What is the reason behind the HQET Lagrangian identity?

In summary, The conversation is about proving an identity in Heavy Quark Effective Theory, specifically the identity stated in Wise's book "Heavy Quark Physics" in section 4.1. The identity, which is $\bar Q_v\sigma^{\mu\nu}v_\mu Q_v=0$, is discussed and it is suggested to use the identity Q_v=P_+Q_v where P_\pm=(1\pm \displaystyle{\not} v)/2 are projection operators. Eventually, it is determined that the identity is always zero by going to the rest frame of the heavy quark. Another proof of this is also presented using the equation of motion.
  • #1
Einj
470
59
Hi everyone. I'm studying Heavy Quark Effective Theory and I have some problems in proving an equality. I'm am basically following Wise's book "Heavy Quark Physics" where, in section 4.1, he claims the following identity:

$$
\bar Q_v\sigma^{\mu\nu}v_\mu Q_v=0
$$

Does any of you have an idea why this is true??

I think that an important identity to use in order to prove that should be [itex]Q_v=P_+Q_v[/itex], where [itex]P_\pm=(1\pm \displaystyle{\not} v)/2[/itex] are projection operators.

Thanks a lot
 
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  • #2
what is ##v_\mu##?
 
  • #3
Is the four velocity of the heavy quark. However, I don't think it really matters. The only important thing is that the P's are projectors. I think I solved it, it's just an extremely boring algebra of gamma matrices
 
  • #4
Yes, that is what it seems. But if you go in the rest frame of the particle, the term you will be having is like ##σ^{4\nu}##,which is a off diagonal matrix in the representation of Mandl and Shaw ( or may be Sakurai).Those projection operators are however diagonal in this representation and hence it's zero.
 
  • #5
Yes, it sounds correct. Do you think this is enough to say that it is always zero?
 
  • #6
Of course, you can always go to the rest frame of a heavy quark. That is how we evaluated the matrix elements in qft in old days.
 
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  • #7
Great sounds good! Thanks
 
  • #8
I thought it was because : (bear with me i don't remember the slash command for the forums right now) the equation of motion:

$$ v^{\mu}\gamma_{\mu} Q_v = Q_v$$
$$ \bar{Q}_v v^{\mu}\gamma_{\mu} = \bar{Q}$$

so
$$ v_{\mu} \bar{Q} \left( \gamma^{\mu} \gamma^{\nu} - \gamma^{\nu} \gamma^{\mu}\right) Q $$
becomes
$$ \bar{Q} \left( \gamma^{\nu} - \gamma^{\nu} \right) Q $$
 

1. What is the HQET Lagrangian identity?

The HQET Lagrangian identity, also known as the Heavy Quark Effective Theory Lagrangian identity, is a mathematical equation that describes the relationship between the Lagrangians of two different quantum field theories - one with heavy quarks and one with light quarks. It helps to simplify calculations involving heavy quarks by allowing them to be treated as static, non-relativistic particles.

2. How is the HQET Lagrangian identity derived?

The HQET Lagrangian identity is derived using the method of “integrating out” heavy quark degrees of freedom in the full QCD Lagrangian. This involves expanding the full QCD Lagrangian in powers of the heavy quark mass and neglecting terms of higher order, resulting in an effective Lagrangian for heavy quarks.

3. What are the applications of the HQET Lagrangian identity?

The HQET Lagrangian identity has many applications in particle physics, particularly in the study of heavy quark systems such as B mesons and bottom quarks. It is used to simplify calculations and make predictions about the behavior of these systems, and has been used to gain insights into phenomena such as heavy quark symmetry breaking and the properties of the Higgs boson.

4. Are there any limitations to the HQET Lagrangian identity?

While the HQET Lagrangian identity is a useful tool in particle physics, it does have some limitations. It assumes that the heavy quark is much more massive than the other particles in the system, which may not always be the case. Additionally, it does not account for certain effects such as electroweak interactions and higher order corrections, which may be important in some situations.

5. How does the HQET Lagrangian identity relate to other theories in particle physics?

The HQET Lagrangian identity is closely related to other theories in particle physics, such as the Standard Model and QCD. It can be seen as a simplification of these theories in certain cases, and is often used in conjunction with them to make calculations more manageable. It also plays a role in the development of new theories, such as effective field theories, which aim to describe physical phenomena at different energy scales.

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