Okay, so I need to find the mean decay distance of a meson resonance X. The given parameters are the production energy E_X = 8 GeV, mass m_X = 1.2 GeV, and width Γ_X = 320 MeV. The options are A to D, each with different values in meters. 

First, I remember that the mean decay distance, often called the decay length, depends on the particle's lifetime and its velocity. The formula for decay length is L = γβcτ, where γ is the Lorentz factor, β is the velocity in terms of the speed of light, c is the speed of light, and τ is the proper lifetime of the particle. The proper lifetime τ is related to the width Γ by the uncertainty principle: Γ = ħ/τ, so τ = ħ/Γ. ħ is the reduced Planck constant, which is approximately 6.582×10^-25 GeV·s. Wait, but since Γ is given in MeV, I need to convert units properly.

Wait, let me start step by step. First, let's compute the proper lifetime τ. The uncertainty principle says that the width Γ (which is the total decay width) is inversely proportional to the lifetime. The relation is Γ = ħ / τ, so τ = ħ / Γ. But Γ is given in MeV, and ħ is in GeV·s. So I need to convert Γ to GeV to match the units. 

Γ_X is 320 MeV. Since 1 GeV = 1000 MeV, so Γ_X = 0.320 GeV. Then τ = (6.582×10^-25 GeV·s) / (0.320 GeV) = 6.582×10^-25 / 0.320 s. Let me compute that: 6.582/0.32 ≈ 20.56875. So τ ≈ 20.56875×10^-25 s. Wait, 6.582 divided by 0.32 is approximately 20.56875. So τ ≈ 20.56875×10^-25 seconds.

Next, I need to find the Lorentz factor γ. The Lorentz factor is given by γ = E_total / m, where E_total is the total energy. But wait, the total energy E_X is given as 8 GeV. However, the rest mass energy is m_X c². The mass is given as 1.2 GeV/c², so the rest mass energy is 1.2 GeV. Therefore, the total energy E_X = 8 GeV. So γ = E_X / (m_X c²) * c²? Wait, no, the definition of Lorentz factor is γ = E_total / (m c²). Since E_total is given as 8 GeV, and m c² is 1.2 GeV, so γ = 8 / 1.2 = 6.666... Approximately 6.6667.

Then, β is the velocity divided by c. β can be found using the relation between γ and β. The relation is γ² = 1 / (1 - β²), so 1 - β² = 1 / γ², so β = sqrt(1 - 1/γ²). Let's compute that. γ is 8/1.2 = 10/1.5? Wait, 8 divided by 1.2 is 6.666..., so γ ≈ 6.6667. Let's compute 1/γ²: (6.6667)^2 ≈ 44.4444. So 1/44.4444 ≈ 0.0225. Then 1 - 0.0225 = 0.9775. So β = sqrt(0.9775) ≈ 0.9887. Let me check that calculation again. Wait, 8 divided by 1.2 is 20/3 ≈ 6.6667. So γ = 20/3. Then γ² = (20/3)^2 = 400/9 ≈ 44.4444. Then 1/γ² = 9/400 = 0.0225. So 1 - 0.0225 = 0.9775. The square root of 0.9775 is approximately 0.9887. Let me compute sqrt(0.9775) more accurately. Let's see, 0.988 squared is 0.976144, which is 0.9761, which is less than 0.9775. 0.9887 squared: (0.9887)^2. Let's compute 0.9887 * 0.9887. Let's compute 0.98 * 0.98 = 0.9604. Then 0.0087 * 0.98 = 0.008526, and 0.0087 * 0.0087 = ~0.00007569. So total would be approximate 0.9604 + 2*0.008526 + 0.00007569 ≈ 0.9604 + 0.017052 + 0.00007569 ≈ 0.977527, which is very close to 0.9775. So β ≈ 0.9887. 

Now, the decay length L is γβcτ. Let's compute each part. First, c is the speed of light, approximately 3×10^8 m/s. τ we found to be approximately 20.56875×10^-25 s. So let's compute γβ first. γ is 20/3 ≈ 6.6667, β ≈ 0.9887. So γβ ≈ 6.6667 * 0.9887 ≈ 6.6667 * 0.9887. Let me compute that: 6.6667 * 0.9887. Let's approximate 6.6667 * 1 = 6.6667. Subtract 6.6667 * (1 - 0.9887) = 6.6667 * 0.0113. 0.0113 * 6.6667 ≈ 0.0753. So 6.6667 - 0.0753 ≈ 6.5914. So γβ ≈ 6.5914. 

Then, multiply by c (3×10^8 m/s) and τ (20.56875×10^-25 s). So L = 6.5914 * 3×10^8 * 20.56875×10^-25 m. Let's compute this step by step. 

First, multiply 6.5914 * 3 = 19.7742. Then multiply by 20.56875: 19.7742 * 20.56875. Let me compute that. Let's approximate 20 * 20.56875 = 411.375, but 19.7742 is 0.2258 less than 20. So 20.56875 * 19.7742 = 20.56875*(20 - 0.2258) = 20.56875*20 - 20.56875*0.2258. 

20.56875*20 = 411.375. 20.56875*0.2258 ≈ 20.56875 * 0.2 = 4.11375, plus 20.56875*0.0258 ≈ 0.53145. So total ≈ 4.11375 + 0.53145 ≈ 4.6452. So 411.375 - 4.6452 ≈ 406.7298. 

So the product of 6.5914 * 3 * 20.56875 ≈ 406.7298. Now, multiply by 10^8 * 10^-25 = 10^-17. So L ≈ 406.7298 × 10^-17 meters. Wait, but wait, 6.5914 * 3 is 19.7742, then 19.7742 * 20.56875 ≈ 406.7298. Then 406.7298 × 10^8 × 10^-25 = 406.7298 × 10^-17. Wait, but 10^8 * 10^-25 is 10^-17. So 406.7298 × 10^-17 m. But that's 4.067298 × 10^-15 m. Let me check that again. Because 406.7298 × 10^-17 is the same as 4.067298 × 10^-15. 

Wait, but looking at the options, B is 4.0655 × 10^-15 m, which is close. But let me check if there was a miscalculation. Let me recompute the numbers more accurately.

First, let's compute τ again. τ = ħ / Γ. ħ is 6.582×10^-25 GeV·s, Γ is 0.320 GeV. So τ = 6.582×10^-25 / 0.320 = (6.582 / 0.320) × 10^-25. Let's compute 6.582 / 0.320. 0.320 × 20 = 6.4, so 6.582 - 6.4 = 0.182. So 0.182 / 0.320 = 0.56875. So total is 20 + 0.56875 = 20.56875. So τ = 20.56875 × 10^-25 s. That's correct.

γ = E_X / m_X = 8 GeV / 1.2 GeV = 20/3 ≈ 6.666666667. 

β = sqrt(1 - (m_X^2 c^4)/(E_X^2 c^4)) = sqrt(1 - (m_X^2)/(E_X^2)) = sqrt(1 - (1.2^2)/(8^2)) = sqrt(1 - (1.44)/(64)) = sqrt(1 - 0.0225) = sqrt(0.9775) ≈ 0.9887. 

So γβ = (20/3) * sqrt(0.9775). Let's compute this more precisely. Let's compute sqrt(0.9775). Let's use a calculator approach. The square of 0.9887 is 0.9775. So sqrt(0.9775) ≈ 0.9887. So γβ ≈ (20/3) * 0.9887 ≈ 6.666666667 * 0.9887. Let's compute that: 6 * 0.9887 = 5.9322, 0.666666667 * 0.9887 ≈ 0.6591. So total ≈ 5.9322 + 0.6591 ≈ 6.5913. So γβ ≈ 6.5913.

Then L = γβ c τ. c is 3×10^8 m/s. τ is 20.56875×10^-25 s. So L = 6.5913 * 3×10^8 * 20.56875×10^-25.

First, multiply 6.5913 * 3 = 19.7739. Then multiply by 20.56875: 19.7739 * 20.56875. Let's compute this precisely. Let's break it down:

20 * 19.7739 = 395.478

0.56875 * 19.7739 ≈ 0.5 * 19.7739 = 9.88695, plus 0.06875 *19.7739 ≈ 1.3564. So total ≈ 9.88695 +1.3564 ≈ 11.24335. So total for 0.56875 *19.7739 ≈11.24335. So total 395.478 +11.24335 ≈406.72135. 

So 6.5913 *3 *20.56875 ≈406.72135. Then multiply by 10^8 *10^-25 =10^-17. So L ≈406.72135 ×10^-17 m. Which is 4.0672135 ×10^-15 m. 

Looking at the options, option B is 4.0655 ×10^-15 m, which is very close. Option D is 4.0655 ×10^-16 m, which is an order of magnitude smaller. So perhaps there was a miscalculation in the exponent. Let me check the exponents again.

Wait, τ is 20.56875 ×10^-25 s. c is 3×10^8 m/s. So L = γβ c τ = (6.5913) * (3×10^8) * (20.56875×10^-25). 

Multiplying the coefficients: 6.5913 *3 =19.7739. Then 19.7739 *20.56875 ≈406.72135. Then the exponents: 10^8 *10^-25 =10^-17. So 406.72135 ×10^-17 m. Which is 4.0672135 ×10^-15 m. 

But the options have 4.0655e-15 (B) and 4.0655e-16 (D). So why is there a discrepancy? Maybe I made a mistake in the calculation of τ. Let me check τ again.

Wait, Γ is given as 320 MeV. So Γ = 0.320 GeV. Then τ = ħ / Γ = (6.582×10^-25 GeV·s) / (0.320 GeV) = 6.582/(0.320) ×10^-25 s = 20.56875 ×10^-25 s. That's correct. 

Alternatively, maybe the formula for decay length uses Γ in a different way. Wait, sometimes Γ is given as the total decay width, but the relation τ = ħ / Γ holds. So that's correct. 

Alternatively, perhaps the problem uses the total width Γ as the sum of all partial widths, but I think that's already accounted for. 

Another possibility is that the energy given is the total energy, but sometimes people use the center-of-mass energy or something else. Wait, the problem says "production energy is E_X =8 GeV". So I think that's the total energy of the particle. 

Wait, but sometimes the width is given as Γ = m_X * τ^{-1}, so τ = m_X / Γ. Wait no, that's not right. The relation is τ = ħ / Γ. Γ is in energy units, so τ has units of seconds. So that's correct. 

Alternatively, perhaps the formula for decay length uses the mean lifetime multiplied by the velocity. So L = βγcτ. Which is what I used. 

Wait, maybe I made an error in the calculation of β. Let me recompute β. β = sqrt(1 - (m_X^2)/(E_X^2)) = sqrt(1 - (1.2^2)/(8^2)) = sqrt(1 - 1.44/64) = sqrt(1 - 0.0225) = sqrt(0.9775). Let me compute sqrt(0.9775) more accurately. 

Using a calculator approach:

0.9887^2 = (0.98 + 0.0087)^2 = 0.98^2 + 2*0.98*0.0087 + 0.0087^2 = 0.9604 + 0.017092 + 0.00007569 ≈0.9604 +0.01716769≈0.97756769. Which is very close to 0.9775. So β is approximately 0.9887. 

So γβ ≈6.6667 *0.9887≈6.5913. 

So L =6.5913 *3e8 *20.56875e-25. Let's compute this step by step. 

First, 6.5913 *3 =19.7739. 

Then multiply by 20.56875: 19.7739 *20.56875. Let me compute this precisely:

20.56875 *19.7739. Let's break it down:

20 *19.7739 = 395.478

0.56875 *19.7739: 

First, 0.5 *19.7739 =9.88695

0.06875 *19.7739: 

0.06 *19.7739=1.186434

0.00875 *19.7739≈0.00875*20=0.175, minus 0.00875*0.2261≈0.00198. So ≈0.175 -0.00198≈0.17302. So total for 0.06875 is 1.186434 +0.17302≈1.35945.

So 0.56875*19.7739≈9.88695 +1.35945≈11.2464.

So total 20.56875*19.7739≈395.478 +11.2464≈406.7244.

So 406.7244 ×10^-17 m =4.067244 ×10^-15 m. 

Looking at the options, option B is 4.0655 ×10^-15 m, which is very close. The difference is about 0.0008 ×10^-15 m, which is about 0.8 femtometers. That could be due to rounding errors in the intermediate steps. Let me check the exact values.

Alternatively, perhaps I should carry more decimal places in the calculations. Let me recompute τ with more precision.

Γ = 0.320 GeV. So τ = 6.582×10^-25 / 0.320 = 6.582 /0.320 =20.56875 exactly. So τ is 20.56875 ×10^-25 s. 

γ = 8 / 1.2 = 20/3 ≈6.666666667. 

β = sqrt(1 - (1.2/8)^2) = sqrt(1 - (0.15)^2) = sqrt(1 -0.0225)=sqrt(0.9775)=approximately 0.9887. Let's compute sqrt(0.9775) more precisely. 

Using a better approximation for sqrt(0.9775):

Let's use the Newton-Raphson method. Let x = sqrt(0.9775). Let's start with an initial guess x0 = 0.9887. 

x0^2 = 0.9775. Since 0.9887^2 = 0.97756769, which is slightly higher than 0.9775. So x0 =0.9887 gives x0^2=0.97756769. The error is 0.97756769 -0.9775=0.00006769. 

Let's apply one iteration of Newton-Raphson:

x1 = x0 - (x0^2 - 0.9775)/(2x0)

=0.9887 - (0.97756769 -0.9775)/(2*0.9887)

=0.9887 - (0.00006769)/(1.9774)

≈0.9887 - 0.00003418≈0.98866582.

Now, x1=0.98866582. Let's compute x1^2:

0.98866582^2 = (0.9886 +0.00006582)^2 =0.9886^2 + 2*0.9886*0.00006582 + (0.00006582)^2

0.9886^2 = (0.98^2) + 2*0.98*0.0086 +0.0086^2=0.9604 +0.016816 +0.00007396≈0.97728996.

Then 2*0.9886*0.00006582≈2*0.9886*0.00006582≈0.0001295.

And (0.00006582)^2≈4.33e-9. 

So total x1^2≈0.97728996 +0.0001295 +0.0000000043≈0.97741946. 

But we need x1^2 =0.9775. The difference is 0.9775 -0.97741946≈0.00008054. 

Another iteration:

x2 =x1 - (x1^2 -0.9775)/(2x1)

=0.98866582 - (0.97741946 -0.9775)/(2*0.98866582)

=0.98866582 - (-0.00008054)/(1.97733164)

=0.98866582 +0.0000407≈0.98870652.

Now, x2≈0.98870652. Let's compute x2^2:

= (0.9887)^2 + 2*0.9887*0.00000652 + (0.00000652)^2

≈0.97756769 + 0.00001293 + 0.0000000000425≈0.97758062.

But we still need to reach 0.9775. The difference is now 0.97758062 -0.9775=0.00008062. Hmm, this seems like it's not converging quickly. Maybe it's better to use a calculator here. Alternatively, accept that β is approximately 0.9887. 

Alternatively, perhaps using a calculator for sqrt(0.9775). Let me check with a calculator.

Calculating sqrt(0.9775):

0.9887^2 =0.97756769

0.9887 -0.00006=0.98864. (0.98864)^2= (0.9886 +0.00004)^2=0.9886^2 +2*0.9886*0.00004 +0.00004^2=0.97728996 +0.000079088 +0.0000000016≈0.97736905. Still less than 0.9775.

0.98864^2=0.97736905

0.9887^2=0.97756769

Difference between 0.9775 and 0.97736905 is 0.00013095. So between 0.98864 and 0.9887. Let's find x where x^2=0.9775.

x=0.98864 + d. 

(0.98864 +d)^2 =0.9775. 

0.98864^2 + 2*0.98864*d +d^2=0.9775. 

We know 0.98864^2=0.97736905. So:

0.97736905 + 2*0.98864*d +d^2=0.9775. 

Subtract 0.97736905:

2*0.98864*d +d^2=0.9775 -0.97736905=0.00013095.

Assuming d is small, d^2 is negligible:

2*0.98864*d ≈0.00013095 → d≈0.00013095/(2*0.98864)≈0.00013095/1.97728≈0.0000662.

So x≈0.98864 +0.0000662≈0.9887062. Which matches our previous result. So x≈0.9887062. 

So β≈0.9887062. Let's use this more precise value. 

Now, γβ = (20/3) *0.9887062≈6.666666667 *0.9887062. Let's compute this:

6 *0.9887062=5.9322372

0.666666667 *0.9887062≈0.6591374

Total γβ≈5.9322372 +0.6591374≈6.5913746.

So more precisely, γβ≈6.5913746. 

Then, L = γβ * c * τ =6.5913746 *3×10^8 m/s *20.56875×10^-25 s.

Multiplying the coefficients:

6.5913746 *3=19.7741238.

Then 19.7741238 *20.56875. Let's compute this accurately.

20 *19.7741238=395.482476

0.56875 *19.7741238: 

0.5 *19.7741238=9.8870619

0.06875 *19.7741238=0.06875*20=1.375; subtract 0.06875*0.2258762≈0.015568. So 1.375 -0.015568≈1.359432.

So total 0.56875 *19.7741238≈9.8870619 +1.359432≈11.2464939.

So total 19.7741238*20.56875≈395.482476 +11.2464939≈406.72897.

So 406.72897 ×10^-17 m =4.0672897 ×10^-15 m. 

Rounded to five significant figures, that's 4.06729 ×10^-15 m. 

Looking at the options:

A. 5.0223 ×10^-15 m

B. 4.0655 ×10^-15 m

C. 5.0223 ×10^-16 m

D. 4.0655 ×10^-16 m

The closest is option B: 4.0655 ×10^-15 m. The difference is about 0.00078 ×10^-15 m, which is 7.8e-19 m. That's a very small difference, possibly due to rounding during intermediate steps. 

Alternatively, perhaps there's a mistake in the calculation. Let me check the units again. The decay length formula is correct: L = γβcτ. All units are in meters, so that's correct. 

Another possible source of error: Maybe the production energy is not the total energy but the kinetic energy? But the problem states "production energy is E_X=8 GeV". Typically, production energy refers to the total energy, so E_X is the total energy. If it were kinetic energy, then E_total = E_kinetic + m_X c². But since m_X is 1.2 GeV, and E_X is 8 GeV, which is much larger than m_X, the difference would be E_kinetic =8 -1.2=6.8 GeV. But the formula for γ would then be E_total / (m_X c²) = (6.8 +1.2)/1.2 =8/1.2=20/3. So same γ. So whether E_X is total or kinetic energy, in this case, since E_X is much larger than m_X, the difference is negligible. So the calculation remains the same.

Alternatively, perhaps the formula uses the natural logarithm? No, decay length formula is well-established as L=γβcτ. 

Another possible mistake: Did I mix up the units for Γ? Γ is given in MeV, but used in GeV. That's correct. 320 MeV =0.320 GeV. 

Wait, but sometimes decay width Γ is related to the mean lifetime via Γ = ħ / τ. So τ = ħ / Γ. But ħ is in GeV·s, Γ in GeV. Yes. So correct. 

Alternatively, perhaps the problem uses Γ in a different context, like partial decay width. But the question says "the width Γ_X=320 MeV", which I assume is the total decay width. 

Alternatively, maybe the formula is L = (Γ_X c τ) / (2π), but no, the standard formula is L = γβcτ. 

Wait, perhaps there's a missing factor of 2π? Let me check. The mean decay distance is the average distance traveled, which is indeed γβcτ. But sometimes, the lifetime τ is given in units of 1/seconds, and when multiplied by c gives meters. So the formula is correct. 

Alternatively, maybe the problem uses Γ in a different way. Let me check the formula again. Decay length L = (c τ) * γβ. Yes, that's correct. 

So after all these checks, the calculation seems correct, leading to approximately 4.067 ×10^-15 m. The closest option is B: 4.0655 ×10^-15 m. The slight difference might be due to rounding in intermediate steps. 

Alternatively, perhaps there's a factor of 1/γ missing? Let me check. No, the formula is L = γβcτ. So that's correct. 

Another possibility: Did I mix up the units for c? c is 3×10^8 m/s. Correct. 

Alternatively, maybe the problem uses the root mean square (RMS) lifetime instead of the mean lifetime? No, the mean lifetime is τ, and the mean decay length is L=γβcτ. 

Alternatively, perhaps the formula uses the total decay width Γ, not the partial width. But the problem states Γ_X=320 MeV, which is the width of the resonance. That's the total width, so the formula applies. 

Alternatively, perhaps the problem uses the full width formula, which involves Γ and lifetimes, but again, the formula is correct. 

Therefore, the answer should be B: 4.0655 ×10^-15 m. However, let me check if there's any step where I might have missed a factor of 10. For example, in converting GeV to eV or meters. Wait, 6.582×10^-25 GeV·s is correct because 1 eV·s = 1.602×10^-19 J·s, and 1 J = 1 eV ×1.602×10^19. Wait, no, 1 GeV = 1e9 eV. So 1 GeV·s =1e9 eV·s. The reduced Planck constant is ħ=6.582×10^-25 GeV·s. So that's correct. 

Alternatively, perhaps there's a factor of 2 missing in the formula. Let me check. No, the formula is L=γβcτ. The c is in m/s, τ in seconds, so units are m. 

Alternatively, perhaps the problem uses the width in a different way. For example, the total width Γ is related to the lifetime via Γ = ħ / τ. So τ=ħ / Γ. Correct. 

Another possible mistake: Perhaps the energy given is not the total energy but the rest mass energy. Wait, the mass m_X is given as 1.2 GeV. Wait, that can't be. Wait, the mass of a particle is typically given in GeV/c², but here it's written as 1.2 GeV. Wait, that's a mistake. Because mass has units of GeV/c². If the mass is given as 1.2 GeV, that's incorrect. Because E=mc² implies that if m is in GeV/c², then m c² is in GeV. So if m_X is 1.2 GeV/c², then its rest energy is 1.2 GeV. But the problem states m_X=1.2 GeV, which would imply m_X c²=1.2 GeV. So perhaps the problem is using units where c=1, which is common in particle physics. Because in particle physics, energies are often expressed in GeV, masses in GeV/c², and c=1. So for example, m_X=1.2 GeV/c², and E=8 GeV. Then γ = E / (m_X c²) =8 /1.2 =20/3. So that's correct. 

But if the problem states m_X=1.2 GeV, without the c², then that would be incorrect, but in particle physics, when E is given in GeV and m_X is given in GeV/c², it's implied that c=1. So perhaps the problem assumes c=1. But in that case, the calculation remains the same. 

Alternatively, if the problem uses non-dimensional units (c=1), then the formula for decay length would be L = β γ τ. But since c=1, then L = β γ τ * c = β γ τ. But in natural units, c=1, so L is in meters if τ is in seconds. But since the problem gives c in m/s, probably c is not set to 1. So our previous calculation is correct. 

Alternatively, if the problem uses c=3×10^8 m/s, then our calculation is correct. 

Therefore, after all these checks, I think the correct answer is B: 4.0655 ×10^-15 m, but the precise calculation gives approximately 4.067 ×10^-15 m. The difference could be due to rounding during intermediate steps. However, since the options are given with five significant figures, perhaps the precise calculation would yield 4.0655 ×10^-15 m. Alternatively, maybe there's a more precise value of β that leads to that. 

Alternatively, let's compute β with more precision. Let's use the exact value of Γ=0.320 GeV. So τ=6.582×10^-25 /0.320 =20.56875 ×10^-25 s. 

γ=8 /1.2=6.666666667. 

β= sqrt(1 - (m_X^2)/(E_X^2))=sqrt(1 - (1.2^2)/(8^2))=sqrt(1 -1.44/64)=sqrt(1 -0.0225)=sqrt(0.9775)=approximately 0.988706. 

So γβ=6.666666667 *0.988706≈6.591374. 

Then L=6.591374 *3×10^8 *20.56875×10^-25. 

Calculating 6.591374 *3=19.774122. 

19.774122 *20.56875= 

Let me compute this more precisely:

19.774122 *20 =395.48244

19.774122 *0.56875= 

First compute 19.774122 *0.5=9.887061

19.774122 *0.06875= 

0.06 *19.774122=1.18644732

0.00875 *19.774122≈0.17300986

Total:1.18644732 +0.17300986≈1.35945718

So total 0.56875 *19.774122≈9.887061 +1.35945718≈11.24651818

So total 19.774122 *20.56875≈395.48244 +11.24651818≈406.72895818

So 406.72895818 ×10^-17 m =4.0672895818 ×10^-15 m. 

Rounded to five significant figures, that's 4.06729 ×10^-15 m. 

But the option B is 4.0655 ×10^-15 m. The difference is about 0.00078 ×10^-15 m. That's about 7.8e-19 m. That's a small discrepancy, likely due to rounding during intermediate steps. For example, if we use more precise values for the constants or carry out more decimal places in the calculation. 

Alternatively, perhaps the problem expects the answer to be 4.0655 ×10^-15 m, which is option B. Therefore, the correct answer is B.
</think>

To determine the mean decay distance of the meson $X$, we use the formula for the decay length:

\[
L = \gamma \beta c \tau
\]

where:
- $\gamma$ is the Lorentz factor, given by $\gamma = \frac{E_X}{m_X c^2}$,
- $\beta = \frac{v}{c}$ is the velocity in terms of $c$, obtained from $\beta = \sqrt{1 - \frac{1}{\gamma^2}}$,
- $c$ is the speed of light ($3 \times 10^8$ m/s),
- $\tau$ is the mean lifetime, related to the total decay width $\Gamma_X$ by $\tau = \frac{\hbar}{\Gamma_X}$.

### Step 1: Calculate $\gamma$
The energy of the meson is $E_X = 8$ GeV, and its mass is $m_X = 1.2$ GeV. The Lorentz factor is:
\[
\gamma = \frac{E_X}{m_X c^2} = \frac{8}{1.2} = \frac{20}{3} \approx 6.6667
\]

### Step 2: Calculate $\beta$
Using $\beta = \sqrt{1 - \frac{1}{\gamma^2}}$, we compute:
\[
\beta = \sqrt{1 - \frac{1}{\left(\frac{20}{3}\right)^2}} = \sqrt{1 - \frac{9}{400}} = \sqrt{0.9775} \approx 0.9887
\]

### Step 3: Calculate $\tau$
The total decay width is $\Gamma_X = 320$ MeV $= 0.320$ GeV. The mean lifetime is:
\[
\tau = \frac{\hbar}{\Gamma_X} = \frac{6.582 \times 10^{-25} \, \text{GeV·s}}{0.320 \, \text{GeV}} = 20.56875 \times 10^{-25} \, \text{s}
\]

### Step 4: Calculate $L$
The decay length is:
\[
L = \gamma \beta c \tau
\]
Substituting the values:
\[
L = \left(6.6667\right) \left(0.9887\right) \left(3 \times 10^8 \, \text{m/s}\right) \left(20.56875 \times 10^{-25} \, \text{s}\right)
\]

First, compute $\gamma \beta$:
\[
\gamma \beta = 6.6667 \times 0.9887 \approx 6.5914
\]

Then, compute $c \tau$:
\[
c \tau = \left(3 \times 10^8 \, \text{m/s}\right) \left(20.56875 \times 10^{-25} \, \text{s}\right) = 6.170625 \times 10^{-17} \, \text{m}
\]

Finally, compute $L$:
\[
L = 6.5914 \times 6.170625 \times 10^{-17} \, \text{m} \approx 4.0673 \times 10^{-15} \, \text{m}
\]

### Step 5: Match with Options
The calculated decay length is approximately $4.0673 \times 10^{-15} \, \text{m}$. Among the given options:
- **A.** $5.0223 \times 10^{-15} \, \text{m}$ (too large),
- **B.** $4.0655 \times 10^{-15} \, \text{m}$ (closest),
- **C.** $5.0223 \times 10^{-16} \, \text{m}$ (incorrect order of magnitude),
- **D.** $4.0655 \times 10^{-16} \, \text{m}$ (incorrect order of magnitude).

The closest match is **B**, with a slight difference likely due to rounding during intermediate steps.

ANSWER: B