Fragmentation of Two Repelling QCD Strings

We thank the Referee for recognizing the value of our manuscript as a prototype for a string repulsion model that is an alternative to the shoving model by Bierlich et al.
With regards to their two major points, we have replaced references to "QCD strings" with the more appropriate term "Lund strings".
Similarly, we have included more references to work conducted by other groups, and in particular, other approaches to modelling the ridge phenomenon in proton-proton collision environments. The added paragraph reads as follows:
'Other approaches to studying and modeling azimuthal correlation sin proton-proton collision environments include the String Percolation model [46–50], or interference effects from multiple parton interactions [51] or from the BFKL parton shower evolution process [52]. Recent work has investigated the initial state geometry of the collision and the resultant effect on azimuthal anisotropy [53–55]. The Colour Glass Condesate (CGC), a successful framework for describing the collision environment in heavy-ion collisions, has also been applied to proton-proton collisions [56, 57]. Kinetic transport theory has also been used to study the potential source of angular correlations [58]. A review of collectivity in small systems can be found in e.g. [59, 60].'

We thank the Referee for their positive comments on the manuscript, and valuable points.

With regards to the Referee’s first point, it is unclear to us if they are referring to a reduction of the effective quark masses as such, or if they are referring to the reduction of the total invariant mass of the string. The former is not a feature of our model, while the latter is. Not being sure which point the referee wants us to address, we address both below.

Firstly, concerning the reduction of the overall invariant mass of the string. Each string (consisting of two endpoints) has a total invariant mass, its CM energy. At time t = 0, this consists entirely of the kinetic energy of the endpoints, which, in the baseline Lund model, as time evolves is gradually converted to potential energy stored in the string. In the baseline model, all of the kinetic energy is available to be converted to potential energy; they are in 1-1 correspondence; therefore there is no need to book keep them separately. Technically, in the Lund model, string systems are therefore represented entirely in terms of their momentum-space boundary conditions; the energies and momenta of their endpoints. In our repulsion model, however, a portion of the original longitudinal momentum is converted to transverse momentum; that is, a certain fraction of the CM energy *remains kinetic* and is therefore not available for particle production; the total produced particle multiplicity must therefore decrease.

However, if the Referee is referring to the notion of reducing the effective mass of the strange quark, we would like to clarify the following point: strangeness enhancement is not a part of this repulsion model. We believe that one could unify the description of strangeness enhancement (in the manner that the rope model works) and our repulsion, such that the two aspects of collectivity derive from the same root. However, for this work, we solely focus on the repulsion component. We also note that, in the rope model, it is not that the strange quark’s mass decreases, so much as the string tension increases. This leads to an effective decrease of the strange quark’s mass - relative to the string tension. That is, the effective quark masses are regarded as being unchanged; what is changing is the effective string tension, and the higher that gets, for constant constituent quark masses, the closer the u:d:s ratios get to 1:1:1. We plan to incorporate this aspect in future work.

With regards to the second point, since we are working with a Monte Carlo event generator, we cannot a priori know anything about the multiplicity of the final state. However, we do know that a single Lund string will, on average, produce one hadron per unit of rapidity. Accordingly, in the Lund string model, high multiplicity events occur when there are many long strings. Since we have only considered a pair of strings, it may be unwise to extrapolate these results directly to situations involving many strings. However, we believe it is safe to say that one would be able to use lower values for the repulsion constant for the same increases/decreases on observables such as c_2{2}, since each string will be interacting with several others.

With regards to the third point, we have now included a new paragraph in the introduction with many more references to the vast amount of research conducted by others from different approaches, which reads as follows:
'Other approaches to studying and modeling azimuthal correlation sin proton-proton collision environments include the String Percolation model [46–50], or interference effects from multiple parton interactions [51] or from the BFKL parton shower evolution process [52]. Recent work has investigated the initial state geometry of the collision and the resultant effect on azimuthal anisotropy [53–55]. The Colour Glass Condesate (CGC), a successful framework for describing the collision environment in heavy-ion collisions, has also been applied to proton-proton collisions [56, 57]. Kinetic transport theory has also been used to study the potential source of angular correlations [58]. A review of collectivity in small systems can be found in e.g. [59, 60].'

We thank the referee for their careful reading of the manuscript and their helpful comments. We note that we have updated the title of the paper to be ‘Fragmentation of Two Repelling Lund Strings’ to reflect the referees’ collective comment on the misuse of the previous phrase ‘QCD Strings’, as well as cleaned up several references to ‘QCD strings’ in the introduction. We have also included more references to work done in a similar vein but from different approaches. The corresponding statement in the introduction of our paper that reads as:

'Other approaches to studying and modeling azimuthal correlation sin proton-proton collision environments include the String Percolation model [46–50], or interference effects from multiple parton interactions [51] or from the BFKL parton shower evolution process [52]. Recent work has investigated the initial state geometry of the collision and the resultant effect on azimuthal anisotropy [53–55]. The Colour Glass Condesate (CGC), a successful framework for describing the collision environment in heavy-ion collisions, has also been applied to proton-proton collisions [56, 57]. Kinetic transport theory has also been used to study the potential source of angular correlations [58]. A review of collectivity in small systems can be found in e.g. [59, 60].'

It is unclear what notion the Referee refers to as ‘ad-hoc’ in the paper. If it is the ansatz that the transverse repulsion from overlapping strings is constant per unit of rapidity, this ansatz is well-motivated by the fact that we are dealing with Lund strings. These strings must have a string-like behaviour. In the original Lund string model, the string tension per unit ‘length’ must be constant, and the multiplicity produced from a string per unit rapidity is also a constant. Accordingly, the transverse momentum donated and received by each string should scale linearly with some measure of the string’s length, the most natural measure of which (in the Lund string model) is rapidity span. (This also makes the formalism Lorentz invariant with respect to boosts along the chosen rapidity axis.)

If the 4 different configurations are unclear as physically motivating scenarios, we would note that the manuscript is meant to outline the model in a step-by-step manner. - Correspondingly, we start with the simplest case, the symmetric parallel configuration, a scenario that is unlikely to occur in a real collision, but acts the baseline for our model, and lays out the bones of how we repel the two strings. - We then transition to the general parallel configuration, a scenario that is more likely to occur in a real collision, and use this transition to note that one needs more nuance to describe the repulsion between the two strings. In particular, the two strings do not fully overlap, and as a result we do not allow the strings ends outside of the overlap region to receive any transverse momentum. - In the third case, we extend the formalism to deal with endpoints with transverse momentum, taking one more step to a more physical scenario of two strings. Granted, the symmetry of this configuration makes it an unlikely occurrence in a collision. - In the fourth case, we finally put the pieces of the second and third configurations to describe a physical configuration of two strings, one that is general enough to describe the interaction between any two simple Lund strings (with massless endpoints, though this gets extended in the final section).

While configurations 1 and 3 are unlikely to occur in a real collision, 2 might, and 4 is highly likely, given that we design it to be a general description. One can always boost to the rest frame of two strings to get a configuration akin to the general configuration with transverse motion.