Hi Michael,
Actually, I used Adrian's crunched ntuple too, except for the MC
variables, which came from the new ntuple, as a friend. However that
does not out rule problems with the way the original ntuple was
crunched. But this brings out the point that we need to look at the
W spectrum after each MC batch.
--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)
_______________________________________
On Apr 21, 2006, at 17:38:29, Michael Kohl wrote:
> 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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