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Guess the function: results January 21, 2009

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Thanks to the many offers for help received a few days ago, when I asked for hints on possible functional forms to interpolate a histogram I was finding hard to fit, I have successfully solved the problem, and can now release the results of my study.

The issue is the following one: at the LHC, Z bosons are produced by electroweak interactions, through quark-antiquark annihilation. The colliding quarks have a variable energy, determined by probability density functions (PDF) which determine how much of the proton’s energy they carry; and the Z boson has a resonance shape which has a sizable width: 2.5 GeV, for a 91 GeV mass. The varying energy of the center of mass, determined by the random value of quark energies due to the PDF, “samples” the resonance curve, creating a distortion in the mass distribution of the produced Z bosons.

The above is not the end of the story, but just the beginning: in fact, there are electromagnetic corrections (QED) due to the radiation of photons, both “internally” and by the two muons into which the Z decays (I am focusing on that final state of Z production: a pair of high-momentum muons from Z \to \mu^+ \mu^-). Also, electromagnetic interactions cause a interference with Z production, because a virtual photon may produce the same final state (two muons) by means of the so-called “Drell-Yan” process. All these effects can only be accounted for by detailed Monte Carlo simulations.

Now, let us treat all of that as a black box: we only care to describe the mass distribution of muon pairs from Z production at the LHC, and we have a pretty good simulation program, Horace (developed by four physicists in Pavia University: C.M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini), which handles the effects discussed above. My problem is to describe with a simple function the produced Z boson lineshape (the mass distribution) in different bins of Z rapidity. Rapidity is a quantity connected to the momentum of the particle along the beam direction: since the colliding quarks have variable energies, the Z may have a high boost along that direction. And crucially, depending on Z rapidity, the lineshape varies.

In the post I published here a few days ago I presented the residual of lineshape fits which used the original resonance form, neglecting all PDF and QED effects. By fitting those residuals with a proper parametrized function, I was trying to arrive at a better parametrization of the full lineshape.

After many attempts, I can now release the results. The template for residuals is shown below, interpolated with the function I obtained from an advice by Lubos Motl:

After multiplying that function by the original Breit-Wigner resonance function, I could fit the 24 lineshapes extracted from a binning in rapidity. This produced additional residuals, which are of course much smaller than the first-order ones above, and have a sort of parabolic shape this time. A couple of them are shown on the right.

I then interpolated those residuals with parabolas, and extracted their fit parameters. Then, I could parametrize the parameters, as the graph below shows: the three degrees of freedom have roughly linear variations with Z rapidity. The graphs show the five parameter dependences on Z rapidity (left column) for lineshapes extracted with the CTEQ set of parton PDF; for MRST set (center column); and the ratio of the two parametrization (right column), which is not too different from 1.0.

Finally, the 24 fits which use the f(m,y) shape, with now all of the rapidity-dependent parameters fixed, are shown below (the graph shows only one fit, click to enlarge and see all of them together).

The function used is detailed in the slide below:

I am rather satisfied by the result, because the residuals of these final fits are really small, as shown on the right: they are certainly smaller than the uncertainties due to PDF and QED effects. The f(m,y) function above will now be used to derive a parametrization of the probability that we observe a dimuon pair with a given mass m at a rapidity y, as a function of the momentum scale in the tracker and the muon momentum resolution.

Guess the function! January 15, 2009

Posted by dorigo in mathematics, personal, physics, science.
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I have a problem today -actually I’ve fiddled with it for a couple of days now. So, since it does not involve particles (at least not directly), I figured I’d bounce it off the mathematically inclined among you: maybe I get an answer before I can figure my problem out by myself!

The problem is simple: find a functional form that can be a good fit, with suitable parameters, to the following graph:

(This is a residual of a Z lineshape fit to a relativistic Breit-Wigner function by the way, but you need not bother with these unnecessary details).

As you can see, we have a negative asymptote and a positive asymptote that have different values, and a central wiggling which has different “width” for the negative and positive component. I have been trying several combinations like f(x) = atan(h(x))*g(x), where g(x) is a gaussian and h(x) some kind of “warping factor” with a different slope in the negative and positive side (with respect to x=91)… But I am getting nowhere. I am sure there is somebody out there that has a good advice, so please shoot!

UPDATE: Marius suggests a function in the comments thread below. I thank him for his input, but as is, the function f(x)=A atan(x)+ B atan(x+C) +D does not work well: see the best fit below (parameters in the upper right legend are A,B,C,D as in the function suggested by Marius):

Maybe with suitable modifications this might work, though. Hmmmm…

UPDATE: Using the hint by Marius that the addition of another arctangent could account for the different height of the two asymptotes, I have cooked up a better fit:

This is better, but I am really not satisfied. The function has 11 degrees of freedom -which is not too troublesome since there are 300 points in the graph to fit anyway; but the function is UGLY:

p_o atan[(p_1-x) e^{-((x-p_3)/p_4)^2}] + p_5 atan[p_6(p_7-x)] + p_8 e^{-((x-p_9)/p_{10})^2}

Any further idea on how to improve it ?

Hmmm, and I should add that having 11 parameters is a curse for me, because what I am going to do after I have a reasonable functional form is to study the parameters as a function of Z rapidity (which modifies the original graph), and parameterize those 11 dependencies… I already have a headache!

UPDATE: Lubos makes a very good attempt with a simple ratio of polynomials in the comments thread, offering f(x) = p_0 ( x-p_1)/(p_2x^2 + p_3 x+p_4) (he even offers some eyeballed parameters). Nice try, but the problem is that the function seems to be very irregular. If one fits the center region, Lubos’ function obtains a good fit (see upper plot below); if one tries to extend the fit further out on the tails, however, the fit rapidly worsens (lower plot).

Despite the shortcomings, I think I will investigate some ways to fix the function offered by Lubos -it has the potential of describing with few parameters the whole shape, once tweaked a bit…

UPDATE: Lubos tried to mend himself the function he proposed above, by adding a hyperbolic tangent. The function fits better the whole range, but it still fails to catch the subtleties of the slopes… Here is a fit using his suggested parameters:

I think I will remove the hyperbolic tangent and work on some warping of the polynomial…

UPDATE: warping the x values above 91 GeV from Lubos’ polynomial with a function f(x)=-p_0+p_1 x +\sqrt{(p_1-1)^2 x^2 + p_0^2} seems to work. The result is below:

The fit is not extremely precise, but these are residuals from a Breit-Wigner, so I guess that the multiplication of this function by the original shape will give a more than adequate parametrization, for my goals. Next up is obtaining 50 different fits like the one above, one per each interval in Z rapidity from 0 to 5.0, and parametrizing each of the seven parameter of the fits…

An exam in Subnuclear Physics January 9, 2009

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Here are the questions asked at an exam in Subnuclear Physics this morning:

  • Draw the strong and electromagnetic coupling constants as a function of Q^2, explain their functional dependence using feynman graphs of the corrections to the photon and gluon propagators, write their formula, and compute the value of the constants at Q^2=M_Z^2, given the values at Q^2=1 MeV^2 (QED) and Q^2=1 GeV^2 (QCD).
  • The GIM mechanism: explain the need for a fourth quark using box diagrams of kaon decays to muon pairs. How does the charm contribution depend on its mass ? What conclusion could be drawn by that dependence in the case of B mixing measurements in the eighties ?
  • Discuss a measurement of the top quark mass. For a dileptonic decay of top quark pairs, discuss the final state and its production rate.
  • Discuss decay modes of W bosons and their branching fraction values. Discuss decay modes of Z bosons and their branching fraction values.

The student answered well all questions and he got 30/30 points.

Some posts you might have missed in 2008 – part II January 6, 2009

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Here is the second part of the list of useful physics posts I published on this site in 2008. As noted yesterday when I published the list for the first six months of 2008, this list does not include guest posts nor conference reports, which may be valuable but belong to a different place (and are linked from permanent pages above). In reverse chronological order:

December 29: a report on the first measurement of exclusive production of charmonium states in hadron-hadron collisions, by CDF.

December 19: a detailed description of the effects of parton distribution functions on the production of Z bosons at the LHC, and how these effects determine the observed mass of the produced Z bosons. On the same topic, there is a maybe simpler post from November 25th.

December 8: description of a new technique to measure the top quark mass in dileptonic decays by CDF.

November 28: a report on the measurement of extremely rare decays of B hadrons, and their implications.

November 19, November 20, November 20 again , November 21, and November 21 again: a five-post saga on the disagreement between Lubos Motl and yours truly on a detail on the multi-muon analysis by CDF, which becomes a endless diatriba since Lubos won’t listen to my attempts at making his brain work, and insists on his mistake. This leads to a back-and-forth between our blogs and a surprising happy ending when Motl finally apologizes for his mistake. Stuff for expert lubologists, but I could not help adding the above links to this summary. Beware, most of the fun is in the comments threads!

November 8, November 8 again, and November 12: a three-part discussion of the details in the surprising new measurement of anomalous multi-muon production published by CDF (whose summary is here). Warning: I intend to continue this series as I find the time, to complete the detailed description of this potentially groundbreaking study.

October 24: the analysis by which D0 extracts evidence for diboson production using the dilepton plus dijet final state, a difficult, background-ridden signature. The same search, performed by CDF, is reported in detail in a post published on October 13.

September 23: a description of an automated global search for new physics in CDF data, and its intriguing results.

September 19: the discovery of the \Omega_b baryon, an important find by the D0 experiment.

August 27: a report on the D0 measurement of the polarization of Upsilon mesons -states made up by a b \bar b pair- and its relevance for our understanding of QCD.

August 21: a detailed discussion of the ingredients necessary to measure with the utmost precision the mass of the W boson at the Tevatron.

August 8: the new CDF measurement of the lifetime of the \Lambda_b baryon, which had previously been in disagreement with theory.

August 7: a discussion of the new cross-section limits on Higgs boson production, and the first exclusion of the 170 GeV mass, by the two Tevatron experiments.

July 18: a search for narrow resonances decaying to muon pairs in CDF data excludes the tentative signal seen by CDF in Run I.

July 10: An important measurement by CDF on the correlated production of pairs of b-quark jets. This measurement is a cornerstone of the observation of anomalous multi-muon events that CDF published at the end of October 2008 (see above).

July 8: a report of a new technique to measure the top quark mass which is very important for the LHC, and the results obtained on CDF data. For a similar technique of relevance to LHC, also check this other CDF measurement.

Gravitons are heavier than 500 GeV! December 23, 2008

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About a year ago I reported here on a search performed by CDF for events featuring two Z bosons, both decaying to electron-positron pairs: I had been an internal reviewer of that analysis, and I discussed it in some detail after we approved it for publication. While the standard model expectation for electroweak production of two Z bosons is of about 1.5 pb, and the process has indeed been put in evidence in CDF and D0 Run II data, the analysis was rather focused on a search for heavy mass resonances decaying to the ZZ final state: new physics, that is, either in the form
of a heavy Higgs boson, or of a graviton (in the Randall-Sundrum scenario), or other still fancier (and improbable) beasts.

CDF has now repeated that search by increasing the dataset size by a factor of three, and by including mixed final states which include muon pairs and even jet pairs. This makes the analysis intrinsically interesting to me, since I have started a similar analysis with the CMS experiment, together with a PhD student in Padova, Mia Tosi. Mia and I will be looking for Higgs bosons in the dilepton plus dijet final state, with particular emphasis on the Z \to b \bar b decay, which is a signal with which we have quite some familiarity.

The new CDF search for high-mass ZZ events configures itself as a “signature-based” one: despite the reference to the Randall-Sundrum graviton, the analysis cuts are kept generic, such that a signal can be found for anything that decays to two Z bosons, and in case no signal is seen, a model-independent limit on the cross section can be set. The only limitation of the search is that the four-body mass is studied only above the minimum value of 300 GeV. Such a requirement allows to steer away from phase space regions where backgrounds dominate.

Once four objects (electrons, muons, and jets, with the specification that at most two jets are present) are selected with loose cuts, a statistical estimator is built to test the hypothesis that they originate from the decay X \to ZZ \to llll (lljj). It is a simple \chi^2 function, which utilizes the expected resolution on the two two-body masses and the resulting four-body mass to estimate how much the event departs from the tentative signal interpretation. Only in the case of jet pairs, an explicit cut is set on the dijet mass to lay between 65 and 120 GeV, to avoid accepting too many random jet combinations.

While the M_x>300 GeV region is the one where the signal is sought, the complementary region of the four-body mass is used as a control sample, to verify that background estimates obtained with Monte Carlo simulations are in agreement with the observed data. The nice thing about such a spectacular signature as the production of two Z bosons is that backgrounds are exclusively of electroweak nature: by having at least one Z \to ll decay in the final state, the signal cannot be mimicked easily by purely quantum chromodynamical processes, which plague most hadron collider searches with high rates. Besides regular ZZ pairs from standard model processes, backgrounds include WZ, WW, and Z+jets production. At high four-body mass, however, all of these are really small, and even in the 3 inverse femtobarns of proton-antiproton collisions analyzed by CDF for this search, they contribute only few events; only the dilepton+dijet signature accepts a few hundred events, because of the large cross-section of Z+2 jet production processes.

In the end, no signal is seen, and a cross-section limit is extracted as a function of the X mass. The limit is shown below, compared to the expected cross section for graviton production and decay to the ZZ final state. The comparison of upper limit (the red curve) with the theory hatched line allows to exclude gravitons with masses below 491 GeV, for a particular choice of model parameters k/M_p=0.1 (k is a warp factor for the extra dimensions, and M_p is the Planck mass).

As a by-product of this analysis, a new set of excellent standard-model-like ZZ decay candidates have been selected. I am unable to show any of the new event displays here, because they have not been approved for public consumption by CDF yet… So please see the lego plot of a ZZ \to eeee candidate below, extracted last year by the same authors. The two pairs of electrons make masses very close to that of the Z boson, as evidenced by the two pink numbers.

To read this graph, you have to know that the greek letter \eta is the pseudorapidity, basically a function of the angle that particles make with the beam axis. A pseudorapidity of zero means that the particle is emitted at 90 degrees from the beam, while positive and negative values indicate the proton and antiproton directions. The other coordinate, \phi, indicates the azimuthal angle in the transverse plane. The z axis (the height of the bars) indicates how much energy is deposited in the \eta - \phi interval span by the bars. In bright pink are shown the four electron candidates, as measured by the CDF calorimeter, and each bar is labeled by the energy in GeV measured for each.

I am only left with the pleasant task of congratulating my colleagues Antonio Boveia, Ben Brau, and David Stuart for this new result, which greatly extends the scope of the analysis I have reviewed last year. During my review I had encouraged them to pursue the other decay modes of ZZ pairs, and so they did. Well done, folks!

More on the Z lineshape at LHC December 19, 2008

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Yesterday I posted a nice-looking graph without abounding in explanations on how I determined it. Let me fill that gap here today.

A short introduction

Z bosons will be produced copiously at the LHC in proton-proton collisions. What happens is that a quark from one proton hits an antiquark of the same flavour in the other proton, and the pair annihilates, producing the Z. This is a weak interaction: a relatively rare process, because weak interactions are much less frequent than strong interactions. Quarks carry colour charge as well as weak hypercharge, and most of the times when they hit each other what “reacts” is their colour, not their hypercharge. Similarly, when you meet  John at the coffee machine you discuss football more often than chinese checkers: in particle physics terms, that is because your football coupling with John is stronger than your chinese-checkers coupling.

(more…)

Result now, explanation later December 18, 2008

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Tonight I feel accomplished, since I have completed a crucial update of the cornerstone of the algorithm which provides the calibration of the CMS momentum scale. I have no time to discuss the details tonight, but I will share with you the final result of a complicated multi-part calculation (at least, for my mediocre standards): the probability distribution function of measuring the Z boson mass at a certain value M, using the quadrimomenta of two muon tracks which correspond to an estimated mass resolution \sigma_M, when the rapidity of the Z boson is Y_Z.

The above might -and should, if you are not a HEP physicist- sound rather meaningless, but the family of two-dimensional functions P(M,\sigma_M)_Y is needed for a precise calibration of the CMS tracker. They can be derived by convoluting the production cross-section of Z bosons \sigma_M at a given rapidity Y with the proton’s parton distribution functions using a factorization integral, and then convoluting the resulting functions with a smearing Gaussian distribution of width \sigma_M.

Still confused ? No worry. Today I will only show one sample result – the probability distribution as a function of M and \sigma_M for Z bosons produced at a rapidity 2.8< |Y| <2.9, and tomorrow I will explain in simple terms how I obtained that curve and the other 39 I have extracted today.

In the three-dimensional graph above, one axis has the reconstructed mass of muon pairs M (from 71 to 111 GeV), the other has the expected mass resolution \sigma_M (from 0 to 10 GeV). The height of the function is the probability of observing the mass value M, if the expected resolution is \sigma_M. On top of the graph one also sees in colors the curves of equal probability displayed on a projected plane. It will not escape to the keen eye that the function is asymmetric in mass around its peak: that is entirely the effect of the parton distribution functions…

The Z mass at a hadron collider November 25, 2008

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The Z boson mass has been measured with exquisite precision in the nineties by the LEP experiments ALEPH, OPAL, DELPHI and L3, and by the SLD experiment at SLAC: we know its value to better than a few MeV precision. The PDG gives M_Z = 91.1876 \pm 0.0021 GeV. Now, a precise Z mass is an important input to our theory, the Standard Model, and through its measurement, as well as that of other Z-related quantities that the four LEP experiments and SLD measured with great precision, a giant leap forward has been made in the understanding of the subtleties of electroweak interactions.

For an experimental physicist, however, the knowledge of the Z mass is more a tool for calibration purposes than a key to theoretical investigations. Indeed, as I have discussed elsewhere recently, I am working at the calibration of the CMS tracker using the decays of Z bosons, as well as of lower-mass resonances. We take Z \to \mu \mu decays, we measure muon tracks, determine the measured mass of the Z boson with them, and compare the latter to the world average. This provides us with precious information on the calibration of the momentum measurement of muon tracks.

In CMS we will quickly collect large numbers of Z bosons, so statistics is not an issue: we will be able to study the calibration of tracks very effectively with those events. However, when statistics is large, experimentalists start worrying about systematic uncertainties. Indeed, there are several effects that cause a difference between the mass value we reconstruct with muon tracks and the true value of the Z boson mass -the one so well determined which sits in the PDG.

I decided to study one of those effects today: the mass shift due to parton distribution functions (PDF). When you collide protons against other protons, what creates a Z boson is the hard interaction between a quark and an antiquark. These constituents of the projectiles carry a fraction of the total proton momentum, but this fraction -called parton distribution function– is unknown on an event-by event basis. By studying proton collisions in different conditions and environments for a long time, we have been able to extract functions f_q(x) which describe how likely it is that a quark q in the proton carries a fraction x of the proton’s momentum. As an example, if the proton travels at 5 TeV as in LHC, an x value of 0.1 means that the quark q will carry 500 GeV by itself.

Now, things are complicated, because each different quark q (u,d,s,c,b) has its own different parton distribution function. The proton contains two valence up-quarks and one valence down-quark: it has a (uud) composition. Those quarks carry a good part of the proton’s momentum, but a large share is due to the rest of partons the proton is made of: sea quark-antiquark pairs, and gluons. Protons do carry antiquarks of all kinds -five in total-, as well as gluons, and these, too, get their own distribution function. A plot of the parton distribution functions of the proton (with a logarithmic x-axis to enhance the low-x behavior) is shown on the right. Note the bumps of u- and d- quark distributions, in blue and green, respectively: those bumps are due to the valence quark contributions.

In reality, things are even more complicated than what I discussed above: you do not simply get away with one function per each of the 11 partons I mentioned thsi far, because these functions have a value which depends on the energy at which you probe the proton, Q^2: in a soft collision (which means a small Q_1), f(x, Q^2_1) is very different from what it is in a harder one, f(x, Q^2_2) (with a larger Q_2).

The reason for the weird behavior of parton distribution functions -their evolution with Q^2– is that quarks have the tendency of emitting gluons, becoming less energetic, and this tendency in turn depends on the energy Q at which they are studied. What is stated above is encoded in very famous functions called DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equations. They are in a sense another consequence of the “asymptotic freedom” exhibited by strongly interacting particles: at high energy they behave as free particles, emitting little color radiation, while at low energy their interaction with the gluon field increases in strength. It is all due to the fact that the coupling constant of the theory, \alpha_s, is large at small Q. That constant is not a constant by any means!

You have every reason to be confused now: I was talking about calibrating the CMS tracker using muons, and now we are deep into Quantum ChromoDynamics. What gives ? Well: Z bosons are created by quark-antiquark annihilations, and those are found inside the colliding protons with probabilities which depend on their momentum fraction x, and on the total collision energy Q. Since the PDF of quarks and antiquarks peak at very small values of x, the probability of a collision yielding a Z boson -which has a respectable mass of 91 GeV- is small.  If the Z was lighter, more of them would be produced. Now, the Z boson is a resonance, and like every resonance, it has a finite width. What that means is that not all Z bosons have exactly the same mass: while the peak is at 91.186 GeV, the width is 2.5 GeV, which means that it is not infrequent for a Z boson to have a mass of  89, or 92 GeV, rather than the average value. This is described by the Z lineshape, a function called Breit-Wigner:

F(\Gamma,M) = \frac{\Gamma/2} {(M-M_Z)^2 + \Gamma^2 /4}.

The function is shown below.

As you can see, there is a non-negligible probability that a Z boson has a mass quite different -even a few GeV off- from 91.19 GeV. Now, since Z bosons can be created at masses lower than M_Z, they will be privileged by parton distribution functions over masses higher than M_Z by the same amount, because .parton distribution functions are larger at lower x. This creates a bias: the perfectly symmetric Breit-Wigner lineshape gets distorted by the preference of partons to carry a lower fraction of the proton momentum.

The distorsion is very small, but it is very important to take it in account when one wants to use measured Z masses to precisely calibrate the track momentum measurement. To size up the effect of the PDF on the Z lineshape, one can compute an integral of the Breit-Wigner weighted with the PDF f(x), by taking into account the different combinations of quarks which give rise to a Z boson in proton-proton collisions.

A Z can be produced by the following quark-antiquark interactions:

  • u \bar u: this can originate from a valence u-quark and a sea anti-u-quark, as well as from a sea u-quark and a sea anti-u-quark. The probability that this quark pair creates a Z depends on the coupling of u-quarks to the Z boson, and this probability is a function of some coefficient predicted by electroweak theory. It is proportional to 0.11784.
  • d \bar d: same as above, but the coupling is proportional to 0.15188.
  • s \bar s: these can only occur through sea-sea interactions. The coefficient is the same as for d-quarks.
  • c \bar c: these are due to the small charm component of the proton sea. They get the 0.11784 coefficient as u-quarks too.
  • b \bar b: these are tiny, but still exist. b-quarks couple to the Z with the 0.15188 factor.
  • t \bar t: these are basically zero.
  • g g: gluon-gluon collisions cannot produce a Z boson, because they are vector particles as the Z (spin 1), and a vector-vector-vector vertex is zero by construction. Note that the same does not hold for the Higgs boson, which is a scalar (spin 0) particle: a vector-vector-scalar vertex is possible, and in fact it is the largest contribution to H production at the LHC.

Putting everything together, one may compute the shift in the lineshape of the Z, and plot it directly (right, on a logarithmic scale to show the effect on the tails), or as a function of the rapidity of the Z boson, a quantity labeled by the letter Y (the dependence is shown in the last graph of this post, below). Rapidity is a measure of how fast is the Z boson moving in the detector reference frame: when one of the partons has a much larger momentum fraction than the one it is colliding against, the produced Z boson has a large momentum in the direction of the more energetic parton.

The rapidity distribution of Z bosons is shown in the graph below, separately for Zbosons produced by valence-sea collisions (in red) and by sea-sea collisions (in blue).

A rapidity Y=0 means that the Z was produced at rest in the detector, +5 is a fast-forward-moving Z, and -5 is a Z moving in the opposite direction with as much speed. As you can see, the valence-sea interactions are the most asymmetric ones, predominantly producing a forward-moving Z boson.

On the right here I also plot with the same color-coding the x distribution of quarks taking part in the Z creation. The red distribution has both a very small-x and a very large-x component, highlighting the asymmetric production.

Despite being in black and white, the most interesting plot is however the following one. It shows the average mass of the Z bosons (on the vertical scale, in GeV) as a function of the Z rapidity. The downward shift from 91.186 GeV is relevant -about 0.25 GeV overall- but it increases at large values of rapidity, when one of the two partons has a very small value of x, so that the collision “samples” a rapidly varying PDF for that parton.

The plot on the left here is what is needed as an input for our calibration program: we will have to study how this dependence affects our determination of the momentum scale. A lot of work ahead, but a very enlightening one!