Mass measurements at the Large Hadron Collider

Transcription

1 Mass measurements at the Large Hadron Collider Priyotosh Bandyopadhyay Helsinki Institute of Physics, Helsinki Seminar Course Lecture-4, Ŝ1/2 min JHEP 0903 (2009) 085, JHEP 1106 (2011) 041 March 30, 2012

2 Plan 1 Ŝ min 2 Comparison between min and other global inclusive variables 3 Dependence of min on the unknown masses of invisible particles 4 Conclusions

3 Ŝ min : An Intro A global variable, fully inclusive. To determining the mass scale of new particles in events with missing energy at hadron colliders. : The minimum center-of-mass parton level energy consistent with the measured values of the total calorimeter energy E and the total visible momentum P. min min E = 2 Pz 2 + ET 2 + M2 inv, where M inv is the total mass of all invisible particles produced in the event.

4 Ŝ min : An Intro The peak in the min distribution is correlated with the mass threshold of the parent particles originally produced in the event. This conjecture allows an estimate of the heavy superpartner mass scale (as a function of the LSP mass) in a completely general and model-independent way. This does not need any exclusive event reconstruction.

5 Ŝ min : An Intro For SUSY at the percent level,the accuracy never worse than 10%. With the effects of initial state radiation and multiple parton interactions, the precision gets worse, but for heavy SUSY mass spectra remains 10%

6 Typical Event at the LHC Figure: Most generic event at the LHC.

7 Most general topology No specification about the production mechanism. No assumption that the BSM particles are pair produced and resulting in identical dark matter particles with equal masses. No attempt to group the observed SM objects X i, i = 1,2,...,n vis, into subsets corresponding to individual decay chains. Allow for the presence of SM neutrinos which could contribute towards the measured MET.

8 Looking for a Global event variable Variable sensitive to the mass scale of the particles that were originally produced in the event of Fig. 1, or more generally, to the typical energy scale of the event. Not requiring any event reconstruction Dependent only on the global event variables; describing the visible particles X i, namely, the total energy E in the event and the total visible momentum P in the event. For the invisible particles, it is only the missing transverse momentum PT P T = (P x e x + P y e y ) = P T, (1) E T P T = P T = P 2 x + P 2 y. (2)

9 Ŝ min ŝ min is simply the minimum value of the parton-level Mandelstam variable ŝ which is consistent with the observed set of E, P z and P T in a event. Its square root min is the minimum parton level center-of-mass energy, which is required in order to explain the observed values of E, P z and E T. min (M inv) E 2 P 2z + E 2 T + M2 inv, (3) where the mass parameter M inv is nothing but the total mass of all invisible particles in the event: n inv M inv m i = i=1 n χ i=1 m i, (4)

10 Ŝ min M inv is the bill that we have to pay for the model -independence of the set up. This is very similar to the set up of M T2 and its various cousins. Cambridge M T2 variable is a much more model-dependent quantity, since it requires the identification of two separate decay chains in the events. M T2 is essentially a purely transverse quantity, unlike to the case of Ŝmin.

11 Derivation of Ŝmin Parton-level Mandelstam variable ŝ for the event depicted in Fig. 1: ŝ = = ( ) n inv 2 ( ) n inv 2 E + mi 2 + p i 2 P + p i i=1 i=1 i=1 ( ) n inv 2 E + mi 2 + p it 2 + p2 iz (5) ( ) n inv 2 ( ) n inv 2 P T + p it P z + p iz. i=1 i=1 Now, n inv p it = PT = P T, (6) i=1

12 Derivation of Ŝmin ( ) n inv 2 ( ŝ = E + mi 2 + p it 2 + p2 iz P z + i=1 n inv ) 2 p iz. (7) i=1 Given that we are missing so much information about the missing momenta p i, it is clear that there is no hope of determining ŝ exactly from experiment.

13 Model independent approach for the solution The function ŝ has an absolute global minimum ŝ min, when considered as a function of the unknown variables p i. Approximate the real values of the missing momenta with the values corresponding to the global minimum ŝ min. Minimize the function (7) with respect to the variables p i, subject to the constraint (6). global minimum, where the parameter p it = m i PT, (8) M inv m i P z p iz = 1 + P2 T E 2 Pz 2 Minv 2, (9) n inv M inv m i = i=1 n χ i=1 m i (10)

14 Derivation of Ŝmin min (M inv) = E 2 Pz 2 + PT 2 + M2 inv E 2 P 2z + ET 2 + M2 inv, (11) Ŝ min is global inclusive variable; does not require event reconstruction. It is independent of number of invisible particles.

15 Derivation of Ŝmin It is invariant under longitudinal boosts, since it depends on the quantities E 2 P 2 z, E T and M inv, all three of which are invariant under such boosts. has units of energy and thus provides some measure of the energy scale in the event. min

16 Total visible energy E = α E α, (12) where E α is the energy deposit in the α tower (ECAL + HCAL) with muon energy correction included. The total visible momentum P are P x = α E α sinθ α cos ϕ α, (13) P y = α E α sinθ α sinϕ α, (14) P z = α E α cos θ α, (15) where θ α and ϕ α are correspondingly polar and azimuthal angular coordinates of the α calorimeter tower.

17 Total visible energy The total transverse energy E T is E T α E α sinθ α, (16) H T : very commonly used in the literature but no universally accepted definition for it. One issue is whether one should use only reconstructed objects or simply sum over all calorimeter towers. Former method has the advantage in reducing the pollution from the underlying event, noise, etc. Then it would introduce dependence on the jet reconstruction algorithm, the ID cuts, etc.

18 Total visible energy Calorimeter-based, all inclusive H T. Another ambiguity; the number of jets to be included, inclusion of leptons Here H T defined as The total visible mass in the event M H T E T + E T. (17) E 2 P 2 x P2 y P2 z = E 2 P 2 T P2 z. (18) In terms of visible mass, we can write, min (M inv) = ET 2 + M2 + ET 2 + M2 inv. (19)

19 min (0) In most conservative approach: we will simply set M inv = 0 and consider the variable min (0) = E 2 P 2 z + E T. (20) For SM processes, where the missing energy is due to neutrinos, this would be the proper variable to use anyway. For BSM processes with massive invisible particles, this is an issue.

20 Numerical Session All numerical results have been obtained with PYTHIA. If not mentioned for the following distributions ISR and MPI are turned off. PGS detector simulation package used for the analysis.

21 min in t t production Figure: Unit-normalized distributions : E (blue), E T (cyan), E T (magenta), H T (green), M (red) and min (0) (black); in (a) single-lepton and (b) dilepton t t events. The dotted (yellow-shaded) histograms are identical in panels (a) and (b) and show the true distribution.

22 min in t t production In each panel of Fig. 2, the dotted (yellow-shaded) histogram shows the true distribution, which is the one we would ideally want to measure. However, due to the missing neutrinos, is not directly observable, unless we make some further assumptions and attempt some kinematical event reconstruction. Which distribution is closest to true distribution? In Fig. 2 it is min (0) defined in (20). We see that the missing transverse energy E T is a very poor estimator of the energy scale of the events.

23 min in t t production While E T, H T and M are doing a little bit better, yet are still quite far off. In contrast, the min (0) distribution is quite sharp, and is thus a better indicator of the relevant energy scale. Since min was defined through a minimization procedure, it is clear that it will always underestimate the true. We see that min (0) is tracking the true ŝ1/2 quite well for the case of semi-leptonic t t events in Fig. 2(a). This is because in semi-leptonic events, we are missing a single neutrino, whose transverse momentum is actually measured through PT.

24 min in t t production The only mistake is in approximating min (0) is due to the unknown longitudinal component p 1z. In the case of dilepton events, however, there are two missing neutrinos, and thus more unknown degrees of freedom. The resulting error is larger and leads to a larger displacement between the true distribution and its min (0) approximation, as can be seen in Fig. 2(b). In the case of t t illustrated in Fig. 2 the missing energy arises from massless SM neutrinos, so that the approximation M inv = 0 is well justified.

25 Example of massive missing particle Low energy SUSY with conserved R-parity. Each SUSY event will be initiated by the pair-production of two superpartners, which will then cascade decay to the lightest supersymmetric particle (LSP), which we shall assume to be the lightest neutralino χ 0 1. Since there are two SUSY cascades per event, two LSP particles in the final state, n inv = n χ = 2. (21) Again, since the two LSPs are identical, we also have m 1 = m 2 m χ, (22)

26 Example of massive missing particle However, the true LSP mass m χ is a priori unknown, therefore, when we construct our variable min (M inv) = min (2m χ) (23) for the SUSY examples, we will have to make a guess for the value of the LSP mass m χ. This situation is reminiscent of the case of the Cambridge M T2 variable Lester:99,

27 g g Production Consider gluino pair-production ( g g) and the corresponding decay, or g jj χ 0 1, (24) g jj χ 0 2 jjjj χ0 1. (25) Events will have 4 jets or 8 jets plus missing energy.

28 g g Production Of course, the actual number of reconstructed jets in such events may be even higher, due to the effects of initial state radiation (ISR) and/or jet fragmentation. With gaugino unification we have, m g = 3m χ 0 2 = 6m χ 0 1 6m χ, (26) Since we assume three-body decays in (24) and (25), we do not need to specify the SUSY scalar mass parameters, which can be taken to be very large. Also two LSPs are gaugino like.

29 g g Production Figure: Gluino pair production events with (a) 2-jet gluino decays as in (24) and (b) 4-jet gluino decays as in (25). The SUSY masses are fixed as follows: m χ 0 1 = 100 GeV, m χ 0 2 = 200 GeV and m g = 600 GeV. Here we also plot the min (2m χ) distribution (dotted line) with the correct value of the invisible mass M inv = 2m χ = 2m χ 0 1.

30 g g Production Fig. 3 shows the outcome is not too different from what we found previously in Fig. 2 for the t t case. Knowing LSP mass M inv = 2m χ = 2m χ 0 1. It helps as min (2m χ) gets closer to the true. The quantity min (0) still does surprisingly well in approximating the true.

31 g g Production When the missing energy in the data is due to massive BSM particles, there are two sources of error in approximating min (0). First, when we take the minimum possible value of in (7), we are underestimating by a certain amount, which can be seen by comparing the cheater distribution min (2m χ) (dotted line) to the truth (yellow shaded). Second, as we do not know a priori the LSP mass, we take conservatively M inv = 0, which leads to a further underestimation, as evidenced by the difference between the min (0) distribution (solid line) and its cheater version min (2m χ).

32 Asymmetric event pp gχ 0 1 All visible particles come from the same side of the event, i.e. from a single decay chain. Nevertheless, as seen in Fig. 4, we find very similar results.

33 pp gχ 0 1 Figure: The same as Fig. 3, but for events of associated gluino-lsp production.

34 Dependence of min on M inv The assumption of M inv = 0 is precisely what one would do if one were to assume that the missing energy is simply due to SM neutrinos, as opposed to some new physics. In case where BSM contribution is excess than SM neutrinos, we would have to make a guess about the mass of the BSM invisible particle. What is the effect of this guess? min (2 m χ) distributions shift to higher energy scales, as we increase the value of the test mass m χ.

35 Figure: Unit-normalized distributions of the min (M inv) variable for several different SUSY mass spectra: (a) m χ 0 1 = 100 GeV, (b) m χ 0 1 = 200 GeV, (c) m 0 = 300 GeV, and (d) m 0 = 400 GeV. M inv dependence for pp g g 4j + 2χ 0 1

36 M inv dependence for pp g g 8j + 2χ 0 1 Figure: The same as Fig. 5, but for 4-jet gluino decays as in (25).

37 Figure: The same as Fig. 6, but for events of associated gluino-lsp production ( g χ 0 ). M inv dependence for pp gχ 0 1

38 Dependence of min on M inv For any given set of E, P z and E T values, min (M inv) is a monotonically increasing function of M inv. This is also obvious as one needs more energy in order to produce heavier invisible particles subject to the mass relations. When the test mass m χ is equal to the true mass m χ (i.e. for the black colored histograms), the corresponding distribution min (2m χ) peaks very close to the true threshold (ŝ1/2 ) thr. The threshold ( ) as the value where the true ŝ1/2 thr distribution (yellow shaded histogram) sharply turns on.

39 Dependence of min on M inv This observation is potentially extremely important, since the threshold ( ) is simply related to the masses of the two thr particles which were originally produced in the event. For gluino pair production events, ( ) thr = 2m g = 12m χ, (27) In case of gluino-lsp production, ( ) thr = m g + m χ 0 1 = 7m χ, (28)

40 ( min (M inv) ) peak ( ) min (M inv) to denote the particular value of peak where we find the peak of the distributions Introducing min [ d d min dn( min (M inv)) d min dn( min (M inv)) d min ] min = min (M inv) peak (29) = 0. (30)

41 ( min (M inv) ) peak Empirical observation above as () ( ) min (2m χ). (31) thr peak This is the main result of the paper. Though mathematical derivation was not possible, numerical results support this. Different mass spectra and different production processes also verify this.

42 Conclusions and drawbacks It is a global inclusive variable. Identification of decay chain or event reconstruction is not needed. ( ) min (M inv) gives the mass scale involved in the process. peak In the case of gluino pair-production in SUSY m g ( m χ ) 1 2 ( ) min (2 m χ) peak (32) In the challenging case of associated gluino-lsp production: ( ) m g ( m χ ) min (2 m χ) m χ. (33) peak

43 Conclusions and drawbacks Only experimental input needed is the location of the peak of our all-inclusive global variable min. One should not be bothered by the fact that we did not get an absolute measurement of the gluino mass, but only obtain it as a function of the LSP mass. This is a well-known drawback of the other common mass measurement methods as well. Unlike other mass variables min applys to asymmetric decay chains as well.

44 Conclusions and drawbacks Omitting relevant particles from the ŝ calculation. This case arises whenever some of the decay products resulting from the HS are not detected. Including irrelevant particles in the ŝ calculation. In general, any given event will contain a certain number of particles which will be seen in the detector, but did not originate from the primary HS. Initial state radiation (ISR), multiple parton interactions (MPI) and pile-up are the main examples of processes contributing to this effect. ISR and MPI can be a serious problem. Including the extra particles will necessarily lead to an increase in the measured value of ŝ. η max = 1.4, which is nothing but the end of the barrel and beginning of HE/HF calorimeters in CMS can reduce these kind of errors.

45 Thank you