Isgur-Wise function
In HQET hadronic matrix elements of semileptonic decays (e.g. $\bar{B}\to D^*$) only depend on a single form factor called the Isgur-Wise function
# Semileptonic decays
In general the hamiltonian weak semileptonic for $b\to d$ transitions is $$H_{EW} = \frac{G_F}{\sqrt{2}}V_{cb}, \bar{c}\gamma_\mu (1-\gamma_5)b\bar{e}\gamma^\mu (1-\gamma_5)\nu_e$$ We can factorize the hadronic part of the matrix element. Taking the example of $\bar{B}\to D$ (pseudoscalar meson) and $\bar{B}\to D^*$ (vector meson) we get 3 non vanishing hadronic matrix elements ($\braket{D|A^\mu|\bar{B}}=0$ due to parity).
$$\braket{D(p^\prime)|V^\mu|\overline{B}(p)} = f_+(q^2)(p+p^\prime)^\mu + f_-(q^2)(p-p^\prime)^\mu$$ $$\braket{D^*(p^\prime,\epsilon^\prime)|V^\mu|\bar{B}(p)}=ig(q^2)e^{\mu\nu\alpha\beta}\epsilon^{\prime }\nu(p+p^\prime)\alpha(p-p^\prime)_\beta$$ $$\braket{D^(p^\prime,\epsilon^\prime)|A^\mu|\bar{B}(p)} = f(q^2)\epsilon^{\prime *\mu} + \epsilon^{\prime *}\cdot p \left[a_+(q^2)(p+p^\prime) + a_-(q^2)(p-p^\prime)^\mu\right]$$
with the vector- and axialvector- currents $V^\mu$ and $A^\mu$ and $q^2 = (p-p^\prime)^2$ the momentum transfer.
Here we have in total 6 real form factors: $f_\pm, g, f, a_\pm$.
# HQ limit
We can use HQET if the momentum transfer among the light d.o.f. is much smaller than the HQ masses $m_b,m_c$ : $$|(\Lambda_{QCD}v - \Lambda_{QCD}v^\prime)^2| \ll m^2_{b,c}$$
- Here $p^{(\prime)\mu} = m_{B(D)} v^\mu$ are the approximate hadron velocities
- The light d.o.f. approximately move along the hadron velocity
- By rearranging and estimating after plugging in $v^\mu = p^\mu / m_B$ one can show that the condition is satisfied quit well.
- HQET is applicable!
# Decay to a pseudo scalar
Definition: Isgur-Wise function
In HQET the hadronic matrix element of $\bar{B}\to D$ is given by $$\braket{D(v^\prime)|\bar{c}_{v^\prime}\gamma^\mu b_v|\bar{B}(v)} = \xi(w)(v + v^\prime)^\mu$$ where $w = v\cdot v^\prime$ and $$\xi(1)=1.$$ $\xi(w)$ is called the Isgur-Wise function
Derivation of the Isgur Wise function
In HQET the general decomposition is in terms of the lorentz vectors $v^\mu$ and $v^{\prime \mu}$: $$\braket{D(v^\prime)|\bar{c}_{v^\prime}\gamma^\mu b_v|\bar{B}(v)} = \xi(v+v^\prime)^\mu + a(v-v^\prime)^\mu,$$ with form factors $\xi$ and $a$.
Contracting with $(v-v^\prime)^\mu$ the left side vanishes due to $\cancel{v}b_v = b_v$ and the $\xi$ term vanishes due to $v^2=v^{\prime 2}=1$. Therefore $a=0$.
The relation $\xi(1) = 1$ follows from identifying the vector current with the generator of the flavor symmetry: $$N_{cb} = \int d^3x J^{0}(x) = \int d^3x c^\dagger_v(x)b_v(x)$$ with (why?) $$N_{cb}\ket{\bar{B}(v)}=\ket{D(v)}.$$ The relation follows from equating $$\int d^3x \braket{D(v)|\bar{c}v\gamma^0b_v|\bar{B}(v)} = \braket{D(v)|N{cb}|\bar{B}(v)}$$
# Decay to a vector meson
For the decay into a vector meson like $\bar{B}\to D^*$ we can decompose the hadronic matrix elements as well.
Hadronic matrix elements of decay into vector meson
The corresponding vector and axialvector matrix elements in the HQET become $$\braket{D^(v^\prime,\epsilon)|\bar{c}_{v^\prime}\gamma^\mu b_v|\bar{B}(v)} = i\xi(w)\epsilon^{\mu\nu\alpha\beta}\epsilon^\nu v^\prime\alpha v_\beta$$ $$\braket{D^*(v^\prime,\epsilon)|\bar{c}_{v^\prime}\gamma^\mu\gamma_5 b_v|\bar{B}(v)} = \xi(w)\left[\epsilon^{\prime * \mu}(1+v\cdot v^\prime) - v^{\prime\mu}(v\cdot \epsilon^{\prime *}) \right]$$
Note that we again have the Isgur-Wise function as only form-factor which is even the same as for the decay into the $D$.
Derivation of the vector meson matrix elements
For the derivation we need the explicit spin operators for the heavy quark fields: $$\vec{S}_Q = \int d^3x h^\dagger_v(x)\vec{S}h_v(x),$$ with $S^i = \frac{1}{2}\gamma_5\gamma^0\gamma^i$ in the hadron rest frame.
With $$2 S^3_c\ket{D(v^\prime)} = \ket{D^(v^\prime,\epsilon_3)}$$ and $$S^3_c\bar{c}{v^\prime} = \bar{c}{v^\prime}S^3$$ We can explicitly relate the $D^$ matrix elements to $D$ matrix elements, e.g. $$\braket{D^*(v^\prime,\epsilon_3)|\bar{c}{v^\prime}\gamma^0 b_v|\bar{B}(v)} = -\braket{D(v^\prime)|\bar{c}{v^\prime}\gamma_5\gamma^3 b_v|\bar{B}(v)},$$ where we used $S^3\gamma^0 = -\frac{1}{2}\gamma_5\gamma^3$.
Using the original form-factor decompositions we can relate the Isgur-Wise function to the full set of form-factors $f_\pm, g, f, a_\pm$.