Fragmentation of $\omega$ and $\phi$ Mesons in $e^+ e^-$ and $p p$ Collisions at NLO

TL;DR

This study performs a combined NLO analysis of $ ho$ and $\phi$ meson production in $e^+ e^-$ and $p p$ collisions, refining fragmentation functions and revealing new insights into gluon contributions.

Contribution

First combined NLO analysis of vector meson production in both $e^+ e^-$ and $p p$ collisions using a broken SU(3) model, with improved gluon fragmentation function determination.

Findings

Gluon fragmentation function is now well constrained with smaller errors.

Universal quark fragmentation functions are consistent with previous $e^+ e^-$ analysis.

New relation between gluon and sea suppression in $K^*$ and $\phi$ production.

Abstract

A combined analysis of both $e^+\,e^-$ (LEP, SLD) and $p\,p$ (RHIC-PHENIX and LHC-ALICE) hadroproduction processes are done for the first time for the vector meson nonet at the next-to-leading order (NLO) using a model with broken SU(3) symmetry. The transverse momentum ($p_T$) and rapidity ($y$) dependence of the differential cross section for $\omega$ and $\phi$ mesons of the $p\,p$ data are also discussed. The input universal quark (valence and singlet) fragmentation functions at a starting scale of $Q_0^2=1.5$ GeV$^2$, after evolution, have values that are consistent with the earlier analysis for $e^+\,e^-$ at NLO. However, the universal gluon fragmentation function is now well determined from this study with significantly smaller error bars, as the $p\,p$ hadroproduction cross section is particularly sensitive to the gluon fragmentation since it occurs at the same order as quark fragmentation, in contrast to the $e^+\,e^-$ hadroproduction process. Additional parameters involved in describing strangeness and sea suppression and octet-singlet mixing are found to be close to earlier analysis; in addition, a new relation between gluon and sea suppression in $K^*$ and $\phi$ hadroproduction has been observed.

Tables

Table 1. Figure 1: Initial fragmentation functions D i ( z , Q 2 D_{i}(z,Q^{2} ) at the starting scale Q 2 = Q 0 2 = 1.5 superscript 𝑄 2 superscript subscript 𝑄 0 2 1.5 Q^{2}=Q_{0}^{2}=1.5 GeV 2 (left) and fragmentation functions at a sample value of Q 2 = 56.3 superscript 𝑄 2 56.3 Q^{2}=56.3 GeV 2 (right), i = V , γ , D g 𝑖 𝑉 𝛾 subscript 𝐷 𝑔 i=V,\gamma,D_{g} , for ρ 𝜌 \rho mesons as a function of the momentum fraction z 𝑧 z .

	$\left(N_{\omega }+N_{\rho }\right)$	$=$	$\left(BR\left(\omega \to \mu \mu \right){\sigma }_{\omega }+BR\left(\rho \to \mu \mu \right){\sigma }_{\rho }\right),$
	$N_{\varphi }$	$=$	$BR\left(\varphi \to \mu \mu \right){\sigma }_{\varphi },$

Equations

$$\frac{1}{{\sigma }_{tot}}\frac{d{\sigma }_{e^{+}{e}^{-}}^h}{dx}(x;Q^2)$$

$$and{\sigma }_{tot}$$

$$E_3\frac{d^3\sigma }{d^3{k}_3}$$

$$E_3\frac{d^3\sigma }{d^3{k}_3}\equiv \frac{1}{p_T}\frac{d^3\sigma }{d{p}_{T}dyd\varphi }=\frac{1}{\pi }\frac{d^2\sigma }{d{p}_T^{2}dy}\text{leavevmode},$$

$$\frac{\delta (x,Q^2)}{3}=\frac{f_1^i}{3}\alpha (x,Q^2)\text{leavevmode},$$

$$F_i(x,Q_0^2)=a_i{x}^{b_i}(1-x{)}^{c_i}{P}_i(x);\text{leavevmode}\text{leavevmode}{P}_i(x)=(1+d_i\sqrt{x}+e_{i}x)\text{leavevmode};$$

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PACS numbers: 13.60.Le, 13.60.Hb, 13.66.Bc, 13.85.Ni

1 Introduction

A number of analyses are available for fragmentation of pseudoscalar mesons and baryons till date; see, for example, Ref.[1] for $\pi$, Refs.[2, 3, 4, 5, 6, 7, 8, 9] for $K$ meson, Refs.[10, 11] for proton, and Refs.[3, 12] for $\eta$ fragmentation with comprehensive reviews in Refs.[4, 5, 6, 13] as well. No such considerable interest has been shown towards vector meson production due to the scarcity of the data available so far.

Hadroproduction of $\phi$ vector mesons in proton-proton collisions is a good candidate signal for Quark Gluon Plasma (QGP) in heavy nucleus-nucleus collisions. This requires a good understanding of $\phi$ hadroproduction in $p\,p$ collisions, which will serve as a baseline for nucleus–nucleus studies. With this motivation, analyses had been done for vector meson fragmentation in $e^{+}\,e^{-}$ scattering at Leading Order (LO)[14] and Next-to-Leading Order (NLO)[15] and in $p\,p$ collisions at LO[14] as well. In this paper, for the first time, this has been extended to a combined investigation for the fragmentation of the entire vector meson nonet in $e^{+}\,e^{-}$ and for ${\omega}$ and $\phi$ meson production in $p\,p$ collisions at NLO using the LEP[16, 17, 18, 19, 20, 21, 22] and SLD[23, 24] data for $e^{+}\,e^{-}$ and RHIC PHENIX[25] data and LHC ALICE[26] data for $p\,p$ hadroproduction. Analysis of individual vector meson production has been done, for instance, analysis of $\phi$ hadroproduction from LHC data was done in Ref. [27].

A key feature of the analysis (described earlier in Refs.[14] and [15] and applied to the present study) is the ability to use the entire nonet vector meson hadroproduction data by defining SU(3)-symmetric fragmentation functions common to the entire set of octet mesons. This drastically reduces the number of independent fragmentation functions (from three quark- and one gluon fragmentation function for each member of the octet) to two universal quark- and one gluon fragmentation function. Some additional parameters are subsequently introduced to account for SU(3) symmetry breaking and singlet-octet mixing to allow the study of the entire vector meson nonet. The definition of input fragmentation functions and other parameters relevant to the model[15] remain the same in this study and have been briefly reviewed here for completeness; some differences in the choice of fragmentation functions for analyses are also mentioned below. The hadroproduction in $p\,p$ collisions at NLO presented here is particularly important in view of the fact that gluons contribute at higher order in $e^{+}\,e^{-}$ collisions but contribute at the same order as quarks in $p\,p$ processes. This study at NLO therefore enables a more precise determination of the gluon fragmentation functions.

Further studies such as gluon and singlet quark suppression, the dependence of the $p\,p$ hadroproduction cross sections on transverse momentum $p_{T}$ as well as rapidity $y$, and inclusion of data on the (branching fraction-weighted) $\phi$ and $\omega$ cross section ratios helps in a more detailed understanding of the hadroproduction process. With this study we complete the program of vector meson nonet fragmentation using this model.

In section 2, we list the relevant cross-section formulae for hadroproduction in $e^{+}\,e^{-}$ and $p\,p$ collisions. In Section 3, we present highlights of the model used to determine the vector meson fragmentation functions. In Section 4 we use the available data to best-fit the parameters involved and show the resultant fits and their quality. We conclude in Section 5 with some remarks and summary.

2 Kinematics and Cross sections

We summarise here for completeness the relevant cross-sections for inclusive hadroproduction in $e^{+}\,e^{-}$ and $p\,p$ collisions (in the c.m. frame).

2.1 Hadroproduction in $e^{+}\,e^{-}$ Collisions

The hadronic cross section for inclusive hadroproduction in $e^{+}\,e^{-}$ collisions to NLO is given by[28]:

[TABLE]

Here $x(=2p_{h}/\sqrt{s})$ is the fraction of the parent quark momentum carried by the hadron $(h)$ having momentum $p_{h}$, $Q=\sqrt{s}$ is the energy scale at which the analysis is carried out (the data is taken at the $Z$-pole, with $Q=91.2$ GeV), functions like $D^{h}_{q_{f}}$, $D^{h}_{\bar{q_{f}}}$ and $D^{h}_{g}$ are the quark, anti-quark and gluon fragmentation functions and $N_{c}$ refers to the number of colours. Terms like $\mathcal{C}^{F(1)}(z)$, $\mathcal{C}^{g(1)}(z)$, ${\lambda}_{B}^{F}$ and ${\lambda}_{B}^{g}$ are the coefficient functions for quarks (F) and gluons (g), whose expressions are given in detail in Refs.[15, 28] and where $\alpha_{s}(Q^{2})$ is also defined to NLO.

2.2 Hadroproduction in $p\,p$ Collisions

The hadronic cross section for inclusive hadroproduction in $p\,p$ collisions at NLO is given in terms of the underlying partonic interaction $p_{i}(x_{1})\,p_{j}(x_{2})\rightarrow p_{l}(x_{3})\,p_{k}(x_{4})$ as[29],

[TABLE]

where the indices $i,j,l,$ sum over all possible flavours of quarks and anti quarks, and gluons. The term $f_{p_{i}/H_{1}}(x_{1},M^{2})$ ($f_{p_{j}/H_{2}}(x_{2},M^{2})$) refers to the parton distribution function of parton $p_{i}(p_{j})$ inside hadron $H_{1}(H_{2})$ with a momentum fraction $x_{1}(x_{2})$ and initial factorization scale $M$. Likewise, $D_{p_{l}/H_{3}}(z,M^{2}_{f})$ is the fragmentation function for a parton $p_{l}$ to fragment into a hadron $H_{3}$ with a momentum fraction $z$ and fragmentation scale $M_{f}$.

The first term within the bracket, $d\sigma^{0}$, is the LO Born cross section term for $p_{i}p_{j}{\rightarrow}p_{l}$ with $s,v$ and $w$ expressed in terms of $x_{1},x_{2}$ and $z$ and hadronic momenta $k^{\prime}s$; for example, $s=x_{1}x_{2}S$ where $S$ is the usual hadronic centre of mass energy (squared).

The second term having $\alpha_{s}(\mu^{2})$ with renormalization scale $\mu$ corresponds to the higher order contribution with its correction factor $K_{p_{i}p_{j}{\rightarrow}p_{l}}(s,v,w;\mu^{2};M^{2},M^{2}_{f})$ for each subprocess. A detailed calculation of the correction factors for various subprocesses is given in Ref.[29]; here we merely note that, unlike in the $e^{+}\,e^{-}$ case, in $p\,p$ processes the gluon fragmentation function contributes at the LO itself through subprocesses such as $q\,g\to q\,g$ and $g\,g\to g\,g$. Hence we expect that inclusion of hadroproduction in $p\,p$ processes will significantly improve our knowledge of gluon fragmentation.

Hence from Eqs. 1 and 2 it is very clear that the gluon fragmentation function appears at a higher order of ${\alpha}(Q^{2})$ as compared to quark fragmentation in $e^{+}\,e^{-}$ processes and at the same order in $p\,p$ processes.

The LHS of Eq. 2 can be expressed in terms of physical observables, the rapidity $y$ and the transverse momentum $p_{T}$, as

[TABLE]

where the last simplification occurs because the cross-section is independent of the azimuthal angle $\phi$. According to the factorization theorem, the cross section for $p\,p$ in Eq. 2 is expressed as a convolution over three parts: parton distribution functions, partonic subprocess cross sections and fragmentation functions. For this study, the initial parton distribution functions are taken from GRV 98 NLO code[30] (Within this limited $x$ region, the CTEQ parton distribution functions can be used as well), the partonic cross sections for hadroproduction in $p\,p$ processes at NLO are taken from Aversa et al.[31], and the fragmentation of the final state parton is obtained using our $x$-real space Fortran code based on the broken SU(3) model.

3 The Model

We now briefly describe the broken SU(3) model that is used to describe the input fragmentation functions at NLO in this paper. The details regarding the model were discussed in detail in Refs.[2, 14, 15] in which the $e^{+}\,e^{-}$ data were fitted to the NLO cross sections using this model. In Ref.[14], a study of hadroproduction in $p\,p$ processes at LO was also taken up. The present study, which includes consistently an analysis of both $e^{+}\,e^{-}$ and $p\,p$ hadroproduction to NLO completes this program.

The model uses SU(3) flavour symmetry to express the quark fragmentation functions $\alpha(x,Q^{2})$, $\beta(x,Q^{2})$ and $\gamma(x,Q^{2})$ corresponding to the underlying quark fragmentation subprocesses $q^{i}\to M^{i}_{j}\,X^{j}$, where $X^{j}$ is a member of 3-, $\overline{6}$-, or 15-plet respectively. Application of charge conjugation symmetry and isospin invariance significantly reduces the number of unknown fragmentation functions. In addition, fragmentation functions of different mesons are related within this model, and this is what allows for the analysis of the otherwise sparse vector meson data.

The fragmentation functions of all octet vector mesons can be written in terms of three universal functions that are named valence $(V)$, sea $(\gamma)$, and gluon $(D_{g})$ fragmentation functions[15] (see Table 3). The model defines the fragmentation functions at an initial scale of $Q_{0}^{2}$, taken to be $Q_{0}^{2}=1.5$ GeV2, for three light quarks $u$, $d$ and $s$, where the charm and bottom flavour contributions are kept zero. The contribution of such heavy flavours are added in at appropriate thresholds during DGLAP evolution. These input fragmentation functions are then evolved to various momentum scales for comparison with available data.

Breaking of SU(3) symmetry due to strangeness suppression is included through an $x$-independent strangeness suppression parameter ${\lambda}$ at the starting scale. For instance, non-strange quark fragmentation into strange mesons such as $K^{*}$ is suppressed by $\lambda$: $D_{u}^{K*}\to\lambda D_{u}^{K*,{\rm pure\leavevmode\nobreak\ SU(3)}}$ while strange quark fragmentation is not suppressed: $D_{s}^{K*}$ = $D_{s}^{K*,{\rm pure\leavevmode\nobreak\ SU(3)}}$ (see Table 3 for the pure SU(3) expressions). The entire sea quark fragmentation into $K^{*}$ is thus suppressed by a factor of $\lambda$ compared to sea quark fragmentation into $\rho$ mesons.

The model is extended to include the SU(3) singlet-octet mixing since it is known that the physical $\omega$ and $\phi$ mesons are admixtures of the octet and singlet states. An angle $\theta$ is used to describe SU(3) singlet–octet mixing. The singlet sector has an additional fragmentation function, $\delta(x,Q^{2})$, due to the single subprocess that contributes: $q^{i}\to M^{0}\,X^{i}$, where $X^{i}$ belongs to a 3-plet, which is taken to be proportional to the octet fragmentation function $\alpha$:

[TABLE]

thus adding only two parameters for $i=\omega,\phi$, viz., $f_{1}^{u,\omega}$ and $f_{1}^{s,\phi}$. Note that $f_{1}^{d,\omega}=f_{1}^{u,\omega}$ and $f_{1}^{s,\omega}=f_{1}^{u,\phi}=f_{1}^{d,\phi}=0$. The former arises from SU(3) and SU(2) symmetry and the latter from the observation that the physical $\phi$ ($\omega$) meson is almost purely an $s\overline{s}$ (non-strange) state since the phenomenological value of the mixing angle $\theta\sim 39^{\circ}$ is very close to the value $\theta=35^{\circ}$ where this is exactly true. Finally the sea suppression factors for $\omega$ and $\phi$ are denoted as $f_{sea}^{\omega}$ and $f_{sea}^{\phi}$; they are expected to be of order unity and $\lambda^{2}$ respectively. Note that no additional singlet fragmentation functions are required.

In toto, we have the fragmentation functions for octet valence, sea and gluon ($V,\gamma$ and $D_{g}$) with strangeness suppression ${\lambda}$, the octet-singlet mixing angle ${\theta}$, and other $x$-independent singlet and suppression factors for the mixed $\omega$-$\phi$ system such as $f_{1}^{u}({{\omega}})$, $f_{1}^{s}({{\phi}})$, $f_{sea}^{{\omega}}$ and $f_{sea}^{{\phi}}$. Finally, we have the gluon suppression factors $f_{g}^{K^{*}}$, $f_{g}^{{\omega}}$, $f_{g}^{{\phi}}$[15], where $D_{g}=f_{g}^{i}D_{g}$, $i=K^{*},\omega,\phi$.

The following modification has been made in the parameter descriptions compared to the earlier analyses, leading to better stability during evolution: We have used upto linear terms in $x$ instead of the choice of a quadratic form in the standard polynomial[32] for the parameterization of input quark and gluon fragmentation functions:

[TABLE]

instead of $P_{i}=(1+d_{i}x+e_{i}x^{2})$ which was the form used in the earlier analysis. This polynomial can have large fluctuations and even go negative during $Q^{2}$ evolution especially in the low-$x$ region, while the current choice shows smooth behaviour at low and intermediate values of $x$. Hence, this polynomial choice helps in obtaining a more stable fit at low-$x$, and a better fit at intermediate-$x$, particularly for $p\,p$ data. Here $F_{i}(x)=V(x),\gamma(x)$ and $D_{g}(x)$ are the corresponding valence, sea and gluon input fragmentation functions and $a_{i},b_{i},c_{i},d_{i}$ and $e_{i}$ are the parameters to be determined for these functions at the starting scale $Q_{0}^{2}$.

4 Combined Analysis of $e^{+}\,e^{-}$ and $p\,p$ Data

4.1 Choice of Data Sets

A combined analysis of both $e^{+}\,e^{-}$ and $p\,p$ data is done in order to fit the vector meson fragmentation functions. The LEP data[16, 17, 18, 19, 20, 21, 22] for $\rho(\rho^{+},\rho^{-},\rho^{0})$ and $\omega$ mesons and SLD “pure uds” data[23, 24] for K∗ and $\phi$ are used for $e^{+}\,e^{-}$ process at the $Z$-pole, $\sqrt{s}=91.2$ GeV. The SLD “pure uds” data (three flavours alone) are used in the case of K∗ and $\phi$ mesons in order to avoid the contamination from heavy flavour meson production such as $B$ and $D$ mesons which decay into one of the strange mesons which will contaminate the data on direct hadroproduction into K∗ or $\phi$ due to large CKM matrix elements $V_{cs}$ and $V_{cb}$. In the case of non-strange mesons like $\rho$ and $\omega$, the contamination is very small, since heavier $b$- and $c$- mesons will decay mostly (vis $s$) to $\pi$, the lightest non-strange pseudoscalar meson, rather than $\rho$ or $\omega$.

Likewise, the 2011 RHIC/PHENIX data[25] at centre-of-mass energy, $\sqrt{s}=200$ GeV, with rapidity (to be considered as pseudorapidity throughtout the paper), $|y|\leq 0.35$ for $p\,p$ collisions is used in the analysis for ${\omega}$ and ${\phi}$ hadroproduction. The data has three types of systematic errors added in quadrature with no statistical errors given in the literature. Effort is taken to add statistical errors from RHIC experimental group paper[25] and thesis[33] for ${\omega}$ and $\phi$ mesons decaying through various channels. Thus care is taken to include both the statistical and systematical errors which are added in quadrature. More recently[34], detailed doubly differential rates in both rapidity $y$ and transverse momentum $p_{T}$ have been measured by RHIC-PHENIX, in the forward rapidity region $1.2\leq|y|\leq 2.2$ for $\omega$ and $\phi$ hadroproduction, as well as their weighted events ratio. Recently, the LHC-ALICE collaboration[26] has also provided $K^{*}$ and $\phi$ hadroproduction data at $\sqrt{s}=2.76$ TeV111We thank the referee for bringing this to our notice. Since the $K^{*}$ data is sensitive to both the valence fragmentation function and the strangeness suppression factor, $\lambda$, we have also included this data in our analysis.

The rapidity and azimuthal acceptances are different in different sets of $p\,p$ data, and we have used the values to match with the experimental data.

4.2 Determining the Best-fit Parameters

Using the standard functional form in Eq. 4, the unknown input fragmentation functions for $V$, $\gamma$ and $D_{g}$ are parameterized at an initial scale of $Q^{2}_{0}=1.5$ GeV2. Contributions of the heavy $c$ and $b$ flavours are included at the appropriate thresholds during evolution. The fragmentation functions of all mesons are evolved to various scales, say, $Q^{2}=(91.2)^{2}$ GeV2 for $e^{+}\,e^{-}$ and $Q^{2}=p_{T}^{2}$ GeV2 for $p\,p$ collision, using the DGLAP evolution equations[35, 36, 37] for $\rho$, $K^{*}$, $\omega$ and $\phi$.

The best fit to the unknown parameters is found by performing a combined $\chi^{2}$ minimization with both $e^{+}\,e^{-}$ LEP[16, 17, 18, 19, 20, 21, 22] and SLD[23, 24] data and $p\,p$ RHIC-PHENIX (for both hadroproduction[25] and branching ratio weighted differential cross section[34]) and LHC-ALICE[26] data.

The best fit values of the parameters $a,b,c,d$ and $e$ for valence, singlet and gluon input fragmentation functions, with $1\sigma$ errors are given in Table 4.2. The errors on the quark parameters are about 5% or less, much better than the earlier[15] studies. However, the fits to the gluon parameters are much better determined than earlier, with errors of 1-4% on the fit parameters. This is due to two reasons, the first that the $p\,p$ cross sections are sensitive to both quark and gluon fragmentation functions at the same order, and the second that a huge energy sale separates RHIC and LHC data, thereby restricting the allowed parameter space considerably.