Heavy quark effective theory (HQET)

Last updated Aug 5, 2023

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Heavy Quark Effective Theory (HQET) is an EFT for low energy QCD to describe hadrons containing one heavy quark (charm, bottom)

Bound states of contain one heavy quark $Q$ and many light d.o.f. (light quarks $q$ and gluons)

• Every hadron is color neutral: Singlet under $SU(3)$
• Heavy quark $Q$ in color triplet $(3)$ can only bind with antitriplet ($\bar{3}$), since $3\otimes\bar{3}=1\oplus 8$:

1. Meson $Q\bar{q}$: $3\otimes\bar{3}=1\oplus 8$ - “Heavy-light meson”
2. Barion $Qqq$: $3\otimes3\otimes 3=1\oplus …$

Hadrons are classified by different properties

1. Total angular momentum $j$ $$\vec{J} = \vec{S}_Q + \vec{S}_l$$ Sum of angular momentum of heavy quark $Q$ and light d.o.f. $l$

• The spin of the heavy quark is $s_Q = \frac{1}{2}$
• Heavy hadrons are always doublets with $j_\pm = |s_l \pm \frac{1}{2}|$

2. Parity $P$

• Mesons have parity $(-1)^{1+l}$ with orbital angular momentum $l$
• Baryons have parity $(-1)^{l}$ with orbital angular momentum $l$

3. Isospin $I_3$

• Quantum number of flavor symmetry $SU(2)$ for up and down quark

4. Electric charge

If $s_l = s_q = \frac{1}{2}$ we have two possible $j$s: $j_+ = 1$ ($D^,B^$), $j_- = 0$ ($D,B$)

We can build the flavor $SU(3)$ diagram (Isosping-Strangeness) for $0^-$ (Pseudoscalar) and $1^-$ (Vector) states ($j^P$)

• First excited heavy mesons ($*$) have orbital angular momentum $l=1$
• Since $s_l = l\pm\frac{1}{2}$, $s_l = \frac{1}{2}$ or $s_l = \frac{3}{2}$
• For $s_l = \frac{1}{2}$ the mesons have spin and parity $0^+$ or $1^+$, e.g. in charm named $D^_0$ and $D^_1$
• For $s_l = \frac{3}{2}$ the mesons have spin and parity $1^+$ or $2^+$, e.g. in charm named $D^_1$ and $D^_2$

• For $l=0$ the subsystem $s_l$ which are the light quarks can have $s_l=0$ or $s_l=1$
• Build flavor $SU(3)$ diagram for $\frac{1}{2}^+$ and $\frac{3}{2}^+$ states

Physical model: One heavy quark $Q$ ($c$ or $b$) surrounded by “cloud” of light d.o.f.:

1. One light (anti)quark $q$ (for mesons) or one pair $qq$
2. Light virtual pairs $q\bar{q}$
3. gluons Only 1. gives the cloud its quantum numbers (spin, color, …)

The radius of the system is $R\approx \frac{1}{\Lambda_{QCD}}$. The energy $\Lambda_{QCD}$ is the order of the momentum and energy of the light d.o.f. The mass of the heavy quark $M$ $$M \gg \Lambda_{QCD}$$

For $m\to\infty$ the interaction of the heavy quark does not depend on flavor or spin. This leads to the heavy quark symmetry $$SU(2)\times U(N_h).$$

1. $SU(2)$: Heavy quark Spin symmetry
2. $U(N_h)$: Heavy quark flavor symmetry with the number of heavy quarks $N_h$ (usually c,b $N_h=2$ ?)

Heavy quark symmetry is only an approximate symmetry for HQET not for the QCD Lagrangian in general. It is broken by $\mathcal{O}(\frac{\Lambda_{QCD}}{m})$ and manifest at order $\mathcal{O}(\frac{1}{m^0})$.

following the EFT recipe for $\mathcal{O}(\frac{1}{m^0})$ (static limit):

1. energy scales: $m \gg \Lambda_{QCD}$
2. degrees of freedom (d.o.f): Heavy quarks, light quarks, “soft” gluons, all momenta $\approx \Lambda_{QCD}$
3. symmetries: Symmetries of QCD, Lorentz inv. not manifest, for $m\to \infty$ heavy quark symmetry
4. Lagrangian: Consider hadron moving with velocity $v^\mu$, in the hadron rest frame $v^\mu=(1,0,0,0)$ The momentum of the heavy quark in the hadron $p^\mu = mv^\mu + k^\mu$ with $k^2\approx \Lambda_{QCD}$ due to interaction with gluons Using the momentum we can approximate the feynman rules by “expanding in $\frac{k^2}{m^2}\ll 1$”:

• Heavy quark propagator: $$\frac{i}{\mathrlap{/}p - m +i\epsilon} = \frac{\mathrlap{/}v+1}{2}\frac{i}{vk+i\epsilon}+\mathcal{O}(\frac{1}{m})$$
• Heavy quark - gluon - vertex: $$\frac{i}{\cancel{p}-m+i\epsilon}(-igT^a\gamma^\mu)\frac{i}{\cancel{p}^\prime-m+i\epsilon}=\frac{i}{vk+i\epsilon}\frac{1+\cancel{v}}{2}(-igT^av^\mu)\frac{i}{vk+i\epsilon}\frac{1+\cancel{v}}{2} + \mathcal{O}(\frac{1}{m})$$

The Lagrangian that results in these rules is the QCD lagrangian with modified term for the heavy quark fields: $$\mathcal{L}=\bar{hv}(iv\cdot D)hv - \frac{1}{4}F^a_{\mu\nu}F^{\mu\nu a}+\sum^{h_l}_l\bar{q}_l(i\mathrlap{/}D - m_l)q_l + \mathcal{O}(\frac{1}{m}),$$ with $iD^\mu = i\partial^\mu - g A^\mu$ The heavy quark field $hv$ satisfies $\mathrlap{/}v hv = hv$ (why ? - because projection on hadron part ?) In the rest frame $\cancel{v} = \gamma_0$, so that from $hv$ only two components really contribute.

• Derivation is tree level (only the TL vertices considered above)
• Feynman rules like described above with $\delta_{ij}$ and $(T^a)_{ij}$ for color indices

Alternative derivation of the HQET Lagrangian

In QCD for the heavy quark: $$\mathcal{L}_\text{heavy} = \bar{\psi}(i\cancel{D} - m)\psi$$ We can define the quark field $$\psi(x) = e^{-imvx}(h_v(x) + H_v(x))$$ with the fields $$h_v(x) = e^{imvx}\frac{1+\cancel{v}}{2}\psi(x) \quad H_v(x) = e^{imvx}\frac{1-\cancel{v}}{2}\psi(x)$$. With $\cancel{v}h_v = h_v$, $\cancel{v}H_v = -H_v$ and $\bar{h}vmH_v=0$ this leads to $$\mathcal{L}\text{heavy} = \bar{h}_v i v\cdot Dh_v + \mathcal{O}(\frac{1}{m}),$$ when applying the equation of motion of the field $H_v$: $(iv\cdot D + 2m)H_v = i(\cancel{D} - \cancel{v} v\cdot D)h_v$

The heavy quark symmetry is satisfied by the static limit HQET lagrangian:

1. Heavy flavor symmetry: Satisfied since the lagrangian does not depend on the mass $m$
2. Heavy quark spin symmetry: Satisfied as one can check when applying an infinitesimal transformation using the $SU(2)$ generator

Example: Spectroscopy implications

Heavy quark symmetry can make predictions for differences of hadron masses e.g. $m_{B^*}-m_B=\mathcal{O}(\frac{\Lambda^2_\text{QCD}}{m_b})$ etc..

In HQET pair creation is forbidden ($hv$ does not create a heavy antiquark). The Lagrangian for a heavy antiquark is obtained from $v \to -v$ since the decomposition of the momentum of the heavy antiquark is $p^\mu = -mv^\mu + k^\mu$. $$\mathcal{L} = \bar{h}{-v}(-ivD)h{-v}+…$$ where $h_{-v}$ is the antiquark field in HQET.

While “normal” relativistic hadronic states $\ket{H(p)}$ have a relativistic normalization $$\braket{H(q)|H(p)}=2E_p(2\pi)^3\delta^3(\vec{p}-\vec{q}),$$ in HQET the hadron states are labeled by the four velocity of the hadron $v^\mu$ and the residual momentum $k^\mu$: $$\braket{H(v^\prime,k^\prime)|H(v,k)}=2v^0\delta_{vv^\prime}(2\pi)^3\delta^3(\vec{k}-\vec{k}^\prime)$$ These states have mass dimension $-\frac{3}{2}$.

Decay constants parametrize hadronic matrix elements:

Definition: Decay constant

The decay constant for a pseudoscalar heavy-light meson $f_p$ is defined by $$\braket{0|(\bar{q}\gamma^\mu\gamma_5 Q)|P(p)}=f_p p^\mu$$ Mass dimension: $[f_p]=1$

The decay constant for a vector heavy-light meson $f_{p^}$ is defined as $$\braket{0|(\bar{q}\gamma^\mu Q)|P^(p,\epsilon)}=f_{p^} \epsilon^\mu$$ Mass dimension: $[f_{p^}]=1$

These are the most general decomposition due to parity and the conservation of the vector current.

In HQET the decompositions becomes $$\braket{0|(\bar{q}\gamma^\mu\gamma_5h_v)|P(v)}=av^\mu$$ $$\braket{0|(\bar{q}\gamma^\mu h_v)|P^(v,\epsilon)}=\tilde a\epsilon^\mu$$ with mass dimension $[a] = [\tilde a] = \frac{3}{2}$. In the hadron rest frame $v^\mu=(1,0,0,0)$ and $\epsilon^\mu =(0,0,0,1)$, using $S^3_Q\ket{P(v)}=\frac{1}{2}\ket{P^(v,\epsilon)}$ we find

$$\boxed{a=\tilde a}$$

Since $\ket{P(p)}=\sqrt{m_p}\ket{P(v)}$ and $\ket{P^(p,\epsilon)}=\sqrt{m_{p^}}\ket{P(v,\epsilon)}$, we can make predictions for fractions of decay constants, e.g. $$f_P = \frac{a}{\sqrt{m_P}} \Rightarrow \frac{f_B}{f_D}=\sqrt{\frac{m_D}{m_B}}$$