If all you need is a quick, reliable, and simple method for calculating MT2 (and no other kinematic variables!) then we recommend you consider using the algorithm contained entirely within the single header file lester_mt2_bisect.h
which is distributed as an Ancillary file attached to arXiv:1411.4312, the paper documenting the implementation. No installation is required as the algorithm is contained within a single header file and there are no external dependencies. At time of writing (Spring 2015) this implemetation is the fastest algorithm known to compute MT2 reliably to machine precision, and it can do both symmetric MT2 (where the two invisible particls are considered to have identical mass) and asymmetric MT2 (where the two invisible particles are considered to have different masses).
You should only need to download and install the Oxbridge Kinetics library if there is something harder or more specialised that you wish to do, for example if you wish to compare different specialist flavours of MT2 or other kinematic variables.
The "Cambridge MT2 Variable", sometimes also called the "Stransverse Mass", is an event variable used to bound the masses of an unseen pair of particles which are presumed to have decayed semi-invisibly into particles which were seen. MT2 is therefore a function of the momenta of two visible particles and the missing transverse momentum in an event. The main references for the definition of MT2 are:
Other papers with interesting contributions to understanding MT2 include:
Here is a full list of papers which use MT2.
The purposes of this library are:
--without-minuit2
as an argument to ./configure
in step 3 below. However if you want to get the full features of our library, you should first make sure you have a working ROOT installation in which minuit2 support has been enabled.
./configure --prefix=/somwhere/in/particular make make installbut users without a ROOT installation will need to do
./configure --prefix=/somwhere/in/particular --without-minuit2 make make install
./example_helloWorld_Mt2_Without_Minuit2_Exampleand/or
./example_helloWorld_Mt2_Exampleif you complied with
minuit2
support enabled.
We recommend that most users wishing to calculate MT2 inspecting the simplest helloWorld_Mt2_Example.cpp
example described below, and adapt it to their purposes.
This example calculates MT2 three times, with the same inputs, and prints out what it is doing.
Unless you specified the "--without-minuit2
" option to configure
in Step 3 above (in which case you should instead look here!) the "helloWorld" example should have compiled automatically.
However, you might might want to compile it manually. Assuming you have installed the library, and have a copy of helloWorld_Mt2_Example.cpp
in the current directory, the simplest way to build it manually would be something like the following:
g++ -o example_helloWorld_Mt2_Example \ -I $PREFIX/include/oxbridgekinetics-1.0 \ -L $PREFIX/lib/ \ -loxbridgekinetics-1.0 \ `root-config --cflags` \ -Xlinker -rpath $PREFIX/lib/ \ helloWorld_Mt2_Example.cppwhere $PREFIX should be replaced by the name of the directory into which the library was installed. (You supplied $PREFIX as the "--prefix=BLAH" argument when you configured the library. The default is equivalent to ./configure --prefix=/usr/local)
If you run it by typing
./example_helloWorld_Mt2_Exampleyou should see the following output:
[lxplus214] > ./example_helloWorld_Mt2_Example ---------------------------------------------------------------------- M_T2 : a variable for measuring masses when missing energy is expected. If you use MT2 or this library, please cite: (o) C.G.Lester, D.J.Summers Phys.Lett.B.463:99-103 (1999) hep-ph/9906349 (o) A.J.Barr, C.G.Lester, P.Stephens J.Phys.G 29:2343-2363 (2003) hep-ph/0304226 If you use MTGEN please also cite: (o) C.G.Lester, A.J.Barr (2007) arXiv:0708.1028 (hep-ph) JHEP 12(2007)102 If you use either of the Elliptic Bisection MT2 calculators, please also cite: (o) H.Chen, Z.Han, arXiv:0810.5178 ---------------------------------------------------------------------- Going to calculate MT2 with ltv_Vis_A = Lor2Vec: x = 410, y = 20, Et = 422.493, mass = 100 ltv_Vis_B = Lor2Vec: x = -210, y = -300, Et = 395.727, mass = 150 pT_Miss = TwoVector: x = -200, y = 280 invis_mass = 100 ANSWER: mt2 = 412.628 for Basic_Mt2_332 algorithm Going to calculate MT2 with ltv_Vis_A = Lor2Vec: x = 410, y = 20, Et = 422.493, mass = 100 ltv_Vis_B = Lor2Vec: x = -210, y = -300, Et = 395.727, mass = 150 pT_Miss = TwoVector: x = -200, y = 280 invis_mass = 100 ANSWER: mt2 = 412.628 for Basic_Mt2_332 algorithm Going to calculate MT2 with ltv_Vis_A = Lor2Vec: x = 410, y = 20, Et = 422.493, mass = 100 ltv_Vis_B = Lor2Vec: x = -210, y = -300, Et = 395.727, mass = 150 pT_Miss = TwoVector: x = -200, y = 280 invis_mass = 100 ANSWER: mt2 = 412.628 for Basic_Mt2_332 algorithm
--without-minuit2
option supplied to configure
like this:./configure --without-minuit2then the first "helloWorld" example won't have been built. However, you will still be able to use an alternative example that is indepentent of minuit2 and which is always built. It is complied from
examples/helloWorld_Mt2_Without_Minuit2_Example.cpp
and generates the following output when it is run:
[lester@LesterHP mt2ExampleWithOptionalRoot]$ ./example_helloWorld_Mt2_Without_Minuit2_Example ---------------------------------------------------------------------- M_T2 : a variable for measuring masses when missing energy is expected. If you use MT2 or this library, please cite: (o) C.G.Lester, D.J.Summers Phys.Lett.B.463:99-103 (1999) hep-ph/9906349 (o) A.J.Barr, C.G.Lester, P.Stephens J.Phys.G 29:2343-2363 (2003) hep-ph/0304226 If you use MTGEN please also cite: (o) C.G.Lester, A.J.Barr (2007) arXiv:0708.1028 (hep-ph) JHEP 12(2007)102 If you use the ChengHanBisection MT2 calculator, please also cite: (o) H.Cheng, Z.Han, arXiv:0810.5178 If you use the LesterNachmanBisection MT2 calculator, please also cite: (o) C.G.Lester, B.Nachman (2014) arXiv:1411.4312 ---------------------------------------------------------------------- Going to calculate MT2 with ltv_Vis_A = Lor2Vec: x = 410, y = 20, Et = 422.493, mass = 100 ltv_Vis_B = Lor2Vec: x = -210, y = -300, Et = 395.727, mass = 150 pT_Miss = TwoVector: x = -200, y = 280 invis_massA = 100 invis_massA = 100 ANSWER: mt2 = 412.628 for LesterNachmanBisect_Mt2_3322 algorithm Going to calculate MT2 with ltv_Vis_A = Lor2Vec: x = 410, y = 20, Et = 422.493, mass = 100 ltv_Vis_B = Lor2Vec: x = -210, y = -300, Et = 395.727, mass = 150 pT_Miss = TwoVector: x = -200, y = 280 invis_massA = 100 invis_massA = 100 ANSWER: mt2 = 412.628 for LesterNachmanBisect_Mt2_3322 algorithm Going to calculate MT2 with ltv_Vis_A = Lor2Vec: x = 410, y = 20, Et = 422.493, mass = 100 ltv_Vis_B = Lor2Vec: x = -210, y = -300, Et = 395.727, mass = 150 pT_Miss = TwoVector: x = -200, y = 280 invis_massA = 100 invis_massA = 100 ANSWER: mt2 = 412.628 for LesterNachmanBisect_Mt2_3322 algorithm
Regardless of whether or not you specified the "--without-minuit2
" option to configure
in Step 3 above, this second "helloWorld" example should always have compiled automatically.
However, you might might want to compile it manually. Assuming you have installed the library, and have a copy of helloWorld_Mt2_Without_Minuit2_Example.cpp
in the current directory, the simplest way to build it manually would be something like the following:
g++ -o example_helloWorld_Mt2_Without_Minuit2_Example \ -I $PREFIX/include/oxbridgekinetics-1.0 \ -L $PREFIX/lib/ \ -loxbridgekinetics-1.0 \ `root-config --cflags` \ -Xlinker -rpath $PREFIX/lib/ \ helloWorld_Mt2_Without_Minuit2_Example.cppwhere $PREFIX should be replaced by the name of the directory into which the library was installed. (You supplied $PREFIX as the "--prefix=BLAH" argument when you configured the library. The default is equivalent to ./configure --prefix=/usr/local).
Note that if you have no ROOT installation and you compiled with the --without-minuit2
option, then above you should omit the line:
`root-config --cflags` \
bash-3.2$ ls example_*
To use the library semi-interactively from within ROOT, follow the instructions at the top of the example ROOT macro demo_1_for_ROOT_usage.C
which shows how MT2 values can be calculated from within ROOT. This example requires the mt2 library to have first been built outside ROOT.
Algorithm | Notes | |||||||
---|---|---|---|---|---|---|---|---|
A well debugged but easy to understand MT2 algorithm. Not the fastest or most accurate. | See the Basic_Mt2_332 Algorithm entry below. | |||||||
Recommended "external" MT2 algorithm for non-asymmetric use-cases. | See the LesterNachmanBisect_Mt2_332 Algorithm entry below. | Does not need minuit2, is the most accurate and fastest. | ||||||
Recommended "external" Asymmetric MT2 algorithm. | See the LesterNachmanBisect_Mt2_3322 Algorithm entry below. | Does not need minuit2, is the most accurate and fastest, and can cope with dissimilar mass invisible particles. | ||||||
Advanced_Mt2_332 | The "Advanced" MT2 332 calculator tries to improve on the "Basic" Mt2 332 algorithm, by avoiding the use of an iterative solution for so-called "unbalanced" cases. In practice, this saves very little time, and may even introduce more potential for instabilities (divide by small) for certain input cases. In short, we recommend users stick with "Basic" MT2 332 algorithm. | |||||||
AlphaT_Multijet | Attempts to figure out what the definition of alphaT really is. | Does not need minuit2 | ||||||
Analytic_Mt2_2220 | There is a closed form (non-iterative) expression for calculating MT2 valid in the case that (1) both visible particles are massless, and (2) the assumed mass of the invisible particles is zero. The "Analytic_Mt2_2220_Calculator" implements that expression. | Does not need minuit2 | ||||||
Analytic_Mt2_330 | There is a closed form (non-iterative) expression for calculating MT2 valid in the case that the missing transverse momentum is exactly balanced by the (combined) visible transverse momentum of the two sets of "important" visible particles. Put another way, this is the case where the "upstream transverse momentum" or "other sources of recoil" are zero. The Analytic_Mt2_330 calculator implements that special case. Note that it does not take PTMISS as an input, since it is deducable (in this special case) from minus the sum of the visible particle momenta. | Does not need minuit2 | ||||||
Basic_MCT2_332 | Calculates Cho and Kim's MCT2 (see arXiv:0912.2354 and arXiv:1006.2568) | |||||||
Basic_MPairProd | MPairProd is a bit like MT2, except that it assumes that neither parent particle had any invisible daughter particles. As a consequence, it does not need to use PTMISS and does not discard z-momenta. It places a bound on the mass of the pair of identical parents that could have lead to the observed final state. | Does not need minuit2 | ||||||
Basic_Mt2_332 | This is the recommended (i.e. most well tested) algorithmm for calculating MT2. You can see an example of its use in the "helloWorld" example described above. It is a very simple implementation that harks back to the original definition of MT2. It numerically minimises the larger of two transverse masses over a two dimensional space of momentum splittings. | |||||||
Basic_Mt2_4441 | This was an evolutionary dead-end for MT2. It came from considering whether there was anything to be gained from adding an additional constraint to the usual space over which the MT2 search over momentum splittins is conducted. That additional constraint was to be the requirement that the rootSHat of the splitting hypothesis should not exceed the collider rootSHat. Reasonable though this requirement is, it turns out to make next to no observable difference to any real MT2 value. So ignore this calculator. | |||||||
Basic_Mt2_AsymmParents332 | This is an algorithm for calculating bounds on parental particle masses, when those parents are produced in pairs, have DIS-SIMILAR masses, but yet decay into the same invisible particle (and some visible particles). Eg A->BC and D->EC with C an invisible particle, B and E visible decay products, and A and D heavy parents that need not share the same mass. | |||||||
Basic_MtGen_330 | Calculates MTGEN under the assumption of UTM=0, using a very "dumb" exhaustive search of every single possible partition of the visible momenta into two sets. Can be very slow. | Does not need minuit2 | ||||||
Basic_MtGen_332 | Calculates MTGEN under the assumption of UTM=0, using a very "dumb" exhaustive search of every single possible partition of the visible momenta into two sets. Can be very slow. | Does not need minuit2 | ||||||
Basic_Nt2_332 | NT2 is like MT2 but "backwards": wheras in MT2 you determine a bound on the mass of the parents given a hypothesis for the mass of the invisible daughters, in NT2 you determine a bound on the mass of the invisible daughter given a hypothesis for the mass of teh parents. | |||||||
LesterNachmanBisect_Mt2_332 | This is the fastest and most accurate algorithm for calculating vanilla (non asymmetric) MT2. It is a wrapper around the minuit2-independent Lester and Nachman Bisection algorthm of arXiv:1411.4312. If you have no preference, we recommend you use this one for vanilla MT2 problems. | Does not need minuit2 | ||||||
LesterNachmanBisect_Mt2_3322 | This is the fastest and most accurate algorithm for calculating MT2 in the so-called AsymmetricMT2 in which the invisible particles have different masses. It is a wrapper around the minuit2-independent Lester and Nachman Bisection algorthm of arXiv:1411.4312. If you have no preference, we recommend you use this one for asymmetric MT2 problems. It can also be used for symmetric MT2 at no computational or accuracy cost. | Does not need minuit2 | ||||||
ChengHanBisect_Mt2_332 | This is the second fastest, and second-most accurate, minuit2-independent Cheng and Han Bisection algorthm. Actually, in our library all there really is is a wrapper around Cheng an Han's own implementation. See this more detailed discussion earlier. | Does not need minuit2 | ||||||
Frugal_MPairProd | This implementation improves on that of the Basic MPairProd Calculator by being economical and intelligent with its search through parititions. | Does not need minuit2 | ||||||
Frugal_MtGen_330 | Calculates MTGEN under the assumption of UTM=0, using the fewest possible number of MT2 evaluations. | Does not need minuit2 | ||||||
Frugal_MtGen_332 | Calculates MTGEN using the fewest possible number of MT2 evaluations. | Does not need minuit2 | ||||||
M2C_332 | Calculates Serna and Ross's M2C (see arXiv:0712.0943). | |||||||
MC_330 | Calculates Tovey's MC -- the Contralinear-Boost Invariant Mass -- (see arXiv:0802.2879). | Does not need minuit2 | ||||||
MCT_330 | Calculates Tovey's MCT -- the Contransverse Mass -- (see arXiv:0802.2879). | Does not need minuit2 | ||||||
SUSYPhys_Mt2_222 | Do not use! This is included only for historical reasons. It is an archive of an algorithm used to calculate MT2 for some very early ATLAS susy studies. The algorithm was only accurate for massless input particles. |
Short lists of number like 332, 4441, 222 in names of MT2 algorithms describe the number of degrees of freedom in the algorighm's inputs. Reading the digits from left to right, the number of degrees of freedom and the nature of those degrees of freedom are as follows:
To get all the features of our library, you need a version of ROOT compiled with minuit2 support. However, it is possible to compile our library without any external dependencies by specifying the "--without-minuit2
" option at the configure stage:
./configure --without-minuit2 --prefix=/somewhere make make installIf you do this, only the algorithms and example programs which are indepentent of minuit2 and ROOT will be compiled. This is not a problem if you want the fastest available implmentation (LesterNachman) as this has no ROOT or minuit2 dependencies. If all you need is MT2 or MCT, then you might want to compile in this mode.
See StransverseMassMaintainers.
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