Searching for odderon in exclusive reactions: $p p \to p p p {\bar p}$, $p p \to p p \phi \phi$ and $p p \to p p \phi$

TL;DR

This paper explores the potential of three exclusive proton-proton reactions to detect odderon exchange, analyzing past data and predicting outcomes at the LHC to clarify the existence of C=-1 exchanges.

Contribution

It introduces a formalism for analyzing three specific exclusive reactions as probes for odderon exchange and provides predictions for LHC experiments based on past measurements.

Findings

Evidence for C=-1 exchanges at high energies.

Predictions for odderon signals at the LHC.

Analysis of past WA102 data for future experiments.

Abstract

There seem to be recently an evidence for $C =$ -1 exchanges in $p p$ and $p \bar p$ elastic scattering at high energies. The analysis there is difficult as the two processes were not measured at the same (large) energies. Here we discuss three different exclusive processes given in the title as a possible source of information for odderon exchange. A sketch of the formalism is presented for each of the reactions. We consider low energy processes measured in the past by the WA102 collaboration and try to make predictions for the LHC. We discuss possible evidences at the low energies and try to make suggestions for the LHC.

Figures

Equations

$$i{\Gamma }_{\mu \nu \kappa \lambda }^{(f_2\varphi \varphi )}(p_3,p_4)$$

$${\Delta }_{{\rho }_1{\rho }_2}^{(\varphi )}(\hat{p})\to {\Delta }_{{\rho }_1{\rho }_2}^{(\varphi )}(\hat{p}){\left(\exp (i\varphi (s_{34}))\frac{s_{34}}{s_0}\right)}^{{\alpha }_{\varphi }({\hat{p}}^2)-1},$$

$$\varphi (s_{34})=\frac{\pi }{2}\exp \left(\frac{s_0-s_{34}}{s_0}\right)-\frac{\pi }{2}$$

$$i{\Delta }_{\mu \nu }^{(\mathbb{O})}(s,t)=-i{g}_{\mu \nu }\frac{{\eta }_{\mathbb{O}}}{M_0^2}(s{\alpha }_{\mathbb{O}}^{\prime }{)}^{{\alpha }_{\mathbb{O}}(t)-1},$$

$${\alpha }_{\mathbb{O}}(t)={\alpha }_{\mathbb{O}}(0)+{\alpha }_{\mathbb{O}}^{\prime }t,$$

$${\eta }_{\mathbb{O}}=-1,{\alpha }_{\mathbb{O}}(0)=1.05,{\alpha }_{\mathbb{O}}^{\prime }=0.25{\mathrm{GeV}}^{-2}.$$

$$i{\Gamma }_{\mu \nu \kappa \lambda }^{(\mathbb{PO}\varphi )}(k^{\prime },k)=i{F}^{(\mathbb{PO}\varphi )}((k+k^{\prime }{)}^2,k^{\prime 2},k^2)\left[2a_{\mathbb{PO}\varphi }{\Gamma }_{\mu \nu \kappa \lambda }^{(0)}(k^{\prime },k)-b_{\mathbb{PO}\varphi }{\Gamma }_{\mu \nu \kappa \lambda }^{(2)}(k^{\prime },k)\right].$$

$${\tilde{A}}^{(2)}(\eta ,{\eta }^{\prime })=\frac{\frac{d^2\sigma }{d{\eta }_{3}d{\eta }_4}(\eta ,{\eta }^{\prime })-\frac{d^2\sigma }{d{\eta }_{3}d{\eta }_4}({\eta }^{\prime },\eta )}{\frac{d^2\sigma }{d{\eta }_{3}d{\eta }_4}(\eta ,{\eta }^{\prime })+\frac{d^2\sigma }{d{\eta }_{3}d{\eta }_4}({\eta }^{\prime },\eta )}.$$

Full text

1 Introduction

The odderon exchange became recently topical again. So far there is no unambiguous evidence for the odderon – the $C=-1$ partner of the pomeron [1]. For a theoretical review of the odderon see e.g. [2]. Recent analysis of the elastic scattering by the TOTEM collaboration [3, 4] allow for the (soft) odderon exchange interpretation of the data [5].

In this talk we discussed three processes mentioned in the title [6, 7, 8]. We try te estimate upper limit of the size of odderon exchanges in these exclusive processes.

2 A sketch of formalism

We discuss shortly the three considered here reactions.

2.1 $pp\to pp\phi\phi$

In Fig.LABEL:fig:diagrams_phiphi we show conventional processes leading to the exclusive production of two $\phi$ mesons.

Let us sketch the formalism to descibe the conventional processes.

For the $f_{2}\phi\phi$ vertex we take the following ansatz, in analogy to the $f_{2}\gamma\gamma$ vertex:

[TABLE]

with $M_{0}=1$ GeV and dimensionless coupling constants $g^{\prime}_{f_{2}\phi\phi}$ and $g^{\prime\prime}_{f_{2}\phi\phi}$ being free parameters. Two free couplings are allowed in general.

The propagator for $\phi$ exchange in diagram (a) must be modified (compared to the corresponding one for meson exchange) at higher $\phi\phi$ subsytem energies. In [6] we proposed the following parametrization:

[TABLE]

where we take $s_{0}=4m_{\phi}^{2}$ and $\alpha_{\phi}(\hat{p}^{2})=\alpha_{\phi}(0)+\alpha^{\prime}_{\phi}\,\hat{p}^{2}$ with $\alpha_{\phi}(0)=0.1$ from [10] and $\alpha^{\prime}_{\phi}=0.9$ GeV*-2*.

The different considered processes are added coherently and therefore interfere. In order to have a correct phase behaviour we introduced the function $\exp(i\phi(s_{34}))$ with

[TABLE]

which role is to interpolate between meson physics close to the $\phi\phi$ threshold, $s_{34}=4m_{\phi}^{2}$, and Regge physics at high energies. Another prescription was considered in [6] in addition.

In addition to the processes discussed above we considered processes shown in Fig.2. We think that the contribution of the diagram with two odderon exchanges is much smaller than that for the diagram with one odderon exchange. Therefore we include only contribution of the diagram with odderon exchange in the middle of the left diagram.

Our ansatz for the effective propagator of $C=-1$ odderon follows [9]

[TABLE]

where we have $M_{0}^{-2}=1$ (GeV)-2 for dimensional reasons. Furthermore, we shall assume representative values for the odderon parameters

[TABLE]

For the $\mathbb{P}\mathbb{O}\phi$ vertex we use an ansatz analogous to the $\mathbb{P}\rho\rho$ vertex. We get then, orienting the momenta of the $\mathbb{O}$ and the $\phi$ outwards, the following formula:

[TABLE]

2.2 $pp\to pp\phi$

For single $\phi$ production we include processes shown in Fig.3.

The Odderon exchange contributions (lower row) may modify the photon-exchange contribution (upper low). More processes will be considered in [6].

2.3 $pp\to ppp\bar{p}$

Several mechanisms of central $p\bar{p}$ production were discussed in [7] and the formalism how to calculate relevant diffractive processes (continuum and resonances) was given there.

The exchange of $C=$ -1 objects leads to specific asymmetries discussed in [7]. In two dimensions (e.g. $\eta_{1},\eta_{2}$) we can define the asymmetry:

[TABLE]

As discussed in [7] the $C=$-1 reggeon exchanges lead to such asymmetries. The odderon exchange also leads to asymmetries. How big are such asymmetries was discussed in detail in [7].

3 Results

It is very difficult to describe the WA102 data for $pp\to pp\phi\phi$ reaction including resonances and $\mathbb{P}\mathbb{P}$ mechanism only [6]. Inclusion of the odderon exchange improves the description of the data. The result of our analysis is shown in Fig.4.

Having fixed the parameters of our quasi fit to the WA102 data we wish to show our predictions for the LHC. In Fig. 5 we show the results for the ATLAS experimental conditions ($|\eta_{K}|<2.5$, $p_{t,K}>0.2$ GeV). The distribution in four-kaon invariant mass is shown in the left panels and the difference in rapidity between the two $\phi$ mesons in the right panels.

In Fig. 6 we show more complete result including the odderon exchange with $\eta_{\mathbb{O}}=-1$ and various values of the odderon intercept $\alpha_{\mathbb{O}}(0)$.

The measurement of large $M_{\phi\phi}$ or $Y_{diff}$ events at the LHC would therefore suggest presence of the odderon exchange.

We will not present here (in the written version) preliminary results for the $pp\to pp\phi$ reaction. Some preliminary results were presented in the talk and details will be presented soon in [8]. We wish to mention only here that the inclusion of $\mathbb{P}\mathbb{O}$ fusion (see Fig.3) leads to a sizeable improvement of the description of the rather old WA102 data [12] for this reaction.

Now we go to the $pp\to ppp{\bar{p}}$ reaction. In Fig.7 we show estimated, allowed at present, asymmetry defined in detail in [7] for two different kinematical conditions specified in the figure description. Rather small asymmetries are allowed. A measurement of such asymmetries would require gigantic statistics for the $pp\to ppp{\bar{p}}$ reaction and may be difficult to reach in planned experiments.

4 Conclusions

Our results/presentation can be summarized in the following way:

• •

The Regge phenomenology was extended to $2\to 3$, $2\to 4$ and $2\to 6$ exclusive processes.

• •

The tensor pomeron/reggeon model was applied in all these reactions.

• •

At lower energies tensor/vector reggeon exchanges must be included.

• •

Three reactions ($pp\to pp\phi\phi$, $pp\to pp\phi$ and $pp\to ppp\bar{p}$ have been studied in the context of identifying odderon exchange.

• •

$pp\to pp\phi\phi$ seems promissing as here the odderon does not couple to protons. An upper limit for the odderon exchange has been etablished based on the WA102 data.

• •

This upper limit for the $\mathbb{P}\mathbb{O}\to\phi$ coupling was used for the $pp\to pp\phi$ reaction, together with the TOTEM estimate. The WA102 data for single $\phi$ production support existence of odderon exchange (not shown in the written version).

• •

Special asymmetries for centrally produced $p\bar{p}$ system have been proposed to identify $C=-1$ exchanges. The asymmetry caused by subleading reggeon exchanges is probably considerably larger than that for the odderon exchange which makes this reaction difficult in searches for odderon exchanges.