#  $\varepsilon^{\prime}/\varepsilon$ in the Standard Model and Beyond: 2021

TUM-IAS, Lichtenbergstr. 2a, D-85748 Garching, Germany,
Technical University Munich, Physics Department, D-85748 Garching, Germany,
E-mail

###### Abstract:

The present status of the ratio $\varepsilon^{\prime}/\varepsilon$ in the
Standard Model (SM) is summarized. In particular we stress the differences
between three approaches that attempt to calculate
$\varepsilon^{\prime}/\varepsilon$ within the SM: Lattice QCD, Dual QCD and
Chiral Perturbation Theory. As presently there is still a significant room
left for new physics (NP) contributions, we summarize the present status of
the technology that has been recently developed with the goal to analyze
$\varepsilon^{\prime}/\varepsilon$ and its correlations with other observables
within the Weak Effective Theory (WET) and the Standard Model Effective Field
Theory (SMEFT). We also make a few comments on the interpretation of the main
dynamics behind the $\Delta I=1/2$ rule in $K\to\pi\pi$ decays.

##  1 Overture

The ratio $\varepsilon^{\prime}/\varepsilon$ measuring the size of the direct
CP violation in $K_{L}\to\pi\pi$ decays ($\varepsilon^{\prime}$) relative to
the indirect one described by $\varepsilon$ is very sensitive to new sources
of CP violation. As such it played a prominent role in particle physics
already for 45 years [1].

Due to the smallness of $\varepsilon^{\prime}/\varepsilon$ its measurement
required heroic efforts in the 1980s and the 1990s on both sides of the
Atlantic with final results presented by NA48 and KTeV collaborations at the
beginning of this millennium. On the other hand, even 45 years after the first
calculation of $\varepsilon^{\prime}/\varepsilon$ we do not know to which
degree the Standard Model (SM) agrees with this data and how large is the room
left for new physics (NP) contributions to this ratio. This is due to
significant non-perturbative (hadronic) uncertainties accompanied by partial
cancellations between the QCD penguin contributions and electroweak penguin
contributions. In addition to the calculation of hadronic matrix elements of
the relevant operators including isospin breaking effects and QED corrections,
it is crucial to evaluate accurately the Wilson coefficients of the relevant
operators. While the significant control over the latter short distance
effects has been achieved already in the early 1990s, with several
improvements since then, different views on the non-perturbative contributions
to $\varepsilon^{\prime}/\varepsilon$ have been expressed by different authors
over last thirty years. In fact even at the dawn of the 2020s the uncertainty
in the room left for NP contributions to $\varepsilon^{\prime}/\varepsilon$ is
still very significant, which I find to be very exciting.

As in addition to [1], this topic is presented in my recent book [2] and in my
recent contribution to Kaon 2019 [3] some overlap with these reviews is
unavoidable but I will attempt to reduce it as far as it is possible without
loosing most important information.

Let us then look first at the effective Hamiltonian relevant for $K\to\pi\pi$
decays at the low energy scales. It has the following general structure:

${\cal
H}_{\text{eff}}=\sum_{i}C_{i}\mathcal{O}_{i}^{\text{SM}}+\sum_{j}C_{j}^{\text{NP}}\mathcal{O}_{j}^{\text{NP}}\,,\qquad
C_{i}=C_{i}^{\text{SM}}+\Delta_{i}^{\text{NP}},$ (1)

where

  * •

$\mathcal{O}_{i}^{\text{SM}}$ are local operators present in the SM and
$\mathcal{O}_{j}^{\text{NP}}$ are new operators having typically new Dirac
structures, in particular scalar-scalar and tensor-tensor ones.

  * •

$C_{i}$ and $C_{j}^{\text{NP}}$ are the Wilson coefficients of these
operators. NP effects modify not only the Wilson coefficients of SM operators
but also generate new operators with non-vanishing $C_{j}^{\text{NP}}$.

The amplitude for the transition $K\to\pi\pi$ can now be written as follows

${\cal
A}(K\to\pi\pi)=\sum_{i}C_{i}\langle\pi\pi|\mathcal{O}_{i}^{\text{SM}}|K\rangle+\sum_{j}C_{j}^{\text{NP}}\langle\pi\pi|\mathcal{O}_{j}^{\text{NP}}|K\rangle\,.$
(2)

The coefficients $C_{i}$ and $C_{j}^{\text{NP}}$ can be calculated in the
renormalization group (RG) improved perturbation theory. The status of these
calculations is by now very advanced, as reviewed in 2014 in [4] including
also rare meson decays and quark mixing. I plan to update this review this
year. The complete NLO corrections to $K\to\pi\pi$ have been calculated almost
30 years ago [5, 6, 7, 8, 9, 10]. The dominant NNLO QCD corrections to
electroweak penguin (EWP) contributions have been presented in [11] and those
to QCD penguins (QCDP) in [12, 13, 14]. These NNLO corrections reducing
various scale and renormalization scheme dependences will play a significant
role when the calculation of hadronic matrix elements will be brought under
control. On the whole, the status of present short distance (SD) contributions
to $\varepsilon^{\prime}/\varepsilon$ is satisfactory.

The evaluation of the hadronic matrix elements is a different story. In
$K\to\pi\pi$ decays, we have presently three approaches to our disposal:

  * •

Lattice QCD (LQCD). It is a sophisticated numerical method with very demanding
calculations lasting many years, even decades. But eventually in the case of
$K\to\pi\pi$ decays and $K^{0}-\bar{K}^{0}$ mixing this method is expected to
give the ultimate results for $\varepsilon^{\prime}/\varepsilon$, $\Delta
I=1/2$ rule and $K^{0}-\bar{K}^{0}$ mixing, both in the SM and beyond it. For
$K\to\pi\pi$ only results for the SM operators are known and they are far from
being satisfactory. The ones for BSM $K^{0}-\bar{K}^{0}$ matrix elements are
already known with respectable precision and interesting results have been
obtained for long distance contributions to $\Delta M_{K}$ [15, 16].

  * •

Dual QCD (DQCD) proposed already in the 1980s [17], significantly improved in
the last decade [18] and also very recently as reported below. This approach,
which opposed to LQCD is fully analytical, allows to obtain results for
$K\to\pi\pi$ decays and $K^{0}-\bar{K}^{0}$ mixing much faster than it is
possible with the LQCD, typically within a few months. Therefore several
relevant results have been obtained already in the 1980s and confirmed within
uncertainties by LQCD in the last decade. While not as accurate as the
expected ultimate LQCD calculations, it allowed already to calculate hadronic
matrix elements for all BSM operators entering $K\to\pi\pi$ decays [19] and
$K^{0}-\bar{K}^{0}$ mixing [20]. Very importantly this approach allows to get
the insight into the QCD dynamics at low energy scales which is not possible
using a purely numerical method like LQCD. A good example are four hadronic
matrix elements of BSM operators entering $K^{0}-\bar{K}^{0}$ mixing [20]. For
a detailed exposition of this approach see [18, 2, 21]. More about it below.

  * •

Chiral Perturbation Theory (ChPT) developed since 1978 [22, 23, 24, 25, 26,
27, 28] and discussed recently in [29, 30] in the contex of
$\varepsilon^{\prime}/\varepsilon$. It is based on global symmetries of QCD
with the QCD dynamics parametrized by low-energy constants $L_{i}$ that enter
the counter terms in meson loop calculations. $L_{i}$ can be extracted from
the data or calculated by LQCD. Presently this framework as applied to non-
leptonic transitions has serious difficulties in matching long distance (LD)
and short distance (SD) contributions, a problem admitted by the authors of
[30, 31]. For instance the expression for the matrix element of the QCDP
operator $Q_{6}$ in terms of $L_{5}$ is only valid in the large $N$ limit,
that is using factorization. On the other hand the dominant QCD dynamics in
Wilson coefficients is given by non-factorizable contributions. This problem
is absent in LQCD and DQCD as we will discuss below. Therefore, while the ChPT
approach is very suitable for leptonic and semi-leptonic Kaon decays, it can
only provide partial information on $\varepsilon^{\prime}/\varepsilon$ and the
$\Delta I=1/2$ rule in the form of isospin breaking effects and final state
interactions (FSI). Yet, in the case of isospin breaking contributions to
$\varepsilon^{\prime}/\varepsilon$ the difficulties in the matching in
question imply a rather significant error as we will see below.

This writing is arranged as follows. In Section 2 I will briefly describe the
DQCD approach. In Section 3 I will summarize my view on the present status of
$\varepsilon^{\prime}/\varepsilon$ in the SM. In Section 4 I will review the
recent progress in the evaluation of $\varepsilon^{\prime}/\varepsilon$ within
the Weak Effective Theory (WET) and the Standard Model Effective Field Theory
(SMEFT). In Section 5 several comments on the dynamics behind the $\Delta
I=1/2$ rule in $K\to\pi\pi$ decays will be made. A brief outlook in Section 6
ends this presentation.

##  2  The Dual QCD Approach: A Grand View

This analytic approach to $K\to\pi\pi$ decays and $K^{0}-\bar{K}^{0}$ mixing
in [17, 32, 18] is based on the ideas of ’t Hooft and Witten [33, 34, 35, 36]
who studied QCD with a large number $N$ of colours. In this limit QCD is dual
to a theory of weakly interacting mesons with the coupling ${\cal O}(1/N)$ and
in particular in the strict large $N$ limit it becomes a free theory of
mesons, simplifying the calculations significantly. With non-interacting
mesons the factorization of matrix elements of four-quark operators into
matrix elements of quark currents and quark densities, used adhoc in the 1970s
and early 1980s, is automatic and can be considered as a property of QCD in
this limit [37]. But the factorization cannot be the whole story as the most
important QCD effects related to asymptotic freedom are related to non-
factorizable contributions generated by exchanges of gluons. In DQCD this role
is played by meson loops that represent dominant non-factorizable
contributions at the very low energy scales. Calculating these loops with a
momentum cut-off $\Lambda$ one finds then that the factorization in question
does not take place at values of $\mu\geq 1\,{\rm GeV}$ at which Wilson
coefficients are calculated, but rather at very low momentum transfer between
colour-singlet currents or densities.

Thus, even if in the large $N$ limit the hadronic matrix elements factorize
and can easily be calculated, in order to combine them with the Wilson
coefficients, loops in the meson theory have to be calculated. In contrast to
chiral perturbation theory, in DQCD a physical cut-off $\Lambda$ is used in
the integration over loop momenta. As discussed in detail in [17, 18] this
allows to achieve a much better matching with short distance contributions
than it is possible in ChPT, which uses dimensional regularization. The cut-
off $\Lambda$ is typically chosen around $0.7\,{\rm GeV}$ when only
pseudoscalar mesons are exchanged in the loops [17] and can be increased up to
$0.9\,{\rm GeV}$ when contributions from lowest-lying vector mesons are taken
into account as done in [18]. These calculations are done in a momentum
scheme, but as demonstrated in [18], they can be matched to the commonly used
naive dimensional regularization (NDR) scheme. Once this is done it is
justified to set $\Lambda\approx\mu$.

The application of DQCD to weak decays consists in any NP model of the
following steps:

Step 1: At $\Lambda_{\text{NP}}$ one integrates out the heavy degrees of
freedom and performs the RG evolution including Yukawa couplings and all gauge
interactions present in the SM down to the electroweak scale. This evolution
involves in addition to SM operators also beyond the SM (BSM) operators. This
is the SMEFT.

Step 2: At the electroweak scale $W$, $Z$, top quark and the Higgs are
integrated out and the SMEFT is matched onto the WET with only SM quarks
except the top-quark, the photon and the gluons. Subsequently QCD and QED
evolution is performed down to scales ${\cal O}(1\,{\rm GeV})$. The presence
of many new operators makes this evolution rather involved. Fortunately as we
will report below this evolution is known by now at NLO level in QCD.

Step 3: Around scales of ${\cal O}(1\,{\rm GeV})$ the matching to the theory
of mesons is performed and the quark evolution is followed by the meson
evolution down to the factorization scale.

Step 4: At the appropriate hadronic scale between the kaon and pion masses,
the matrix elements of all operators are calculated in the large $N$ limit,
that is using factorization of matrix elements into products of currents or
densities.

I do not claim that these are all QCD effects responsible for non-leptonic
transitions, but these evolutions based entirely on non-factorizable QCD
effects, both at short distance and long distance scales, appear to be the
main bulk of QCD dynamics responsible for the $\Delta I=1/2$ rule,
$\varepsilon^{\prime}/\varepsilon$ and $K^{0}-\bar{K}^{0}$ mixing. Past
successes of this approach have been reviewed in [2, 21, 38]. They are related
in particular to the non-perturbative parameter $\hat{B}_{K}$ in
$K^{0}-\bar{K}^{0}$ mixing and $\Delta I=1/2$ rule [18]. In fact DQCD allowed
for the first time to identify already in 1986 the dominant mechanism behind
this rule [17]. It is simply the quark evolution from short distance scales,
possibly involving NP, down to scales of ${\cal O}(1\,{\rm GeV})$, followed by
meson evolution down to the factorization scale around the kaon and pion
masses.

In summary DQCD turns out to be an efficient approximate method for obtaining
results for non-leptonic decays, years and even decades, before useful results
from numerically sophisticated and demanding lattice calculations could be
obtained. This will be particularly clear in Section 4, where we will turn our
attention to the results in the WET and the SMEFT which are to date available
only within DQCD approach.

##  3 $\varepsilon^{\prime}/\varepsilon$ in the SM

The situation of $\varepsilon^{\prime}/\varepsilon$ as of March 2022 can be
summarized as follows.

The experimental world average from NA48 [39] and KTeV [40, 41] collaborations
reads

$\boxed{(\varepsilon^{\prime}/\varepsilon)_{\text{exp}}=(16.6\pm 2.3)\times
10^{-4}\,.}$ (3)

The most recent result from LQCD, that is from the RBC-UKQCD collaboration
[42], reads

$\boxed{(\varepsilon^{\prime}/\varepsilon)_{\rm SM}=(21.7\pm 8.4)\times
10^{-4}\,,\qquad(\text{RBC-UKQCD}-2020),}$ (4)

where statistical, parametric and systematic uncertainties have been added in
quadrature. The central value is by an order of magnitude larger than the
central 2015 value presented by this collaboration but is subject to large
systematic uncertainties which dominate the quoted error. It is based on the
improved values of the hadronic matrix elements of QCDP, includes the Wilson
coefficients at the NLO level but does not include isospin breaking effects,
charm contributions and NNLO QCD effects.

The most recent estimate of $\varepsilon^{\prime}/\varepsilon$ in the SM from
ChPT [30, 31] reads

$\boxed{(\varepsilon^{\prime}/\varepsilon)_{\text{SM}}=(14\pm 5)\times
10^{-4}\,,\qquad(\text{ChPT}-2019).}$ (5)

The large error is related to the problematic matching of LD and SD
contributions in this approach which can be traced back to the absence of
meson evolution in this approach. Here the isospin breaking corrections are
included but as discussed below they are subject to significant uncertainties
again related to the problematic matching of LD and SD contributions.

Finally, based on the insight from DQCD obtained in collaboration with Jean-
Marc Gérard [43, 44] one finds [1]

$\boxed{(\varepsilon^{\prime}/\varepsilon)_{\rm SM}=(5\pm 2)\cdot
10^{-4},\qquad(\text{DQCD}-2020).}$ (6)

While the results in (4) and (5) are fully consistent with the data in (3),
the DQCD result in (6) if confirmed by other groups would one day imply a
significant anomaly and NP at work.

Let us then have a look at the phenomenological expression for
$\varepsilon^{\prime}/\varepsilon$ in the SM in order to understand the origin
of these differences. This formula presented in [45] reads

$\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)_{\text{SM}}={\rm
Im}\lambda_{\rm t}\cdot\left[\,\big{(}1-\hat{\Omega}_{\rm
eff}\big{)}\big{(}-2.9+15.4\,B_{6}^{(1/2)}(\mu^{*})\big{)}+2.0-8.0\,B_{8}^{(3/2)}(\mu^{*})\,\right].$
(7)

It includes NLO QCD corrections to the QCD penguin (QCDP) contributions and
NNLO contributions to electroweak penguins (EWP). The coefficients in this
formula and the parameters $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$, conventionally
normalized to unity at the factorization scale, are scale dependent. Their
values for different scales are collected in Table 1 of [45]. Here we will set
$\mu^{*}=1\,{\rm GeV}$ because at this scale it is most convenient to compare
the values for $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ obtained in the three non-
perturbative approaches in question. The four contributions in (7) are
dominated by the following operators:

  * •

The terms involving the non-perturbative parameters $B_{6}^{(1/2)}$ and
$B_{8}^{(3/2)}$ contain only the contributions from the dominant QCDP operator
$Q_{6}$ and the dominant EWP operator $Q_{8}$, respectively. There are two
main reasons why $Q_{8}$ can compete with $Q_{6}$ here despite the smallness
of the electroweak couplings relative to the QCD one. In the basic formula for
$\varepsilon^{\prime}/\varepsilon$ its contribution is enhanced relative to
the $Q_{6}$’s one by the factor ${{\rm Re}A_{0}/{\rm Re}A_{2}}=22.4$ with
$A_{0,2}$ being isospin amplitudes. In addition its Wilson coefficient is
enhanced for the large top-quark mass which is not the case of the $Q_{6}$’s
one [46, 47]. The expressions for these two operators and the remaining
operators mentioned below are well known and can be found in [1, 2].

  * •

The term $-2.9$ is fully dominated by the QCDP operator $Q_{4}$.

  * •

The term $+2.0$ is fully dominated by EWP operators $Q_{9}$ and $Q_{10}$.

  * •

The quantity $\hat{\Omega}_{\rm eff}$ represents the isospin breaking
corrections and QED corrections beyond EWP contributions. It is not in the
ballpark of a few percent as one would naively expect, because in
$\varepsilon^{\prime}/\varepsilon$ it is again enhanced by the factor ${{\rm
Re}A_{0}/{\rm Re}A_{2}}=22.4$.

  * •

${{\rm Im}\lambda_{\rm t}}$ is the CKM factor that within a few percent is in
the ballpark of ${1.45\cdot 10^{-4}}$.

Now at $\mu=1\,{\rm GeV}$ the values of $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ in
the three non-perturbative approaches are found to be:

$B_{6}^{(1/2)}(1\,{\rm GeV})=1.49\pm 0.25,\quad B_{8}^{(3/2)}(1\,{\rm
GeV})=0.85\pm 0.05\,,\quad(\text{RBC-UKQCD}-2020).$ (8)

$B_{6}^{(1/2)}(1\,{\rm GeV})=1.35\pm 0.20,\quad B_{8}^{(3/2)}(1\,{\rm
GeV})=0.55\pm 0.20\,,\quad(\text{ChPT}-2019).$ (9)

$B_{6}^{(1/2)}(1\,{\rm GeV})\leq 0.6,\quad B_{8}^{(3/2)}(1\,{\rm GeV})=0.80\pm
0.10\,,\quad(\text{DQCD}-2015).$ (10)

Let us begin with the good news. There is a very good agreement between LQCD
and DQCD as far as EWP contribution to $\varepsilon^{\prime}/\varepsilon$ is
concerned. This implies that this contribution to
$\varepsilon^{\prime}/\varepsilon$, that is unaffected by leading isospin
breaking corrections, is already known within the SM with acceptable accuracy:

$\boxed{(\varepsilon^{\prime}/\varepsilon)^{\text{EWP}}_{\text{SM}}=-(7\pm
1)\times 10^{-4}\,,\qquad(\text{LQCD~{}and~{}DQCD}).}$ (11)

Because both LQCD and DQCD can perform much better in the case of EWPs than in
the case of QCDPs I expect that this result will remain with us for the coming
years.

On the other hand the value from ChPT $B_{8}^{(3/2)}\approx 0.55$ [30] implies
on the basis of the last two terms in (7) the EWP contribution roughly by a
factor of 2 below the result in (11). This lower value originates in the
suppression caused by FSI which seems to be not supported by the results from
the RBC-UKQCD collaboration. Yet, the uncertainty in the ChPT estimate is
large. It would be good to clarify this difference in the future.

The case of QCDPs is a different story. Here the LQCD value of $B_{6}^{(1/2)}$
overshoots the DQCD one by more than a factor of two and consequently despite
the agreement on EWP contribution the result in (4) from RBC-UKQCD differs by
roughly a factor of four from the one of DQCD in (6).

This difference will surely decrease when the RBC-UKQCD collaboration will
include isospin breaking and QED corrections. As far as these corrections are
concerned LQCD, ChPT, DQCD use in their estimates respectively

$\hat{\Omega}_{\rm eff}=0,\qquad\hat{\Omega}_{\rm eff}^{(8)}=(17\pm
9)\,10^{-2},\qquad\hat{\Omega}_{\rm eff}^{(9)}=(29\pm 7)\,10^{-2},$ (12)

where the index “(9)” indicates that the full nonet of pseudoscalars has been
taken into account in the case of DQCD. This means that also
$\eta-\eta^{\prime}$ mixing has been taken explicitly into account [48]. In
the octet scheme, necessarily used in ChPT, this mixing is buried in a poorly
known low-energy constant $L_{7}$ and I expect that including the effect of
this mixing in $\varepsilon^{\prime}/\varepsilon$ will remain a challenge for
ChPT for some time. Yet, it is evident from the analysis in [48] that this
mixing enhances significantly $\hat{\Omega}_{\rm eff}$, the fact known already
for 35 years [49, 50].

There is still another reason why the DQCD result in (6) is much lower than
the ones in LQCD and ChPT. We include NNLO QCD corrections to both QCDP and
EWP contributions that together provide a downward shift of
$\varepsilon^{\prime}/\varepsilon$ in the ballpark of $3\times 10^{-4}$. While
ChPT experts criticized us for ignoring FSI, this is really a fake news. We
devoted a full paper to FSI [44]. While we agreed that in DQCD precise
estimate of FSI is difficult, we gave arguments why the effect of the
enhancement of $B_{6}^{(1/2)}$ by FSI is likely to be smaller than its
suppression by the meson evolution. In fact FSI are included in (6) with
$B_{6}^{(1/2)}\approx 1.0$ corresponding to the upper limit when subleading
FSI would fully cancel the suppression by the leading meson evolution.

In Table 1 I summarize the main SM dynamics which in my view is responsible
for the strong suppression of $\varepsilon^{\prime}/\varepsilon$ below the
experimental value. The question marks in the case of ChPT mean that presently
it is not clear how well the low energy constants $L_{5}$ and $L_{7}$ will
describe meson evolution and $\eta-\eta^{\prime}$ mixing, respectively. To be
fair one should add that while RBC-UKQCD and ChPT experts claim to have a
satisfactory treatment of FSI for penguin operators, in the case of DQCD it
can only be argued at present that the subleading FSI can at best cancel the
suppression of QCDP contribution by the leading meson evolution [44].

Table 1 summarizes my view of the controversy between the three approaches. In
other words my message for unbiased non-experts to take home is the following

  * •

RBC-UKQCD collaboration and ChPT experts do not claim that there is no NP in
$\varepsilon^{\prime}/\varepsilon$. But as of March 2022 their methods are not
sufficiently powerful to see the anomaly in
$\varepsilon^{\prime}/\varepsilon$. I expect that RBC-UKQCD collaboration will
see this anomaly after they reached their goals for coming years summarized
recently in [51].

  * •

DQCD approach, even if approximate, can much faster see the underlying
dynamics responsible for possible anomaly in
$\varepsilon^{\prime}/\varepsilon$ because it is an analytical approach and
important QCD dynamics is not hidden in the numerical values of hadronic
matrix elements evaluated by LQCD nor hidden in low energy constants $L_{i}$
of ChPT.

I am truly delighted that at least one group strongly believes in NP in
$\varepsilon^{\prime}/\varepsilon$ at work. How boring would be a situation of
$\varepsilon^{\prime}/\varepsilon$ if also DQCD found
$\varepsilon^{\prime}/\varepsilon$ the ballpark of the values presented by
RBC-UKQCD and ChPT. Simultaneously I understand that both groups are, in
contrast to me, not motivated to study NP in
$\varepsilon^{\prime}/\varepsilon$. I wish them therefore luck in improving
their calculations and will now go beyond the SM.

| DQCD | RBC-UKQCD | ChPT
---|---|---|---
Large $m_{t}$ | $\star$ | $\star$ | $\star$
Meson evolution | $\star$ | $\star$ | $L_{5}(?)$
QCDP at NNLO | $\star$ | $-$ | $-$
EWP at NNLO | $\star$ | $-$ | $-$
IB (Octet) | $\star$ | $-$ | $\star$
$\eta-\eta^{\prime}$ mixing | $\star$ | $-$ | $L_{7}(?)$
Table 1: Comparision of various suppression mechanism of
$\varepsilon^{\prime}/\varepsilon$ in the SM taken ($\star$) or not taken
($-$) into account in a given approach. See text for more details.

##  4 $\varepsilon^{\prime}/\varepsilon$ beyond the SM

A new era for $\varepsilon^{\prime}/\varepsilon$ began in 2015, when the RBC-
UKQCD collaboration [52, 53] presented their first results for
$\varepsilon^{\prime}/\varepsilon$ with the central value by rougly a factor
of 15 below their 2020 value in (4). This inspired many authors to search for
NP which would explain this anomaly. I will not review these analyzes again as
I have done it already in [1, 2, 3]. But I would like to stress that all these
analyses with a few exceptions involved only the SM operators, that is the
first term in (2) in wich NP modifies the Wilson coefficients of the SM
operators, in particular the one of $Q_{8}$.

Here I would like to confine first the discussion to the second term in (2)
and subsequently to $\varepsilon^{\prime}/\varepsilon$ in the framework of the
WET and the SMEFT which include all operators contributing to
$\varepsilon^{\prime}/\varepsilon$ in these effective theories.

In fact since 2018 a significant progress towards the general search for NP in
$\varepsilon^{\prime}/\varepsilon$ with the help of DQCD and more generally
has been made. I will just list all papers written by us in this context and
describe in more details only the most recent ones. Here we go.

  * •

The first to date results for the $K\to\pi\pi$ matrix elements of the chromo-
magnetic dipole operators [54] that are compatible with the LQCD results for
$K\to\pi$ matrix elements of these operators obtained earlier in [55].

  * •

The first to date calculation of $K\to\pi\pi$ matrix elements of all four-
quark BSM operators, including scalar and tensor operators, by DQCD [19].

  * •

The derivation of a master formula for $\varepsilon^{\prime}/\varepsilon$
[56], which can be applied to any theory beyond the SM in which the Wilson
coefficients of all contributing operators have been calculated at the
electroweak scale. The relevant hadronic matrix elements of BSM operators used
in this formula are from the DQCD, as lattice QCD did not calculate them yet,
and the SM ones from LQCD.

  * •

This allowed to perform the first to date model-independent anatomy of the
ratio $\varepsilon^{\prime}/\varepsilon$ in the context of the $\Delta S=1$
effective theory with operators invariant under QCD and QED and in the context
of the SMEFT with the operators invariant under the full SM gauge group [57].

  * •

Finally the insight from DQCD [20] into the values of BSM $K^{0}-\bar{K}^{0}$
elements obtained by LQCD made sure that the meson evolution is hidden in
lattice calculations.

The main messages from these papers are as follows:

  * •

The inclusion of the meson evolution in the phenomenology of any non-leptonic
transition like $K^{0}-\bar{K}^{0}$ mixing and $K\to\pi\pi$ decays with
$\varepsilon^{\prime}/\varepsilon$ and the $\Delta I=1/2$ rule is mandatory!

  * •

Meson evolution is hidden in LQCD results, but among analytic approaches only
DQCD takes this important QCD dynamics into account. Whether meson evolution
is present in the low energy constant $L_{5}$ of ChPT is an interesting
question, still to be answered.

  * •

Most importantly, the meson evolution turns out to have the pattern of
operator mixing, both for SM and BSM operators, to agree with the one found
perturbatively at short distance scales. This allows for a satisfactory, even
if approximate, matching between Wilson coefficients and hadronic matrix
elements.

The BSM part of $\varepsilon^{\prime}/\varepsilon$ in our 2018 papers was
performed at LO in QCD RG quark evolution. In the more recent papers we made
progress by including NLO QCD corrections and in this context we derived
master formulae not only for $\varepsilon^{\prime}/\varepsilon$ and
$K\to\pi\pi$ but generally for non-leptonic $\Delta F=1$ and $\Delta F=2$
transitions. Let me briefly summarize these papers.

In [58] we reconsidered the complete set of four-quark operators in the WET
for non-leptonic $\Delta F=1$ decays that govern $s\to d$ and $b\to d,s$
transitions in the SM and beyond, at NLO in QCD. In particular we discussed
the issue of transformations between operator bases beyond leading order to
facilitate the matching to high-energy completions or the SMEFT at the
electroweak scale. As a first step towards a SMEFT NLO analysis of
$K\to\pi\pi$ and non-leptonic $B$-meson decays, we calculated the relevant WET
Wilson coefficients including two-loop contributions to their renormalization
group running, and expressed them in terms of the Wilson coefficients in a
particular operator basis, the so-called JMS basis [59] for which the one-loop
matching to SMEFT is already known [60].

In [61] we have presented for the first time the NLO master formula for the
BSM part of $\varepsilon^{\prime}/\varepsilon$ expressed in terms of the
Wilson coefficients of all contributing operators of WET evaluated at the
electroweak scale. To this end we use again JMS basis and the results from
[58]. The relevant hadronic matrix elements of BSM operators at the
electroweak scale are taken from Dual QCD approach and the SM ones from
lattice QCD. It includes the renormalization group evolution and quark-flavour
threshold effects at NLO in QCD from hadronic scales, at which these matrix
elements have been calculated, to the electroweak scale.

This updated master formula for the BSM part of
$\varepsilon^{\prime}/\varepsilon$ reads reads [61]

$\displaystyle\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)_{\text{BSM}}$
$\displaystyle=\;\;\sum_{b}P_{b}({\mu_{\text{ew}}})\text{Im}\big{[}C_{b}({\mu_{\text{ew}}})-C^{\prime}_{b}({\mu_{\text{ew}}})\big{]}\times(1\,\text{TeV})^{2},$
(13)

with the sum over $b$ running over the Wilson coefficients $C_{b}$ of all
operators in the JMS basis and their chirality-flipped counterparts denoted by
$C_{b}^{\prime}$. The relative minus sign accounts for the fact that their
$K\to\pi\pi$ matrix elements differ by a sign. Among the contributing
operators are also operators present already in the SM, but their WCs in (13)
are meant to include only BSM contributions. The numerical values of
$P_{b}({\mu_{\text{ew}}})$ are collected in [61]. Calculating
$C_{b}({\mu_{\text{ew}}})$ and $C^{\prime}_{b}({\mu_{\text{ew}}})$ in a given
BSM scenario this master formula allows in no time to calculate the BSM
contribution to $\varepsilon^{\prime}/\varepsilon$.

Another master formula, this time for $\Delta F=2$ transitions in the SMEFT
has been presented in [62]. We illustrated it on a number of simplified models
containing colourless heavy gauge bosons ($Z^{\prime}$) and scalars and models
with coloured heavy gauge bosons ($G_{a}^{\prime}$) and scalars. Also the
cases of vector-like quarks and leptoquarks are briefly considered. See brief
summary of this work in [63].

The next important step is the inclusion of NLO QCD corrections to the RG
evolution in the SMEFT. The important benefit of these new contributions is
that they allow to remove renormalization scheme dependences present both in
the one-loop matchings between the WET and SMEFT and also between SMEFT and a
chosen UV completion. For $\Delta F=2$ transitions this step has been already
made very recently in [64].

##  5 Comments on the $\Delta I=1/2$ Rule

I have described the history of this rule, the ratio ${{\rm Re}A_{0}/{\rm
Re}A_{2}}=22.4$, in Section 7 of [1] and in particular in Section 7.2.3 of my
recent book [2]. From this it is evident that the credit for the
identification of the basic dynamics behind this rule should go to the authors
of [17] who demonstrated that the current-current operators and not QCDP
operators are dominantly responsible for this rule. This has been confirmed
more than 30 years later by the RBC-UKQCD collaboration [65, 42] basically
ignoring our work.

Leaving this issue here aside what is really puzzling for me is there
interpretation of the dynamics behind this rule as they state from first
principles. According to them this rule originates in the relation between two
leading contractions

$\boxed{2}=-K~{}\boxed{1}$ (14)

induced by QCD. The factor $K$ depends on the scale at which these
contractions are evaluated. In 2012 they were evaluated at $\mu=2.15\,{\rm
GeV}$, in 2020 at $\mu=4.0\,{\rm GeV}$.

In order to make my point let us follow the exercise performed by us in
Section 9 of [18] and extract the values of these contractions from the
experimental values for ${\rm Re}A_{0}$ and ${\rm Re}A_{2}$ [66]:

${\rm Re}A_{0}=27.04(1)\times 10^{-8}~{}\,{\rm GeV},\quad\quad{\rm
Re}A_{2}=1.210(2)\times 10^{-8}~{}\,{\rm GeV}.$ (15)

Adjusting the expressions for the isospin amplitudes in [65] to our
normalization we have

$({\rm
Re}A_{0})=\frac{G_{F}}{\sqrt{2}}V_{ud}V_{us}^{*}\left(\frac{\sqrt{2}}{3}\right)\left[z_{1}\left(2~{}\boxed{2}-\boxed{1}\right)+z_{2}\left(2~{}\boxed{1}-\boxed{2}\right)\right],$
(16)

$({\rm
Re}A_{2})=\frac{G_{F}}{\sqrt{2}}V_{ud}V_{us}^{*}(\frac{2}{3})(z_{1}+z_{2})\left(\boxed{1}+\boxed{2}\right)~{},$
(17)

where $z_{1,2}$ are the real parts of the Wilson coefficients of the current-
current operators $Q_{1,2}$. With the NDR-${\rm\overline{MS}}$ values
$z_{1}=-0.199$ and $z_{2}=1.090$ at $\mu=4.0\,{\rm GeV}$, the data in (15) can
be reproduced with

$\boxed{1}=0.0865\,{\rm GeV}^{3},\quad\boxed{2}=-0.0752\,{\rm
GeV}^{3},\quad\boxed{2}=-0.87~{}\boxed{1}.$ (18)

Thus at $\mu=4.0\,{\rm GeV}$ we need $K=0.87$ to reproduce the data. In 2012
RBC-UKQCD found $K\approx 0.7$ at $\mu=2.15\,{\rm GeV}$ which was not
sufficient to reproduce the data but in 2020 their value of $K$ is in the
ballpark of $0.85$ with a significant error. Certainly it is an important
result.

So far so good. Now using this relation in (17) the RBC-UKQCD finds strong
suppression of ${\rm Re}A_{2}$ and concludes that this suppression is the main
origin of the $\Delta I=1/2$ rule.

This is a surprizing interpretation because until the 2012 RBC-UKQCD paper
[65], the particle community thought that it is the strong enhancement of
${\rm Re}A_{0}$ and some moderate suppression of ${\rm Re}A_{2}$ responsible
for this rule.

This very original interpretation from first principles, also repeated at this
workshop, differs totally from the one of DQCD as presented in our papers, in
particular in [17, 18] but also recently in [1, 2]. I do not want to repeat
this interpretation again because it is easy to demonstrate in a few lines
without using super computers or even calculating meson loops that the RBC-
UKQCD interpretation is incorrect.

The point is that in order to find out the impact of the QCD dynamics on these
two amplitudes, that is responsible for this rule, one has to ask first the
question what would be the values of these two amplitudes and of the
corresponding contractions in the absence of QCD. Of course we cannot fully
switch off QCD because we want to have quarks confined in mesons but we can
switch off non-factorizable contributions both in SD and LD by going to the
strict large $N$ limit. In this limit only the operator $Q_{2}$ contributes,
its coefficient is $z_{2}=1$ and its matrix element can be calculated
precisely using factorization valid in QCD in this limit. One finds then111see
[18, 2] for a detailed presentation.:

${\rm Re}A_{0}=3.59\times 10^{-8}\,{\rm GeV},\quad{\rm Re}A_{2}=2.54\times
10^{-8}\,{\rm GeV}~{},\quad\frac{{\rm Re}A_{0}}{{\rm Re}A_{2}}=\sqrt{2},$ (19)

in plain disagreement with the experimental value of 22.4. It should be
emphasized that the explanation of the missing enhancement factor of $15.8$
through some dynamics must simultaneously give the correct values for the
experimental values of ${\rm Re}A_{0}$ and ${\rm Re}A_{2}$ in (15). This means
that these dynamics should suppress ${\rm Re}A_{2}$ by a factor of $2.1$, not
more, and enhance ${\rm Re}A_{0}$ by a factor of $7.5$. This tells us that
while the suppression of ${\rm Re}A_{2}$ is an important ingredient in the
$\Delta I=1/2$ rule, it is $\underline{not}$ the main origin of this rule. It
is the enhancement of ${\rm Re}A_{0}$, as already emphasized in [67], even if,
in contrast to this paper, as demonstrated in [17], the current-current
operators are responsible dominantly for this rule and not QCD penguins.

One can also do this exercise using contractions to find in the factorization
limit [18]

$\boxed{1}=0.0211~{}\,{\rm GeV}^{3},\qquad\boxed{2}=0,\qquad(\mu\approx 0)$
(20)

which drastically differ from the values of contractions in (18). In other
words the main impact of QCD is to change the values of contractions from (20)
to the ones in the ballpark of the ones in (18) and this change have
significantly larger effect on ${\rm Re}A_{0}$ than on ${\rm Re}A_{2}$.

The simple discussion above shows that the RBC-UKQCD interpretation of the
dynamics behind $\Delta I=1/2$ rule cannot be correct. But working numerically
at $4\,{\rm GeV}$ and being not able to switch-off non-factorizable QCD
interactions in LQCD one simply cannot see properly what is really going on.
Yet, their numerical result is very important and one should congratulate them
for it. In particular because their central value for the ratio in question,
$19.9\pm 5.0$, is closer to the data than present one from DQCD, $16\pm 2$
[18]. In the latter approach an additional enhancement could come from FSI
which this time are not compensated by meson evolution as was the case of
$\varepsilon^{\prime}/\varepsilon$. It could also come from NP [68].

Let me then repeat. The dominant dynamics behind the $\Delta I=1/2$ rule for
$K\to\pi\pi$ isospin amplitudes is our beloved QCD as also confirmed by RBC-
UKQCD. But the correct physical interpretation differs drastically from the
one provided by them. It is simply the quark evolution from $M_{W}$ down to
scale ${\cal O}(1\,{\rm GeV})$ as analysed first by Altarelli and Maiani [69]
and Gaillard and Lee [70], followed by the meson evolution down to very low
scales at which QCD becomes a theory of weakly interacting mesons and free
theory of mesons in the strict large $N$ limit [33, 34, 35, 36]. This non-
perturbative evolution within the Dual QCD approach dominates by far the
enhancement of ${{\rm Re}A_{0}}$ over ${{\rm Re}A_{2}}$ as demonstrated by
Bardeen, Gérard and myself in [17, 18]. The relation of contractions in (18)
cannot provide this interpretation because at $\mu=4\,{\rm GeV}$ the computer
only tells us what the values of these contractions are.

##  6 Outlook

Let me finish this presentation by stating what should be improved in the
three non-perturbative approaches in order to clarify the situation of
$\varepsilon^{\prime}/\varepsilon$ in the SM.

RBC-UKQCD: Here a very important progress would be the calculation of isospin
breaking and QED corrections including also $\eta-\eta^{\prime}$ mixing.
Another important issue are charm contributions still missing in RBC-UKQCD
calculations. The prospects for improvements in the coming years are good
[51].

ChPT: First of all matching to short distance which would amount, as far as I
understand, at least to the determination of $L_{5}$. Better inclusion of
$\eta-\eta^{\prime}$ mixing with the help of $L_{7}$.

DQCD: Here certainly one should have a look at the subleading FSI.

It would also be very important if other LQCD groups contributed to these
investigations and in particular several LQCD groups calculated BSM hadronic
matrix elements. Here some promising results are signalled by two LQCD groups
[71, 72].

However, in order to identify possible NP also correlations of
$\varepsilon^{\prime}/\varepsilon$ with other processes, in particular rare
Kaon decays, have to be studied as summarized recently in [73]. Definitely
there are exciting times ahead of us!

##  Acknowledgements

I would like to thank Jean-Marc Gérard for invaluable comments on the
manuscript and him, Jason Aebischer, Christoph Bobeth, Jacky Kumar, Mikolaj
Misiak and David Straub for exciting time we spent together analyzing the
topics discussed in this talk. I would like to thank the organizers of WG3 for
inviting me to present these results. The research presented in this report
was dominantly financed and done in the context of the ERC Advanced Grant
project “FLAVOUR” (267104). It was also partially supported by the DFG cluster
of excellence “Origin and Structure of the Universe”.

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# Levitation of superconducting micro-rings for quantum magnetomechanics

Carles Navau1,∗, Stefan Minniberger2, Michael Trupke2, and Alvaro Sanchez1 1
Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona, Catalonia, Spain  2 Vienna Center for Quantum Science and
Technology, Universität Wien, Boltzmanngasse 5, 1090 Vienna, Austria

###### Abstract

Levitation of superconductors is becoming an important building block in
quantum technologies, particularly in the rising field of magnetomechanics. In
most of the theoretical proposals and experiments, solid geometries such as
spheres are considered for the levitator. Here we demonstrate that replacing
them by superconducting rings brings two important advantages: Firstly, the
forces acting on the ring remain comparable to those expected for solid
objects, while the mass of the superconductor is greatly reduced. In turn,
this reduction increases the achievable trap frequency. Secondly, the flux
trapped in the ring by in-field cooling yields an additional degree of control
for the system. We construct a general theoretical framework with which we
obtain analytical formulations for a superconducting ring levitating in an
anti-Helmholtz quadrupole field and a dipole field, for both zero-field and
in-field cooling. The positions and the trapping frequencies of the levitated
rings are analytically found as a function of the parameters of the system and
the field applied during the cooling process. Unlike what is commonly observed
in bulk superconductors, lateral and rotational stability are not granted for
this idealized geometry. We therefore discuss the requirements for simple
superconducting structures to achieve stability in all degrees of freedom.

##  I Introduction

Levitation of superconductors (SCs) has been an active topic of research in
the last decades due to its significant scientific and technological
potential. Until recently, most of the research has been carried out for
large-scale systems and in areas like transportation and energy storage Brandt
(1989); Moon (1994); Ma et al. (2003); Hull (2000); Wang et al. (2002). A
typical levitation system consists of a source of the magnetic field (e.g. one
or several permanent magnets) and one (or several) superconducting objects.
The performance and optimization of such devices are based mainly on the
choice and control of the magnetic field sources, on the optimization of the
superconducting properties and on the election of an advantageous geometry for
the superconductor. Typically, a set of bulk high-$T_{c}$ superconductors are
used as levitators, each of them with sizes on the order of a few centimetres
Navau et al. (2013). As a rule of thumb, levitation forces are enhanced by
increasing the critical-current density of the SCs and the gradients of
external fields; stability is increased by an adequate tuning of the field
distribution and the cooling procedure Del Valle et al. (2007). At these
scales, apart from the geometry, the key parameter of the superconductor is
the critical-current density $J_{c}$, which is related to the internal vortex
structure as a macroscopic averaged quantity. The critical-state model is
commonly used to describe the superconductor response Navau et al. (2013). The
calculation of the current-density distribution inside the superconducting
materials is the basic step from which one can evaluate all the relevant
parameters such as force, stability, or energy.

Recently, a new application of levitating superconductors has emerged in the
context of quantum magnetomechanics, where magnetically trapped
superconducting objects can create low-loss mechanical oscillators with long
coherence times Romero-Isart et al. (2012); Cirio et al. (2012); Johnsson et
al. (2016); Pino et al. (2018); Timberlake et al. (2019). A typical system
consists of a superconducting object levitated in a specifically designed
magnetic field, which creates a magnetic trap for the superconductor. The
objective of these proposed experiments is to perform ground-state cooling of
these objects, allowing for the generation of non-classical center-of-mass
motional states. In these systems, the superconductor sizes are expected to be
on the order of a few tens of micrometers or less. At these scales, the basic
principles of superconductivity still hold, but some assumptions generally
taken for granted macroscopically should be addressed in more detail. In
particular, macroscopic approximations, such as the critical-state model, are
not applicable and other approaches need to be studied.

In magnetomechanical experiments, the cooling process of the center-of-mass
motion typically starts when the SC is already stabilized in the magnetic
trap. Thus, the first step is to levitate a superconductor in a given
configuration of the external magnetic field. The presence of external
magnetic field during the cooling below $T_{c}$ (field cooling process, FC)
can result in a substantially different current distribution in the SC with
respect to the case where the superconductor is cooled in the absence of an
external magnetic field (zero-field cooling process, ZFC), yielding different
levitation forces and stability. Also, the displacement of the superconductor
from the cooling position to the final trapping position can modify the
current distribution and, thus, the force. At macroscopic scales, the effects
of cooling procedures have previously been considered in several works Navau
et al. (2004); Dias et al. (2010).

An important property of superconductors with non simply connected geometries,
like rings, is that the flux that threads a hole surrounded by a
superconducting region has to be conserved. At the macroscopic scale, one
needs to evaluate the current distribution inside the rings; the flux
conservation through the hole is a consequence of the resulting flux
distribution Navau et al. (2005); Bartolomé et al. (2005).

At the microscopic scale, the exact current distribution inside the ring can
become irrelevant and the flux conservation can be used as the condition for
evaluating the total circulating current. The latter becomes then the relevant
quantity.

While the microscopic details of the current distribution throughout the
superconducting material are key for understanding the lateral and angular
stability, this circulating current becomes the relevant quantity for the
motion along the axis of the ring. In most experiments, this is the direction
of interest.

In this work, we develop a comprehensive analytical theory for the levitation
of superconducting micro-rings and demonstrate how their use can be very
advantageous for applications in quantum magnetomechanics. One advantage for
using rings in magnetomechanical experiments, with respect to single-connected
geometries such as spheres Romero-Isart et al. (2012); Hofer and Aspelmeyer
(2019), arises from the fact that a ring with the same radius as the sphere
would have much less weight. However, the current induced to expel the field
from the levitator is not greatly reduced, resulting in a similar force.

We study different scenarios for cooling and trapping small superconducting
rings in the presence of external fields. The flux conservation equation is
used to derive analytical expressions for the circulating currents and to
evaluate the interaction energy between the magnetic field source and the
levitating superconductor. We focus on two cases of interest. The first, a
magnetic landscape created by a pair of anti-Helmholtz coils, is introduced in
Section III. This has practical importance since it has been proposed for
actual quantum magnetomechanics experiments Romero-Isart et al. (2012). The
second studied case is a superconducting ring coaxial with the field created
by a point dipole. We find that for a given cooling position, a single
parameter is enough to characterize the dynamics of the system along the
rings’ axis in this case. We derive the equilibrium positions and study its
vertical stability for the ZFC (section IV) and FC (section V) cases. We
discuss more qualitatively the lateral and angular stability, as well as other
possible configurations in section VI. Conclusions are presented in Section
VII.

##  II Model for the superconductor ring

Consider a superconducting ring of mean radius $R$ through which a total
current $I$ is circulating, so that the cross-section is a circle of radius
$a\ll R$. The self-inductance of the ring can be regarded as Landau et al.
(1984)

$L=\mu_{0}R\left[\log\left(\frac{8R}{a}\right)-b\right],$ (1)

where $\mu_{0}$ is the vacuum permeability, and $b$ is a dimensionless
constant that accounts for the particular current distribution across the
cross-section of the superconductor: if the current is uniformly distributed
over the cross-section, $b\simeq 1.75$, and if it flows over the surface,
$b\simeq 2$. Since $a\ll R$, the particular distribution of current inside the
ring hardly affects the self-inductance. Thus, the total circulating current
$I$ is the relevant parameter. Note that the inductance $L$ depends on the
radius of the ring as $L\sim R\log R$. In this paper, we assume that the
superconducting ring is in the Meissner state with a London penetration depth
$\lambda$ smaller than all other relevant dimensions. This allows us to set
$b=2$.

Whilst the ring is in the superconducting state, the magnetic flux threading
it is maintained Tinkham (2004). We shall distinguish between the ZFC regime,
in which the ring has reached the superconducting state far from any field
source, so that the flux that threads it is zero and will be zero after any
subsequent displacement; and the FC regime, where the SC has been cooled when
some flux was threading it, a flux that will remain after movements. Except
otherwise indicated, the superconducting ring has the axis coincident with the
$z$-axis. The levitated superconductor has a total mass of $M$. Gravity has
direction $-\hat{\bf z}$. We use standard cylindrical coordinates
$(\rho,\phi,z)$.

In the limit $\lambda\ll a$ assumed throughout this work, the flux threading
the superconducting ring is quantized in multiples of the flux quantum
$\Phi_{0}$. If one is dealing with smaller dimensions such that $\lambda$ is
no longer negligible, it is the fluxoid (which includes not only the flux
threading the ring but also a contribution of the currents in the
superconductor volume) the quantized magnitude Brandt and Clem (2004).

##  III A small ring in the field of an anti-Helmholtz coil

The first case we analyze corresponds to a superconducting ring levitating in
the field created by a pair of anti-Helmholtz coils (AHC). Such field
configuration, using a levitated superconducting sphere, has been recently
proposed for improving the performance in quantum systems with respect to
optical counterparts (e. g. longer coherence times), enabling in this way
access to a completely new parameter regime of macroscopic quantum physics
Romero-Isart et al. (2012); Hofer and Aspelmeyer (2019).

We consider AHC coils with radii $C$ (and, thus separation between parallel
coils $C$), coaxial with the $z$-axis and centered at the origin, and with a
circulating current $I_{0}$. The magnetic field created by the AHC, ${\bf
B}_{AHC}$ and the vector potential, ${\bf A}_{AHC}$ at the central region
($\rho,|z|\ll C$) can be approximated by

$\displaystyle{\bf B}_{AHC}(\rho,z)$ $\displaystyle=$
$\displaystyle\mu_{0}\frac{24}{25\sqrt{5}}\frac{I_{0}}{C^{2}}(-\rho\hat{\boldsymbol{\rho}}+2z\hat{\bf
z}),$ (2) $\displaystyle{\bf A}_{AHC}(\rho,z)$ $\displaystyle=$
$\displaystyle\mu_{0}\frac{24}{25\sqrt{5}}\frac{I_{0}}{C^{2}}z\rho\hat{\boldsymbol{\phi}}.$
(3)

We begin our discussion by assuming that the superconducting ring of radius
$R\ll C$ is located at the origin, in the absence of any applied field or
current. The flux threading the SC ring is zero. We also consider that the
ring can be moved in the vertical direction, maintaining the condition $|z|\ll
C$. The current induced in the ring can be evaluated from the flux-
conservation equation as

$2\pi RA_{\phi}(R,z)+LI=0.$ (4)

Here $A_{\phi}(R,z)$ is the $\phi$ component of the magnetic vector potential
evaluated at the position $(R,z)$. The first term in Eq. (4) is the flux
threading the superconducting ring due to the field produced by the AHC coils
and the second one the flux due to the current circulating in the ring.

From Eqs. (3)-(4), the current in the ring is

$I=-\Theta I_{0}\frac{z}{R},$ (5)

where we have defined the dimensionless constant

$\Theta\equiv\mu_{0}\frac{48\pi}{25\sqrt{5}}\frac{R^{3}}{C^{2}\,L},$ (6)

that geometrically characterizes the system.

The $z$-component of the Lorentz force, $F_{z}$, acting on a ring centred on
the axis of the field is

$F_{z}=-\Theta^{2}I_{0}^{2}\frac{L}{R^{2}}z\,\hat{{\bf z}},$ (7)

yielding a spring constant for vertical displacement

$\kappa_{z}\equiv-\frac{\partial F_{z}}{\partial
z}=\Theta^{2}I_{0}^{2}\frac{L}{R^{2}}.$ (8)

$\kappa_{z}$ is always positive, which indicates a vertical stable levitation.
Both the force and the spring constant depend on $R$ as $\sim R^{3}(\log
R)^{-1}$, and on the AHC radius as $\sim C^{-4}$. The dependencies for a solid
sphere are $\sim R^{3}$ and $\sim C^{-4}$, respectively Romero-Isart et al.
(2012), showing no significant decrease of the force value despite the large
reduction in weight for a ring.

The vertical frequency of this trap is obtained from the stiffness coefficient
as

$\omega_{z}=\frac{I_{0}}{R}\Theta\sqrt{\frac{L}{M}}.$ (9)

Following Eqs. (1) and (6), $\omega_{z}$ depends on $R$ as $\omega_{z}\sim
R^{3/2}(\log R)^{-1/2}$.

The vertically stable position of the levitation force can be found either by
equating the magnetic force to the gravity or by minimizing the energy of the
system $E$ with respect to the vertical position $z$. $E$ can be described
from the interaction of the field of the AHC with an induced dipole of
magnetic moment $I\pi R^{2}\hat{\bf z}$. Setting zero potential gravity at
$z=0$ we have

$E=Mgz-\frac{1}{2}I\pi R^{2}\hat{{\bf z}}\cdot{\bf B}_{AHC}(0,z),$ (10)

where $g$ is the gravity constant. The equilibrium position is found to be

$z_{eq}=-\frac{M\,g\,R^{2}}{I_{0}^{2}\Theta^{2}L}.$ (11)

If the ring is cooled below the critical temperature at a vertical position
$z_{FC}\neq 0$ where some flux is threading it before cooling (FC case), this
flux is maintained after subsequent movements. The flux conservation equation
for this case is

$2\pi RA_{\phi}(R,z)+LI=2\pi RA_{\phi}(R,z_{FC}).$ (12)

Owing to the linear dependence of the potential vector on the vertical
position (within the approximations considered here), the above expressions
for the ZFC case remain valid with the change $z\rightarrow z-z_{FC}$ or
$z_{eq}\rightarrow z_{eq}-z_{FC}$. Actually, this result permits selecting the
desired value for the levitation position $z_{eq}$ by adjusting the cooling
position $z_{FC}$, in an actual experimental setting.

##  IV Dipole and zero-field cooled superconductor

We now apply the theoretical framework to the case of a ring levitating in the
field of a magnetic dipole. We start here with the ZFC case and treat the FC
case in the next section.

Consider a point dipole ${\bf m}=m{\hat{\bf z}}$ located at the origin of
coordinates and the superconducting ring axially symmetric with the $z$-axis,
its center located at a position $d{\hat{\bf z}}$. In the ZFC case, the
cooling distance is very large, so that the ring becomes superconducting in
the absence of any applied field and the flux initially threading the SC ring
is zero. This zero flux value is maintained when the SC is vertically
descended towards the point dipole. The current $I$ that flows in the
superconducting ring in order to maintain this flux is found from the zero-
flux threading condition, as in Eq. (4),

$2\pi RA_{\phi}(R,d)+LI=0.$ (13)

Here $A_{\phi}(R,d)$ is the $\phi$ component of the magnetic vector potential
created by the dipole and evaluated at the position ${\bf
r}=R\,\boldsymbol{\hat{\rm\rho}}+d\,\boldsymbol{\rm\hat{z}}$.

The vector potential created by the point dipole is

${\bf
A}(R,d)=A_{\phi}(R,d)\,\boldsymbol{\hat{\phi}}=\frac{\mu_{0}}{4\pi}\frac{mR}{(d^{2}+R^{2})^{3/2}}\boldsymbol{\hat{\phi}}.$
(14)

Thus, the current flowing in the superconducting ring will depend on the
distance $d$ from the dipole as

$I=-\frac{\mu_{0}mR^{2}}{2L(d^{2}+R^{2})^{3/2}}.$ (15)

The energy of the levitated ring can be evaluated from the Lorentz force
acting over the current in the ring due to the field created by the dipole or,
alternatively, from the interaction between the field created by the
superconducting ring ${\bf B}_{SC}$ and the dipole. Including the
gravitational potential (set as zero at the $z=0$ plane), we obtain

$E=Mgd+\frac{2\pi^{2}R^{2}}{L}(\Delta A_{\phi})^{2}=Mgd-\frac{1}{2}{\bf
m}\cdot{\bf B}_{SC},$ (16)

where $\Delta A_{\phi}$ represents the variation (with respect to the cooling
position –infinite in the present case) of the vector potential generated by
the dipole and evaluated at the ring position. Note that the term
$\frac{2\pi^{2}R^{2}}{L}(\Delta A_{\phi})^{2}$ equals
$\frac{1}{2L}(\Delta\Phi)^{2}$, where $\Delta\Phi$ is the variation of the
magnetic flux threading the ring due to the field created by the dipole. In
the present case

$E=Mgd+\frac{\mu_{0}^{2}m^{2}R^{4}}{8L(d^{2}+R^{2})^{3}}.$ (17)

Normalizing the positions to the radius of the ring, $\zeta=d/R$, and the
energy to $E_{0}=\mu_{0}^{2}m^{2}/8LR^{2}$, one can describe the above system
with a normalized energy $e=E/E_{0}$ as

$e=\alpha\zeta+\frac{1}{(1+\zeta^{2})^{3}},$ (18)

where the dimensionless constant

$\alpha=\frac{8LMgR^{3}}{\mu_{0}^{2}m^{2}}$ (19)

is a _single_ parameter characterizing the system. $E_{0}$ is the energy of
the superconducting ring when moved from infinity (ZFC) to the $z=0$ position.

The energy $e$ has a minimum as a function of $\zeta$ only if

$\alpha<\frac{1029\sqrt{7}}{2028}\equiv\alpha_{c}\simeq 1.329.$ (20)

This inequality sets a condition for the stable levitation of ZFC rings with
the considered dipole. If the radius of the ring is fixed, the mass of the
ring should be less than $\simeq 0.33m^{2}\mu_{0}^{2}/(gLR^{3})$. If the
levitating material is fixed (with a given density), then the above condition
sets a maximum radius for the levitated ring.

For a given $\alpha<\alpha_{c}$, the normalized position of the stable
levitation point, $\zeta_{eq}$ can be found as the largest of the two
solutions of the equation

$\alpha=\frac{6\zeta_{eq}}{(1+\zeta_{eq}^{2})^{4}},$ (21)

which has to be solved numerically. The vertical oscillation spring constant
in this potential well can be obtained from the second derivative of the
energy, evaluated at the equilibrium position,
$\kappa_{z}=(1/2)\frac{\partial^{2}E}{\partial d^{2}}\big{|}_{d=R\zeta_{eq}}$.
With Eq. (17), the oscillation frequency can be expressed as

$\omega=\omega_{0}\sqrt{\frac{3(7\zeta_{eq}^{2}-1)}{(1+\zeta_{eq}^{2})^{5}}},$
(22)

being
$\omega_{0}=\sqrt{E_{0}/2R^{2}M}=\frac{\mu_{0}m}{4R^{2}}\sqrt{\frac{1}{M\,L}}$
($\zeta_{eq}$ is the largest solution of Eq.(21)).

Figure 1: (a) Equilibrium position and (b) frequency of the potential well for
a superconducting ring levitating in the field of a magnetic dipole, as a
function of the $\alpha$ parameter characterizing the levitation (see text).
Normalization values are defined in the text.

In Fig. 1 we plot the equilibrium positions $\zeta_{eq}$ and the frequency of
the trap $\omega$, as a function of $\alpha$. When $\alpha\rightarrow 0$, the
equilibrium position tends to infinity as $\sim\alpha^{-1/7}$.
$\alpha\rightarrow 0$ could be interpreted as a massless object or as a very
large-radius ring. In addition, the frequency tends to zero as
$\sim\alpha^{-4/7}$. When $\alpha\rightarrow\alpha_{c}$ the equilibrium
position tends to $\zeta_{eq}\rightarrow 1/\sqrt{7}\simeq 0.378$ and the
frequency of the trap goes to zero, corresponding to the limit in which the
levitation becomes unstable. Interestingly, between these two limits there is
a maximum in the stability of the force as a function of the parameter
$\alpha$. This maximum is achieved when the equilibrium position is
$z_{eq}=\sqrt{3/7}\simeq 0.655$, corresponding to $\alpha_{\rm max}\simeq
0.935$ (after Fig. 1). The maximum value for the trapping frequency is
$\omega_{max}=\omega_{0}(49\sqrt{21})/(50\sqrt{20})\simeq 1.004\omega_{0}$.

The fact that there exists an optimum frequency (maximum stability) means
that, for a given superconducting material (a given density), there is an
optimum radius that can be found from the definition of $\alpha$.
Alternatively, if the geometry is also fixed, for a mass less than that needed
for achieving the optimum $\alpha$, it could be interesting in practice to
ballast the ring for increasing the stability.

##  V Dipole and field-cooled superconductor

We consider now that the superconducting ring has been cooled below $T_{c}$ at
a given position $d_{FC}{\bf\hat{z}}$. Now there is a flux threading the ring
$\Phi_{FC}=2\pi RA_{\phi}(d_{FC})$ that will be maintained after subsequent
movement of the SC ring when already cooled. The flux conservation equation
becomes

$2\pi RA_{\phi}(d)+LI=\Phi_{FC},$ (23)

from which the current circulating in the ring is (defining
$\zeta_{FC}=d_{FC}/R$ and $I_{0}=\mu_{0}m/2LR$)

$I=I_{0}\left(\frac{1}{(1+\zeta_{FC}^{2})^{3/2}}-\frac{1}{(1+\zeta^{2})^{3/2}}\right).$
(24)

Eq. (16) also holds in this case. The energy of the levitated ring can now be
written in terms of the $\alpha$ parameter and on the $\zeta_{FC}$ value. It
is found that

$e=\alpha\zeta+\frac{1}{(1+\zeta^{2})^{3}}-\frac{2}{(1+\zeta^{2})^{3/2}(1+\zeta_{FC}^{2})^{3/2}}.$
(25)

Figure 2: For a superconducting ring levitating in the field of a magnetic
dipole, critical values for the $\alpha$ parameter for achieving positive
equilibrium position ($\zeta>0$, red) and negative one ($\zeta<0$, black). The
shadowed area corresponds the ($\alpha,\zeta_{FC}$) values that results in two
equilibrium positions.

Without loss of generality, we consider only positive field cooling positions
($\zeta_{FC}>0$). Now, depending on the values of $\alpha$ and $\zeta_{FC}$,
the energy [Eq. (25)] can have one minimum at $\zeta<0$, one minimum at
$\zeta>0$, two minima (one at $\zeta>0$ and another at $\zeta<0$), or no
minima at all. Actually, for each $\zeta_{FC}$, one can define a critical
$\alpha$ for having positive equilibrium positions, $\alpha_{c+}$ and another
critical $\alpha$ for achieving equilibrium positions at negative $\zeta$’s,
$\alpha_{c-}$. These values can be found numerically. The functions
$\alpha_{c+}(\zeta_{FC})$ and $\alpha_{c-}(\zeta_{FC})$ are plotted in Fig. 2.
We observe that for $\zeta_{FC}\rightarrow\infty$ we recover the ZFC case
where only equilibrium at positive $\zeta$’s is possible when
$\alpha<\alpha_{c+}(\infty)=\alpha_{c}\simeq 1.329$. $\alpha_{c-}\rightarrow
0$ in this limit. However, when the cooling position is zero (that is, when
the ring is cooled with the dipole in its centre), there can be only one
equilibrium position at negative $\zeta$’s. This is because the current in the
ring creates an attractive force to the dipole after any movement. The
equilibrium can be achieved only when the movement is made to negative
positions, where the attraction to the dipole opposes gravity. Interestingly,
for some values of $\zeta_{FC}$ and $\alpha$ (shadowed region in Fig. 2) one
can have two equilibrium positions, one at positive and the other at negative
$\zeta$’s. In Fig. 3 we show the numerically evaluated equilibrium positions
as a function of the $\alpha$ parameter, for different cooling distances
$\zeta_{FC}$. The shadowed area in Fig. 3 includes all possible equilibrium
positions for all possible $\alpha$’s and $\zeta_{FC}$’s.

Figure 3: For a superconducting ring levitating in the field of a magnetic
dipole, equilibrium positions as a function of $\alpha$ for different cooling
positions (indicated in the plot). The shadowed regions correspond to all the
possible equilibrium positions for any $\zeta_{FC}$ value. Note that for some
values of $\zeta_{FC}$ and $\alpha$ corresponding to the shadowed region in
Fig.2 there are two equilibrium positions.

To study the stability of such solutions one can follow the same procedure as
above. To simplify, we split the treatment in two parts: for the positive and
for the negative equilibrium positions. In Fig. 4 we present the results for
the numerically evaluated frequency of the equilibrium position. For any
$\zeta_{FC}$ there is an optimum $\alpha$ such that the equilibrium position
is the most stable. We see also that the total maximum frequency is achieved
in the ZFC case [$\omega_{opt}(\zeta_{FC}=\infty)\simeq 1.004\omega_{0}$],
although for the other cooling positions, the optimum value of the frequency
is above $0.85\omega_{0}$. In other words, the field cooling of the
superconductor does not degrade substantially the stability of the equilibrium
while it adds a new parameter to control the equilibrium position allowing a
larger region for stabilizing the levitated ring.

Figure 4: For a superconducting ring levitating in the field of a magnetic
dipole, frequency of the trap around the equilibrium position as a function of
the $\alpha$ parameter for different cooling positions (indicated in the
plot). For simplicity, we show in the left (right) side the frequencies for
the negative (positive) equilibrium positions. Note the decreasing scale in
the left part ($\alpha$ is always positive).

##  VI Discussion

###  VI.1 Lateral movements and rotational degrees of freedom

One of the main objectives of the present work is to describe the forces
resulting from the flux-conservation property of a levitated superconductor.
In this sense, we have focused on the main characteristics and properties that
can be derived from this property and have concentrated on the vertical
movements and vertical stability, this being the direction of the rings’ axis.
Levitation is, however, only truly stable if it is so in all degrees of
freedom, aside from rotation around the ring’s axis. In a real ring with
finite width and thickness, stability against rotation and lateral movements
is provided by the current distribution induced on the ring’s whole surface.
This setting has recently been described and calculated numerically Lat . Even
though the finite thickness of the ring in any actual implementation may lead
to stable levitation, it is still worthwhile studying the stability of the
idealized case, and to explore further stabilizing mechanisms for levitation.
This matter is discussed in the two following subsections.

###  VI.2 Stability in the idealized thin ring case

For the (idealized) example of an infinitely thin ring, energy in a purely
magnetic potential is minimized when the circulating current is zero. This is
equivalent to preservation of the initial flux through the ring. In the ZFC
case, the initial flux (zero) can be obtained at every position in space by
tilting the ring until the local field is parallel to the surface. The zero-
flux condition can similarly be achieved in regions of the trapping field with
an arbitrary field gradient or curvature: Rotation of a ring by 180 degrees
about an axis in its plane will lead to a sign change of the flux in the
ring’s coordinate system. Since flux is a continuous quantity in free space,
it follows immediately that an angle exists at which the flux must be zero.
For field cooling, the entire volume in which the average field is at least
$\Phi_{FC}/\pi R^{2}$ is accessible, again by adjusting the orientation
accordingly. The volume where the field is smaller than this value can not be
reached without inducing a current. This means that in either case, an
idealized thin ring can not be trapped at the minimum of, for example, the
field provided by anti-Helmholtz coils.

By extension, the above arguments are equally valid for an infinitely thin SC
forming an arbitrary closed curve, no matter how intricate its shape.

One might conversely consider seeking trapping configurations which rely on a
maximum of the flux, such that it decreases for any displacement or rotation.
It follows from Earnshaw’s theorem for the magnetic field that such a
situation is impossible for static fields: Since there is no maximum of the
magnetic field in free space, there can also be no maximum of the flux in a
ring which is disconnected from all field sources.

###  VI.3 Stabilizing mechanisms

Beyond finite-thickness effects, trapping of the levitator in the other
degrees of freedom can be provided by other means, such as gyroscopic effects,
time-orbiting potentials or more complex topologies.

Spin-stabilized magnetic levitation is well known from the physics of spinning
devices like the Levitron Jones et al. (1997). We expect that similar
considerations apply to a rotating ring.

A further path towards stable levitation of thin rings is the use of time-
averaged potentials. The combination of three non-orthogonal anti-Helmholtz
coil pairs makes it possible to create a time averaged trapping potential, in
which the flux through the ring is proportional to the distance from the trap
center, regardless of its orientation. This configuration is equivalent to a
ring rotating around both its planar axes in a static AHC field. The rich
dynamics of such quasi-static potentials presents fertile ground for future
work.

In general, the idea of flux-conservation makes the levitation of a
superconducting system ’rigid’, in the sense that any variation of the flux in
any element would make the superconductor react to counteract this variation.
This idea of rigid levitation has been exploited in several works of
macroscopic levitation of superconductors, where the ’rigidity’ comes from the
hysteretic current penetration into the superconductor Brandt (1990); Del-
Valle et al. (2008, 2009). In a ring, it is the flux conservation which yields
this ’rigidity’, albeit only along one axis. While a single ring cannot be
trapped by the same mechanisms, the behaviour of a macroscopic superconductor
can be approached by a rigidly connected arrangement of rings. A planar
arrangement of electrically isolated loops will be pinned to its initial
vertical and lateral position, albeit only in the FC case, since the zero-
current condition can only be fulfilled there for all loops. Lowering the
symmetry of the trap potential by applying different gradients in all
directions can furthermore provide rotational stability.

A minimal assembly of rings which allows stable levitation is found by using
two non-parallel, rigidly connected rings. In the ZFC case, the flux through
both rings can only be minimized when both are parallel to the local magnetic
field, or both are in a plane in which the field components cancel over their
areas. The first condition can be met, for example, by moving the rings along
an axis until they reach the zero-current condition. As an example, if the
rings are orthogonal to each other, their axes could lie along the lines
$|x|=2z$ in an AHC field. Once again, applying a different field gradient
along $y$ could provide rotational stability around the $z$-axis, however not
against combinations of translation and rotation. These would be suppressed by
the addition of gravity. The use of multiple rings in a similar arrangement,
e.g. placed on the three angled surfaces of a tetrahedron or the four angled
surfaces of a pyramid, would be fully stable in an AHC-like field with
different gradients along all axes. Complex, microscopic three-dimensional
geometries made of superconducting materials pose a major challenge for
fabrication, although recent work indicates that the creation of such
structures is within reach of cutting-edge nano-assembly techniques Shani et
al. (2020).

##  VII Conclusion

We have introduced a theoretical framework for the levitation of
superconducting rings over magnetic field sources, considering the current
flowing in the superconductor resulting from the flux-conservation conditions
(assumed as 1D circuit) as the main parameter. These conditions provide a
novel set of properties with respect to those arising from the induced
currents in a single-connected superconductor.

Our approach has been applied to both a quadrupolar anti-Helmholtz field, a
case recently proposed for quantum magnetomechanics experiments, as well as to
the case of a superconducting ring levitating in the field of a magnetic
dipole. A complete set of analytical formulas are derived for describing these
cases. For the force acting on a ring in the $z$-direction, we have found
that, depending on the cooling position, a continuous range of equilibrium
positions can be achieved. We have also found the conditions that the system
should satisfy in order to have this equilibrium; an optimum for having the
largest possible vertical stability is obtained. This condition is expressed
in terms of a single parameter that combines information from the levitating
object (mass, radius, cross-section) as well as from the field source
(magnetic moment). We have seen that even in the simple dipole-SC ring system,
there is the possibility of having one, two, or zero vertical equilibrium
positions. The stability of the ring with respect to translation and rotation
has been qualitatively discussed. Several strategies are proposed to achieve
full stability. We have argued that although closed curved geometries are not
trapped in DC fields, rigidly connected arrangements of several rings can be
trapped, but only if they are three-dimensional or their symmetry is
sufficiently low. In actual implementations, the finite thickness of the
superconducting rings will lead to current distributions that result in stable
levitation.

In quantum magnetomechanical experiments, the relevant dynamics of the system
are those along a single axis. Our model takes the flux conservation through
the ring into account and provides an analytical toolkit for the design and
optimization of levitating rings. The present work will guide future
experiments of such levitation systems, by incorporating the degree of freedom
provided by flux-conservation conditions as an extra key element for
characterizing and enhancing the magnetic levitation.

##  VIII Acknowledgements

We thank the members of the MaQSens Project for useful discussions. We thank
financial support from Catalan project 2014-SGR-150, Spanish project
MAT2016-79426-P of Agencia Estatal de Investigación / Fondo Europeo de
Desarrollo Regional (UE), the ESQ Discover Programme of the Austrian Academy
of Sciences (project ELISA), and the European Union Horizon 2020 Project FET-
OPEN MaQSens (grant agreement 736943). AS acknowledges support from Catalan
ICREA Academia program.

∗ Corresponding author<EMAIL_ADDRESS>
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