Probing novel scalar and tensor interactions from (ultra)cold neutrons to the LHC

Scalar and tensor interactions were once competitors to the now well-established $V-A$ structure of the standard model weak interactions. We revisit these interactions and survey constraints from low-energy probes (neutron, nuclear, and pion decays) as well as collider searches. Currently, the most stringent limit on scalar and tensor interactions arise from $0^{+}\to 0^{+}$ nuclear decays and the radiative pion decay $\pi \to e\nu \gamma$, respectively. For the future, we find that upcoming neutron beta decay and LHC measurements will compete in setting the most stringent bounds. For neutron beta decay, we demonstrate the importance of lattice computations of the neutron-to-proton matrix elements to setting limits on these interactions, and provide the first lattice estimate of the scalar charge and a new average of existing results for the tensor charge. Data taken at the LHC is currently probing these interactions at the ${10}^{-2}$ level (relative to the standard weak interactions), with the potential to reach the $\lesssim {10}^{-3}$ level. We show that, with some theoretical assumptions, the discovery of a charged spin-0 resonance decaying to an electron and missing energy implies a lower limit on the strength of scalar interactions probed at low energy.

Figure 1

The allowed region in the two-dimensional plane ${ϵ}_P^{ee}-{ϵ}_P^{ex}$ determined by $R_{\pi }$ is given by an annulus of thickness $1.38\times {10}^{-6}$. In the absence of information on ${ϵ}_P^{ex}$, the 90% C.L. bound on ${ϵ}_P^{ee}$ is $-1.4\times {10}^{-7}<{ϵ}_P^{ee}<5.5\times {10}^{-4}$.

Figure 2

90% C.L. allowed regions in the ${ϵ}_S-{ϵ}_T$ plane implied by (i) the existing bound on $b_{0^{+}}$ (green horizontal band); (ii) projected ${10}^{-3}$-level limits on $b$ (red band), $b_{\nu }-b$ (blue band, left panel), and $b_{\nu }$ (blue band, right panel). The hadronic form factors are taken to be $g_S=g_T=1$ in the ideal scenario of no uncertainty. The impact of hadronic uncertainties is discussed in Sec. 6.

Figure 3

90% C.L. allowed regions in the ${ϵ}_S-{ϵ}_T$ plane implied by (i) the existing bound on $b_{0^{+}}$ (green horizontal band); (ii) projected ${10}^{-4}$-level limits on $b$ (red band), $b_{\nu }-b$ (blue band, left panel), and $b_{\nu }$ (blue band, right panel). The hadronic form factors are taken to be $g_S=g_T=1$ in the ideal scenario of no uncertainty. The impact of hadronic uncertainties is discussed in Sec. 6.

Figure 4

(Upper row) The axial charge versus $M_{\pi }^2$ from $N_f=2$ [73, 74, 75, 76, 77] (left) and $N_f=2+1$ [78, 80, 82, 83, 84] (right) calculations with different types of $O(a)$-improved fermion actions. The filled symbols and solid error bar (open symbols and dashed errorbar) denote results taken from published papers (the latest lattice proceedings). (Lower panel) Comparison of the published values of $g_A$ with experimental measurements [98] (vertical band). The solid lines indicate statistical error while the dashed lines include systematic errors. The analysis of systematic errors does not include all sources of uncertainty. For example, only the QCDSF and ETMC calculations have been done at multiple values of the lattice spacing. Also, the chiral extrapolation is different in the different calculations. Lin et al. [99] find that an SU(3)-constrained fit to the $g_A$ for octet baryons reduces the statistical error in the chiral extrapolation. This is illustrated by the larger errors in the LHPC result [82] compared to those in Ref. 99, which are obtained using similar lattice parameters.

Figure 5

(Left) The induced-pseudoscalar charge of the nucleon $g_P^{*}$ from experimental measurements [101, 102, 104, 105, 106] and an earlier estimation from $\mathrm{HB}\chi \mathrm{PT}$ [103]. (Right) Comparison of lattice estimates of $g_P^{*}$ using the DWF fermion action [77, 107, 138] with MuCap data.

Figure 6

Summary of LQCD estimates of $g_{\mathrm{T}}$ using $N_f=2$ [74, 77, 115] (top) and $N_f=2+1$ [79, 82] (bottom) $O(a)$-improved fermion actions. Two chiral extrapolations of $g_T$ are shown using the combined RBC/UKQCD DWF data on their 2.7-fm ensemble [79] and the LHPC mixed-action data [82]. The value at the physical pion mass from the linear fit is shown by the red diamond. Fits using the $\mathrm{HB}\chi \mathrm{PT}$ ansatz are very sensitive to removing points at large $M_{\pi }$ so no error band is shown. For the $g_T^{2\mathrm{f}}$ data, the filled symbols and solid error bars (open symbols and dashed error bars) denote results taken from the published papers (the latest lattice proceedings).

Figure 7

The scalar charge $g_{\mathrm{S}}$ from $N_f=2+1$ anisotropic clover and DWF/asqtad lattices. We also show a linear fit to the full data set and a constant fit to the data at the five smallest values of $M_{\pi }^2$. The extrapolated values are shown using red diamonds.

Figure 8

Left panel: 90% C.L. allowed regions in the ${ϵ}_S-{ϵ}_T$ plane implied by (i) the existing bound on $b_{0^{+}}$ (green horizontal band); (ii) projected measurements of $b$ and $b_{\nu }-b$ in neutron decay (red bow-tie shape and blue region) at the ${10}^{-3}$ level; (iii) hadronic matrix elements taken in the ranges $0.25<g_S<1.0$, $0.6<g_T<2.3$ [23]. Right panel: same as left panel but with scalar and tensor charges taken from lattice QCD: $g_S=0.8(4)$ and $g_T=1.05(35)$. Note that by reducing the uncertainty in $g_S$ the constraint on ${ϵ}_S$ from $b_{0^{+}}$ becomes stronger, independent of any future neutron measurement. The effective couplings ${ϵ}_{S,T}$ are defined in the $\overline{\mathrm{MS}}$ scheme at 2 GeV.

Figure 9

Combined 90% C.L. allowed regions in the ${ϵ}_S-{ϵ}_T$ plane based on: (i) existing limit on $b_{0^{+}}$ from $0^{+}\to 0^{+}$ nuclear decays; (ii) future neutron decay measurements with projected sensitivity of ${10}^{-3}$ in $b$ and $b_{\nu }-b$. The four curves correspond to four different scenarios for the hadronic matrix elements: quark model estimates $0.25<g_S<1.0$, $0.6<g_T<2.3$ [20, 23]; lattice results with current central values from Eq. (38) and $\delta g_S/g_S=50\%$, 20%, 10% with $\delta g_T/g_T=2/3\delta g_S/g_S$ (this choice assumes that the ratio of fractional uncertainties in $g_S$ and $g_T$ will remain approximately constant as these uncertainties decrease). The effective couplings ${ϵ}_{S,T}$ are defined in the $\overline{\mathrm{MS}}$ scheme at 2 GeV.

Figure 10

Joint 90% CL limit on ${ϵ}_S$ and ${ϵ}_T$ implied by: (i) current bounds from nuclear $\beta$ decay $0^{+}\to 0^{+}$ and radiative pion decay (blue, dashed); (ii) CMS search [132] in the channel $pp\to e\nu +X$ at $\sqrt{s}=7\mathrm{TeV}$ with $1.03{\mathrm{fb}}^{-1}$ of data. The limit is obtained by requiring less than 3.7 $e\nu$-produced events having $m_T>1\mathrm{TeV}$ (red, solid). LO MSTW 2008 [134] parton distribution functions are used; (iii) projected LHC searches in the channel $pp\to e\nu +X$ at $\sqrt{s}=7\mathrm{TeV}$ with $10{\mathrm{fb}}^{-1}$ of data (gold, dotted). The limit is obtained by requiring less than 3 $e\nu$-produced signal events with $m_T>1.5\mathrm{TeV}$ and assuming that no events are observed. The cut is chosen to reduce the expected leading background to be below 1 event. The effective couplings ${ϵ}_{S,T}$ are defined in the $\overline{\mathrm{MS}}$ scheme at 2 GeV.

Figure 11

Projected joint 90% credibility level limit on ${ϵ}_S$ and ${ϵ}_T$ from the LHC at $\sqrt{s}=14\mathrm{TeV}$, obtained from requiring less than 3 $e\nu$-produced signal events with: (i) $m_T>2.5\mathrm{TeV}$ and $10{\mathrm{fb}}^{-1}$ of integrated luminosity (solid, red ellipse); and (ii) $m_T>4\mathrm{TeV}$ and $300{\mathrm{fb}}^{-1}$ (dashed, yellow ellipse). Cuts are chosen to reduce the expected leading background to be below 1 event. To obtain the projection it is assumed no events are found. Same PDFs are used as in Fig. 10. Note the change in scale between these two figures. Anticipated bounds from low-energy experiments and reduced LQCD uncertainties, redrawn from Fig. 9, are shown for comparison. The effective couplings ${ϵ}_{S,T}$ are defined in the $\overline{\mathrm{MS}}$ scheme at 2 GeV.

Figure 12

Projected lower bound on $|{ϵ}_S|$ (at $\mu =2\mathrm{GeV}$), for a discovery cross section of $\sigma (pp\to e+\mathrm{MET}+X)=10\mathrm{fb}$ (blue, solid), 1 fb (red, dashed) and 0.1 fb (black, dotted), as a function of $\tau =m^2/s$. Shaded region (green) shows the current experimental exclusion on ${ϵ}_S$ from $0^{+}\to 0^{+}$ nuclear $\beta$ decay.

Figure 13

Projected lower bound on $|{ϵ}_S|$ (at $\mu =2\mathrm{GeV}$) for $\sqrt{s}=7\mathrm{TeV}$ and a discovery cross section of $\sigma (pp\to e+\mathrm{MET}+X)=5\mathrm{fb}$ (blue, solid), 1 fb (red, dashed) and 0.5 fb (black, dotted). Shaded region (green) shows the current experimental exclusion on ${ϵ}_S$ from $0^{+}\to 0^{+}$ nuclear $\beta$ decay. The bound scales linearly with $\sigma ·\mathrm{BR}$.

Figure 14

Same as Fig. 13 but for $\sqrt{s}=14\mathrm{TeV}$ and $\sigma ·\mathrm{BR}=100\mathrm{fb}$ (blue, solid), 10 fb (red, dashed) and 2 fb (black, dotted).