Hi Chris,
thank you very much for the elaborate explanations of your slides!
On your reported shift of the reconstructed unradiated events:
Am I right that you use Adrian's MC (=coda events) and analyze them with
the most recent lrn? If that's true, then the shift may be explainable:
Adrian's MC was generated before the geometry update; but you are using
up-to-date geometry for reconstruction.
One should generate a new MC with the latest calibrations and see if the
unradiated events reconstruct to the proton mass in the W spectrum.
Note that in Eugene's plots in blast_anaware from 3/23/2006 there was no
noticeable shift between unradiated reconstructed and tossed momenta
(plot attached, the black Gaussian is a fit to pwl-pml with Eloss and
Mascarad turned off).
Regards,
Michael
On Fri, 21 Apr 2006, Christopher Crawford wrote:
> Here is a summary of my talk at the BLAST analysis meeting,
> http://blast.lns.mit.edu/PRIVATE_RESULTS/USEFUL/ANALYSIS_MEETINGS/meeting_060419/rad_blast-2006-04-19.ppt
> . Since it caused some controversy, I am including extra details so anyone
> interested can verify my arguments. I have tried to present it in a more
> coherent manner, and I apologize in advance for the length of it, but please
> read it in full before responding.
>
> I. Mascarad Radiative Tail
> A. Kinematics -- inelasticity v = W^2 - M^2, W^2 = M^2 + 2 M \nu -
> Q^2. The two other photon variables (tau, phi_k) are integrated over.
> B. Cutoffs -- In integrating the radiative tail, there are two cutoffs in
> 'v' to consider:
> 1) Upper cutoff, used to match the experimental cuts.
> 2) Lower cutoff, used historically to avoid the infrared divergence
> (copious soft photon emission). This cutoff is now avoided in MASCARAD by
> proper renormalization, in which the infinite vertex correction (I.C.2) at
> the pole cancels the divergent integral of the tail at low v (I.C.3).
> However, because the infinities are in different places, they only cancel in
> the integral over 'v'. Therefore, in a Monte Carlo generator, the elastic
> pole must also include the radiative tail up to some low cutoff 'v', which
> should be less than the BLAST resolution.
> C. Details -- MASCARAD calculates the cross section in different parts:
> 1) the Born (tree level) amplitude.
> 2) virtual elastic -- contribution to the elastic cross section (same
> kinematics) from vertex corrections and loop diagrams.
> 3) the soft radiative part -- basically the part same as used in Mo &
> Tsai. This includes the infrared divergent part.
> 4) finite radiative correction -- the remaining contributions to the
> exact calculation of the radiative tail. This part is a function of \vec
> p_k, the photon 3-momentum. It is actually a small negative correction to
> (3), and cannot be used alone. (see values in attachment)
> D. Cross section -- Mascarad outputs one final number: delta(v) =
> (integral of radiative cross section from elastic peak up to 'v') / (Born
> cross section). The radiative invariant mass spectrum including the tail
> equals: sigma_Born * d(delta)/d(v), (after transforming to W). Discretely,
> delta(v0)=pole+soft photons<v0; delta(v1)-delta(v0) = first bin of tail,
> etc. This is the blue fill histogram in slide 2. The pole (100x greater
> than the first bin of the tail) is omitted, but its area is equivalent to the
> yellow fill histogram. I attached a data-file of delta(v) for 5 Q^2 bins.
> It was generated by adding two parts: delta_soft(v) (I.C.1-3) calculated on
> the same grid, and delta_hard(v) (I.C.4) calculated on a much coarser grid to
> save computation time.
>
> II. BLAST Invariant Mass Spectrum
> A. Data -- shown as the black histogram, simply a plot of W from the
> elastic data with the minimal cut of "qwl==-1 && qwr==1 && 25 < twl && twl <
> 35" (left sector). I only go to 1050 MeV to avoid inelastic contributions to
> the cross section (pion threshold ~ 1070 MeV).
> B. Resolution (generalized) -- shown as the yellow fill curve; don't
> worry for the moment how it was obtained. This is actually the BLAST
> response function to a delta pole, \delta(W-M). For example, this would be
> what we measured if there were no radiation. It has been offset by M=.938
> for visual effects, but is actually centered very close to 0 MeV. The shift
> from zero (W_0-M) is just the kinematic offsets we normally talk about, and
> the width is the BLAST resolution. But these are just two characteristics of
> the BLAST response; another might be the strength of the tail. The integral
> must be less than unity (i.e. the conversion factor between yield and cross
> section) and equals the BLAST efficiency.
> C. Convolution -- shown as the red curve, the theoretical cross section
> convoluted with the BLAST response should equal the measured W-spectrum
> according to the definition of (B).
> D. Assumptions -- there was very good agreement of the convolution (red
> curve) with data (black histogram). However, there were a number of
> assumptions made:
> 1) The response is independent of momentum (there's no way around this;
> otherwise you can't de-convolute the W-spectrum).
> 2) I also assumed that it is symmetric in W. In theory there is no
> problem with relaxing this constraint, although it would be more difficult to
> extract the strength of the radiative tail, and one would just have to
> blindly trust the MASCARAD calculation.
> E. Computer Code -- this is all implemented in
> 'blast/exp/analysis/macros/fit_invmass.C' In particular, there are two
> functions defined:
> 1) 'res_fn' -- implements the BLAST response (II.B), but shifted by
> 'M=.938'. See below for details of the free parameters.
> 2) 'rad_fn' -- calculates the convolution of 'res_fn' with the
> radiative cross section (I.D).
>
> III. Shift in Elastic Peak due to Radiative Tail
> A. Result -- 0.8 MeV. This is just the difference in the peaks of the
> blue curve (II.B) and the red curve (II.C). Or in other words, the
> difference between the BLAST response to the elastic peak (note f(x)
> convoluted with delta(x-x0) = f(x0)) and the BLAST response to the radiative
> cross section. And actually for this analysis, the shape of the response
> function is immaterial; the shift of the elastic peak _ONLY_ depends on the
> width of the response function you use (the BLAST resolution). In
> particular, clearly the offset 'W_0' and amplitude 'A' have no effect on the
> shift, as seen from the properties of convolution.
> B. Resolution dependence -- the shift of the convolution was repeated for
> three resolutions: 25,50,100 MeV (side 4), resulting in shifts in W of about
> 1,2,4 MeV.
> C. Caveats -- there are three things which I can think of which may
> affect the results, none of which are the above methodology. I would prefer
> to deal with these issues before trying different response functions, as
> different people have suggested.
> 1) MASCARAD calculates radiative corrections for fixed Q^_l, defined by
> Q^2_l = 4 E E' sin^2(\theta_e/2). I histogrammed the invariant mass spectrum
> with a cut on \theta_e instead, which only coincides with Q^2_l on the
> elastic ridge.
> 2) I binned the radiative tail starting at W=940MeV in steps of 2MeV.
> You see that most of the contributions come from the first bin, and it is
> very steep. So calculating the radiative tail with finer bins can
> potentially have a big impact. I also note that DGen starts it's tail at
> v=0.10 ~ dW=5.3 MeV, so it may also be affected by the same issue.
> 3) In order to account for multiple photon emission, MASCARAD
> exponentiates the integral of the soft part of (I.C.3). So if your lower
> cutoff is too small, multiple photon emission will not be properly accounted
> for. My analysis actually did not use a lower cutoff (only DGen), but I'm
> not sure what effect this has on the derivative d(delta)/dv.
> D. Shift of Mean -- this can be much larger, but depends on the exact
> details of cuts and fitting. However, note that the shift of the mean with
> respect to the mode (peak) can be extracted from the data themselves, and
> does not need to be simulated. The only important thing is to remain
> consistent with your analysis.
>
> < interlude: The above is fairly straight-forward and we already have the
> radiative correction, but I went one step farther and extracted the BLAST
> response function from the W-spectrum of the data. This is the controversial
> part, explained below. >
>
> IV. Extraction of the BLAST response function (resolution).
> A. Approach -- the basic idea is to de-convolute the radiative cross
> section from the BLAST resolution. The caveats in (II.C) and (III.B) apply.
> Of course one could de-convolute by dividing the Fourier transforms, but you
> would end up with an ugly function, and I'm not sure how reliable this method
> is. I chose to parameterize the response with a simple analytic function,
> and fit the convolution with the radiative cross section for the free
> parameters, as discussed below. The whole process is computed in the code
> 'fit_invmass.C'. I would like to emphasize that this step is just as
> important for testing MASCARAD against our data as it is for actually
> extracting the response function. It is the ONLY way to compare our data
> against MASCARAD.
> B. Left Tail Symmetric -- (black dotted histogram) I mention it in
> passing because it was used to determine general features of the response
> function. The idea is that radiation is mostly on the right side of the
> peak. However, this is NOT the BLAST resolution, since the radiation also
> bleeds in from smearing out the tail; just compare it with the yellow fill
> curve to see how much!
> C. Response Function -- ('res_fn') I chose the parametrization
> '[A]/(1+(W-[W_0])/[sigma]))^[n]'. I tried a Gaussian, but it had the wrong
> shape in the tails (as expected). A pure Lorentzian had good tails, but
> could not reproduce the peak. Adding combinations of the two or multiplying
> by '(1-k*gaus)' produced funny-looking functions with extra wiggles. So this
> is purely phenomenological, but matches the data real good, ant least for
> small theta. At higher theta, the momentum resolution is a mess, even
> double-valued, so not much you can do there. No constant offset was needed,
> as there is essentially no background.
> D. Convolution Function -- ('rad_fn') This is just a numerical
> convolution of 'res_fn' with the elastic pole and each bin of the radiative
> tail (blue). However, I added one extra parameter, [alpha_rc], a scale
> factor for the radiative tail only (not the elastic peak). The purpose of
> this parameter was to test the validity of MASCARAD. A fit of close to '1'
> indicates that MASCARAD calculates the proper strength of the tail or,
> turning the argument around, that the fit was done properly. For final
> results, one should really fix 'alpha_rc' to 1.
> E. Results -- the red curve (IV.D). The parameters of this curve are
> shown at the right, but most of these parameters were directly passed to the
> resolution function, the yellow fill curve (IV.C). This is the unique
> function, which can be convoluted with the radiative cross section to match
> up with the BLAST yield, and is NOT a circular argument.
>
> < the end. now miscellaneous issues: >
>
> V. Investigation of MC Reconstruction
> A. Source -- I used Adrian's MC file generated with DGen + Mascarad. See
> his email in BLAST_TALK, 2006-04-13.
> B. 'lrn' bug -- The problems I reported to BLAST_TALK, 2006-04-17 were
> caused by 'lrn' booking a photon instead of the electron or proton. I fixed
> it to preferentially book charged particles, and checked it in.
> C. MC reconstruction -- after this fix, I was able to compare the
> generated (thrown) kinematical variables (red, top left figure, slide 5) with
> the reconstructed ones: with a cut on the elastic part (blue), or all events
> (black). The elastic pole was peaked at W=948 MeV, and the complete
> reconstructed spectrum at W=958 MeV. This is dramatically greater than the
> shift reported above. I did check my calculation for an obvious error and
> found none. Part of it is definitely due to reconstruction (ie. the blue
> curve should have no shift), but another explanation may be the magnitude of
> the tail (VI.B). Repeating (III.A) with a 3.3x larger tail, I get a shift of
> the peak of 2.3 MeV instead of 0.9 MeV.
> D. Log file -- attached as 'mc_gen_recon.log'
>
> VI. Comparison of original MASCARAD with translation into DGen.
> A. Test -- I compared the generated radiative tail (V.A, red) with the
> radiative tail calculated by the original MASCARAD (I.D, black), shown in the
> lower left (left sector, \theta_e=30 deg), and right (left,right sector;
> \theta_e=30,40,50,60,70 deg) panels of slide 5. The plots are in units of
> d(delta)/dv. The black curve was already normalized; the the red was scaled
> by normalizing the pole (v<0.010) to the value 'delta(0.010)', calculated
> from the original Fortran code. The dotted red histogram has been scaled to
> best match the black curve. This scale factor is reported in the results.
> B. Results -- the DGen code generates a radiative tail 2.4--3.3 times
> larger than expected. The left and right sectors were consistent.
> C. Log file -- attached as 'dgen_masc.log'
>
> VII. Comparison of ELOSS calculations.
> A. Aaron's calculation -- see BLAST_TALK, 2004-12-03
> B. Eugene's calculation -- see plot in BLAST_TALK, 2006-02-22 12:52, or
> parametrization
> C. Computer code -- 'eloss_aaron_eugene.C' compares parametrizations
> D. Results -- slides 6 and 7: Aaron's plot agrees with Eugene's, but the
> parametrizations look off by a factor of 2.
>
> I will continue to pursue radiative corrections, energy loss, and report on
> the final geometric offsets after someone (not me!) has resolved these
> issues.
> --Chris
> _______________________________________
>
> TA-53/MPF-1/D111 P-23 MS H803
> LANL, Los Alamos, NM 87545
> 505-665-9804(o) 665-4121(f) 662-0639(h)
> _______________________________________
>
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