Tracing QCD-Instantons in Deep Inelastic Scattering
ABSTRACT We present a brief status report of our broad and systematic study of QCD-instantons at HERA.
arXiv:hep-ph/9607238v1 4 Jul 1996
Tracing QCD – Instantons
in Deep Inelastic Scattering∗
A. Ringwald and F. Schrempp
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
We present a brief status report of our broad and systematic study
of QCD-instantons at HERA.
∗Talk given at the Workshop DIS96 on “Deep Inelastic Scattering and Related Phe-
nomena”, Rome, Italy, April 15-19, 1996; to be published in the proceedings.
Instantons  are well known to represent tunnelling transitions in non-
abelian gauge theories between degenerate vacua of different topology. These
transitions induce processes which are forbidden in perturbation theory, but
have to exist in general  due to Adler-Bell-Jackiw anomalies. An exper-
imental discovery of such a novel, non-perturbative manifestation of non-
abelian gauge theories would clearly be of basic significance.
Searches for instanton-induced processes received new impulses during
recent years: First of all, it was shown  that the natural exponential sup-
pression of these tunnelling rates, ∝ exp(−4π/α), may be overcome at high
energies. Furthermore, deep inelastic scattering (DIS) at HERA now offers a
unique window [4, 5, 6] to experimentally detect processes induced by QCD-
instantons. Here, a theoretical estimate of the corresponding production
rates appears feasible as well [4, 7], since a well defined instanton contribu-
tion in the regime of small QCD-gauge coupling may be isolated on account
of the photon virtuality Q2.
In this brief status report we concentrate on a first, preliminary esti-
mate of the rate for instanton-induced events  and some characteristics
of the instanton-induced final state along with new search strategies .
These new results are based on our instanton Monte-Carlo generator [6, 9]
2Instanton-Induced Cross Sections
The instanton (I) contribution to the nucleon structure functions is described
in terms of the standard convolution of parton-structure functions, e.g. F2g,
with corresponding parton densities. The I-contribution to the (dominating)
gluon-structure function F2garises from the γ∗g matrix element as displayed
in Fig. 1. The apparent structure of an I-subprocess, denoted by “I” in
Fig. 1, is due to the fact that the virtual photon only couples to instantons
via it’s quark content. We find , that the I-contribution to the gluon-
structure functions may be expressed in terms of the I-subprocess total cross
? ?Q2 x′
current quark jet
in final state!
Instanton-induced contribution to the cross section of γ∗g scat-
The integrations in Eq. 1 extend over the Bjorken variables Q′2= −q′2and
x′= Q′2/(s′+ Q′2) ≥ x ≥ xBj, referring to the I-subprocess.
I-Subprocess Cross Section
A standard evaluation  leads to the following result ,
The running scale µ(Q′) in αs, satisfying µ(Q′) = κQ′αs(Q′)/(4π) with κ =
O(1), plays the rˆ ole of an effective renormalization scale. The x′dependence
resides in the functions Σ(x′) and the so-called “holy-grail” function F(x′) ≤
1, which are both known as low-energy expansions in s′/Q′2= (1−x′)/x′≪ 1
within conventional I-perturbation theory. Their form implies a rapid growth
relevant region of small x′, the perturbative expressions are of little help and
we have to ressort to some extrapolation.
A distinguished possibility to go beyond instanton perturbation theory is
the II-valley approximation [11, 10] which we have adopted. It amounts to
the identification of the holy-grail function with the known II-valley action.
It appears reasonable to trust this method down to x′= 0.2, where F(0.2) ≡
q∗gfor decreasing x′. Unfortunately, in the phenomenologically most
SII−valley(0.2) ≃ 1/2, a value sometimes advocated  as the lower bound
for the holy-grail function. An important phenomenological/experimental
task will be to make sure (e.g. via kinematical cuts to the final state) that
x′does not become too small.
Note the following important feature of σ(I)
Q′dependences from the high inverse power of αs and the exponential in
Eq. 2 compete to produce a strong peak far away from the IR region, e.g.
over Q′2(c.f. Eq. 1), is dominated by this peak and hence Q independent
(scaling) in the Bjorken limit. The predicted approach to this scaling limit
resembles a “fractional twist” term, where the twist is sliding with x: the
scaling violations vanish faster for increasing x.
q∗gas a function of Q′: The
peak(x′= 0.2) ≈ 31Λ. This implies that F(I)
2g, which involves the integral
2.2HERA Cross Section
In Fig. 2 (left) we present the resulting I-induced total cross section for
HERA for two values (0.2,0.3) of the lower x′cut (c.f. discussion in Sec. 2.1),
as a function of the minimal Bjorken x, xBj min, considered. It is surpris-
ingly large. So far, only the (dominating) gluon contribution has been taken
into account. The inherent uncertainties associated with the renormaliza-
tion/factorization scale dependences may be considerable and are presently
being investigated. Therefore, Fig. 2 is still to be considered preliminary.
3Final-State Signatures and New Search
The typical event (Fig. 2 (right)) from our Monte-Carlo generator 
QCDINS 1.3 based on HERWIG 5.8, illustrates most of the important fea-
tures characteristic for the underlying instanton mechanism: A current-quark
jet along with a densely populated hadronic “band” of width △η = ±0.9 in
the (ηlab,φlab)-plane . The band reflects the isotropy in the I-rest system.
The total ET= O(20) GeV is large as well as the multiplicity, Nband= O(25).
Finally, there is a characteristic flavor flow: All (light) flavors are democrati-
cally represented  in the final state. Therefore, strongly enhanced rates of
K0’s and µ’s (from strange and charm decays) represent crucial signatures
for I-induced events.
HERA(x’ ≥ x’min, xBj ≥ xBj min, yBj ≥ 0.1 ) [ nb ]
various cuts. Right: Lego plot (ηlab,φlab,ET[GeV]) of a typical I-induced
event in the HERA-lab system at xBj= 10−3.
Left: I-induced total cross section for HERA (preliminary) with
A first, preliminary 95% CL upper limit of 0.9 nb on the I-induced cross
section at HERA has been obtained by the H1 collaboration by searching for
an excess in the K0rate .
Let us finally mention some recent attempts  to improve the sensitiv-
ity to I-induced events by adding in characteristic information on the event
shapes. The first step consists in boosting to the γ∗-proton c.m. system and
looking for events with high ET (c.f. Fig. 3 (left)). We note that in this
system 1 and 2 jet (hard) perturbative QCD processes deposit their energy
predominantly in a plane passing through the γ∗-proton direction. In con-
trast, the energies from I-induced events are always distributed much more
spherically (isotropy in the I-rest system!). Therefore, one may substantially
reduce the normal DIS background by looking at
i.e. by minimizing Eoutby choice of ? n, normal to the γ∗-proton direction. For
standard boson-gluon fusion 2 jet events, Eoutis given by the jet widths. In
contrast, for I-induced events Eout≃√s′/2 is large. The quantitative results
from the Monte-Carlo simulation, subject to additional cuts in η which are