Is broken supersymmetry compatible with a small cosmological constant? I understand that we can find the energy of a bosonic field in its vacuum state via
$E_{vac}^{(B)} = \sum_{\vec{k},s} \frac{1}{2}\hbar\omega_{\vec{k},s}^{(B)}$
and a fermionic one similarly,
$E_{vac}^{(F)} = \sum_{\vec{k},s} -\frac{1}{2}\hbar\omega_{\vec{k},s}^{(F)}$
since anticommutators and commutators are as a rule swapped for fermionic fields.
If these fields have masses $m_B$ and $m_F$ respectively, we can estimate the total energy density of both fields in their vacuum state from dimensional arguments as
$\Lambda \propto m_B^4 - m_F^4$
As I understand it, part of the appeal of unbroken supersymmetry is that if each excitation has a partner of opposite statistics/spin ("commutativity"?) but identical mass, then $\Lambda = 0$.
But experimentally, we don't observe equal-mass superpartners, or even similar mass ones, so I see no reason $\Lambda$ is required to be "small".
The current lower bound on the selectron mass is on the order of $10$ GeV, so it seems to me that this contribution is still required to be some $10^{51}$ orders of magnitude too larger than the value known value from cosmology, which is admittedly better than $10^{120}$ but still not very convincing.
How is spontaneously broken supersymmetry salvaged?
 A: Yes, you're right, we say that broken supersymmetry – as we understand it today – does not solve the cosmological constant problem. The cosmological constant may still be $O(m_{SUSY}^4)$ or even $O(m_{Pl}^2 m_{SUSY}^2)$ after SUSY breaking.
There are respects in which SUSY "improves" the problem with the cosmological constant; there is a sense in which it makes it worse, too.
Concerning the first thing, non-supersymmetric theories predict $\Lambda\sim O(m_{Pl}^4)$ which is 123 orders of magnitude too high. Broken SUSY theories predict $O(m_{SUSY}^4)$ which is "just" 60 orders of magnitude too high so broken SUSY "solves one-half of the problem", so to say. On the other hand, in non-SUSY theories, the cosmological constant problem is in some sense even sharper because the cosmological constant may be linked to the gravitino mass etc. by more universalist formulae while in non-SUSY theories the C.C. may be thought of as being an independent parameter of anything else.
I personally believe that SUSY is the first step towards solving the C.C. problem (why C.C. is so small), after all. Various no-scale supergravity models present this expectation including some technical details. But the "second step" that solves the second part of the C.C. problem – the remainder of the huge hierarchy – isn't understood today and many people believe it can't exist. That's why the anthropic principle and the multiverse became such a widely believed paradigm among the professionals. It's ugly but at least, it has the potential to solve the C.C. problem in an ugly way; competing proposals have been found dysfunctional even if they're allowed to be ugly.
