On the kinematic limit of the charm production in fixed-target experiments with the intrinsic charm from the target

TL;DR

This paper derives the kinematic limit for charm production in fixed-target experiments with intrinsic charm and discusses the first measurement of charm quark production at the LHC in this setup.

Contribution

It provides a detailed derivation of the kinematic limit and analyzes the first experimental measurement of charm production in fixed-target conditions at the LHC.

Findings

Derived the kinematic limit for charm production with intrinsic charm.

Discussed the first measurement of charm production in fixed-target mode at the LHC.

Provided insights into the feasibility of charm detection in fixed-target experiments.

Abstract

In this note we provide a detailed derivation of the kinematic limit of the charm production in fixed-target experiments with the intrinsic charm coming from the target. In addition, we discuss the first measurement of charm quark production in the fixed-target configuration at the LHC.

Equations

$$x\simeq \frac{2m_c}{\sqrt{s_{NN}}}\exp (-y^{\ast }),$$

$$s=(p_P+p_T{)}^2=p_P^2+2p_P{p}_T+p_T^2=m_P^2+2p_P{p}_T+m_T^2$$

$$s=m_P^2+m_T^2+2E_P^{\mathrm{lab}}{m}_T$$

$$\sqrt{m_P^2+\vec{p}{}^2}+\sqrt{m_T^2+\vec{p}{}^2}=\sqrt{s},$$

$$2\sqrt{m_P^2+\vec{p}{}^2}\sqrt{m_T^2+\vec{p}{}^2}$$

$$4(m_P^2+\vec{p}{}^2)(m_T^2+\vec{p}{}^2)$$

$$4\vec{p}{}^4+4\vec{p}{}^2(m_P^2+m_T^2)+4m_P^2{m}_T^2$$

$$4s\vec{p}{}^2$$

$$\vec{p}{}^2$$

$$\left(\Lambda^{\mu}{}_{\nu}(E_{T};-\vec{p}\,)\right)=\Lambda(p_{T})=\pmatrix{E_{T}/m_{T}&\vec{p}\,^{T}/m_{T}\cr\vec{p}/m_{T}&\hbox{\rm 1\kern-3.0ptl}+\vec{p}\vec{p}\,^{T}/m_{T}/(E_{T}+m_{T})\cr}$$

$$E_{\psi }^{\mathrm{lab}}$$

$${\vec{p}}_{\psi }^{\mathrm{lab}}$$

$$E_{\psi }^{\mathrm{lab}}=\frac{1}{2m_T}\left(m_{\psi }^2+m_T^2\right),{\vec{p}}_{\psi }^{\mathrm{lab}}=\frac{\vec{p}}{2m_T|\vec{p}|}\left(m_{\psi }^2-m_T^2\right)$$

Full text

1 Introduction

The production of charmed hadrons via the intrinsic charm mechanism from the target was proposed in Ref. [1]111see chapter 6.1 and Ref. [2]. Utilizing this idea, in Refs. [3, 4] production of double charmed baryons and the $J/\psi$ meson from the target was investigated at the STAR and LHCb/SMOG fixed-target programs [5, 6]. By a straightforward calculation it was shown that in case of the production of intrinsic charm from the target, charm hadrons have a maximum momentum in the laboratory frame of only a few $\mbox{\,GeV}/c$ and a rapidity $y<2$. This result is in perfect agreement with predictions from the intrinsic charm model about the $\Delta y$ region for the charm state, $\Delta y=y-y_{\rm tar}(y_{\rm tar}=0)<2$.

Recently, the LHCb collaboration presented the first measurement of charm production in the fixed-target configuration at the LHC [7]. The production of $J/\psi$ and $D^{0}$ mesons is studied with beams of protons of different energies colliding with gaseous targets of helium and argon with nucleon–nucleon centre-of-mass energies of $\sqrt{s}=86.6$ and $110.4\mbox{\,GeV}$, respectively. The $J/\psi$ and $D^{0}$ production cross-sections in $p$He collisions in the rapidity range $[2,4.6]$ are found to be $\sigma(J/\psi)=652\pm 33\pm 42\mbox{\,nb}$ per nucleon and \sigma(D^{0})=80.8\pm 2.4\pm 6.3\mbox{\,\mub} per nucleon, where the first uncertainty is statistical and the second is systematic. Using an approximation for the fraction $x$ of the nucleon momentum carried by the target parton,222We provide the equation in the form it was used in the LHCb publication [7]. However, it is easy to see that this equation is not fully correct. Using the standard Feynman-$x$ notation in the center-of-mass system $x\approx\frac{2m_{T}}{\sqrt{s}}\sinh(y^{*})$ (cf. Eq. (47.43) in Ref. [8]) and taking into account that $\sinh(y^{*})\approx-\exp(-y^{*})/2$ for large negative values of $y^{*}$, one finally obtains $x\approx-\frac{m_{T}}{\sqrt{s}}\exp(-y^{*})$.

[TABLE]

where $m_{c}=1.28\mbox{\,GeV}/c^{2}$ is the mass of the $c$ quark and $y^{*}$ is centre-of-mass rapidity, Providing an analysis of the production of $D^{0}$ mesons with the intrinsic charm from the target, LHCb reported no evidence for a substantial intrinsic charm content of the nucleon [7].

In this short note we provide a detailed derivation of the kinematic limit of the charm production in the fixed-target experiments with the intrinsic charm coming from the target. In addition, we show that based on a misunderstanding of the kinematics of the intrinsic charm from the target in the laboratory framem the LHCb/SMOG result is in contradiction to the properties of the intrinsic charm model.

2 Derivation of the kinematic limit

2.1 Calculation of the Center of Mass System energy

Given an invariant energy $\sqrt{s}$, one can easily calculate the energies in the laboratory frame, the center-of-mass frame and of course in a lot of different other frames. Starting from

[TABLE]

we first can calculate in the laboratory system where $p_{T}=(m_{T};0,0,0)$. One obtains

[TABLE]

For the center-of-mass system, however, we have $\vec{p}_{P}+\vec{p}_{T}=\vec{0}$ ($\Rightarrow\vec{p}_{P}=-\vec{p}_{T}=:\vec{p}$) and, therefore, $\sqrt{s}=E_{P}+E_{T}$ where $E_{P}=(m_{P}^{2}+\vec{p}\,^{2})^{1/2}$, $E_{T}=(m_{T}^{2}+\vec{p}\,^{2})^{1/2}$. Inserting these energies we obtain

[TABLE]

and this equation can be solved by two-fold squaring,

[TABLE]

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$ is Källén’s function. Given $s$ by Eq. (3), one can proceed to calculate the square of the three-momentum to be

[TABLE]

2.2 Lorentz transformations

In order to get from one system to another, four-vectors have to be multiplied by Lorentz matrices. The Lorentz matrix which puts the general target four-vector $p_{T}=(E_{T};-\vec{p}\,)$ in the CMS to rest is given by

[TABLE]

where $m_{T}^{2}=E_{T}^{2}-\vec{p}\,^{2}$. The upper index $T$ stands for transposition of the (column) three-vector $\vec{p}\,$. It can easily be seen that $\Lambda(p_{T})p_{T}=(m;\vec{0})^{T}$. Assuming that $J/\psi$ takes over a ratio $x\in[0,1]$ of the three-momentum $\vec{p}\,$ (for $x>0$ from the projectile, for $x<0$ from the target) and applying the Lorentz transformation to the corresponding four-vector $p_{\psi}=(E_{\psi};x\vec{p}\,)$ with $E_{\psi}^{2}=m_{\psi}^{2}+x^{2}\vec{p}\,^{2}$, one obtains a value for the four-vector in the laboratory frame, i.e. the frame where $p_{T}$ is at rest. The result reads

[TABLE]

Taking into account that the three-momenta are usually much larger than the masses, we can expand into $m_{T}^{2}/\vec{p}\,^{2}$ and $m_{\psi}^{2}/\vec{p}\,^{2}$ to $x<0$ (target) and obtain the maximum values of energy and momentum of the charm in the laboratory frame

[TABLE]

This last expression depend solely on the two masses $m_{\psi}$ and $m_{T}$ and no longer on the beam energy.

3 A comment on the LHCb/SMOG result

Let us remind the reader that in the fixed-target configuration, the LHCb acceptance gives access to the large Bjorken-$x$ region of the nucleon target (up to $x\sim 0.37$ for $D^{0}$ mesons) and to rapidity region $2<y<4.6$ in the laboratory frame. Using Eq. (9) we can obtain the kinematic limit (i.e. $x_{F}=1$) of the charm produced by the intrinsic charm mechanism from the target in the laboratory frame, $y<0.6$ and $p_{L}\sim 1.2-1.3\mbox{\,GeV}/c$. In other words, the LHCb collaboration provided an analysis outside the signal region.

4 Conclusion

In this note we provide a detailed derivation of the kinematic limits of the charm production in fixed-target experiments with the intrinsic charm coming from the target. We have also shown that the LHCb/SMOG result is based on a misunderstanding of the kinematics of the intrinsic charm. Moreover, the investigation of the production of charmed hadrons with intrinsic charm from the target at the LHCb/SMOG fixed-target program is fundamentally unfeasible.

Acknowledgements

We would like to thank Stanley J. Brodsky for pointing out the LHCb/SMOG result to us and stimulating discussions. We would also like to thank Ramona Vogt, Paul Hoyer and Jean-Philippe Lansberg for useful comments. This research was supported by the Estonian Research Council under Grants No. TK133 and PRG356.