Proceedings TH2002 Supplement (2003) 245 – 252
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/040245-8 $ 1.50+0.20/0
Proceedings TH2002
Topologically Nontrivial Events in High Energy
Hadron Collisions
Dmitry Ostrovsky
Abstract.
I describe a hadron collision event in which color field undergoes classically prohibited forced tunneling (analogous to instanton tunneling) and appear as classical
chromo-magnetic configuration (analogous to electroweak sphaleron). This classical
state then evolves according to classical Yang-Mills equations in real (Minkowski)
space producing a few gluons and light quarks. Phenomenological relevance to
hadron and nuclear collisions is discussed.
The talk is based on: G. W. Carter, D. M. Ostrovsky, E. V. Shuryak, Instantoninduced Semi-hard Parton Interactions and Phenomenology of High Energy Hadron
Collisions. Phys. Rev. D66 (2002) 036004
I have organized this note with the attempt to preserve the style of the original
presentation at the Conference. Text originally appeared on transparencies is given
here in a bold face.
1 Preface: in electroweak theory
We begin with recalling a few points about electorweak sphaleron, the development
which took place in the early 90s. QCD analogue, which we are interested in, in
many ways is similar to its electroweak kin. It will be useful also to emphasize the
differences between two constructions.
• Vacuum tunneling (instanton) in electroweak theory leads to nonconservation of barion number. The possibility of such spectacular event was
the primary source of interest in topologically nontrivial quasiclassical solutions in electroweak theory [1]. In QCD, observables, which can serve as a
smoking gun for tunneling events have yet to be revealed. However,
• Probability of 1-instanton event in EW is e−2S ≈ e−100 and therefore
is impossible to observe.
• Ringwald and Espinosa (early 90s): if tunneling occurs with nonzero
energy, the probability may be much higher due to final particles
phase space[2]. The original papers started a several years search for the
energy dependence of quasiclassical tunneling probability. The findings are
summarized in [3] and
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• The answer: probability (cross section) has a maximum at sphaleron
energy and is about e−S . This result is a balance of two competing tendencies. The higher is the energy of the collision the more particles are produced
and the larger is a final phase space. On the other hand, the higher is the
energy the more difficult is the task of packing it into the particular quasiclassical state. Thus, the most probable configuration is the
• Sphaleron – non-stable time independent solution of Yang-Mills-Higgs
equations, the top of minimal pass of magnetic-type fields connecting
topologically different vacua
2 From EW to QCD
Going from EW to QCD we should address several points, such as how to implement color SU(3) group with topologically nontrivial projection in 4 dimensions,
how to deal with scale invariance of QCD, what is our primary aim in terms of
observables.
• We are working with SU (2) subgroup of colour SU (3) group, it has
nontrivial topological projection on 4D space. The rest of the SU (3)
group can be taken to account by embeddings.
• Scale invariance is broken by instanton size parameter ρ ∼ 0.3fm.
There is a vast literature about instanton properties of QCD. We are taking
the parameters of instanton liquid model presented in [4].
• Probability of the tunneling is defined by instanton gas diluteness
parameter n0 = nρ4 ∼ 0.01 In instanton calculations the diluteness parameter refers to tunneling amplitude. Probability is ∼ n20 . However, the
same mechanism as in EW case brings down one power of n0 for nonzero
energy events.
• Main point to look at: final state. It should be quite different from those
proposed by string model or perturbative QCD. It has to bear a very high
degree of coherence resembling the production of a resonance. We will not
come here to further details.
3 Instanton-induced events vs. pQCD
• Characteristic momentum transferred for instanton-induced processes
Q ∼ 2GeV. Roughly coincides with minijet scale. Which implies that
instanton-induced events are playing in the same phenomenological field as
minijets, at the borderline between perturbative and nonperturbative QCD.
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Topologically Nontrivial Events in High Energy Hadron Collisions
247
• Multiparticle instanton-induced production is NOT suppressed by
powers of αs . This is quite clear because instanton field is proportional
to 1/g. Now we can expect relatively more instanton-induced events among
those with high multiplicities. It can be one of the key points which reveal
to us the final state of interest.
• Final state in instanton-induced processes is highly coherent. And
therefore, can be distinguished in a background of other sources of particles.
4 Three stages of quasiclassical evolution
E
classical barrier
turning state
real time
evolution
forced tunneling
initial state
0
n
1
CS
We will now describe instanton-induced event expected in a high energy collision as a three stage process.
1. Forced tunneling. Quasiclassical field undergoes virtual path under
the potential barrier. Theoretical description of this process is still highly
speculative. One way to look at it is instanton-antiinstanton pair describing
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2 sides of the cut in cross section calculations [5]. Another approach is based
on analytical continuation of quasiclassical fields to the singular point in
which an energy jump can occur [6]. Tunneling ends at the
2. Turning state. The minimal potential energy state with Chern-Simons
number and characteristic size fixed. Kinetic energy equals 0. At this
point quasiclassical field leaves clasically prohibited region and returns to
Minkowski space. Chern-Simons number serves as a topological coordinate.
Minimum of energy insures maximum probability for the event of a characteristic size with fixed Chern-Simons number.
3. Turning state explosion. Occupation number diminishes and problem
becomes essentially quantum. However, in the early stage it can be studied
by means of classical Yang-Mills equation, which provides an input for further
(quantum) considerations
In this report we will discuss the second and the third stages in some details.
5 Turning states from constrained minimization
We now find the continuum of possible turning states parameterized by ChernSimons number and characteristic size. These states will further serve as initial
conditions for the Cauchy problem for Yang-Mills equation.
Let us take space components of gauge field Ai in A0 = 0 gauge as coordinates. Then, chromo-electric field E serves as conjugate momentum, and at the
turning point they must vanish. Chromo-magnetic field B plays rôle of potential
energy.
Problem: find the minimum of energy
1
E= 2
4g
d3 xB 2
(chromo-electric field E = 0) for fixed
a) Chern-Simons number
NCS =
d3 xK0 =
1
16π 2
1
d3 x Bia Aai − abc ijk Aai Abj Ack
6
b) mean radius of the field
r
2
=
d xr B
3
2
2
d3 xB 2
The last condition is necessary because of (classical) scale invariance of QCD
Lagrangean and Chern-Simons number. We are fixing turning state scale in order
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Topologically Nontrivial Events in High Energy Hadron Collisions
249
to comply with phenomenological observations (see [4]) and obtain finite solution
for our minimal energy problem.
The Lagrange function
EL = E + κNCS +
Solution:
Bia = 4ρ2
E=
3π 2
(1 − κ2 )2 ,
g2 ρ
E 2
r
ρ2
1 − κ2 a
δ
(r 2 + ρ2 )2 i
r 2 = ρ2 ,
NCS =
sign(κ)
(2 + |κ|)(1 − |κ|)2
4
E/g2ρ
30
20
10
0
0
0.2
0.4 0.6
Ncs
0.8
1
6 Explosion
• Starting from turning state one solves Yang-Mills equation in
Minkowski space
• For initial states (turning points) in hand the solution can be found an-
alytically and appear to be the elliptic solution (Lüscher and Schechter,
1977)[7]
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D. Ostrovsky
• Energy profile rapidly acquires asymptotic form
8π
r E(r, t) = 2 2 (1 − κ2 )2
g ρ
2
Proceedings TH2002
ρ2
ρ2 + (r − t)2
3
• Chern-Simons number changes on non integer amount. NCS (0) is non-
integer and NCS (∞) is non-integer
r2E(r,t)
3
t=0
t=10
2
t=20
1
0
0
5
10
r/ρρ
15
20
25
Energy spectrum is gauge dependant. For the gauge with minimal
energy per particle
32
E(ω) = 2 ((1 − κ2 )ωρK1 (ωρ)))2
g
which means that occupation number diverges as
Ng ∼ ln(1/ωρ)
the problem is solved by introduction of saturation limit. For gluon effective
mass Mg ∼ 0.4GeV it gives Ng ≈ 4.5
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Topologically Nontrivial Events in High Energy Hadron Collisions
251
7 Conclusions
• High energy hadron collision may lead to the tunneling through topo-
logical barrier of a quasiclassical field
• Turning states can be obtained by constrained minimization of po-
tential energy of Yang-Mills field
• This state evolves into an expanding shell, which subsequently turns
to several gluons
8 Questions
I have selected the questions that are not only of interest, but also can be successfully addressed in the near future.
• What is the cross section? How to describe forced tunneling?
• How to include quarks?
• What is the phenomenological relevance?
Thank you!
References
[1] F. R. Klinkhamer and N. S. Manton, Phys. Rev. D 30, 2212 (1984).
[2] A. Ringwald, Nucl. Phys. B 330, 1 (1990).
O. Espinosa, Of Unitarity In The Standard Model,” Nucl. Phys. B 343, 310
(1990).
[3] M. P. Mattis, Phys. Rept. 214, 159 (1992).
[4] T. Schafer and E. V. Shuryak, Rev. Mod. Phys. 70, 323 (1998) [arXiv:hepph/9610451].
[5] A. V. Yung, Nucl. Phys. B 297, 47 (1988); M. A. Nowak, E. V. Shuryak and
I. Zahed, Phys. Rev. D 64, 034008 (2001); G. W. Carter, D. M. Ostrovsky and
E. V. Shuryak, Phys. Rev. D 65, 074034 (2002)
[6] D. Diakonov and V. Petrov, Phys. Rev. D 50, 266 (1994); R. Janik, E. Shuryak,
I. Zahed [arXiv:hep-ph/0206005]
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[7] M. Luscher, Phys. Lett. B 70, 321 (1977). B. M. Schechter, Phys. Rev. D 16,
3015 (1977).
Dmitry Ostrovsky
State University of New York at Stony Brook