Okay, let's try to figure out this physics problem. So the question is about the decay modes of a boson X with a mass of 6 GeV. The interaction Lagrangian is given, and we need to determine which kinematically allowed decays are possible. The options are A, B, C, D, each listing different fermion pairs that the boson X can decay into.

First, I remember that for a boson to decay into fermions, the decay process must satisfy certain conditions. The main thing that comes to mind is the conservation of energy and momentum, but more importantly, the mass of the boson must be greater than or equal to the sum of the masses of the decay products. Since it's a decay, the boson's mass must be greater than the sum of the masses of the fermions it's decaying into. But wait, in this case, the boson X has a mass of 6 GeV. So each fermion pair must have a combined mass less than or equal to 6 GeV for the decay to be kinematically allowed.

But wait, the fermions here are the ones being created in pairs, like c c-bar, s s-bar, etc. So each of these pairs would have a mass equal to twice the mass of the individual quark or lepton. For example, a pair of up-type quarks (u and u-bar) would have a combined mass of 2m_u. Similarly for down-type quarks (d and d-bar) at 2m_d, strange at 2m_s, charm at 2m_c, bottom at 2m_b, tau, mu, and electron pairs.

So the first step is to check the masses of the possible fermion pairs. Let me recall the approximate masses of these particles:

- Up-type quarks (u, c): Up quark is very light, around 2-5 MeV. Charm is about 1.5 GeV.
- Down-type quarks (d, s): Down is around 4-5 MeV, strange is about 100 MeV.
- Bottom quark is about 4.5 GeV.
- Tau is around 1.77 GeV.
- Mu is about 0.105 GeV (105.7 MeV).
- Electron is about 0.511 MeV.

Wait, but the pairs are made of quark and antiquark, so their combined mass is twice each individual mass. Let me list each possible pair and their combined mass:

1. c c-bar: 2 * 1.5 GeV = 3 GeV
2. s s-bar: 2 * 100 MeV = 200 MeV = 0.2 GeV
3. u u-bar: 2 * 2-5 MeV ≈ 4-10 MeV ≈ 0.004-0.01 GeV (negligible)
4. d d-bar: similar to u, but perhaps a bit higher, maybe 0.01 GeV
5. t t-bar: 2 * 4.5 GeV = 9 GeV
6. tau tau-bar: 2 * 1.77 GeV ≈ 3.54 GeV
7. mu mu-bar: 2 * 0.105 GeV ≈ 0.21 GeV
8. e e-bar: 2 * 0.511 MeV ≈ 1.022 MeV

Now, the boson X has a mass of 6 GeV. So the sum of the masses of the fermion pair must be less than or equal to 6 GeV. Let's check each option:

Looking at the choices:

Option A includes c c-bar (3 GeV), s s-bar (0.2 GeV), u u-bar (negligible), d d-bar (negligible), t t-bar (9 GeV), tau tau-bar (3.54 GeV), mu mu-bar (0.21 GeV), e e-bar (1.022 MeV). Wait, but t t-bar is 9 GeV, which is way above 6 GeV. So decay into t t-bar is impossible. So option A includes t t-bar, which can't happen. Therefore, A is incorrect.

Option B: X→b b-bar, s s-bar, u u-bar, d d-bar, tau tau-bar, mu mu-bar, e e-bar. Let's check each pair's mass:

b b-bar: 2 * 4.5 GeV = 9 GeV. Again, 9 GeV > 6 GeV. So decay into b b-bar is not allowed. So option B is also incorrect.

Option C: X→b b-bar (9 GeV), which again is not allowed. So option C is out.

Option D: X→c c-bar (3 GeV), s s-bar (0.2), u u-bar (negligible), d d-bar (negligible), tau tau-bar (3.54), mu mu-bar (0.21), e e-bar (1.022 MeV). All of these pairs have combined masses less than 6 GeV. So the only possible issue here is whether these pairs are allowed by the interaction Lagrangian, not just the mass.

Wait, the interaction Lagrangian is given as L = -λ f ψ-bar f (y) (v + X(y)) ψ_f(y). Hmm, I need to parse this. The interaction term is between the fermion field ψ and the boson X. The term is ψ-bar (v + X) ψ. Wait, but the exact form might matter. Let me think.

The Lagrangian term is L = -λ f ψ-bar f (y) (v + X(y)) ψ_f(y). Wait, maybe it's more like ψ-bar (v + X) ψ. Wait, the notation might be a bit ambiguous. Let me think in terms of Yukawa coupling. In the Standard Model, the Yukawa interaction is something like L_Yukawa = - (λ / √2) ψ-bar ψ H, where H is the Higgs field. So the coupling is between fermions and the Higgs field. In this case, the interaction is between ψ and X, the boson. So the term is something like L_int = -λ f ψ-bar (v + X) ψ. Wait, but the exact form is important. If X is a scalar boson (since it's interacting with fermions via a Yukawa-like term), then for each fermion species, there would be a term in the Lagrangian that allows it to couple to X. The interaction would be proportional to X multiplied by the mass term. Wait, but the mass term for fermions is m ψ-bar ψ. So if the interaction term is λ X ψ-bar ψ, then the coupling constant would be related to the mass. For example, m = λ v, where v is the vacuum expectation value (VEV). So in this case, the interaction term would be something like L_int = -λ X ψ-bar ψ. But in the given problem, the Lagrangian is written as L = -λ f ψ-bar f (y) (v + X(y)) ψ_f(y). Wait, maybe I'm getting confused with the notation here. Let's parse the given Lagrangian again.

The given Lagrangian is written as:

\mathcal{L}\left(y\right) = -\lambda f \bar{\psi} f \left(y\right) \left(\upsilon + X\left(y\right)\right) \psi_{f}\left(y\right)

Hmm, maybe it's a product of ψ-bar, f, (v + X), and ψ_f. Wait, that seems off. Wait, perhaps the notation is that ψ_f is a flavor eigenstate, and ψ is the same as ψ_f? Or maybe the f here is a typo. Alternatively, maybe the term is ψ-bar (v + X) ψ, with some other factors. Let's assume that the interaction term is of the form L_int = -λ (v + X) ψ-bar ψ. Then, the coupling would allow the boson X to decay into fermion-antifermion pairs, with a coupling proportional to λ. But the key point is that the mass of the boson X must be greater than the sum of the masses of the decay products. However, in this case, the fermions are massless (or very light) except for the top quark, which is about 1.5 GeV. Wait, but in the problem statement, the decay products are fermions and their antiparticles. The fermions here are the standard model fermions, which have masses. But the mass of the boson is 6 GeV. So when considering decay into a pair of a fermion and its antiparticle, the combined mass of the pair is twice the fermion's mass. For example, a pair of up-type quarks would have a combined mass of 2m_u, which is negligible. Similarly, for down-type, strange, charm, bottom, tau, mu, and electron. Except for the top quark, which is 1.5 GeV. So the top quark pair would have a combined mass of 3 GeV. So the boson's mass of 6 GeV is more than enough to decay into top quark pairs. But earlier I thought that the Lagrangian term might not allow certain decays. Wait, but the interaction term would allow any fermion pair, provided that the mass of the boson is sufficient. So even for the top quark, since 6 GeV > 3 GeV, it's allowed. Wait, but looking back at the options, none of the options include top quark pairs. Let's check the options again.

Option A: Includes t t-bar. But as I calculated, the combined mass is 3 GeV, which is less than 6 GeV, so allowed. But option A includes t t-bar, which the user initially thought was possible, but according to the given Lagrangian, why would it not be allowed?

Wait, maybe I need to consider the interaction coupling. The interaction term is proportional to (v + X), so the coupling strength depends on the mass of the fermion. Wait, in the Standard Model, the Yukawa coupling is proportional to the mass of the fermion. So in the Lagrangian, the term would be something like L_int = - (m_f / v) X ψ-bar ψ. So the coupling strength is proportional to m_f. Therefore, the decay rate into a fermion pair is proportional to m_f squared. So for heavier fermions, the decay rate is higher. However, the question is about kinematical allowed decays, not the rate. So kinematically allowed means that the mass of the boson is greater than or equal to the sum of the masses of the decay products. So even if the decay is kinematically allowed, if the coupling is too weak (like for the top quark, since it's heavy, the coupling is strong), but decay rate is a separate issue. But the question here is about kinematically allowed, so we don't need to worry about coupling strength. So the main condition is that the combined mass of the fermion pair is less than or equal to the boson's mass.

But then why do some of the options include certain fermions and not others? For example, option A includes t t-bar, which should be allowed. But the user originally thought that the answer was option B, but according to mass considerations, option A might be correct. However, perhaps there's another constraint. Let me check the interaction term again. The interaction is given as L = -λ f ψ-bar f (y) (v + X(y)) ψ_f(y). Wait, maybe the f here is a typo. Perhaps it's ψ-bar (v + X) ψ. If that's the case, then the coupling is allowed for any fermion species. But if the interaction term is proportional to the mass of the fermion, then perhaps only certain fermions are allowed. Wait, in the Standard Model, the Yukawa interaction is L = - (m_f / v) ψ-bar ψ H, where H is the Higgs field. The coupling is proportional to the mass of the fermion. So if the interaction here is L = -λ (v + X) ψ-bar ψ, then the coupling is the same for all fermions, but if the coupling is proportional to the mass, then the decay rates would differ. However, the question is about kinematics, not the coupling strength. So kinematically allowed decays are those where the fermion pair's mass is less than or equal to the boson's mass. So all the pairs in the options where the combined mass is ≤ 6 GeV should be allowed. Let's check each option again.

First, the problem states that the mass of the boson X is 6 GeV. So each fermion pair's combined mass must be ≤6 GeV.

Possible fermion pairs and their combined masses:

1. c c-bar: 2 * 1.5 GeV = 3 GeV ≤6 GeV → allowed
2. s s-bar: 2 * 0.1 GeV = 0.2 GeV → allowed
3. u u-bar: negligible → allowed
4. d d-bar: negligible → allowed
5. t t-bar: 2 * 4.5 GeV =9 GeV >6 GeV → not allowed
6. tau tau-bar: 2*1.77 GeV ≈3.54 GeV ≤6 GeV → allowed
7. mu mu-bar: 2*0.105 GeV ≈0.21 GeV → allowed
8. e e-bar: 2*0.511 MeV ≈1.022 MeV → allowed

So the pairs that are kinematically allowed are:

c c-bar, s s-bar, u u-bar, d d-bar, tau tau-bar, mu mu-bar, e e-bar.

Now, looking at the options:

Option A: includes t t-bar (9 GeV), which is not allowed. So A is incorrect.

Option B: includes b b-bar, which is 9 GeV (since b is 4.5 GeV). So B is incorrect.

Option C: includes b b-bar (9 GeV), so incorrect.

Option D: includes c c-bar, s s-bar, u u-bar, d d-bar, tau tau-bar, mu mu-bar, e e-bar. All of these except t t-bar are allowed. So D is the correct option.

Wait, but the original question's interaction Lagrangian is written as:

\mathcal{L}\left(y\right)=-\lambda{f}\bar{\psi}{f}\left(y\right)\left(\upsilon+X\left(y\right)\right)\psi_{f}\left(y\right)

Wait, maybe I misread the interaction. Let me check again. The term is:

-λ f ψ-bar f (y) (v + X(y)) ψ_f(y)

Wait, perhaps the indices are different. For example, ψ_f(y) might be the field for a specific flavor f, and ψ-bar f (y) might be the adjoint. But regardless, the key point is that the interaction term is proportional to (v + X) times ψ-bar ψ. So the coupling is the same for all fermion species. Therefore, all fermion pairs are allowed kinematically, provided their combined mass is ≤6 GeV. So the only fermion pair not allowed is t t-bar. Therefore, the correct answer should include all the pairs except t t-bar. Looking at the options:

Option D includes all except t t-bar. So the answer should be D.

Wait, but let me double-check. The problem states that the mass of X is 6 GeV. The possible fermion pairs are:

c c-bar: 3 GeV → allowed
s s-bar: 0.2 GeV → allowed
u u-bar: negligible → allowed
d d-bar: negligible → allowed
b b-bar:9 GeV → not allowed
tau tau-bar:3.54 GeV → allowed
mu mu-bar:0.21 GeV → allowed
e e-bar:1.022 MeV → allowed

Therefore, the allowed pairs are all except b and t. But looking at the options:

Option D includes all except t. So D is correct.

But wait, in the given options, option D includes c, s, u, d, tau, mu, e. So that's all except t and b. Wait, but option D does not include b. Let me check the options again:

Option D: c c-bar, s s-bar, u u-bar, d d-bar, tau tau-bar, mu mu-bar, e e-bar.

Yes, no b. So that's correct. So the answer is D.

But wait, another thought: in the Standard Model, the Higgs couples to all fermions, but the coupling strength is proportional to mass. So the Higgs can decay into any fermion pair as long as the mass of the fermion pair is less than the Higgs mass. But in this problem, the boson X has a mass of 6 GeV. The Higgs is 125 GeV, so the decay into tau pairs is allowed (since 2*1.77 ≈3.54 <125). But in this case, the boson is much lighter. So the same logic applies: all fermion pairs with combined mass ≤6 GeV are kinematically allowed. Therefore, the answer is D.
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