= Solution
Neglect radiation, take $E_\tau=M_Z/2\simeq45.6\,\mathrm{GeV}$, $m_\tau\simeq1.777\,\mathrm{GeV}$, and treat a fixed hadronic <invariant mass> $m_h=1.5\,\mathrm{GeV}$ as a single recoil system in $\tau^-\to h^-\nu_\tau$. With negligible <neutrino mass>,
$$
E_h^*=\frac{m_\tau^2+m_h^2}{2m_\tau},\quad p_h^*=\frac{m_\tau^2-m_h^2}{2m_\tau},\quad
E_h=\gamma(E_h^*+\beta p_h^*\cos\vartheta),\quad
\gamma=\frac{E_\tau}{m_\tau}.
$$
Thus $E_h^{\min,\max}=\gamma(E_h^*\mp\beta p_h^*)$, numerically approximately $32.50$ to $45.60\,\mathrm{GeV}$. In the ultrarelativistic limit the range is $E_\tau m_h^2/m_\tau^2\leq E_h\leq E_\tau$. A spread of hadronic <invariant masses> broadens these endpoints; an average <mass> alone is not a strict endpoint specification for the whole sample.
Let $P_\tau=+1$ denote a positive-helicity $\tau^-$, and measure $\vartheta$ from its laboratory flight direction in its rest frame. For a spin-zero hadronic recoil, the left-helicity <tau neutrino> and <angular momentum> conservation give
$$
\frac1\Gamma\frac{d\Gamma}{d\cos\vartheta}=\frac12(1+P_\tau\cos\vartheta).
$$
For example a <tau lepton> with <spin> along $+z$ can emit a left-helicity <neutrino> along $-z$, carrying <spin> along $+z$, with the spin-zero recoil along $+z$. A positive-helicity <tau lepton> then gives a harder visible recoil. If $x_h=E_h/E_\tau$, $r=m_h^2/m_\tau^2$ and $\beta\simeq1$, the more general <tau hadronic polarimetry> formula is
$$
\frac1\Gamma\frac{d\Gamma}{dx_h}=\frac1{1-r}\left[1+\alpha_hP_\tau\frac{2x_h-1-r}{1-r}\right],\quad r<x_h<1,\qquad
\langle E_h\rangle=\gamma E_h^*+\frac{\gamma\beta p_h^*}{3}\alpha_hP_\tau.
$$
For spin-zero recoil $\alpha_h=1$. For a spin-one system produced by the <V-A interaction>, the longitudinal and transverse weights are $m_\tau^2:2m_h^2$, with opposite angular slopes; summing gives $\alpha_h=(m_\tau^2-2m_h^2)/(m_\tau^2+2m_h^2)$. At $m_h=1.5\,\mathrm{GeV}$ this is about $-0.175$, so the negative-helicity <tau lepton> has the higher mean total hadronic <energy>, about $39.43$ versus $38.67\,\mathrm{GeV}$, within that spin-one model. <Tau polarization and correlated decays in neutrino experiments (2023)>[Polarized tau decay: longitudinal and transverse spin-one recoil distributions, Eqs. (5)–(6)]{ref}
The supplied mean <invariant mass> does not specify the recoil's <spin> composition or its <helicity amplitudes>. Consequently the last sign cannot be universal: positive helicity is harder for a spin-zero recoil, negative helicity is harder for the specified $1.5\,\mathrm{GeV}$ spin-one recoil, and a mixture requires its measured spectral weights. Polarization of an intermediate vector or axial-vector <meson> also changes the division of <energy> among its daughter <hadrons>; that information is not fixed by the total jet <invariant mass> alone.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2015-angular-and-tau-energy.png]
{title=Chiral annihilation and <tau lepton> recoil <energy> distributions}
{description=Original theoretical curves. Left: normalized pure-QED and illustrative Z-pole distributions for sin-squared weak angle 0.231. Right: spin-zero and spin-one recoil models at fixed hadronic mass 1.5 GeV; the opposite polarization slopes show why the spin assumption matters. These are not reconstructions of either damaged source graph.}
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