STAR HardProbes PWG

[Sooraj Radhakrishnan <skradhakrishnan AT lbl.gov>]

Hi Rongrong,

    Your contention is that we cannot get the momentum loss by measuring px, and v1 and <px> comes from the shift in the pT spectra. This is true only for the small subset with py = 0, otherwise we can measure the momentum loss. And <px> gives the average momentum asymmetry per jet in our selected sample. 

//One can see this with an extreme case of flat underlying spectrum which yields <px> = 0 regardless of the energy loss since N(pT+pT_+) = N(pT+pT_-) due to flat spectrum. //

--- This is true only for the selection with py = 0. Because we impose px = -px with our selection of pT. If you look at the azimuthal distribution you can still measure v1 and <px>. This is what we do in all v1 measurements and v1 = <px/pT> gives the asymmetry in momentum flow along the x direction 

//Now, we can look at a more realistic case, i.e. jets traveling in all directions and a physical underlying spectrum. Again, we measure <px> at a fixed pT. It will become: 

<px> = pT*\sum(N(pT+dpT_i) cos(dphi_i))/\sum(N(pT+dpT_i))

where dpT_i is the energy loss along dphi_i direction. //

----- In this case also, v1 will be given by your previous _expression_ (same as the formula I have on the slides), as v1 gives the peak modulation of the cosine distribution. This definition is not capturing the physical quantity we measure. This will also depend on bin and bin width where you measure. This is not the quantity we measure as I demonstrated 

If you can demonstrate your definition gives a well defined quantity and the physical quantity we measure, we can discuss further

Best,

Sooraj

On Wed, Sep 18, 2024 at 8:00 AM Ma, Rongrong <marr AT bnl.gov> wrote:

Hello Sooraj

Thanks for your summary. However, as I mentioned a few times during our private email exchange, I am not saying we should calculate v1 using jets with py = 0 in data analysis. I was just setting up a simple example to make my point. 

The simple example was that if there are only jets traveling along +x and -x directions, then for the measurement at a fixed pT, <px> becomes:

<px> = pT(N(pT+pT_+) - N(pT+pT_-))/(N(pT+pT_+) + N(pT+pT_-)), 

where pT_+ and pT_- are the average energy lost by jets traveling along +x and -x directions, and N(pT+pT_+) and N(pT+pT_-) are the number of jets at pT+pT_+ and pT+pT_- before energy loss, i.e. from underlying jet spectrum. With this example, my point is: <px> value depends on underlying spectrum shape even though it is also related to the energy loss different (pT_+ - pT_-). One can see this with an extreme case of flat underlying spectrum which yields <px> = 0 regardless of the energy loss since N(pT+pT_+) = N(pT+pT_-) due to flat spectrum. 

Now, we can look at a more realistic case, i.e. jets traveling in all directions and a physical underlying spectrum. Again, we measure <px> at a fixed pT. It will become: 

<px> = pT*\sum(N(pT+dpT_i) cos(dphi_i))/\sum(N(pT+dpT_i))

where dpT_i is the energy loss along dphi_i direction. 

We can define the average energy loss of the selected jet sample as <dpT> = \sum(dpT_i N(pT+dpT_i) cos(dphi_i))/\sum(N(pT+dpT_i))

My point is that measured <px> value depends on pT, <dpT>, and the underlying spectrum. Since it depends on pT which is what we measure experimentally, it is also affected by detector effects. 

If I understand you correctly, you are saying <px> is the energy loss asymmetry for the selected jet sample, i.e. <px> = <dpT>. Please correct me if I am wrong. It is not clear to me how one arrives at that conclusion for this measurement. (As I mentioned before, <px> = <dpT> only holds if we can select on true jet pT before energy loss.)

Nihar: I am on shift now, so it is unlikely that I can join the HP meeting tomorrow. 

Best

Rongrong 

On Sep 18, 2024, at 3:39 AM, Sooraj Radhakrishnan <skradhakrishnan AT lbl.gov> wrote:

Dear All,

    Rongrong raised a question on relating <px> and pT shift from energy loss, pointing to jets with the case where py = 0. His contention was since px = -px for the sample with py = 0 , the v1 (and thus px) should be relatable to the pT shift of the spectra 

However, this is a special subset of jets and the condition px = -px is coming from our selection of jet sample requiring a particular pT. This is a limitation of our selection, we impose no energy loss for these jets from our selection. This constraint exists only for a small fraction of jets which have py very close to 0. On average we are still measuring the asymmetry in px distribution of the jets and <px> gives the mean momentum loss per jet along the x direction

The calculated v1 is a well defined and well behaved quantity as you can see in the slides attached below

On the other hand, defining a v1 from the pT shift of spectra for the case with py = 0 does not give a well defined quantity and the value depends on the bin chosen, as shown in the slides below. This is not the quantity we measure

Thoughts are welcome 

thanks,

Sooraj

On Thu, Sep 12, 2024 at 7:44 PM Sooraj Radhakrishnan <skradhakrishnan AT lbl.gov> wrote:

Hi Rongrong,

   Its not clear to me what you are trying to say. By definition, (N(pT+pT_+) + N(pT+pT_-))/2 is N(pT) we measure. So this gives the asymmetry at the measured pT

//<px> = pT(N(pT+dpT+pT_+) - N(pT+dpT+pT_-))/(N(pT+dpT+pT_+) + N(pT+dpT+pT_-)). I think dpT does affect the measured <px> value. But again, dpT would not matter if we select on jets based on their pT before energy loss.//

---- Its not that we take the difference and divide by a yield along the spectra. N(pT+dpT) is what we measure N(pT) as. So other than a shift on the x-axis, this shouldnt matter 

Let me know if I am missing something

thanks,

Sooraj

On Thu, Sep 12, 2024 at 5:42 PM Ma, Rongrong <marr AT bnl.gov> wrote:

Hello Sooraj

Thanks for the responses. However, it seems like we are viewing <px> is a very different way. 

Let's say we can measure the true jet pT (no detector effects, no background), and we measure <px> at a given pT. Further, let's assume there are only jets along +x and -x directions (py = 0 for all jets). 

<px> = <pTcos(dphi)> = pT<cos(dphi)> = pT(N_+ - N_-)/(N_+ + N_-), where N_+ and N_- are the number of jets along +x and -x directions with pT after energy loss. 

Then we can assume jets traveling along +x direction loss momentum of dpT_+, and those traveling along -x lose dpT_-. Then <px> becomes: <px> = pT(N(pT+pT_+) - N(pT+pT_-))/(N(pT+pT_+) + N(pT+pT_-)), where N(pT+pT_+) and N(pT+pT_-) are the number of jets at pT+pT_+ and pT+pT_- before energy loss. 

If I understand you correctly, you are saying <px> is directly corrected to (pT_+ - pT_-). I agree there is a connection, but the jet spectrum also plays a role here. 

If we could select jets of the same pT BEFORE they lose energy, then <px> is a direct measure of (pT_+ - pT_-). Because in this case:

<px> = <pTcos(dphi)> = [(pT-pT_+)N - (pT-pT_-)N]/2N = (pT_- - pT_+)/2 (because jets are produced without asymmetry, we can assume N_+ = N_- = N)

However, we are selecting jets of the same pT AFTER energy loss. 

Now, if we assume there is a shift in pT between true and reconstructed jets. Then: 

<px> = pT(N(pT+dpT+pT_+) - N(pT+dpT+pT_-))/(N(pT+dpT+pT_+) + N(pT+dpT+pT_-)). I think dpT does affect the measured <px> value. But again, dpT would not matter if we select on jets based on their pT before energy loss. 

Best

Rongrong

On Sep 12, 2024, at 7:09 PM, Sooraj Radhakrishnan <skradhakrishnan AT lbl.gov> wrote:

Hi Rongrong,

   Thanks for follow up thoughts and comments. Please find my responses below 

//If the centroid of the medium is offset by <x>, then the jets traveling along +x direction will see a medium length of R+<x>, while jets traveling along -x direction will see R-<x>. The different is 2<x>. Do I miss something?//

---- But the average difference a jet sees is still <x>. We are also comparing to <px>, a jet along +x will have px - delta px and along -x will have px + delta px, and per jet average would be delta px. This is from the convention how we define flow amplitudes 

//I think I now understand your argument of very narrow pT range, but I still think shifting and smearing in jet pT matter. 

Let's say there are 10 jets with pT = 10 GeV/c, and 6 of them go along +x direction and 4 of them go along -x direction. Then <px> = 20/10 = 2 GeV/c, due to that there are 2 more jets along +x than alone -x direction. Here, <px> reflects the asymmetry of jet yields along +x and -x directions. 

- If all jet pT is shifted by 10%, <px> will also shift by 10%.

- If there is only smearing in jet pT (no shifting), <px> does not change if we can collect all the jets after smearing. However, what smearing does in real data analysis is to move jets in and out of 10 GeV/c bin, so it is not clear that <px> should stay the same.//

---- 'If all jet pT is shifted by 10%, <px> will also shift by 10%.' --> This is not true. <px> from this will be zero as there is no azimuthal preference for the shift. Lets say the true jet pT gets shifted by delta pT. The physics is driven by the true jet pT, <px> could have a dependence on true jet pT (not necessarily a scale factor). The delta pT would be azimuthally symmetric and delta px will average to zero. As long as the x-axis (pT axis) is corrected, we are still measuring the <px> from the underlying physics 

//I think whether we need to correct for <px> depends on how we interpret it. We can view it as a reflection of jet yield asymmetry at a given pT (<px> = pT*v1?), then it is OK not to correct it. But we already have v1 for that. If we want to connect it to energy loss, then I think shifting and smearing matter. //

 

--- The bin migration will matter and this is what we have been evaluating with the pT smearing studies. The impact is small, as the pT dependence of v1 is weak. So we can quote an uncertainty on the measurement from this, or correct for the impact, mainly feed-in from low jet pT and quote a systematic uncertainty on it. The current uncertainties evaluate the impact of both detector effect and background fluctuations. But it is using raw jet spectra and not the unfolded one. We will evaluate with the unfolded jet spectra for the paper 

//Thinking more on this, it is actually not clear if we can interpret <px> as the energy loss difference. As the example above shows, <px> is generated by jet yield asymmetry at a given jet pT bin. It means there are more jets along +x than -x direction. Given that jets traveling along +x direction lose less energy (smaller pathlength) than those along -x direction, so they come from jets with smaller initial energies which naturally have larger yield due to steeply falling jet spectrum. So <px> should also depend on jet spectrum shape. Furthermore, what we measure is <px> for jets in a given pT bin AFTER energy loss. <px> might be related to the energy loss those jets experienced, but the relation is not straightforward to me. Smearing in jet pT moves jets in and out of the bin, making the interpretation even more complicated, I think. //

----- 'Given that jets traveling along +x direction lose less energy (smaller pathlength) than those along -x direction, so they come from jets with smaller initial energies which naturally have larger yield due to steeply falling jet spectrum' -- why do you say so?

Here we are measuring the asymmetry in energy loss from the path length difference, not the overall energy loss. The measured pT is reflecting the average suppression from energy loss from the medium interactions and the v1 is measuring the asymmetry on this along +x and -x. So this still gives access to energy loss from the path length asymmetry. For <px> what we measure is the momentum loss in a given pT bin from the asymmetry. This would have bin edge effects and we would need a pT dependent v1 or <px> measurement to evaluate the energy loss. But these do directly relate to the energy loss induced by the path length asymmetry. We can say <px> is the asymmetry in momentum loss in a given pT bin 

thanks,

Sooraj

On Thu, Sep 12, 2024 at 11:32 AM Ma, Rongrong <marr AT bnl.gov> wrote:

Hello Sooraj

I have some further comments and thoughts inline. 

On Sep 11, 2024, at 5:58 PM, Sooraj Radhakrishnan <skradhakrishnan AT lbl.gov> wrote:

Hi Rongrong,

    Thanks for the follow up

//If the centroid of the medium is offset by <x>, the average path length difference along positive and negative x is 2*<x>, right?//

---- But since we are comparing to <px>, this should cancel out, right? We are looking at the average momentum difference along +x and -x 

If the centroid of the medium is offset by <x>, then the jets traveling along +x direction will see a medium length of R+<x>, while jets traveling along -x direction will see R-<x>. The different is 2<x>. Do I miss something?

//In the D0 v1 paper, it is argued that the large D0 v1 slope is driven by the drag from the titled bulk. Would jets also experience such a drag? If so, that should also contribute to v1, right? Are there any other effects that would induce jet v1? //

----- D0 we measured was at low pT where collisional energy loss is dominant. So the drag from the bulk would be more important there. For the jet pT we are looking at, this should be less important. But is a quantitative question to be taken into account in a detailed model calculation, including other aspects like expansion of medium etc. Thats why we dont want to quote a dE/dL for energy loss in QGP. But I believe the measurements offer an important quantity in getting towards that

I agree that this is a quantitative question. I think it is important to point out that there are other effects that could contribute to jet v1. 

//Still, it is not clear to me how the reconstruction efficiency cancels in <px>. What is the definition of <px>?//

---- What is the efficiency effect you are referring to here? The jet reconstruction efficiency should cancel out as this is averaged over all jets in the pT bin. <px> is the the mean of pTcos(phi -Psi). 

I think I now understand your argument of very narrow pT range, but I still think shifting and smearing in jet pT matter. 

Let's say there are 10 jets with pT = 10 GeV/c, and 6 of them go along +x direction and 4 of them go along -x direction. Then <px> = 20/10 = 2 GeV/c, due to that there are 2 more jets along +x than alone -x direction. Here, <px> reflects the asymmetry of jet yields along +x and -x directions. 

- If all jet pT is shifted by 10%, <px> will also shift by 10%.

- If there is only smearing in jet pT (no shifting), <px> does not change if we can collect all the jets after smearing. However, what smearing does in real data analysis is to move jets in and out of 10 GeV/c bin, so it is not clear that <px> should stay the same.

I think whether we need to correct for <px> depends on how we interpret it. We can view it as a reflection of jet yield asymmetry at a given pT (<px> = pT*v1?), then it is OK not to correct it. But we already have v1 for that. If we want to connect it to energy loss, then I think shifting and smearing matter. 

Thinking more on this, it is actually not clear if we can interpret <px> as the energy loss difference. As the example above shows, <px> is generated by jet yield asymmetry at a given jet pT bin. It means there are more jets along +x than -x direction. Given that jets traveling along +x direction lose less energy (smaller pathlength) than those along -x direction, so they come from jets with smaller initial energies which naturally have larger yield due to steeply falling jet spectrum. So <px> should also depend on jet spectrum shape. Furthermore, what we measure is <px> for jets in a given pT bin AFTER energy loss. <px> might be related to the energy loss those jets experienced, but the relation is not straightforward to me. Smearing in jet pT moves jets in and out of the bin, making the interpretation even more complicated, I think. 

Best

Rongrong

//Regarding the smearing, I am not sure the leading pT bias would reduce the smearing since the smearing is driven by background fluctuation, not jets. //

--- This is what I am confused about a bit. If you look at the rho distributions (Fig 1.10 in the note above), the rms is about 3 - 4 GeV/(c sr). With a jet area of 0.5 for R = 0.4 jets, the background smearing should be around 2 GeV/c, isnt? But the rms of the delta pt distributions for R = 0.4 jets from background smearing (Fig. 1.16) is closer to 6 - 7 GeV/c. What causes this enhanced smearing? 

//When you perform smearing, do you take all the jets with a leading track above 4 GeV/c?//

--- We apply the smearing to all the jets 

 //Also, for the detector effects, this is a significant tail in the response, which I assume is not included in the smearing. With this exercise, is the intent to say that the v1 signal is not induced by smearing or that our v1 results can be directly compared to theoretical calculations?//

---- The impact of smearing is small from the checks we have done. We will check with the increased smearing from background fluctuations and can also directly use the delta pT distribution with the tails to see the impact. If the impact is small, the measurements, with the uncertainties, can be compared to model calculations. Its not just the smearing, but the pT dependence of the signal also what decides the impact here. With the data driven check we could evaluate this and if it is small, can have as part of uncertainties. Else can have as a correction and quote systematics on it

//In any case, I think it is important to stress that these results are not corrected for detector response and background fluctuation, and a biased jet population is used. //

--- We will quote the jet pT as jet pT^raw to state it is uncorrected jet pT. But the detector response and background fluctuations shouldn't be a major factor here, from the checks so far. Also, we see the significant signal after these smearing in the data, they are not washed out. In the final results we might have to quote a correction to account for these effects 

thanks,

Sooraj

On Wed, Sep 11, 2024 at 12:09 PM Ma, Rongrong <marr AT bnl.gov> wrote:

Hello Sooraj

Thanks for the explanation. I understand much better now. 

If the centroid of the medium is offset by <x>, the average path length difference along positive and negative x is 2*<x>, right?

In the D0 v1 paper, it is argued that the large D0 v1 slope is driven by the drag from the titled bulk. Would jets also experience such a drag? If so, that should also contribute to v1, right? Are there any other effects that would induce jet v1? 

Still, it is not clear to me how the reconstruction efficiency cancels in <px>. What is the definition of <px>?

Regarding the smearing, I am not sure the leading pT bias would reduce the smearing since the smearing is driven by background fluctuation, not jets. When you perform smearing, do you take all the jets with a leading track above 4 GeV/c? Also, for the detector effects, this is a significant tail in the response, which I assume is not included in the smearing. With this exercise, is the intent to say that the v1 signal is not induced by smearing or that our v1 results can be directly compared to theoretical calculations? 

In any case, I think it is important to stress that these results are not corrected for detector response and background fluctuation, and a biased jet population is used. 

Thanks. 

Best

Rongrong

On Sep 10, 2024, at 5:51 PM, Sooraj Radhakrishnan <skradhakrishnan AT lbl.gov> wrote:

Hi Rongrong,

   Thanks for the email and the questions. Please find my responses below 

//"Mean momentum loss = 0.232 +/- 0.068 +/- 0.03 for R = 0.2 jets with 10 < pT,jet < 12 GeV/c ... for an estimated initial part length asymmetry of 0.2 fm". 

- 0.232 is the slope of <px> vs. eta. Why is it related to energy loss? Should I think of 0.232 as the average energy loss within the eta range of the measurement (|eta| < 1-R) or at a given eta? 

- Does the initial length asymmetry of 0.2 fm correspond to the slope of <x> vs. eta or <x> at a given eta? //

--- The <px> would be zero if jets along positive and negative impact parameter direction (x) see the same amount of medium. But at finite rapidity, the centroid of the medium along the impact parameter direction is offset, leading to different path lengths for jets along +x and -x. This results in more energy loss in one direction vs the other and <px> reflects this difference in energy loss

   

We use slope to quantify the eta dependence as we see the dependence is linear. Alternatively, we could quote the measured value in the rapidity bin at |eta| = 0.6. That would then give the <px> at a given eta, which could then be related to <x> in the same bin. Taking the slope, with the observed linear dependence, gives these values at eta = 1.0. <x> of 0.2 is also the slope or value at eta = 1.0

//- If both 0.232 GeV/c and 0.2 fm are slopes, do you implicitly assume that energy loss does not depend on eta?//

----- No, this has an eta dependence as we see, largely proportional to <x> at a given eta. In the region we measure, the dependence is linear. But with more precision for differential measurements along eta we could check if this is strictly linear or not. 

//- You mentioned that "The impact of efficiency correction on <px> is also small like in the case of v1 as it is self normalized". What do you mean by self-normalized? I usually think of v1 as the modulation of jet yield w.r.t. the first-order even plane, so it is dimensionless and self-normalized. Should one think of <px> that way? It is not clear to me since <px> is not dimensionless.//

---- Yes, <px> is also modulation relative to the first order event plane. Since its the mean each jet comes in both the numerator and denominator and the reconstruction efficiency cancels out. 

 

//- You also mentioned "The pT windows are narrow, so the impact from pT dependence should also be small". There is a large smearing in jet pT due to background fluctuation. So it is not clear to me why a narrow selection in jet pT would help. I would think the other way around. While it is probably true that the impact of tracking efficiency on <px> is small due to survival bias (it is easier to find jets whose leading particles are not missing), the effect of background smearing is still there, right?//

----- For R=0.2 jets in particular, the background smearing is not so large. You could see for example here (https://drupal.star.bnl.gov/STAR/system/files/STAR_InclJet_AN_Master_v5-5_0.pdf), Fig 1.16. For 0-10% centrality, the width is about 15% for 11 GeV jets. This should be smaller for 10-40% centrality. These are also jets without leading pT selection. We did a smearing with 6% resolution and didnt see impact on the results. We can try increasing this by a factor of 2 to include impact of background smearing as well. 

//- If there is an asymmetry in pathlength, does one need to take into account possible background variation due to the asymmetry? I kind of remember this was looked into for the jet v2 measurement in Isobar.//

----- The bulk v1 is very small, at sub-percent level. For hard probes, it is an offset of the hard production profile and bulk distribution than variation of bulk density in azimuth. This also means that expansion from bulk v1 is also very small, unlike the case of v2. This should make it easier to evaluate the impact of medium expansion on path length and interactions as well

Let me know if you have further questions or comments

thanks,

Sooraj

On Tue, Sep 10, 2024 at 1:37 PM Ma, Rongrong <marr AT bnl.gov> wrote:

Hello Sooraj

I still have a few questions regarding to this statement "Mean momentum loss = 0.232 +/- 0.068 +/- 0.03 for R = 0.2 jets with 10 < pT,jet < 12 GeV/c ... for an estimated initial part length asymmetry of 0.2 fm". 

- 0.232 is the slope of <px> vs. eta. Why is it related to energy loss? Should I think of 0.232 as the average energy loss within the eta range of the measurement (|eta| < 1-R) or at a given eta? 

- Does the initial length asymmetry of 0.2 fm correspond to the slope of <x> vs. eta or <x> at a given eta? 

- If both 0.232 GeV/c and 0.2 fm are slopes, do you implicitly assume that energy loss does not depend on eta?

- You mentioned that "The impact of efficiency correction on <px> is also small like in the case of v1 as it is self normalized". What do you mean by self-normalized? I usually think of v1 as the modulation of jet yield w.r.t. the first-order even plane, so it is dimensionless and self-normalized. Should one think of <px> that way? It is not clear to me since <px> is not dimensionless. 

- You also mentioned "The pT windows are narrow, so the impact from pT dependence should also be small". There is a large smearing in jet pT due to background fluctuation. So it is not clear to me why a narrow selection in jet pT would help. I would think the other way around. While it is probably true that the impact of tracking efficiency on <px> is small due to survival bias (it is easier to find jets whose leading particles are not missing), the effect of background smearing is still there, right?

- If there is an asymmetry in pathlength, does one need to take into account possible background variation due to the asymmetry? I kind of remember this was looked into for the jet v2 measurement in Isobar. 

Best

Rongrong

On Sep 9, 2024, at 3:04 AM, Sooraj Radhakrishnan <skradhakrishnan AT lbl.gov> wrote:

Dear All,

   Please find the Preliminary request slides for the jet v1, <px> measurements here https://drupal.star.bnl.gov/STAR/system/files/JetV1PreliminaryRequest.pdf

thanks,

Sooraj

--

Sooraj Radhakrishnan

Research Scientist,

Department of Physics

Kent State University

Kent, OH 44242

Physicist Postdoctoral Affiliate

Nuclear Science Division
Lawrence Berkeley National Lab
MS70R0319, One Cyclotron Road

Berkeley, CA 94720

Ph: 510-495-2473

Email: skradhakrishnan AT lbl.gov

--

Sooraj Radhakrishnan

Research Scientist,

Department of Physics

Kent State University

Kent, OH 44242

Physicist Postdoctoral Affiliate

Nuclear Science Division
Lawrence Berkeley National Lab
MS70R0319, One Cyclotron Road

Berkeley, CA 94720

Ph: 510-495-2473

Email: skradhakrishnan AT lbl.gov

--

Sooraj Radhakrishnan

Research Scientist,

Department of Physics

Kent State University

Kent, OH 44242

Physicist Postdoctoral Affiliate

Nuclear Science Division
Lawrence Berkeley National Lab
MS70R0319, One Cyclotron Road

Berkeley, CA 94720

Ph: 510-495-2473

Email: skradhakrishnan AT lbl.gov

--

Sooraj Radhakrishnan

Research Scientist,

Department of Physics

Kent State University

Kent, OH 44242

Physicist Postdoctoral Affiliate

Nuclear Science Division
Lawrence Berkeley National Lab
MS70R0319, One Cyclotron Road

Berkeley, CA 94720

Ph: 510-495-2473

Email: skradhakrishnan AT lbl.gov

--

Sooraj Radhakrishnan

Research Scientist,

Department of Physics

Kent State University

Kent, OH 44242

Physicist Postdoctoral Affiliate

Nuclear Science Division
Lawrence Berkeley National Lab
MS70R0319, One Cyclotron Road

Berkeley, CA 94720

Ph: 510-495-2473

Email: skradhakrishnan AT lbl.gov

<check.pdf>

--

Sooraj Radhakrishnan

Research Scientist,

Department of Physics

Kent State University

Kent, OH 44242

Physicist Postdoctoral Affiliate

Nuclear Science Division
Lawrence Berkeley National Lab
MS70R0319, One Cyclotron Road

Berkeley, CA 94720

Ph: 510-495-2473

Email: skradhakrishnan AT lbl.gov