The previous class was ambitious and threw out a lot of ideas. Let's step back and see how hard scattering fits into the big picture so far developed in this class.

- Last Tuesday we discussed nucleon-nucleon scattering and found that the number of particles produced in the collision increased logarithmically with center of mass energy.
- Understanding this theoretically can help in giving an estimate of what would be expected from nucleus on nucleus collisions. If the experimental data deviate from this, it would provide a clue that something more than we expected is happening in the collisions.
- The hard processes give energetic collisions which are
responsible for particle production. There are two opposing models
which try to describe this phenomenon:
**Parton Cascade**:- The probability of finding a parton inside the nucleon with a certain momentum fraction (x_Bjorken) has been measured in deep inelastic scattering. Deviations from this for very low and high momentum fractions can be described in terms of a cascade of partons branching out, as developed in last class.
- When considering two colliding nucleons, these distributions of partons interact. This very complicated process can be described in terms of free moving partons colliding in pairs.
- As things cool down, the partons must come together into hadrons again. This entire process determines the final particle multiplicity.
- We will briefly discuss how this model compares with data.

**QCD String Model**:- A
*string*is just a way to describe the concentrated strong force interaction which confines the quarks inside the nucleon. - If two nucleons collide, this model approximates the interaction by connecting a string between the wounded nucleon and a parton that is pushed out in the collision. As they pull apart, at some point the energy density of the string is so high it is favorable to break apart into a quark-antiquark pair, thereby producing more particles. This is a nice theory with few inputs that describes a lot of low energy heavy ion collision data.
- At higher energies, though, the string density becomes large and there is a possibility the strings can interact with each other. This leads to more unknowns in the theory, but after fitting these the theory agrees relatively well with medium energy heavy ion collisions.
- We will briefly look at how this model compares with data and then explore some background on strings.

- A

Chapters 5 through 8 in Wong review a variety of phenomenological models and approaches used to describe soft QCD processes relevant for NN collisions. Much of this discussion is less relevant at the very high energies that will be typical at RHIC than was apparent several years ago when Wong wrote his text. However, we can understand many of the qualititative features of NN collisions that we have seen in the figures of Chapter 3 of Wong by considering strings.

These chapters contain much more detail than we have time to consider. So we will focus on some basic string physics from Chapters 5 and 7.

We've seen from Chapter 3 that we do produce lots of particles in high-energy NN (and nucleus-nucleus of course!) collisions. We observed various characteristics of the collisions:

- The transverse momentum distributions are typically exponential (or gaussian) with <p_T> of order 300 to 400 MeV/c.
- The multiplicity distribution in rapidity, dN/dy, has a characteristic plateau shape at large energies.

In this chapter we consider a mechanism for particle production originally due to Schwinger. The idea is to have a concrete model of how particles are produced and how a characteristic p_T distribution of the produced particles can arise.

- Basic idea: Between a quark and anti-quark being pulled apart will be a color electric field. A particle--anti-particle pair is produced when a particle tunnels from the negative-energy sea to a positive-energy state (both in the continuum).
- Confined color flux tube between quark and anti-quark
- Recall the cartoon from the first lecture, of the color electric flux lines between a quark and an anti-quark.
- The flux tube is the analog of the tube of magnetic field lines that forms in the interior of a superconductor (which expels a magnetic field: the Meissner effect).
- We'll focus on q qbar pairs for simplicity, but we'll see an analogous tube if we pull a quark out of a nucleon bag (leaving behind a diquark).
- We said the first day that if we pulled the quark too far, it wouldn't come free but that the string would "break" with the production of a quark on one end and an anti-quark on the other. Thus we have produced a new meson. How does this work?

- Schwinger Particle Production Mechanism
- Originally applied to the production of e+e- pairs in a strong uniform electric field, such as between capacitor plates
- Apply here in an analogous way: the q and qbar act as the capacitor plates with a uniform electric field between.
- Assume an abelian color electric field (meaning we neglect self-interactions of gluons that complicate the picture), which will behave like an ordinary QED electric field.
- Schwinger showed that a particle-antiparticle pair is produced when a particle tunnels from the negative-energy continuum (the Dirac sea) to the positive-energy continuum (a real particle).

- Flux tube picture (Figure 5.1)
- Color charge q and -q at ends, which sit at z=0 and z=L
- Transverse cross section is taken to be circle, with radius about 0.5 fm according to Wong (this seems more like the diameter; we'll return to this when we do lattice gauge theory).
- Just as in a capacitor, Gauss' law implies that the
density of flux lines is constant, so there is a constant
magnitude color electric field
*E*running the length of the tube. - What is the energy contained in the flux tube?

E = 1/2*E*^2 A L

so the energy increases*linearly*with L. - So we can write E = \kappa L, which implies
\kappa = 1/2
*E*^2 A. The force on a charge q is q*E*= 2\kappa, so it makes sense to call \kappa the "string tension" (we'll see what happens to the 2 below). - Given \kappa we can find A from \kappa = 1/2 q^2/A.
- How do we find \kappa?

- String tension \kappa.
- A linear potential arises as a quark and anti-quark are pulled apart. The coefficient of the potential is called the "string tension" \kappa. This idea dates back to the sixties and the analysis of hadronic spectra (that is, the pattern of hadron masses).
- Postulated a family of states corresponding to
*orbital*excitations of some underlying system. - The trajectory rises with increasing angular momentum, which implies a strong long-range force => "string" => nowadays associated with flux tube of color electric field
- Model hadrons as rotating string with constant string
tension (see figure 7.3)
- Use relativistic kinematics: massless quarks at the end of a rigid string of length 2L spinning about its center.
- Velocity of segment dr is r/L (where 1 means the speed of light). The quarks have velocity = c (=1 here).
- Identify the mass of the baryon with the sum of the
energy in each string segment: \gamma \kappa dr, where
\gamma = 1/sqrt(1-(r/L)^2). Overall factor of 2 for each
side of the string:

M = 2 \int_0^L \gamma\kappa dr = \pi \kappa L - Similarly, the angular momentum of the hadron J is
found from:

J = 2 \int_0^L \gamma\kappa v r dr = \kappa L^2 \pi/2 - Eliminating L, J = (1/2\pi\kappa) s, where s=M^2.

- So if we plot J versus M^2 for hadrons, we should find that
they lie on straight lines.
- These are called "Regge trajectories".
- See figure 7.2 for an example. The slope of the line \alpha' gives the string tension \kappa = 2\pi\alpha'.
- \kappa estimated from Regge trajectories to be about 1 GeV/fm. You'll do this for one of the homework problems using kaon resonances.

- The string tension is one of the standard inputs to lattice gauge calculations to calibrate a calculation.

- Basic picture of tunneling
- We'll ignore edge effects and treat the system like a capacitor.
- The electrostatic potential A_0(z) is taken to be zero for z<0 (where it is constant since no electric field). A_0(z) = -\kappa z in between the quarks [0<z<L], and it is again constant for z>0, A_0(z) = -\kappa L
- So we have three regions as in Fig. 5.3.
- There is no factor of 2 from q
*E*= 2\kappa because a charge added between the quarks will not be added by itself, but with an antiquark. This antiquark will cancel the force from the quark on the end, so the net force on the added quark is just \kappa. - Now consider the
*energy levels*for a quark in the presence of this static field.- Assume a rest mass m and fixed transverse momentum p_T, so m_T = \sqrt(m^2+p_T^2).
- We use the Dirac sea picture of fermion particles and antiparticles. The negative energy sea has completely filled levels. An excited quark from the sea to an empty positive energy level corresponds to the creation of a quark-antiquark pair.
- If A_0(z) is added to every level, we'll get a linear change in the energy of a given level as we move across z from 0 to L. So some negative-energy states have the same energy as positive-energy states at a larger value of z => tunneling is possible => production of q\qbar pair.
- Must have energy E < 0 (at left end) to be a negative energy state but E > -\kappa L + m_T (at right end). This implies a minimum L: L_min = 2m_T/\kappa which is about 0.7 fm if p_T=0 and 1.0 fm if p_T is 0.35 GeV.

- Tunneling calculation
- There is a nice calculation in the book, which derives an effective Schrodinger equation. We'll shortcut the discussion by motivating a generalization of the usual nonrelativistic tunneling calculation to a relativistic energy-momentum relation.
- We recall that the transmission probability for tunneling
through a classically forbidden region is

P = exp( -2 \int_zL^zR \bar k(z) dz )

where zL and zR are the classical turning points, and \bar k(z) is found from the nonrelativistic energy-momentum relation:

E_eff = (i\bar k(z))^2/2m + V_eff(z) [note the i!] - [Draw a picture and indicate \bar k(z) = \sqrt(2m(V(z)-E))] > 0
- We generalize to the case at hand by using
the relativistic energy momentum relation in the presence of
a electrostatic potential A_0(z) to define \bar k(z):

[E-A_0(z)]^2 = (i\bar k(z))^2 + m_T^2

[note: in general, p^\mu -> p^\mu - A^\mu] - The turning points are [E-A_0(zL)]^2 = m_T^2 and similarly with zR, so A_0(z)=-\kappa z => zL = (-E-m_T)/\kappa and zR = (-E+m_T)/\kappa.
- The integral becomes:

\int_zL^zR \sqrt(m_T^2-[E+\kappa z]^2) dz

= 1/\kappa \int_{-m_T}^m_T \sqrt(m_T^2-u^2) du = (\pi/2\kappa)m_T^2. - Note that E drops out.

- The end result is the tunneling probability:

P = exp(-\pi m_T^2/\kappa) = exp(-\pi [p_T^2+m^2]/\kappa)- So the probability of tunneling decreases rapidly with increasing transverse mass.
- The scale is set by the string tension \kappa.
- The production rate requires some analysis of phase space, but it will also be proportional to P.
- The consequence is a transverse mass distribution of the produced
particle that has the same gaussian dependence as the tunneling
probability:

\sqrt(<p_T^2>) = \sqrt(\kappa/\pi) = 0.25 GeV/c for a quark and about 0.35 GeV/c for a pion (multiply by \sqrt 2). - Good estimate compared to experiment!

- How many kaons are produced compared to the number of pions?
- Empirically, K+/\pi+ in e+e- is 0.16 at sqrt(s)=10 GeV and 0.14 at sqrt(s)=29 GeV. Is this consistent?
- We can make a rough, semi-quantitative estimate using the tunneling probability P, which should tell us dN/dp_T^2 up to a constant. Essentially the ratio K+/\pi+ should reflect the extra cost of producing a \sbar instead of a \dbar quark. (Or, more precisely, an s\sbar pair instead of a d\dbar pair.)
- We use
*constituent*quark masses for the estimate, with m_u = m_d = .325 GeV, and m_s = .450 GeV. In this model, \pi+ => u\dbar and K+ => u\sbar. - P(K+)/P(\pi+) \approx 0.18 with \kappa = 0.9 GeV/fm, so we're in the right ballpark!
- A much more careful calculation is done in Supplement 5.1 of the book.
- The ratio is about 0.08 in nucleon-nucleus collisions at 14.5 GeV. Why? (leading particle effect?)

In this chapter, a model field theory, QED in 1+1 dimensions (1 space plus 1 time dimension) is explored. The last chapter featured a static model; in actual collisions a quark and antiquark will separate at large velocity (close to c). We'd like a model of the production and subsequent hadronization (how the quarks form into hadrons).

Why is QED_2 of interest? Because of the analogs to QCD and because it can be solved!

- QED_2 with massless fermions can be solved exactly.
- The fermion and anti-fermion in the theory are a positive and negative charged electron. The electromagnetic interaction in 1+1 dimensions goes like r (compare 1/r in 3+1), so it is like the string potential we have talked about for QCD.
- As a consequent, the electrons are confined, while the interaction is also asymptotically free (becomes weak at short distances).
- A neutral boson exists as a nonperturbative bound state, like mesons in QCD.
- The model features vacuum polarization with charge screening as in QCD.

It can be shown that QED_2 with interacting fermions is equivalent to a free field theory of a boson field. So we can model the production process by mapping an initial condition of the fermions to simulate a quark-antiquark pair separating at high speed onto the boson field and then evolving it in time.

- We can define the initial charge density as delta functions
at positions z=t and z=-t for t=0 to infinity.

j^0_ext(z,t) = -e \delta(z+t) + e \delta(z-t)

Note that the plus charge is at z=t. - The bound state mass is m=e/sqrt(\pi), so we can choose e so that M=0.40 GeV, which is the average transvers mass for pions produced in high-energy processes.
- Figure 6.2 shows j^0_prod(z,t) [they use x^1 instead of z] at
several fixed times t as a function of z (spatial distribution).
- The + quark goes to the right, the - quark to the left. The produced charges of opposite sign trail behind.
- These in turn make secondary charges of opposite sign, and so on. Screening!

- The boson field \phi_pro(\tau)
- In general would be a function of time t and space z, but in fact only a function of the proper time \tau = \sqrt(t^2-z^2)
- So it has the same value at all (t,z) points with the same \tau.
- The "rise time" of the field determines the production time, which is \tau_pro = 2.405 \hbar/mc^2

- Figure 6.4 shows emergence of particles
- "inside-outside cascade"

- Rapidity distribution is predicted to be a plateau
- Goes like Fourier transform of the initial fermion current
- If close to delta function (all momentum components essentially equal), then constant in y
- A quantitative example is given in supplement 6.7.

- Can we understand this result qualitatively?
- j^\mu(z,t) = -m \epsilon^{\mu\nu} \partial_\nu \phi(z,t)
- \phi(z,t) = 1/sqrt(2\pi) \int d^2p \theta(p^0) \delta(p^2-m^2)
[c(p_z) e^{-ip\cdot x} + c^*(p_z) e^{ip\cdot x}]

1/sqrt(2\pi) \int dp_z 1/(2 p^0) [ ... ]

where in the second equation p^0 = \sqrt(p_z^2 + m^2) - This is a free boson field, so we determine the fourier components c for each p_z at t=0 and this determines the boson field (and hence the charge density) for all subsequent times.
- The key to flatness is the calculation of the energy P_0 as a sum over all p_z modes.
- c(p_z) = -i\sqrt(\pi)/e \tilde j_z(p_z)
- P^0= \int dp_z |c(p_z)|^2/2

= \int dp_z p^0 dN/dp_z

which says it is the sum of the individual energies. - Thus dN/dp_z = |c(p_z)|^2/2p^0
- Thus dN/dp_z = (\pi/2p^0 e^2) |\tilde j_z(p_z)|^2
- pz = m sinh y => dN/dy = dN/dp_z dp_z/dy
- dp_z/dy = m cosh y = p_0
=> dN/dy = \pi/2e^2 |\tilde_z(m sinh y)|^

which is flat if initial distribution is.

The important aspects here are the discussion of strings, which we have moved earlier under Chapter 5, and the overview of the Lund model, which we summarize below.

- Figure 7.1 shows meson-meson scattering in the string picture showing s-channel/t-channel duality (the string diagrams have the same topology hence the same amplitude) => dual resonance model.
- Mesons are described as string segments undergoing longitudinal expansion and contraction with superposed translational motion. There is a quark at one end and an antiquark at the other. "Yo-yo motion."
- Particle production is represented by the fragmentation of a stretching string as the quark and antiquark move apart.
- Lund model particle production indicated in Figure 7.6.
- The text describes the motivation for a splitting function f and a vertex probability function \rho, which combine to describe string fragmentation, and hence particle production.

This chapter extends the discussion of soft (and hard) particle production models. We'll skip it for now.

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Copyright © 1997,1998 Richard Furnstahl and James Steele.