That's what is. Theoritically there should be none. I can transmit an AM signal that is less that 100 Hz wide. Frequency bandwidth is not a function of AM it is a consequence of the electronics being used to transmit the signal. Consider the active device being used such as a modulator. What happens is the junction will expand and contract with varying levels of power. This is in effects a varying capacitance which causes an accidental frequency swing. Any deviation from center frequency is actually FM. So AM as it is today is actually AM and FM. But we detect the amplitude variations because the signal is primarily AM.
Forget about the modulator, the junctions, transistors.
Think about math only.
For example, a DSB-SC signal:
s(t) = m(t)*cos(2*pi*f*t), makes sense for you? You have a base-band signal, m(t), that controls a carrier cos(2.pi.f.t);
When m(t) = 0, then s(t) =0.
When carrier = 0, then s(t) is also 0.
Now let's just take the spectrum with fourier transform:
Fourier is a linear operator, so:
S(f) = M(f)->convolution->1/2 * [D(f - fc) + D(f + fc)], where D is the Dirac Impulse. As a cosine is a pure wave it will have only 1 component at fc, that's why you have an impulse deslocated to fc.
A convolution of any function with an impulse results in that function with the argument replaced by the impulse function's argument.
S(f) = 1/2 * [M(f+fc) + M(f-fc)].
Observe that you have a copy of the baseband signal, M(f), deslocated to +fc and -fc. (In any moment I considered junctions, transistors...).
Think about another POV, you have the fixed frequency carrier, but this fixed frequency carrier has its aplitude varying on the rate of m(t), so actually you have 2 signals over there, not just 1.
When you see an DSB on a oscilloscope, you can actually see the carrier and the modulating 2 signals.
So you have the fixed frequency, and an ampliude that changes with time.
It does not happens with FM, you just can't see the modulating signal, just the frequency changing.
Because you have a fixed amplitude, and a frequency that changes with the integral of the modulating signal amplitude and a modulation index k Hz/V.
s(t) = cos[2.pi.(fc.t + k.∫m(t)dt)]
Plotting a FM spectrum is complex, because it is infinite (in theory) you need Bessel functions to plot. Because you just have the fixed amplitude, but the phase and the frequency keeps changing all the time.