Four-momentum transfers study in simulations of %$pp\to p\Delta^+\to e^+e^-$% reaction

We are interested in the %$\Delta$% Dalitz decay process in pp reactions: %$pp \to p\Delta^+ \to e^+e-$%. To distinguish the two protons in the final state, we will call the one coming from the decay of Delta resonance the "decay" proton and the other one the " scattered " proton .

The %$pp\to p\Delta^+$% reaction is a very peripheral reaction resulting from two graphs. Graph 1 (direct graph) gives a forward peaked proton angular distribution in the center of mass, taking the projectile proton momentum as reference axis and graph 2 (exchange graph) a backward peaked distribution.

graph 1: direct diagram for pp->pDelta:	graph 2: exchange diagram for pp->pDelta:	momentum transfers definition
		%$t_{scatt}=(p_{proj}-p_{scatt})^2 $% %$t_{decay}=(p_{proj}-p_{decay})^2 $% u%$_{scatt}$%=%$(p_{cib}-p_{scatt})^2 $% %$u_{decay}=(p_{cib}-p_{decay})^2 $%

The resulting symmetrical angular distribution (taking also interference of the two graphs into account) as described in Pluto is displayed in fig.1 in the case of pp at 1.25 GeV. The four-momentum transfer %$t_{scatt}$% defined by %$(p_{proj}-p_{scatt})^2 $% (squared of projectile and scattered proton four-momentum transfers difference ) is strongly correlated to the scattered proton angle in cm (fig.2). The width of the correlation is due to the Delta width. %$t_{scatt}$% presents a very steep distribution (fig.3) corresponding to the direct term (forward center of mass angles) and a broader bump at large momentum transfers corresponding to the exchange term (backward center of mass angles). In the backward region, the large %$t_{scatt}$% four-momentum transfers correspond to small %$u_{scatt}$% four-momentum transfers, %$u_{scatt}$% being defined by %$(p_{target}-p_{scatt})^2$%.
E=1.25 GeV:
fig1:cos(%$\theta$%_scatt_cm)             fig.2:cos(%$\theta$%_scatt_cm)%%$t_{scatt}$%       fig.3:%$t_{scatt}$%

The %$pp\to p\Delta^+$% reaction is therefore dominated by either small t%$_{scatt}$% or small u%$_{scatt}$% four-momentum transfers. The suggestion was made at the last analysis meeting to use this caracteristic to distinguish scattered and decay protons, since the four-momentum transfers t%$_{decay}$% and u%$_{decay}$% calculated with the decay proton are not expected to present the same very steep distributions. Practically, once Dalitz decay events are selected, it would consist in calculating the four possible momentum transfers using the two protons in the exit channel (either detected or reconstructed by momentum conservation) and the two protons in the entrance channel (last column in table 1). The proton which produces the smallest of the four values could be identified to the scattered proton. This method had been used in some analysis of pp reactions at 5.5 GeV/c (4.64 GeV) and 6.6 GeV/c (5.73 GeV). We applied this criterium to events generated in PLUTO simulations at 1.25 GeV and 2.2 GeV and checked the efficiency of the method (rate of well identified scattered protons using the minimum for-momentum criterium). The results are shown in table 2, both whith 4%$\pi$% acceptance and with HADES acceptance ( detection of at least one proton and one %$(e^+,e^-)$% pair with opening angle larger than 4).

table 2: rate of well identified scattered protons using the minimum four-momentum transfer difference:

energy	4%$\pi$% acceptance	HADES acceptance
1.25 GeV	70%	63 %
2.2 GeV	76%	63 %

In order to explain these results, we first plot for scattered protons emitted in the forward hemisphere, the four-momentum transfer u%$_{decay}$% calculated with the decay proton as a function of the cm angle of the scattered proton. The correlation is much broader than for the "real" four-momentum transfer t%$_{scatt}$% calculated with the scattered proton, due to the momentum taken by the dilepton pair. In the backward hemisphere, the same trend could be obtained for respectively t%$_{decay}$% and u%$_{scatt}$%. In the following pictures, we consider the difference mintrscatt-mintrdecay between the minimum of the two transfers t%$_{scatt}$% and u%$_{scatt}$% calculated with the scattered proton and the corresponding one for the decay proton. The "minimum transfer" criterium is fullfilled when the scattered proton leads to the minimum transfer, i.e. when the difference mintrscatt-mintrdecay is negative.
This difference is in average negative, but there is a sizeable contribution at positive values, in agreement with the numbers in table 1. From the third row of pictures, it is seen that the difference is strictly negative at scattering angles close to zero and goes to positive values when the angle increases. The difference of transfers is also correlated to the angle of decay proton in the Delta reference frame (fourth row), which is taken isotropic at the moment in the PLUTO simulation. The direction to define backward and forward emission is taken as the 3-momentum transfer of the N%$\to\Delta$% transition calculated in the pp cm. The positive values of the difference correspond to decay protons emitted backwards in this frame. This trend results from the definition of the different four-momentum transfers and the kinematics of scattering and decay graphs. In the expression of mintrscatt-mintrdecay, the decay angle dependent term is indeed proportionnal to cos(%$\theta$%_scatt_cm}). abs(cos(%$\theta$%_decay_%$\Delta$% ref frame)). It vanishes for decay angles equal to 90 where mintrscatt-mintrdecay has an almost symmetrical distribution around negative values and where the contribution of positive values is very small. This term takes positive (respectively negative) values for backward (respectively forward) decay angles. At a given decay angle, the most probable value corresponds to scattering angles close to 0 or 180 and the asymetric tails can be explained by the steep scattering angle distribution. This behaviour yields the two peaks in the integrated mintrscatt-mintrdecay distribution. As a summary, the positive values of mintrscatt-mintrdecay are therefore due to the backward emission in the Delta reference frame and to scattering angles far from 0 or 180.
The second column shows the same plots when the HADES acceptance and the 4 opening angle cut are applied. Both the detected proton and the reconstructed missed proton are taken into account. The acceptance cuts reduce scattering angles around 180 and reduce more drastically forward than backward emission in the %$\Delta$% reference frame. This results in a larger proportion of events with positive four-momentum difference mintrscatt-mintrdecay, when acceptance cuts are applied.
At 2.2 GeV, the same trend is observed, but with a steeper dependance on scattering angle. This explains that the amount of events with mintrscatt-mintrdecay positive is lower than at 1.25 GeV, when acceptance cuts are not applied. The effect of acceptance cuts is more drastic than at 1.25 GeV in such a way that the amount of events with positive mintrscatt-mintrdecay is the same than at 1.25 GeV when acceptance cuts are applied.

t%$_{scatter}$%-u%$_{decay}$% as a function of the scattered proton angle in the center of mass and the integrated t%$_{scatter}$%-u%$_{decay}$% distributions. When HADES acceptance is included, the plot includes both detected protons and "missed" protons reconstructed using four-momentum conservation.

E=1.25 GeV
*                             4%$\pi$%  acceptance                                                                HADES acceptance
cos(%$\theta$%_scatt_cm)%(tscatt-tdecay)                (tscatt-tdecay)
| | |
E=2.2 GeV scattered proton forward in cm:
*                             4%$\pi$%  acceptance                                                                HADES acceptance
cos(%$\theta$%_scatt_cm)%(tscatt-tdecay)                  (tscatt-tdecay)
| ||
It shows that at forward angles, for most of the events the difference is negative, while when the angle increases, the distribution of the difference is broad and stretches over positive values. The criterium is therefore valid only at very small angles. As HADES acceptance cuts the smallest proton angles, the efficiency of the criterium is further reduced. At 2.2 GeV, the t%$_{scatt}$% distribution of the %$pp\to p\Delta^+$% reaction is steeper and the criterium works better, although the same effect of HADES acceptance is seen. Its capacity to identify the scattered proton seems however too low to be useful. -- BeatriceRamstein - 01 Jun 2007

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