following standard practice in the field [24], by placing constraints on each independent component taken one at a time with all others set to zero.

Different experiments may prepare mesons in distinct ways. Some use uncorrelated mesons from various production processes, while others use correlated ones obtained from decays of quarkonia at rest or boosted. Measurements of coefficients with $d=3$ have been performed with uncorrelated mesons using the KTeV [7], D0 [11], FOCUS [8], and LHCb [12] detectors, while ones with unboosted correlated $K^{0}$ have been completed at KLOE [10] and ones with boosted correlated $B_{d}^{0}$ mesons at BaBar [9]. Other experiments could also achieve interesting sensitivities to coefficients for CPT violation. For example, the Belle II experiment [41] also involves correlated mesons and in principle could obtain competitive constraints [19]. Theoretical asymmetries that isolate CPT violation are discussed in Refs. [13, 14, 15, 16, 17, 18, 19], and investigations of these observables and other techniques have been adopted in the various experimental analyses. Analogous methodologies can be applied for other values of $d$. In particular, since experiments have already studied the zeroth and first harmonics of the sidereal time for all meson species, the results reported for coefficients with $d=3$ can be used to deduce constraints for coefficients with higher $d$. Note that the general result (9) reveals that a factor of the boost $\beta$ accompanies each appearance of a spatial index $J$ in any component of $(k_{a}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$, whereas components of the coefficients with $d=3$ are limited to at most a single such factor. Inferring new results in this way thus requires some care.

These ideas can be illustrated using the case of $d=5$, for which the zeroth and first harmonics are controlled by the amplitudes (13)-(LABEL:amp3). While many components of $(k_{a}^{(5)}){}_{\lambda\mu\nu}$ appear in these amplitudes, only the components $(k_{a}^{(5)}){}_{TTT}$ and $(k_{a}^{(5)}){}_{TTJ}$ are accompanied with zero or one power of the boost. We can therefore convert published measurements of coefficients with $d=3$ into measurements of $(k_{a}^{(5)}){}_{TTT}$ and $(k_{a}^{(5)}){}_{TTJ}$, provided we assume that all other observable coefficients entering the amplitudes (13)-(LABEL:amp3) vanish. At a cruder level with the boost factors disregarded, constraints could in principle be extracted also on the component combinations $(k_{a}^{(5)}){}_{TYZ}$, $(k_{a}^{(5)}){}_{TXX}+(k_{a}^{(5)}){}_{TYY}$, and $(k_{a}^{(5)}){}_{XXZ}+(k_{a}^{(5)}){}_{YYZ}$ via the amplitude ${\cal A}_{0}$, and on the component combinations $(k_{a}^{(5)}){}_{TXZ}+(k_{a}^{(5)}){}_{TYZ}$, $(k_{a}^{(5)}){}_{XXX}+(k_{a}^{(5)}){}_{XYY}$, and $(k_{a}^{(5)}){}_{XXY}+(k_{a}^{(5)}){}_{YYY}$ via the amplitudes ${\cal A}_{1}$ and ${\cal B}_{1}$. A complete coverage of all $TJK$ and $JKL$ components of $(k_{a}^{(5)}){}_{\lambda\mu\nu}$ would require a study of the second and third sidereal harmonics using the amplitudes (16)-(19), which to date is lacking in the literature for every meson species. Here, we focus on extracting constraints on the components $(k_{a}^{(5)}){}_{TTT}$ and $(k_{a}^{(5)}){}_{TTJ}$, leaving other prospective investigations open for future research.

In the theoretical analysis above, we considered CPT violation in a generic neutral-meson system using notation that makes no distinction between the different species. However, the behavior of each meson species $P^{0}=K^{0}$, $D^{0}$, $B_{d}^{0}$, $B_{s}^{0}$ can in principle be governed by distinct coefficients $(k_{a,P}^{(d)})_{\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ controlling CPT violation. For instance, in the case $d=5$ with coefficients $(k_{a,P}^{(5)}){}_{\lambda\mu\nu}$, experiments can report bounds on a total of 64 independent observables for CPT violation, corresponding to the independent traceless components of the coefficients $(k_{a,K}^{(5)}){}_{\lambda\mu\nu}$, $(k_{a,D}^{(5)}){}_{\lambda\mu\nu}$, $(k_{a,B_{d}}^{(5)}){}_{\lambda\mu\nu}$, and $(k_{a,B_{s}}^{(5)}){}_{\lambda\mu\nu}$ that are required to characterize $d=5$ CPT violation completely across the four neutral-meson systems. In particular, in extracting results for coefficients with $d=5$ from existing experimental results on coefficients with $d=3$, we can in principle access the 16 components $(k_{a,P}^{(5)}){}_{TTT}$ and $(k_{a,P}^{(5)}){}_{TTJ}$ among the 64 independent observable components of $(k_{a,P}^{(5)}){}_{\lambda\mu\nu}$.

The equations implementing the conversion from the published $d=3$ results to $d=5$ ones can be obtained by comparing the amplitudes for $\xi^{(3)}$ obtained from the expression (9) with the amplitudes (13)-(LABEL:amp3) for $\xi^{(5)}$. The explicit form of the match is

[TABLE]

Some results for $(k_{a,P}^{(5)}){}_{TTT}$ and $(k_{a,P}^{(5)}){}_{TTJ}$ extracted from the existing literature using this match are displayed in Table 1. The table has four parts, one for each of the four mesons $K^{0}$, $D^{0}$, $B_{d}^{0}$, and $B_{s}^{0}$. Within each part, the rows are organized according to the chronological appearance of the original constraint for the $d=3$ case. The first column contains the combinations of components of $(k_{a,P}^{(5)}){}_{\lambda\mu\nu}$ for which results can be deduced. The quantities $N^{P}$ appearing in some of the combinations are defined as $N^{P}=\Delta m/\Delta\gamma$, with $\Delta m=|m_{b}-m_{a}|$, $\Delta\gamma=|\gamma_{a}-\gamma_{b}|$ evaluated for the $P^{0}$-meson system in question. The second column lists the results expressed in units of inverse GeV. In some cases the result is a bound on a magnitude, in others it is a measurement with standard deviation, and in yet others it is a range of allowed values, in accordance with the presentation of the original measurements for $d=3$ in the literature. In obtaining these results, we adopted mean values of the boost factors from the published meson spectra. This procedure could in principle be refined by the corresponding experimental collaborations via detailed reanalysis of the original data incorporating the full meson spectra. The final column provides citations to the source literature reporting the original measurements of coefficients with $d=3$.

The results in Table 1 represent the first reported sensitivities to nonminimal coefficients for CPT violation in neutral-meson oscillations. They extend and complement sensitivities to $d=5$ spin-independent CPT violation obtained from analyses of experiments with other systems, including charged leptons, protons, neutrons, and neutrinos [42]. No comparable effects for photons or gravity are possible in the context of effective field theory, where the $d=5$ CPT-violating operators are necessarily spin dependent. However, potential sensitivities to quark-sector SME coefficients controlling $d=5$ spin-independent CPT violation have been proposed for processes such as deep inelastic and Drell-Yan scattering [35, 43] and are expected to modify top-quark production and decay in analogy to known effects in the minimal SME [44]. Spin-independent quark-sector coefficients with $d=5$ can also induce phenomenologically viable baryogenesis in thermal equilibrium [45], potentially bypassing the necessity for CP violation beyond the SM and the Sakharov requirement of nonequilibrium processes [46]. The results obtained in the present work are competitive, thereby reconfirming the exquisite sensitivity of meson interferometry to CPT violation. The prospects are excellent for further advances and potential discovery in searches for CPT violation using the neutral-meson systems.

Acknowledgments

This work was supported in part by the United States Department of Energy under grant number DE-SC0010120 and by the Indiana University Center for Spacetime Symmetries.