Electroweak Physics with Polarized Electron Beams in a SuperKEKB Upgrade

TL;DR

Upgrading SuperKEKB with polarized electron beams will enable high-precision electroweak measurements at 10.58 GeV, testing the Standard Model and probing new physics with unprecedented accuracy.

Contribution

This paper proposes a detailed plan for upgrading SuperKEKB to include polarized electron beams, enabling new precision electroweak physics measurements at a previously unexplored energy scale.

Findings

Precision of electroweak measurements will match LEP/SLC averages.

Measurements of neutral current couplings will be more accurate.

Potential to resolve existing discrepancies in electroweak data.

Abstract

Consideration is being given to upgrading the SuperKEKB e$^+$e$^-$ collider with polarized electron beams, which would open a new program of precision electroweak physics at a centre-of-mass energy of 10.58GeV, the mass of the $\Upsilon(4S)$. These measurements include $\sin^2\theta_W$ obtained via left-right asymmetry measurements of e$^+$e$^-$ transitions to pairs of electrons, muons, taus, charm and b-quarks. The precision obtainable at SuperKEKB will match that of the LEP/SLC world average and will thereby probe the neutral current couplings with unprecedented precision at a new energy scale sensitive to the running of the couplings. At SuperKEKB the measurements of the individual neutral current vector coupling constants to b-quarks and c-quarks and muons in particular will be substantially more precise than current world averages and the current 3$\sigma$ discrepancy between the SLC A$_{LR}$ measurements and LEP A$_{FB}^b$ measurements of $\sin^2\theta_W^{eff}$ will be addressed. This paper will include a discussion of the necessary upgrades to SuperKEKB. This program opens an exciting new window in searches for physics beyond the Standard Model.

Tables

Table 1. Table 1: For each fermion pair that can be cleanly identified in Belle II for the given efficiency in column 1: column 2 gives the SM value of the left-right asymmetry; column 3 gives the expected relative error on an upgraded SuperKEKB/Belle II measurement of A L ​ R subscript 𝐴 𝐿 𝑅 A_{LR} calculated as the quadratic sum of the statistical error (based on 20 ab -1 , the assumed selection efficiencies given in column 1 and a beam polarization at Belle II of 70%) and a systematic error of ± 0.5 % plus-or-minus percent 0.5 \pm 0.5\% on the 70% beam polarization; column 4 gives the current world average value of its neutral current vector coupling; column 5 gives the projected error on g V f superscript subscript 𝑔 𝑉 𝑓 g_{V}^{f} for an upgraded SuperKEKB/Belle II measurement using 20 ab -1 of data; column 6 gives the projected error on g V f superscript subscript 𝑔 𝑉 𝑓 g_{V}^{f} using 40 ab -1 of data; and column 7 gives the projected SuperKEKB/Belle II error on sin 2 ⁡ θ W e ​ f ​ f superscript 2 superscript subscript 𝜃 𝑊 𝑒 𝑓 𝑓 \sin^{2}\theta_{W}^{eff} with 40 ab -1 of polarized beam data.

Final State	$A_{LR}^{SM}$	Relative	$g_V^f$ W.A.LEPSLDReport2006	$\sigma \left(g_V^f\right)$	$\sigma \left(g_V^f\right)$	$\sigma \left({\sin }^2{\theta }_W^{eff}\right)$
Fermion		$A_{LR}$ Error (%)		for 20 ab-1	for 40 ab-1	for 40 ab-1
b-quark	-0.020	0.5	-0.3220	0.002	0.002	0.003
(eff.=0.3)			$\pm 0.0077$	improves x4
c-quark	-0.005	0.5	+0.1873	0.001	0.001	0.0007
(eff.=0.3)			$\pm 0.0070$	improves x7
tau	-0.0006	2.3	-0.0366	0.0008	0.0006	0.0003
(eff.=0.25)			$\pm 0.0010$
muon	-0.0006	1.5	-0.03667	0.0005	0.0004	0.0002
(eff.=0.5)			$\pm 0.0023$	improves x5
electron	-0.0006	1.2	-0.3816	0.0004	0.0003	0.0002
(1 nb acceptance)			$\pm 0.00047$

Equations

$$A_{LR}^f=\frac{{\sigma }_L-{\sigma }_R}{{\sigma }_L+{\sigma }_R}=\frac{s{G}_F}{\sqrt{2}\pi \alpha Q_f}{g}_A^e{g}_V^f⟨Pol⟩$$

$$⟨Pol⟩=\frac{1}{2}\left[{\left(\frac{N_{eR}-N_{eL}}{N_{eR}+N_{eL}}\right)}_{\mathbf{R}}-{\left(\frac{N_{eR}-N_{eL}}{N_{eR}+N_{eL}}\right)}_{\mathbf{L}}\right]$$

Full text

With its high design luminosity of $8\times 10^{35}$ cm*-2* s*-1*, the SuperKEKB e+e- collider operating at a centre-of-mass energy of 10.58 GeV can access new windows for discovery with the Belle II experiment if it is upgraded to have a longitudinally polarized electron beam. The target integrated luminosity for SuperKEKB/Belle II is 50 ab*-1*BelleIITDR and currently Belle II is projected to collect that amount of data, which will not have beam polarization without an upgrade to SuperKEKB, in 2027. Upgrading SuperKEKB to have electron beams with left and right longitudinal polarization of approximately 70% at the Belle II interaction point creates a unique and versatile facility for probing new physics with precision electroweak measurements that no other experiments, current or planned, can achieve.

In particular, a data sample of 20 ab*-1* with a polarized electron beam enables Belle II to measure the weak neutral current vector coupling constants of the $b$-quark, $c$-quark and muon at significantly higher precision than any previous experiment. With 40 ab*-1* of polarized beam data, the precision of the vector couplings to the tau and electron can be measured at a level comparable to current world averages, which are dominated by LEP and SLD measurements at the Z0-pole.

Within the framework of the Standard Model (SM) these measurements of $g_{V}^{f}$, the neutral current vector coupling for fermion $f$, can be used to determine the weak mixing angle, $\theta_{W}$, through the relation: $g_{V}^{f}=T_{3}^{f}-2Q_{f}\sin^{2}\theta_{W}$, where $T_{3}^{f}$ is the $3^{rd}$ component of weak isospin of fermion $f$, $Q_{f}$ is its electric charge in units of electron charge and the notational conventions of Reference LEPSLDReport2006 are used.

As described in Reference SuperBTDR , with polarized electron beams an e+e- collider at 10.58 GeV can determine $g_{V}^{f}$ by measuring the left-right asymmetry, $A_{LR}^{f}$, for each identified final-state fermion-pair in the process $e^{+}e^{-}\rightarrow f\overline{f}$:

[TABLE]

where $g_{A}^{e}=T_{3}^{e}=-\frac{1}{2}$ is the neutral current axial coupling of the electron, $G_{F}$ is the Fermi coupling constant, $s$ is the square of the centre-of-mass energy, and

[TABLE]

is the average electron beam polarization, where $N_{eR}$ is the number of right-handed electrons and $N_{eL}$ the number of left-handed electrons in the event samples where the electron beam bunch is left polarized or right polarized, as indicated by the ‘${\mathbf{R}}$’ and ‘${\mathbf{L}}$’ subscripts. These asymmetries arise from $\gamma-Z$ interference Bernabeu and although the SM asymmetries are small ($-6\times 10^{-4}$ for the leptons, $-5\times 10^{-3}$ for charm and $-2\%$ for the $b$-quarks), unprecedented precisions can be achieved because of the combination of both the high luminosity of SuperKEKB and a $70\%$ beam polarization measured with a precision of better than $\pm 0.5\%$. Note that higher order corrections are ignored here for simplicity, although studies at higher orders have recently begun Aleskejevs .

The upgrade to SuperKEKB involves three hardware projects:

• •

A low-emittance polarized electron source in which the electron beams would be produced via a polarized laser illuminating a “strained lattice” GaAs photocathode as was done for SLD LEPSLDReport2006 . The source would produce longitudinally polarized electron bunches whose spin would be rotated to be transversely polarized before it enters the SuperKEKB electron storage ring;

• •

A pair of spin-rotators, one positioned before and the other after the interaction region, to rotate the spin to longitudinal prior to collisions and back to transverse following collisions. One configuration under consideration for the spin-rotator is a combined function magnet that replaces an existing dipole in the SuperKEKB electron beam lattice with a magnet that is both a dipole and solenoid UliWienands . The challenge is to design the rotators to minimize couplings between vertical and horizontal planes and to address higher order and chromatic effects in the design to ensure the luminosity is not degraded.

• •

A Compton polarimeter that measures the beam polarization before the beam enters the interaction region.

The high precisions are possible at such an upgraded SuperKEKB because with 20 ab*-1* of data Belle II can identify between $10^{9}$ and $10^{10}$ final-state pairs of b-quarks, c-quarks, taus, muons and electrons with high purity and reasonable signal efficiency, and because all detector-related systematic errors can be made to cancel by flipping the laser polarization from ${\mathbf{R}}$ to ${\mathbf{L}}$ in a random, but known, pattern as collisions occur. $\langle Pol\rangle$ would be measured in two ways. The first method uses a Compton polarimeter, which can be expected to have an absolute uncertainty at the Belle II interaction point of $<1\%$ and provides a ‘bunch-by-bunch’ measurement of $\left(\frac{N_{eR}-N_{eL}}{N_{eR}+N_{eL}}\right)_{\mathbf{R}}$ and $\left(\frac{N_{eR}-N_{eL}}{N_{eR}+N_{eL}}\right)_{\mathbf{L}}$. The uncertainty will be dominated by the need to predict the polarizatoin loss from the Compton polarimeter to the interaction point. The second method measures the polar angle dependence of the polarization of $\tau$-leptons produced in $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events using the kinematic distributions of the decay products of the $\tau$ separately for the ${\mathbf{R}}$ and ${\mathbf{L}}$ data samples. The forward-backward asymmetry of the tau-pair polarization is linearly dependent on $\langle Pol\rangle$ and therefore can be used to determine $\langle Pol\rangle$ to $0.5\%$ at the Belle II interaction point in a manner entirely independent of the Compton polarimeter. The $\tau$ polarization forward-backward asymmetry method avoids the uncertainties associated with tracking the polarization losses to the interaction point and also automatically accounts for any residual positron polarization that might be present. In addition, it automatically provides a luminosity-weighted beam polarization measurement.

Table 1 provides the sensitivities to electroweak parameters expected with polarized electron beams in an upgraded SuperKEKB from $e^{+}e^{-}\rightarrow b\bar{b}$, $e^{+}e^{-}\rightarrow c\bar{c}$, $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$, $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$, and $e^{+}e^{-}\rightarrow e^{+}e^{-}$ events selected by Belle II. From this information the precision on the b-quark, c-quark and muon neutral current vector couplings will improve by a factor of four, seven and five, respectively, over the current world average valuesLEPSLDReport2006 with 20 ab*-1* of polarized data.

This is of particular importance for $g^{b}_{V}$, where the measurement of -0.3220$\pm$0.0077 is 2.8$\sigma$ higher than the SM value of -0.3437 LEPSLDReport2006 . That discrepancy is a manifestation of the 3$\sigma$ difference between the SLC $A_{LR}$ measurements and LEP $A^{b}_{FB}$ measurements of $\sin^{2}\theta_{W}^{eff}$ LEPSLDReport2006 . A measurement of $g^{b}_{V}$ at an upgraded SuperKEKB that is four times more precise and which avoids the hadronization uncertainties that are a significant component of the uncertainties of the measurement of $A^{b}_{FB}$ at LEP, will be able to definitively resolve whether or not this is a statistical fluctuation or a first hint of a genuine breakdown of the SM.

Table 1 also indicates the uncertainties on $\sin^{2}\theta_{W}^{eff}$ that can be achieved with 40 ab*-1* of polarized beam data - the combined uncertainty at Belle II would be comparable to the Z0 world average measured uncertainty of $\pm 0.00016$ from LEP and SLDLEPSLDReport2006 but made at a significanlty lower energy scale. Figure 1 shows the determinations of $\sin^{2}\theta_{W}$ in the $\overline{MS}$ renormalization scheme as a function of energy scale ErlerPDG2018 at present and future experimental facilities including SuperKEKB upgraded with a polarized electron beam delivering 40 ab*-1* of data to Belle II.

This electroweak program with polarized electron beams in SuperKEKB would provide the highest precision tests of neutral current vector coupling universality. In addition, right-handed $b$-quark couplings to the $Z$ can be experimentally probed with high precision at Belle II with polarized beams. Also, measurements with the projected precision will enable Belle II to probe parity violation induced by the exchange of heavy particles such as a hypothetical TeV-scale $Z^{\prime}$ boson(s). If such bosons only couple to leptons they will not be produced at the LHC. Moreover, the SuperKEKB machine will have a unique possibility to probe parity violation in the lepton sector mediated by light and very weakly coupled particles often referred to as “Dark Forces”. Such forces have been entertained as a possible connecting link between normal and dark matter Pospelov:2008jd ; ArkaniHamed:2008qn . SuperKEKB with polarization would be uniquely sensitivity to “Dark Sector” parity violating light neutral gauge bosons – especially when $Z_{dark}$ is off-shell and with a mass between roughly 10 and 35 GeV Davoudiasl or even up to the Z0 pole, or couples more to the 3rd generation.

The enhancement of parity violation in the muon sector has been an automatic consequence of some models Batell:2011qq that aim at explaining the unexpected result for the recent Lamb shift measurement in muonic hydrogen Pohl:2010zza . The left-right asymmetry of the $e^{-}e^{+}\to\mu^{-}\mu^{+}$ in such models is expected to be enhanced at low-to-intermediate energies, and SuperKEKB with polarized beams may provide a conclusive test of such models, as well as impose new constraints on a parity-violating dark sector.

An electron beam polarization upgrade at SuperKEKB opens an exciting and unique discovery window with precision electroweak physics. A growing international team has begun to study the feasibility of such an upgrade with the goal to begin taking Belle II data with polarized SuperKEKB electron beams in the mid-2020’s.