Guide to transverse projections and mass-constraining variables

This paper seeks to demonstrate that many of the existing mass-measurement variables proposed for hadron colliders ($m_T$, $m_{\mathrm{eff}}$, $m_{T2}$, missing ${\overrightarrow{p}}_T$, $h_T$, ${\sqrt{\widehat{s}}}_{\min }$, etc.) are far more closely related to each other than is widely appreciated, and indeed can all be viewed as a common mass-bound specialized for a variety of purposes. A consequence of this is that one may understand better the strengths and weaknesses of each variable, and the circumstances in which each can be used to best effect. In order to achieve this, we find it necessary first to revisit the seemingly empty and infertile wilderness populated by the subscript “$T$” (as in “${\mathrm{p̸}}_T$”) in order to remind ourselves what this process of transversification actually means. We note that, far from being simple, transversification can mean quite different things to different people. Those readers who manage to battle through the barrage of transverse notation distinguishing “$\top$” from “$\vee$” or from “$\circ$,” and “early projection” from “late projection,” will find their efforts rewarded towards the end of the paper with (i) a better understanding of how collider mass variables fit together, (ii) an appreciation of how these variables could be generalized to search for things more complicated than supersymmetry, (iii) will depart with an aversion to thoughtless or naïve use of the so-called transverse methods of any of the popular computer Lorentz-vector libraries, and (iv) will take care in their subsequent papers to be explicit about which of the 61 identified variants of the “transverse mass” they are employing.

Figure 1

The stretched, webbed limbs of the Glaucomys volans have been adapted by generations of natural selection to provide an ideal visual illustration of the various different, yet related, transverse mass variables (and incidentally provide an appropriate aerodynamic shape for gliding flight).

Figure 2

The standard geometry of a collider experiment. The $z$ axis (in blue online) is oriented along the beam, while the $x$ and $y$ axes (in red online) define the transverse plane. Any 3-dimensional vector $\overrightarrow{P}$ can be uniquely decomposed into a longitudinal component $p_z$ and a transverse component ${\overrightarrow{p}}_T$.

Figure 3

A pictorial representation of the three transverse projections discussed in Sec. 3. The colored (online) arrows represent the mappings under the $\top$, $\vee$ and $\circ$ projections. The blue and green (online) dotted lines represent the equivalence classes of the projected points under the $\top$ and $\vee$ projections, respectively.

Figure 4

The same as Fig. 3, but plotted in the $(M^2,\mathrm{sign}(p_z)p_z^2)$-plane.

Figure 5

The event topology for new physics searches and measurements used in this paper.

Figure 6

This figure illustrates the notation used to label physics objects and their assignments to parent hypotheses. The figure shows a hypothesis in which six ($N_{\mathcal{V}}=6$) visible physics objects $\mathcal{V}=\{{\mathbb{V}}_1,{\mathbb{V}}_2,{\mathbb{V}}_3,{\mathbb{V}}_4,{\mathbb{V}}_5,{\mathbb{V}}_6\}$ and five ($N_{\mathcal{I}}=5$) invisible physics objects $\mathcal{I}=\{{\mathbb{I}}_1,{\mathbb{I}}_2,{\mathbb{I}}_3,{\mathbb{I}}_4,{\mathbb{I}}_5\}$ have been assigned to three ($N=3$) parents $\mathcal{P}=\{{\mathbb{P}}_1,{\mathbb{P}}_2,{\mathbb{P}}_3\}$ according to the assignments ${\mathcal{V}}_1=\{{\mathbb{V}}_2,{\mathbb{V}}_4\}$, ${\mathcal{V}}_2=\{{\mathbb{V}}_3\}$, ${\mathcal{V}}_3=\{{\mathbb{V}}_1,{\mathbb{V}}_5,{\mathbb{V}}_6\}$, ${\mathcal{I}}_1=\{{\mathbb{I}}_1\}$, ${\mathcal{I}}_2=\{{\mathbb{I}}_2,{\mathbb{I}}_3\}$, and ${\mathcal{I}}_3=\{{\mathbb{I}}_4,{\mathbb{I}}_5\}$. The number of visible physics objects assigned to each parent in turn are therefore $|{\mathcal{V}}_1|=2$, $|{\mathcal{V}}_2|=1$, and $|{\mathcal{V}}_3|=3$ and the number of invisible physics objects assigned to each parent in turn are $|{\mathcal{I}}_1|=1$, $|{\mathcal{I}}_2|=2$, and $|{\mathcal{I}}_3|=2$.

Figure 7

This figure is provided for the benefit of readers unable to imagine a simpler version of Fig. 6. (Readers finding this figure helpful need not admit this to close friends, relatives, or colleagues.) The figure shows a hypothesis in which four ($N_{\mathcal{V}}=4$) visible physics objects $\mathcal{V}=\{{\mathbb{V}}_1,{\mathbb{V}}_2,{\mathbb{V}}_3,{\mathbb{V}}_4\}$ and two ($N_{\mathcal{I}}=2$) invisible physics objects $\mathcal{I}=\{{\mathbb{I}}_1,{\mathbb{I}}_2\}$ have been assigned to two ($N=2$) parents $\mathcal{P}=\{{\mathbb{P}}_1,{\mathbb{P}}_2\}$ according to the assignments ${\mathcal{V}}_1=\{{\mathbb{V}}_1,{\mathbb{V}}_3\}$, ${\mathcal{V}}_2=\{{\mathbb{V}}_2,{\mathbb{V}}_4\}$, ${\mathcal{I}}_1=\{{\mathbb{I}}_1\}$, and ${\mathcal{I}}_2=\{{\mathbb{I}}_2\}$. The number of visible physics objects assigned to each parent in turn are therefore $|{\mathcal{V}}_1|=2$, $|{\mathcal{V}}_2|=2$ and the number of invisible physics objects assigned to each parent in turn are $|{\mathcal{I}}_1|=1$, $|{\mathcal{I}}_2|=1$. An explicit physics example corresponding to this figure is discussed in Sec. 11b.

Figure 8

This figure is provided for the benefit of readers unable to imagine an even simpler version of Fig. 6 than was shown in Fig. 7. (Readers finding this figure helpful are advised to seek gainful employment in some other field.) The figure shows a hypothesis in which two ($N_{\mathcal{V}}=2$) visible physics objects $\mathcal{V}=\{{\mathbb{V}}_1,{\mathbb{V}}_2\}$ and two ($N_{\mathcal{I}}=2$) invisible physics objects $\mathcal{I}=\{{\mathbb{I}}_1,{\mathbb{I}}_2\}$ have been assigned to one ($N=1$) parent $\mathcal{P}=\{{\mathbb{P}}_1\}$ according to the assignments ${\mathcal{V}}_1=\{{\mathbb{V}}_1,{\mathbb{V}}_2\}$ and ${\mathcal{I}}_1=\{{\mathbb{I}}_1,{\mathbb{I}}_2\}$. For completeness, we note $|{\mathcal{V}}_1|=|{\mathcal{I}}_1|=2$. An explicit physics example corresponding to this figure is discussed in Sec. 11a.

Figure 9

Transverse vector decomposition onto the direction $T_{\parallel }$ specified by the UVM transverse momentum vector ${\overrightarrow{u}}_T$ and the direction $T_{\perp }$ orthogonal to it [25, 34]. All vectors shown are in the plane perpendicular to the beam axis.

Figure 10

Illustration of an example extremal configuration for the $N=1$ variables when applied to the $h$ example.

Figure 11

(a) Unit-normalized distribution of the five $N=1$ mass-bound variables $M_{\mathcal{F}}$, $\mathcal{F}\in \{1,1\top ,\top 1,1\circ ,\circ 1\}$ for the inclusive Higgs production process $h\to W^{+}{W}^{-}\to {\ell }^{+}{\ell }^{-}+{\mathrm{p̸}}_T$ at a 7 TeV LHC, with $m_h=200\mathrm{GeV}$ and $\cancel{\mathbf{M}}=0$. The dotted (yellow-shaded) histogram gives the true $\sqrt{\widehat{s}}$ distribution, which in this case is given by the Breit-Wigner $h$ resonance. (b) Unit-normalized distributions of the variables $m_{\mathrm{eff}}$ and $2{\mathrm{p̸}}_T=2{\mathrm{p̸}}_T$ (solid lines), contrasted with $M_{\circ 1}$ and $M_{1\circ }$ (dotted lines).

Figure 12

The same as Fig. 11, but for the $t\overline{t}$ example. The yellow-shaded distribution now gives the true invariant mass of the $t\overline{t}$ pair.

Figure 13

The same as Fig. 12, but for $N=2$ variables: (a) the unprojected $M_2$ and the singly projected variables $M_{2\top }$, $M_{\top 2}$, $M_{\circ 2}$, and $M_{2\circ }$; and (b) the doubly-projected variables $M_{2\top \perp }$ (black), $M_{\top 2\perp }$ (cyan), $M_{\top \perp 2}$ (magenta), $M_{2\circ \perp }$ (red), $M_{\circ 2\perp }$ (green), and $M_{\circ \perp 2}$ (blue). The yellow-shaded distribution now gives the average top quark mass in the event. In panel (b), “$\top$”-projected quantities are denoted with solid lines and are evaluated with ${\cancel{\mathbf{M}}}_1={\cancel{\mathbf{M}}}_2=0$, while “$\circ$”-projected quantities are denoted with dotted lines.

Figure 14

Illustration of an example extremal configuration for the $N=2$ variables when applied to the $t\overline{t}$ example.