  A new citation index $h_{PI}$ for principal investigators (PIs) is defined in
analogy to Hirsch's index $h$, but based on renormalized citations of a PI's
papers. To this end, the authors of a paper are divided into two groups: PIs
and non-PIs. A PI is defined as an assistant, associate or full professor at a
university who supervises an individual research program. The citations for
each paper of a certain PI are then divided by the number of PIs among the
authors of that paper. Data are presented for a sample of 48 PIs who are senior
faculty members of physics and physics-related engineering departments at a
private research-oriented U.S. university, using the ISI Web of Science
citations database. The main result is that individual rankings based on $h$
and $h_{PI}$ differ substantially. Also, to a good approximation across the
sample of 48 PIs, one finds that $h_{PI} = h \,/ \sqrt{<N_{PI}>}$ where
<$N_{PI}$> is the average number of principal investigators on the papers of a
particular PI. In addition, $h_{PI} = \frac{1}{2} \sqrt{C_{tot}\,/<N_{PI}>}$,
where $C_{tot}$ is the total number of citations. Approaches to broadening the
scope of $h$ or $h_{PI}$ are discussed briefly, and a new metric for highly
cited papers called $h_x$ is introduced which represents the average number of
citations exceeding the minimum of $h^2$ in the $h$-core.
