# What does the direct sum symbol (i.e. $\oplus$) mean in the context of uncertainties

I've noticed the symbol $⊕$ used in a context I'm unfamiliar with. In several papers about the the calculation of the uncertainty of quantities measured with hadron colliders.

For example the uncertainty in transverse momentum given by:

$$\frac{ \sigma_{P_{T}} }{ P_{T} } = 0.038\% P_{T}\,({\rm in\,GeV}) ⊕ 1.5\%$$

What exactly does $⊕$ mean in this context?

• It's possibly supposed to be +/-; that is, a typographical error. Or a font substitution error if this is from online. – Peter Diehr Apr 9 '16 at 19:07
• Which several papers? Which pages? – Qmechanic Apr 9 '16 at 19:37
• It is used once here arxiv.org/abs/1406.0076 and i have seen it used more often in presentations like becher.itp.unibe.ch/LHC/CalorimeterAndJets.pdf So i don't think it's a typo – a.orwell Apr 11 '16 at 9:27
• So your only serious reference is a 100-page paper where it's used once, and you can't even provide the page number, even when asked directly? – Emilio Pisanty May 2 '16 at 17:53
• Presumably the example you quote is from this paper? – Emilio Pisanty May 2 '16 at 18:09

This seems to be addition in quadrature of multiple independent uncertainties in a measurement. In particular, if you have a measurement which depends on two quantities $a$ and $b$ whose uncertainties $\delta a$ and $\delta b$ are completely independent and uncorrelated, then their uncertainties will often be combined as
$$\delta(a+b)=\sqrt{\delta a^2+\delta b^2}=:\delta a\oplus\delta b,$$
where the symbol $\oplus$ is sometimes used to indicate this method of uncertainty combination.
For more on this procedure see e.g. this tutorial or simply google for 'addition in quadrature'; for an explicit example of the $\oplus$ notation used in this sense, see this CMS page.