A general method for spin measurements in events with missing energy (part II) Konstantin Matchev In collaboration with: M. Burns, K. Kong, M. Park arXiv:0808.2472 [hep-ph]

Outline Why is it so difficult to measure the spin? Problems with previous proposals A couple of examples What is our method? Results from 6 idealized exercises

Why is it difficult to measure the spin? How do we measure mass? We cannot measure spin the same way! spin Events/unit of spin

Why is it difficult to measure the spin? Missing energy signatures arise from something like: Several alternative explanations:

Spin measurements – I production cross-section Kane,Petrov,Shao,Wang 2008 G. Kane et al: ”Basically, if the mass and the production cross-section are measured, the spin is then determined” Are we really measuring the production cross-section? How can we be sure that There is no contribution from indirect production of particle Y? Think of W pair production from top quarks The branching fraction B(X->SM) is 100 %?

Spin measurements – I production cross-section Even if we were able to measure It depends on the mass of X (OK, it will be measured) It depends on the mass of heavy t-channel paticles (?) It depends on the coupling of X to gluons (?) It depends on the coupling of X to quarks (?) It depends on the representation (Number of colors) of X (?) It depends on the spin of X (Yes! That’s what we want) Conclusion: it is very unlikely that by measuring a single number (such as a cross-section) we will be able to determine the spin. We have to look at distributions. What about those question marks?

What is a good distribution to look at? Invariant mass distributions! Advantages: well studied, know about spin. For adjacent SM particles Plot versus m2! For an intermediate BSM particle of spin s, the highest order term is m2s For non-adjacent BSM particles, there are log terms as well. Disadvantage: know about many other things (hidden in the coefficients a), not all of which are measured! Masses MA, MB, MC, MD (x,y,z) Couplings and mixing angles (gL and gR) Particle-antiparticle (D/D*) fraction (f/f*) (f+f*=1)

Most likely chronology Measure the SM backgrounds (rediscover SM) Find a missing energy signal Convince yourself it is a real signal Measure the cross-section (times BR) Measure the overall mass scale Observe structures, measure individual masses Measure spins? Then perhaps also measure Chirality of the couplings (L versus R) Mixing angles Particle-antiparticle fraction

What is the relevant question? We ask: Given the data, which spin configuration gives a good fit for arbitrary values of the yet unknown parameters? We fix: mass spectrum We let spins, couplings, mixing angles, particle/antiparticle fraction f, etc. to float Previously people had asked: Given the data, which spin configuration gives a good fit for fixed values (the true ones) of the yet unknown parameters? They fix: everything but the spins Then let spins to float What is wrong with the latter approach? It’s the wrong chronological order To measure the chirality of the couplings, we will probably need to measure the spins first It’s not a pure spin measurement, i.e. it is a spin measurement under certain model assumptions which still need to be verified experimentally

Does this really make any difference? Yes! Dilepton invariant mass distribution. Data from SPS1a. Athanasiou, Lester, Smillie, Webber 06 Burns, Kong, KM, Park 08 Spins vary Everything else fixed to SPS1a values Easy to distinguish! Mass spectrum fixed to SPS1a values Everything else varies Difficult to distinguish!

Does this really make any difference? Yes! Lepton charge (Barr) asymmetry. Data: “UED” with SPS1a mass spectrum. Athanasiou, Lester, Smillie, Webber 06 Burns, Kong, KM, Park 08 Spins vary Everything else fixed to SPS1a values Easy to distinguish! Mass spectrum fixed to SPS1a values Everything else varies Difficult to distinguish!

How to let the couplings float? Scan by brute force. Mobilize an army of graduate students, consider all (105?) possible values of MSSM parameters, fit each set of (105?) parameter values to the data. Decide if it fits or not. Continue until all possible values (or graduate students) are exhausted. Obviously impractical. Mobilize some theorists to figure out which of the 105 parameters are really relevant for the invariant mass distributions. Consider a “constrained” MSSM where only the relevant parameters are varied. Still too many parameters (15?). Ask the theorists to figure out which combinations of parameters are really relevant for the invariant mass distributions. Our claim: For a three-step (two-step) decay chain there are only three (one) parameters which encode all of the model-dependence

Model-independent approach Most general parameterization of the couplings Each vertex has 4 real parameters, |aL|, |aR|, 2 phases The phases are unobservable The product |aL|.|aR| absorbed in the normalization The ratios |aR|/|aL| defined as new parameters

Helicity combinations We cannot measure the fermion helicity Previous work considered only certain helicity combinations We also consider all remaining possibilities Each block gives rise to a specific function FS;IJ The functions FS;IJ provide a model independent basis for the spin analysis

Final parameter count How many parameters altogether? No. Only three! The quark vertex is parameterized by a combination of f and At the Tevatron f=0.5 and there are only 2 parameters A completely model-independent approach requires only up to 3 continuous parameters!

Choice of parameters ~ What is the best parameterization? No. Even better: change basis Write any invariant mass distribution as

The advantage of the new basis Nice (intermediate) result The jlf distribution is also predicted! Note the opposite signs for beta and gamma: those terms will cancel in the sum!

What are all these functions?

What is our method? Construct and then fit the three invariant mass distributions to

Some comments To reduce combinatorial background OF subtract the leptons ME subtract the jets Normalization is arbitrary. May choose to normalize N(l+l-), N(jl+), N(jl-) to unity. L+-= N(l+l-) only depends on the parameter alpha Each jet-lepton distribution N(jl+) and N(jl-) depends on all three parameters alpha, beta and gamma, but S+- =N(jl+)+N(jl-) depends only on alpha! D+- =N(jl+)-N(jl-) depends only on beta and gamma! Use L+- to measure alpha, then S+- is predicted!

What about the Barr asymmetry? It is nothing but It is essentially just D+-. Instead, we make use of all the available information, including L+- and S+- Both A+- and D+- are identically zero and do not help if f=0.5 Always true at the Tevatron Whenever the initial partons are gg or qq-bar A+- depends on all three parameters alpha, beta and gamma, rather model dependent

Some examples Choose SPS1a parameters Take data from each of the 6 spin configurations, then try to fit the remaining 5 models to it.

SFSF example S+- L+- D+- Color code: data, perfect fit L+- eliminates only one possibility S+- eliminates two D+- is very useful in this case D+-

FSFS example L+- S+- D+- Color code: data, perfect fit L+- alone is sufficient S+- alone is sufficient D+- is useless

FSFV example L+- S+- D+- Color code: data, perfect fit L+- and S+- cannot rule out FSFS! D+- is useless First example of exact duplication (FSFV-FSFS) D+-

FVFS example L+- S+- D+- Color code: data, perfect fit L+- alone eliminates all but one alternative S+- alone is sufficient D+- alone is sufficient

FVFV example S+- L+- D+- Color code: data, perfect fit L+-, S+- and D+- cannot rule out FVFS! Second example of exact duplication (FVFV-FVFS) D+-

SFVF example L+- S+- D+- Color code: data, perfect fit L+- alone is sufficient S+- eliminates all but one alternative D+- alone is sufficient D+-

Summary FSFS can always mimic FSFV FVFS can always mimic FVFV The reverse may also be true, depending on the mass spectrum

Coupling measurements The fitted values of alpha, beta, gamma represent measurements of certain combinations of couplings and mixing angles The sign ambiguity corresponds to the chirality exchange

Parameter fitting The three parameters cannot always be measured Good news for parameter fitters! Typically each distribution requires a 1-parameter fit (at worst, 2-parameter fit).

Summary and outlook Make plots versus m2 Spins can be measured from invariant mass distributions in a general and model-independent way A side benefit of the method is the measurement of the couplings and mixing angles encoded in the parameters alpha, beta and gamma CMS and ATLAS should measure alpha, beta and gamma and let the theorists figure out what are the underlying values of the model parameters that correspond to those The map is many (105) to a few (3). Generalize to (longer, other?) decay chains? With Ronny now trying to see how well the method will work with “real” data/simulation?