Precision tau physics: Challenge for Theory, on and off the lattice

Amarjit Soni

Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA *
<EMAIL_ADDRESS>
16th International Workshop on Tau Lepton Physics (TAU2021),
September 27 – October 1, 2021 10.21468/SciPostPhysProc.?

## Abstract

$\tau$ is playing an important role in the current B-physics indications from
experiments of lepton flavor universality violations(LFUV). This suggests it
be given increasing attention theoretically in the coming years, given also
the fact that Belle-II will have much larger data samples to study; similar
comments also apply to LHCb as well as ATLAS and CMS. The fact that $\tau$
decays provide valuable information on its spin is an added advantage. This
gains special significance if the current indications of new physics are
upheld since naturalness arguments then strongly suggest that new physics
should be accompanied by new CP-odd phase(s). Moreover, the fact that the
$\tau$ mass is around 1.8 GeV, i.e a lot less than the B-meson mass makes it
much easier candidate for lattice studies in great detail and consequently
very likely with greater precision. This should help us test the SM and or
BSMs with greater precision as needed.

###### Contents

  1. 1 Introduction and motivation
  2. 2 More on current anomalies
  3. 3 BSM possibilties
  4. 4 New physics repercussions for the tau
  5. 5 Summary & Conclusion
  6. 6 Acknowledgements

##  1 Introduction and motivation

For almost a decade now, there have been persistent indications from all three
major B-facilities of lepton universality violations (LUV) [1], [2, 3, 4, 5,
6, 7, 8, 9, 10], with a combined significance of $4.5\sigma$ [11]. In tree-
level charge current decays, $B\to D(*)\tau(l)\nu$, the reactions involving
$\tau$ seem to differ more from those involving muon or electron than
predicted by the SM. Another class of reaction that is presenting an even
stronger indications of lepton flavor universality violations is flavor
changing neutral current decays which, in the SM are loop-level. At the quark-
lepton level this is $b\to sl^{+}l^{-}$ for $l=\mu$ or e. The corresponding
ratios of such reactions, $R_{K(*)}$ etc which in the SM are supposed to be
unity within O(few %), experimentally at LHCb [9] are found to be
significantly different from unity i.e. O($\approx 30\%$) at around 3 or even
possibly $\approx 4\sigma$.

Moreover, another rather strong indication of new physics (NP) that has been
strengthened this year is the muon (g-2) anomaly. For well over a decade this
was already around $3.5\sigma$ being the difference between the experimental
measurements at BNL from about 2002 [12] to around 2006 [13] and the theory
predictions based on dispersion relations, data driven approach [14]. Recent
confirmation by the Fermilab (g-2) Collaboration [15] of the old BNL result
has now increased the deviation from the data-driven theory prediction from
the SM to about 4.2$\sigma$.

The main point to bear in mind is that there are three anomalies, $R_{D(*)}$,
$R_{K(*)}$ and muon (g-2), and each of these is over three $\sigma$.

So, the chances of at least one of these surviving the test of time and
leading us to new physics is rather high, possibly LUV new physics. Then,
natuarlness arguements strongly suggest CP-violating BSM-phase(s). In the
context of CP-violation searches, $\tau$ is especially important due to the
fact that its decay products provide information on its polarization.

Belle-II with its promise of significantly improved luminosity in an
inherently cleaner environment of $e^{+}-e^{-}$ collisions as well as LHC
experiments ATLAS, CMS and LHCb with RUN2 and later runs can all provide much
larger data samples of $\tau$ with greater precision which should be extremely
valuable.

In so far as theory goes, while for many aspects of $\tau$ decays, $m_{\tau}$
is too large for chiral perturbation theory (ChPT), and not sufficiently large
for QCD perturbation theory to work too well, it is much better suited than
say the B mass, for lattice calculations for precise studies both for testing
the SM as well as serving as a very importantly laboratory for precise studies
at the B-mass.

##  2 More on current anomalies

I.Two B-anomalies

There are for now two rather serious (over 3$\sigma$ each) deviations from the
SM.

  * •

Charge current semi-leptonic tree-level B decays to pseudo-scalar (D) or
vector ($D^{*}$) exclusive hadronic final state accompanied by either
$\tau+\nu_{\tau}$ or $l+\nu_{l}$ (for $l=\mu$ or e). To test lepton
universality as predicted by the SM a simple ratio, of rates is formed,

$\displaystyle R_{D^{(}*)}^{\tau/\ell}={\rm BR}(B\to
D^{(*)}\tau\bar{\nu})/{\rm BR}(B\to D^{(*)}\ell\bar{\nu})(with~{}\ell=e,\mu)$
(1)

Completely analogously,

$\displaystyle R_{\psi}=B_{c}\to J/\psi\
\ell\nu~{}(with~{}\ell~{}=~{}\mu,~{}\tau)$ (2)

  * •

Then there are FCNC decays, at the quark level $b\to s$, that occur in the SM
at 1-loop order wherein comparison of muons to electrons with exclusive
hadronic final states, $B^{+}\to K^{+}(K^{+*})$,

$\displaystyle R_{K^{(*)}}^{\mu/e}={\rm BR}(B\to K^{(*)}\mu^{+}\mu^{-})/{\rm
BR}(B\to K^{(*)}e^{+}e^{-})$ (3)

show significant deviation over 3$\sigma$, O(30%) from unity.

It is important to note here that since experimentally dilepton invariant mass
is restricted to be greater than $\approx$ 500 MeV, the difference in
radiative corrections between electrons and muons in the SM is expected to be
only O(few%); this is negligible compared to the difference in $R_{K^{(*)}}$
that LHCb finds. It is clearly experimentally very important to extend this
test to the $\tau^{+-}$ pair but the expected BR is too small for the
currently available luminosities at LHCb or at BELLE-II.

In so far as the ratios $R_{D^{(*)}}$ and $R_{\psi}$ are concerned they are
completely independent of the mixing angle $V_{cb}$ but they do require
precise knowledge of the $q^{2}$ dependence of the relevant form factors
although because they are ratios some dependence does get partly cancelled.

II. muon (g-2) anomaly

For a long time, the anomalous magnetic moment of the muon has been a target
of search for physics beyond the SM [16]. Modern era started at Brookhaven
National Lab about 30 years ago leading to BNL experiment E821.

E821 published two key results. In 2002 in a Phys. Rev. Lett. [12] and there
final result in 2006 [17]. These two experimental results although one was for
$\mu^{-}$ and the other for $\mu^{+}$ were completely consistent with each
other and in fact agreed to 6 decimal places.

Comparison of the experimental results with the data-driven theory prediction
from the SM, which uses well-grounded dispersion relation [18, 19, 20] showed
that the BNL result of 2002 incompatible to about 2.7$\sigma$. Later in 2006
the final E821 result disagreed with the SM prediction by about 2.2 to 2.7
$\sigma$ [17].

For the past eight or so years there has been a new major experimental effort,
E989 at Fermilab with the express goal of improving the BNL result on muon
(g-2) by a factor of four in accuracy. Along with that experimental effort,
there has been also an extensive theory effort to improve the accuracy of
predicted value of muon (g-2) based on the SM.

Early in 2021, the FermiLab (g-2) collaboration announced their 1st result
[15] which they found to be completely consistent with the old BNL result, in
fact the two results agree again to 6 decimal places. The weighted average of
the two experimental results disagrees with the data driven, "ratio method"
[21] by $\approx$ 4.2 $\sigma$. In our search for new physics this is the
major new positive development. This is because the data driven ratio method
is very reliable.

Indeed these two anomalies, $R_{K(*)}$ and muon (g-2) are very serious,
plausible each one of them is already over 4$\sigma$. Fermilab (g-2)
collaboration is scheduled to announce their result with the addition of Run 2
data in about a year and likewise we expect update from LHCb with more data
and including also more channels, e.g FCNC decays of b-baryons in a similar
time-frame. So, one or both of these anomalies could be over 5 $\sigma$ in a
year.

The conclusion thus far is that chances of at least one of these two signals
of new physics being genuine are very high. That provides us rationale to
think of theoretical interesting possible new physics scenarios.

##  3 BSM possibilties

Taking the experimental hints for lepton flavor violations (LFV) seriously,
many have offered lepto-quark solutions [22, 23, 24, 25, 26]. However, it
seems to us [27, 28, 29] that it is very likely that when nature is taking
such a dramatically different turn from the past, it will also very likely use
the opportunity to address a very glaring and pressing problem in the SM,
namely the extreme radiative instability of the Higgs to radiative
corrections. The simplest and most elegant solution for this problem remains
the one offered by supersymmetry. But, in the current context of LFV the more
natural version of SUSY is in fact RPV-SUSY. Recall that conventionally,
R-parity conservation has to be imposed to prevent proton decay. As Raman et.
al have recently emphasized, the more natural setting of supersymmetry is in
fact RPV-SUSY [30].

Indeed, in our [27] first paper on the subject, to address the experimental
hint in $B\to D(*)\tau(e,\mu)\nu$, for simplicity and minimality we assumed
that the 3rd generation superpartners were the lightest.To emphasize the
crucial role of the 3rd generation, we termed our version as "RPV3-SUSY". We
then explicitly showed in that 1st work that even with one relatively light
generation of superpartners, one of the most attractive feature of SUSY,
namely gauge coupling unification stays intact along with, of course, the
radiative stability of the Higgs.

Another point that we have never seen emphasized before is that in fact as an
inherent consequence of Bose-Fermi symmetry of the RPV-SUSY, it follows from a
simple generalization of the Yang-Mills idea [31] that all interactions
allowed by the internal symmetry leads naturally to lepton flavor violations
and removes the accidental symmetry of individual lepton number conservation
that is there in the SM.

Notable constraints and key experimental consequences of RPV3-Susy

It took significant effort on our part [27, 28] to seriously study and impose
dozens of constraints on the RPV3 set up we had assumed. See, e.g, Table IV
and figs. 6, 7 and 8 in [28]

A very important consequence of seriously imposing all the existing
constraints is that the key super-partner relevant for B-LFV-anomalies, the
sbottom has to lie in the range of $\approx$ 3 to about 12 TeV. In turn, this
has the important consequence that the contribution of the sbottom to solution
of the muon (g-2) anomaly gets severely limited. Indeed, in our RPV3 scenario,
most of the contribution to the muon (g-2) anomaly arises from the exchange of
a super-partner of $\nu_{\tau}$ which is relatively quite light, around 1 TeV.
Not only this should be readily available at the LHC; in fact it leads to a
spectacular signature of 4 charged muons!

##  4 New physics repercussions for the tau

BSM-CP odd phase(s) and $\tau$ decays

As discussed in the preceding sections currently there are very strong
indications of new physics from B-decays as well as from muon (g-2) anomalies.
The key point is that BNL 1964 Nobel prize winning experiment [32] of Cronin-
Fitch et al demonstrated for the 1st time that CP is not a symmetry of
nature.Thus naturalness arguments strongly suggest that new physics should be
accompanied by new BSM-CP-odd phase(s).

The point we want to emphasize is that $\tau$ decays are an extremely
sensitive probe of CP violation effects. In part this is because decay
products of the $\tau$ provide information on its spin. In part also because
tau has an appreciable Br into multi-body final states, The reason is that
once one has at least 4 linearly independent momenta (including spin) all CP
violating effects whether they are odd under naive Time-Reversal ($T_{N}$-odd)
or even under naive Time-Reversal become accessible [33].

With at least 4-linearly four momenta, one can construct an anti-symmetrical
tensor a CP-odd, $T_{N}$-odd observable. Then because of the CPT theorem one
does not need a CP-even phase. This has the advantage that such a CP violating
observable can arise at tree-level. This is unlike the case of a CP-violating
observable that is even under naive time reversal such as is case for an
energy asymmetry or a partial rate asymmetry when one compares particle decays
to a final state with the anti-particle decays to a conjugate final state.
Then a non-vanishing CP-violating asymmetry can only arise if there is also a
CP-even phase. The presence of such non-vanishing CP-even phase requires the
Feynman amplitude be complex. This in turn requires the calculation be at
least at one-loop order, so that will carry a suppression factor of
$\alpha_{s}/\pi$ [34]. Thus, as a rule a tree-level CP violating effect may be
appreciably larger. However, there is another non-trivial requirement to
manifestly demonstrate that this tree-level effect originating from an
observable such as a triple product $T_{N}$-odd observable is genuinely CP-
violating. This requires experimentally comparing particle decays with an
antiparticle to ensure that the observable is non-vanishing in both cases and
in fact the asymmetries will have to be added [33].

Possible CP violation in $\tau^{-}\to\nu_{\tau}\pi^{-}K_{S}(\geq\pi^{0})$

This is an extremely interesting final state that the BABAR Collaboration
focused on to search for CP violation [35]. Having 3 particles in the final
state has the advantage that, in principle, one can also search for an energy
asymmetry between the decays of a particle versus an antiparticle. In their
experiment BABAR chose to focus only on the partial rate asymmetry (PRA). They
found a 2.8 $\sigma$ deviation from the predictions of the SM. As it is
explicitly mentioned in the BABAR paper the final state may have an unknown
number of $\pi^{0}$’s, since there are testing for CP asymmetry so long as
decays of $\tau^{-}$ as well as those of $\tau^{+}$ include similar neutrals
that go undetected and included, it remains a valid test of CP violation.
However, since the final state is known to have at least a neutrino (through
lepton number conservation), a charged pion and a $K_{S}$, both relatively
clearly and cleanly identifiable, therefore, it would be extremely useful to
search for an energy asymmetry also in the same final state. This can be done
with relative easy by focusing on the energy distribution of the $\pi^{-}$ in
$\tau^{-}$ decay and comparing with the energy distribution of the $\pi^{+}$
in $\tau^{+}$ decays. It is useful to stress here two points. First an energy
asymmetry can easily be larger in a certain localized region of energy and be
smaller in another region than the rate asymmetry. In this context, it is very
important to emphasize that the PRA are severely constrained by the CPT
theorem. When you add up all the non-vanishing PRA’s in different channels
there have to cancellations among them so that the life-time of $\tau^{+}$
equals that of $\tau^{-}$. As already mentioned below there is no good reason
to believe that energy asymmetry will not be energy-dependent as a rule and
consequently cancellations will arise as you integrate over the energy-
distribution. Second, because $\pi^{0}$ and such neutrals can escape detection
in a rate asymmetry, an energy asymmetry is likely a much safer test of CP-
asymmetry, especially given that the final state readily allows one to do
that. BABAR Collaboration (and Belle and Belle-II) are strongly urged to do so
with existing and with increasing data sets.

Illustrative examples of multi-particle final states in $\tau$ decays

Here are some examples that should be distinctive experimentally and have
enough number of momenta that all different types of asymmetries can be
searched for:
$\tau^{+}\to\bar{\nu}+\pi^{+}+\pi^{+}+\pi^{-}$; to
$\bar{\nu}+\pi^{+}+K_{S}+K^{+}+K^{-}$;
$\bar{\nu}+\pi^{+}+K_{S}+\pi^{+}+\pi^{-}$ ;$\bar{\nu}+K(*)^{+}\pi^{+}+\pi^{-}$
etc.

In this list energies of conjugate mesons or invariant mass of conjugate pair
of mesons as well as PRA can be used but all three of this type will require
non-vanishing CP-even "strong" phase and there is no good reason to think that
those phases will be not there or will be too small. In fact since different
(possibly overlapping) resonances are likely to feed the final state those
strong phases should be sizeable. Therefore they should help in the search for
SM-CP as well as any BSM CP-odd phase(s). In fact the current hints of new
physics in LFV and or in muon (g-2) anomaly should motivate searches for a BSM
CP-odd phase. In $\tau$ decays final states such as $K_{S},\pi^{+-},K^{+-}$
should allow construction of naive Time Reversal ($T_{N}$)-odd observables. If
any of these are found to be non-vanishing then the decays of the conjugate
$\tau$ into conjugate final states should also be studied. A genuine violation
of CP-symmetry requires both sets to be non-vanishing and consistent with each
other including the sign.

$\tau_{edm}$ may be enhanced: Should be a high priority at BELLE-II

Prominent BSM explanation of the current anomalies are LQ models, RPV-SUSY or
Z’. In all of these cases the electric dipole moment of the $\tau$ can be
significantly enhanced over the expectations from the SM. Direct measurements
of the edm of the $\tau$ may be very challenging or simply not possible
because of its very short life-time, i.e. $O(10^{-13})$ sec. Fortunately
$\tau$ decays to multi-body final states can be a very powerful probe. This
should be a very high priority at Belle-II where $tau$ production and decays
can be studied with relative easy. A traditional method is to examine the
production cross-section near threshold to extract the dipole moment. But
actually use of the multi-body final states is a much more sensitive probe of
all types of CP asymmetries [36]. From that information one can try to extract
the value of the edm. In fact since a large number of possible observables are
available for extracting edm in that work, for the first time a simple theorem
is proved to choose an "optimal observable", namely that will necessarily give
you best bound or the value that will have the minimum statistical error.
Indeed, in recent years, this theorem forms the basis for machine learning
techniques. (See e.g footnote 3 (Ref. 27) in [37] or Ref. 43 in [38]).

Signals of BSM may be (easier) in (CP-conserving) magnetic moment, absolute
rates, CKM unitarity etc: importance of $m_{\tau}$

Since the SM-CKM-phase is not expected to cause any CP-asymmetry in $\tau$
decays except possibly the induced indrect CP violation originating from the
the mixing of the two neutral kaons, any significant detection of sizeable
asymmetry would be a strong sign of a BSM-CP-odd phase. Consequently it also
serves as a very strong indicator of new physics as well. However, it does
depend crucially on the size of the BSM phase. While there is no good reason
for the new CP-odd phase to be too small, clearly it can easily order 0.3 or
0.1. Even such a difference as O(3) can cause years of delay in experimental
detection in the case it is about 0.1 compared to if it is around 0.3.

Therefore, it is very important also to keep on searching for new physics via
CP-conserving tests as well. One important example is the magnetic moment of
the $\tau$; new physics can cause a significant enhancement over the SM. In an
$e^{+}+e^{-}$ machine, the magnetic moment of the $\tau$ can be readily
searched along very similar lines as the edm search discussed in the preceding
paras, namely production via $e^{+}+e^{-}$ machine leading to weak decays into
multi-particle final states. Again a multitude of observables can be used
amongst which an "optimal observable" can be constructed following [36].

Precise comparison of $\tau$ decays of states containing muons versus those
containing electrons can be a very valuable search for new physics especially
lepton universality tests.

In the class of CP-conserving tests, testing the SM via precise predictions of
the BR against the experimental measured value can be immensely useful. To
facilitate such tests lattice calculation of the partial widths can play a
significant role. For such purposes the fact that mass of the $\tau$ is only
about 1.8 GeV, ı.e. it is substantially less than the mass of the b-quark is
very significant. This is because lattice frameworks that preserve essentially
the full chiral symmetry of the continuum theory, such as "domain wall quarks"
(DWQ), and as a result are more continuum-like (thus needing simpler
renormalization as well as a simpler effective chiral Lagrangian for
extrapolation purposes) tend to be more computationally expensive and for them
as well as many other lattice formulations the b-quark of mass 4.5 GeV is
computationally too demanding and they have to resort to approximations. This
is no longer the case for the $\tau$ as simulations with lattice cut-offs of
around 2 GeV are becoming quite common even with DWQ both because of increased
computational resources as well as technical and algorithmic advances.

For precisely extracting weak mixing angles and testing CKM unitarity new
promising ideas have been proposed for using inclusive B-decays. This could be
enormously usefulin B-physics. For one thing there are long-standing few
$\sigma$ tensions between inclusive (which are based on non-lattice continuum
methods) and exclusive (a lot of these come from use of lattice techniques)
extractions. Moreover, if one could use inclusive modes then because the BRs
are a lot bigger then that would make the test for CKM unitarity and or the
search for new physics become as much more sensitive.

Using the $\tau$ as a testing ground for these new ideas has at least two
significant advantages over the B-decays. First as already mentioned use of
$\tau$ does not require an approximation in so far as the lattice cut-off
goes. In addition, for the $\tau$ already a huge amount of data exists which
is not the case especially for the suppressed b $\to$ u transitions. For using
$\tau$ decays inclusively one can subdivide as those going with
$V_{ud}=\cos_{\theta}$ and those going with $V_{us}=\sin_{\theta}$, where
$\theta$ is the Cabibbo angle. Experimentally, they can be distinguished
obviously as those events with 0 strangeness (s) and those with
$s=+~{}or~{}-1$.

##  5 Summary & Conclusion

Experiments at major B-facilities studying B-decays for almost a decade have
been providing hints of lepton universality violations. Relative strong
indications (over $3\sigma$) have persisted for almost two decades also in
muon (g-2) measurements. These original BNL results have now been confirmed by
measurements done in an entirely new, major experiments effort at Fermilab
increasing the tension with the Standard Model to over 4$\sigma$. These
experiments strongly suggest that $\tau$ lepton is being influenced by new
physics. Then naturalness reasoning strongly suggests that the new physics
will be accompanied by BSM-CP-odd phase(s). Spin of the $\tau$ and its decays
into multi-particle final states as well as its electric dipole moment should
be vigorously studied. CP-conserving tests such as rate measurements and
magnetic moment measurements will also be very timely. BELLE-II with its
anticipated increase in luminosity can especially be very valuable. But of
course LHCb as well as ATLAS and CMS can also play a very important role and
continual increase in luminosity are strongly urged. The fact that the mass of
the $\tau$ is a lot less than the b-quark mass can be very useful both for
testing the Standard Model as well as valuable laboratory for testing new
theoretical ideas in non-perturbative lattice efforts.

##  6 Acknowledgements

I wish to thank the organizers and in particular Emilie Passemar for the
excellent organization and for the equally good program of Tau2021. Also i
must thank Bhupal Dev for discussions and other help. This research was
supported in part by USDOE Contract # DE-SC0012704.

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