Extracting the pomeron-pomeron- coupling in the reaction through angular distributions of the pions
We discuss how to extract the pomeron-pomeron- () coupling within the tensor-pomeron model. The general coupling is a combination of seven basic couplings (tensorial structures). To study these tensorial structures we propose to measure the central-exclusive production of a pair in the invariant mass region of the . An analysis of angular distributions in the rest system, using the Collins-Soper (CS) frame, turns out to be particularly relevant for our purpose. Both, and distributions are discussed. We find that the azimuthal angle distributions are quite sensitive to the choice of coupling. One of the couplings leads to four-oscillations in compared to two-oscillations for the six other tensorial couplings. We show results for the resonance case alone as well as when the dipion continuum is included. We show the influence of the experimental cuts on the angular distributions in the context of dedicated experimental studies at RHIC and LHC energies. We discuss also how the absorption effects modify our results.
The pomeron is an essential object for understanding diffractive phenomena in high-energy physics. Within QCD the pomeron is a color singlet, predominantly gluonic, object. The spin structure, of the pomeron, in particular its coupling to hadrons, is, however, not yet a matter of consensus. In the tensor-pomeron model for soft high-energy scattering formulated in Ewerz:2013kda () the pomeron exchange is effectively treated as the exchange of a rank-2 symmetric tensor. The diffractive amplitude for a given process with soft pomeron exchange can then be formulated in terms of effective propagators and vertices respecting the rules of quantum field theory.
It is rather difficult to obtain definitive statements on the spin structure of the pomeron from unpolarised elastic proton-proton scattering. On the other hand, the results from polarised proton-proton scattering by the STAR collaboration Adamczyk:2012kn () provide valuable information on this question. Three hypotheses for the spin structure of the pomeron, tensor, vector, and scalar, were discussed in Ewerz:2016onn () in view of the experimental results from Adamczyk:2012kn (). Only the tensor-ansatz for the pomeron was found to be compatible with the experiment. Also some historical remarks on different views of the pomeron were made in Ewerz:2016onn ().
In the last few years we have undertaken a scientific program to analyse the production of light mesons in the tensor-pomeron model in exclusive processes Lebiedowicz:2013ika (), Lebiedowicz:2014bea (); Lebiedowicz:2016ioh (), () Lebiedowicz:2016ryp (), Lebiedowicz:2018eui (), Lebiedowicz:2016zka (), and in the reaction Lebiedowicz:2018sdt (). Some azimuthal angle correlations between the outgoing protons can verify the couplings for scalar , , , and pseudoscalar , mesons Lebiedowicz:2013ika (); Lebiedowicz:2018eui (). The couplings being of nonperturbative nature are difficult to obtain from first principles of QCD. The corresponding coupling constants were fitted to differential distributions of the WA102 Collaboration Barberis:1998ax (); Barberis:1999cq (); Barberis:1999zh () and to the results of Kirk:2000ws (). As was shown in Lebiedowicz:2013ika (); Lebiedowicz:2018eui (), the tensorial , , and vertices correspond to the sum of two lowest orbital angular momentum - spin couplings, except for the meson. The tensor meson case is a bit complicated as there are seven (!) possible pomeron-pomeron- couplings in principle; see the list of possible couplings in Appedix A of Lebiedowicz:2016ioh ().
It was shown in Barberis:1996iq (); Barberis:1999cq () that the cross section for the undisputed tensor mesons, , , peaks at and is suppressed at small 222Here, is the azimuthal angle between the transverse momentum vectors of the outgoing protons and (the so-called ’glueball-filter variable’ Close:1997pj ()) is defined by their difference , . in contrast to the tensor glueball candidate ; see e.g. Barberis:2000em (). In Lebiedowicz:2016ioh () we gave some arguments from studying the and distributions that the coupling may be preferred. We roughly reproduced the experimental data obtained by the WA102 Collaboration Barberis:1999cq () and by the ABCDHW Collaboration Breakstone:1990at () with this coupling. It was demonstrated in Lebiedowicz:2016ioh () that the relative contribution of resonant and dipion continuum strongly depends on the cut on four-momentum transfer squared in a given experiment.
Now, we ask the question whether the and couplings can be studied in central-exclusive processes. In the present letter we discuss such a possibility: analysis of angular distributions of pions from the decay of , respectively , in the Collins-Soper (CS) system of reference. As a first example we will consider diffractive production of the resonance which is expected to be abundantly produced in the reaction; see e.g. Lebiedowicz:2016ioh (). We will try to analyse whether such a study could shed light on the couplings. We refer the reader to Aaltonen:2015uva (); Khachatryan:2017xsi (); Sikora:2018cyk () for the latest measurements of central production in high-energy proton-(anti)proton collisions. In the future the corresponding couplings could be adjusted by comparison with precise experimental data from both RHIC and the LHC.
We study central exclusive production of in proton-proton collisions at high energies
where , and , denote the four-momenta and helicities of the protons, and denote the four-momenta of the charged pions, respectively.
We should take into account two main processes shown by the diagrams in Fig. 1. For the resonance (the diagram (a)) we consider only the fusion. The secondary reggeons , , , should give small contributions at high energies. We also neglect contributions involving the photon. In the case of the non-resonant continuum (the diagrams (b)) we include in the calculations both and exchanges. For an extensive discussion we refer to Lebiedowicz:2014bea (); Lebiedowicz:2016ioh ().
The kinematic variables for the reaction (1) are
The -exchange (Born-level) amplitude for production via the tensor -meson () exchange can be written as
Here and denote the effective propagator and proton vertex function, respectively, for the tensor-pomeron exchange. For the explicit expressions, see Sect. 3 of Ewerz:2013kda (). and denote the tensor-meson propagator with a simple Breit-Wigner form (see (4.18) of Lebiedowicz:2016ioh ()) and the vertex (see Eqs. (3.37), (3.38) and Sect. 5.1 of Ewerz:2013kda ()), respectively. For a more detailed analysis we should use a model for the propagator considered in Eqs. (3.6) - (3.8) and (5.19) - (5.22) of Ewerz:2013kda (). More details related to the amplitude (3) are given in Lebiedowicz:2016ioh (). In practice we work with the amplitudes in the high-energy approximation; see Eqs. (3.19) - (3.21) and (4.23) of Lebiedowicz:2016ioh ().
The main ingredient of the amplitude (3) is the pomeron-pomeron- vertex
Here is a form factor for which we make a factorised ansatz (see (4.17) of Lebiedowicz:2016ioh ())
We are taking here the same form factor for each vertex with index (). In principle, we could take a different form factor for each vertex. We take
In (8) to (14) GeV and the are dimensionless coupling constants. The values of the coupling constants () are not known and are not easy to be found from first principles of QCD, as they are of nonperturbative origin. At the present stage these coupling constants should be fitted to experimental data.
Considering the fictitious reaction of two “real tensor pomerons” annihilating to the meson, see Appendix A of Lebiedowicz:2016ioh (), we find that we can associate the couplings (8) - (14) with the following values 333Here, and denote orbital angular momentum and total spin of two fictitious “real pomerons” in the rest system of the meson, respectively. , , , , , , , respectively.
We are interested in the angular distribution of the in the centre-of-mass system of the pair. Various reference systems are commonly used; see e.g. Bolz:2014mya () for a discussion of such systems for the reaction. We choose the Collins-Soper basis Collins:1977iv (); Soper:1982wc () for the reaction (1) with unit vectors defining the axes as follows:
Here , , where , are the three-momenta of the initial protons in the rest system. There we have and . Now we denote by and the polar and azimuthal angles of (the meson momentum) relative to the coordinate axes (16). We have
Having defined these angles we can now examine the differential cross sections , , and .
As discussed in the introduction, very good observables which can be used for visualizing the role of the couplings, given by Eqs. (8) - (14) (cf. also Appendix A of Lebiedowicz:2016ioh ()) could be the differential cross sections and . In Figs. 2 - 6 we show such angular distributions for the meson in the rest frame.
In Fig. 2 we show angular distributions for all (seven) independent couplings for TeV, GeV and for two different cuts on pseudorapidities of pions (the top panels), (the bottom panels), that will be measured in the LHC experiments. In Fig. 3 we show results for the STAR experimental conditions with extra cuts on the leading protons from Sikora:2018cyk () specified in the caption of the figure. Quite different distributions are obtained for different couplings. Note that the shape of the angular distributions depends on the coverage in . From the left top panel in Fig. 2 we see that the condition leads to a reduction of the cross sections mostly at compared to the results with shown in the left bottom panel. To our surprise, particularly interesting are the distributions in azimuthal angle in the CS system. The distributions for the resonance contribution alone can be approximated as
for (as will be shown below), where and depend on experimental conditions. For most of the couplings but for the coupling it is . The reader is asked to note the different number of oscillations for the coupling. The shape of distributions depends also on the cuts on . Therefore, we expect these differences to be better visible when one compares the results related to different regions of pion pseudorapidity. Let us note that the LHCb Collaboration can measure production for and GeV.
In Fig. 4 we show the two-dimensional distributions in () for TeV and . We can observe interesting structures for the reaction. We show results for the individual coupling terms and for the continuum production. Different tensorial couplings generate very different patterns which should be checked experimentally.
In Fig. 5 we show the two-dimensional distributions in () for TeV and , GeV. In the panel (a) we show the result for the continuum term, in the panel (b) for the term, and in the panels (c) and (d) for their coherent sum. Adding the amplitudes of the resonant and continuum terms changes a bit the oscillations; see also the right panels in Fig. 6. We can see that the complete result indicates an interference effect of the continuum and the term as is, of course, expected. The interference effect depends crucially on the choice of the coupling. Here we examine only the coupling. In the panel (c) we show the result with a simple Breit-Wigner ansatz for the propagator, but in the panel (d) we show the result for the model of the propagator considered in Ewerz:2013kda (). Different interference patterns can be seen in the panels (c) and (d) since the highest local maxima are in different positions. Comparing the results of panel (b) to those of panels (c) and (d) we see that the differences in position of the local maxima are a measure of the relative strength of the resonant and background terms. This is not necessarily easy to fix when studying e.g. only the distribution. Combined analysis of the and angular distributions would, therefore, help to pin down the underlying reaction mechanism.
Finally we discuss whether the absorption effects 444For the details how to include the absorption effects (the -rescattering corrections) in the eikonal approximation see e.g. Sect. 3.3 of Lebiedowicz:2014bea (). may change the angular distributions discussed so far in the Born approximation. In the top panels of Fig. 6 we show results in the Born approximation for the and the continuum terms separately and for their coherent sum. The interference term of with the continuum is also shown. In the bottom panels, we show results when absorption effects are included in calculation. In general, a different size of absorption effects may occur for the resonant terms. One can see that absorption effects lead to a significant reduction of the cross section. However, the shapes of the polar and azimuthal angle distributions are practically not changed. This indicates that the absorption effects should not disturb the determination of the type of the coupling. Absorption may, however, influence the extraction of the strengths of the couplings.
In the present letter we have concentrated on a possibility to extract the couplings from the analysis of pion angular distributions in the Collins-Soper system of reference. We have shown that the shape of such distributions depends on the choice of coupling. In particular, we have shown that the azimuthal angle distributions may have different numbers of oscillations. The corresponding distributions can be approximately represented by the formula (18): , where . Two-dimensional distributions in () and () will give even more information and could also be useful in understanding the role of experimental cuts. Can such distributions be used to fix the coupling ? It is too early to answer this question now. The answer will require dedicated experimental studies by the COMPASS, STAR, ALICE, ATLAS, CMS and LHCb Collaborations.
In the diffractive process considered the resonance cannot be completely isolated from the continuum background as the corresponding amplitudes interfere strongly Lebiedowicz:2016ioh (). We have discussed how the interference of the resonance and the continuum background may change the angular distributions and . We have shown that the absorption effects change the overall normalization of such distributions but leave the shape essentially unchanged. This is in contrast to the distributions where absorption effects considerably modify the corresponding shapes; see e.g. Lebiedowicz:2015eka (); Lebiedowicz:2016zka ().
In the present analysis we have concentrated on the pronounced resonance, clearly seen in the channel. We have discussed methods how to pin down the pomeron-pomeron- coupling. The analysis presented may be extended also to other resonances seen in different final state channels. We strongly encourage experimental groups to start such analyses. We think that this will bring in a new tool for analysing exclusive diffractive processes and will provide new inspirations in searching for more exotic states such as glueballs, for instance. The exclusive diffractive processes were always claimed to be a good area to learn about the physics of glueballs. The extension of our methods to the production of glueballs, to be identified in suitable decay channels, should shed light on the pomeron-pomeron-glueball couplings. These represent very interesting quantities: the coupling of three (mainly) gluonic objects.
Acknowledgements.This research was partially supported by the Polish National Science Centre Grant No. DEC-2014/15/B/ST2/02528 and by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge in Rzeszów.
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