Longitudinal string decay.

Hadron production by string fragmentation.

A string is stretched between flying away constituents: quark and antiquark or quark and diquark or diquark and antidiquark or antiquark and antidiquark. From knowledge of the constituents longitudinal $p_{3i}=p_{zi}$ and transversal $p_{1i}=p_{xi}$, $p_{2i}=p_{yi}$ momenta as well as their energies $p_{0i} = E_{i}$, where $i=1,2$, we can calculate string mass squared:

$$M^2_S=p^{\mu}p_{\mu}=p^2_0-p^2_1-p^2_2-p^2_3,$$ (22.29)

where $p_{\mu} = p_{\mu 1}+p_{\mu 2}$ is the string four momentum and $\mu = 0,1,2,3$.

The fragmentation of a string follows an iterative scheme:

$$string \Rightarrow hadron + new \ string,$$ (22.30)

i. e. a quark-antiquark (or diquark-antidiquark) pair is created and placed between leading quark-antiquark (or diquark-quark or diquark-antidiquark or antiquark-antidiquark) pair.

The values of the strangeness suppression and diquark suppression factors are

$$u:d:s:qq = 1:1:0.35:0.1.$$ (22.31)

A hadron is formed randomly on one of the end-points of the string. The quark content of the hadrons determines its species and charge. In the chosen fragmentation scheme we can produce not only the groundstates of baryons and mesons, but also their lowest excited states. If for baryons the quark-content does not determine whether the state belongs to the lowest octet or to the lowest decuplet, then octet or decuplet are choosen with equal probabilities. In the case of mesons the multiplet must also be determined before a type of hadron can be assigned. The probability of choosing a certain multiplet depends on the spin of the multiplet.

The zero transverse momentum of created quark-antiquark (or diquark-antidiquark) pair is defined by the sum of an equal and opposite directed transverse momenta of quark and antiquark.

The transverse momentum of created quark is randomly sampled according to probability ($\ref{LSE4}$) with the parameter $a = 0.25$ GeV$^{-2}$. Then a hadron transverse momentum ${\bf p_t}$ is determined by the sum of the transverse momenta of its constituents.

The fragmentation function $f^h(z,p_t)$ represents the probability distribution for hadrons with the transverse momenta $\bf {p_t}$ to aquire the light cone momentum fraction $z=z^{\pm}=(E^h \pm p^h_z/(E^q \pm p^q_z)$, where $E^h$ and $E^q$ are the hadron and fragmented quark energies, respectively and $p^h_z$ and $p^q_z$ are hadron and fragmented quark longitudinal momenta, respectively, and $z^{\pm}_{min} \leq z^{\pm} \leq z^{\pm}_{max}$, from the fragmenting string. The values of $z^{\pm}_{min,max}$ are determined by hadron $m_h$ and constituent transverse masses and the available string mass. One of the most common fragmentation function is used in the LUND model [12]:

$$f^h (z, p_t) \sim \frac{1}{z}(1-z)^a \exp{[-\frac{b(m_h^2+p_t^2)}{z}]}.$$ (22.32)

One can use this fragmentation function for the decay of the excited string.

One can use also the fragmentation functions are derived in [13]:

$$f^{h}_q (z, p_t)=[1+\alpha^h_q(<p_t>)] (1-z)^{\alpha^h_q(<p_t>)}.$$ (22.33)

The advantage of these functions as compared to the LUND fragmentation function is that they have correct three-reggeon behaviour at $z\rightarrow 1$ [13].

The hadron formation time and coordinate.

To calculate produced hadron formation times and longitudinal coordinates we consider the $(1+1)$-string with mass $M_S$ and string tension $\kappa$, which decays into hadrons at string rest frame. The $i$-th produced hadron has energy $E_i$ and its longitudinal momentum $p_{zi}$, respectively. Introducing light cone variables $p^{\pm}_{i}= E_{i} \pm p_{iz}$ and numbering string breaking points consecutively from right to left we obtain $p^{+}_{0} = M_{S}$, $p_{i}^{+}=\kappa (z^{+}_{i-1}-z_i^{+})$ and $p_{i}^{-} = \kappa x^{-}_i$.

We can identify the hadron formation point coordinate and time as the point in space-time, where the quark lines of the quark-antiquark pair forming the hadron meet for the first time (the so-called 'yo-yo' formation point [12]):

$$t_i = \frac{1}{2\kappa}[M_S - 2 \sum_{j=1}^{i-1}p_{zj} + E_i - p_{zi}]$$ (22.34)

and coordinate

$$z_i = \frac{1}{2\kappa}[M_S - 2 \sum_{j=1}^{i-1}E_{j} + p_{zi}- E_i].$$ (22.35)