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Confronting QCD Instantons with HERA Data

The sensitivity of existing HERA data on the hadronic final state in deep-inelastic scattering (DIS) to processes induced by QCD instantons is systematically investigated. The maximally allowed fraction of such processes in DIS is found to be on the percent level in the kinematic domain 0.0001<x-Bjorken<0.01 and 5<Q squared<100 GeV squared. The best limits are obtained from the multiplicity distribution.


I INTRODUCTION
The standard model contains hard processes which cannot be described by perturbation theory, and which violate classical conservation laws like baryon and lepton number B + L in the case of the electroweak sector and chirality in the case of strong interactions [1]. Such anomalous processes are induced by instantons [2].
Observable instanton effects in electroweak processes are only expected to be sizable in the multi-TeV region and are therefore out of the present experimental reach. QCD instantons, however, could be observed in deep-inelastic scattering (DIS) at HERA which collides 27.5 GeV positrons on 820 GeV protons [3][4][5][6].
The theoretical background of the cross-section calculation and aspects of instanton phenomenology based on a Monte Carlo simulation [7] for the QCD instanton have been discussed by F. Schrempp and A. Ringwald [8,9] at this conference. In this contribution limits on instanton production are derived from the most sensitive observables in the hadronic final state of DIS events, namely the multiplicity distributions [10], the transverse energy flows [11] and hard particle production [12].

II INSTANTON PRODUCTION AT HERA
At HERA, events induced by QCD instantons predominantly invoke a quark-gluon fusion process. The total cross-section is given by a convolution of the probability to find a gluon in the proton P g/p , the cross-section σ q * g (x ′ , Q ′2 ) of the instanton induced sub-process and the probability that a photon splits in a quark-antiquark pair in the instanton background P (I) q * /γ [5,6]. Besides the squared transverse momentum transfer Q 2 and the Bjorkenx scaling variable, this scattering process is characterized by Q ′2 , the virtuality of the quark (q * ), and x ′ , the Bjorken scaling variable associated with the q * g sub-process. σ q * g (x ′ , Q ′2 ) decreases with increasing Q ′2 and exponentially grows with decreasing x ′ . Due to inherent ambiguities -in particular the exponential behaviour at low x ′ -the calculation is only reliable for x ′ 0.2 and Q ′2 25 GeV 2 . Since the highest event rate is expected in the low-x regime, DIS data down to x ≈ 10 −4 are analyzed in this study. When comparing to theory predictions, the phase space is restricted to the theoretically controllable regime by imposing limits on (x ′ , Q ′2 ). The characteristics of the hadronic final state of instanton induced events are considered to be more robust than predictions of the absolute event rate [5]. To investigate the sensitivity of the event topology to x ′ and Q ′2 , different instanton scenarios with varying cut-off parameters are examined.

III EXPERIMENTAL SIGNATURE
In its own rest frame the instanton isotropically decays into a multi-parton state, consisting of gluons and all quark flavours which are kinematically allowed. After fragmentation of the semi-hard partons, a densely and uniformly populated band of particles with characteristic flavours is expected in a certain pseudo-rapidity (η) region.
This band of high hadronic activity can be identified in the mean transverse energy flow (dE T /dη) (see Fig. 1). While the data exhibit a plateau of about  For large M ins also the transverse momentum spectrum of charged particles becomes harder than for normal DIS (see Fig. 2). A distinct signature of instanton events is a large particle multiplicity. In Fig. 2 the distribution of the charged particles multiplicity in 1 < η < 5 for large W ≈ Q 2 /x is shown. The mean of this distribution logarithmically depends on M 2 ins .

IV BOUNDS FROM HERA DATA
The standard DIS Monte Carlo simulation program ARIADNE [13] provides an excellent description of all available HERA data on the hadronic final state [14]. Upper bounds on the cross-section of instanton induced events (σ lim ) in DIS can be determined from their distinct event topology. Assuming that the data on dE T /dη consist of normal DIS together with a certain fraction of instanton events, the maximally allowed fraction of instanton events (f lim ) can be extracted using a χ 2 procedure. An example of a superposition of normal DIS with the fraction of instanton events which can be excluded at 95% confidence level (C.L.) is shown in Fig. 1 (dashed dotted line). In Fig. 3 the results for instantons with x ′ > 0.2 and Q ′2 > 25 GeV 2 are summarized (open fields). For 10 −4 x 10 −2 and 10 < Q 2 < 50 GeV 2 , f lim values of about 5 − 10% can be excluded. This corresponds to cross-section limits of ≈ 200 − 800 pb. The best limits are obtained in the domain of higher x and Q 2 . These results can vary by at most a factor 2 when using different choices in the details of the DIS model 2 . Here, however, we used the most conservative choices.
Any dependence on the DIS model can be avoided by exploiting the fact that no events with very high charged particle multiplicity have been observed in the data. For such high multiplicies, the detection efficiency for instanton events is still reasonably high (8 − 20%), see Fig. 2). At 95% C.L. upper limit, f lim , is then obtained from f lim = 3/(ǫ i N DIS ) where ǫ i is the efficiency to detect instanton events at large multiplicity and N DIS is the total number of observed DIS events. The obtained limits in different W bins are displayed in Fig. 3 (shaded areas). For 80 < W < 220 GeV, values f lim ≈ 0.5% and 40 σ lim 80 pb are obtained.
These bounds are not yet stringent enough to test the theory which predicts an instanton fraction of O(0.01%). In the theoretically unsafe region (low x ′ ), where the instanton fraction is expected to be large, limits to constrain the theory can also be obtained.

V CONCLUSION
The observation of instanton effects in DIS events at HERA would be a novel, non-perturbative manifestation of QCD and would furthermore provide valuable indirect information about B + L violation in the multi-TeV region induced by electroweak instantons. The distinct event topology of instanton induced events allows to discriminate them from normal DIS events. Using existing HERA data on the hadronic final state corresponding to an integrated luminosity of O(1 pb −1 ), the maximally allowed fraction of instantons in DIS is found to be of O(1%) for 80 < W < 220 GeV and x ′ > 0.2 and Q ′2 > 25 GeV 2 . The predicted instanton fraction is ≈ 0.01 − 0.02%, i.e. still below the level excluded by existing HERA data. The higher luminosity of O(10 pb −1 ) already delivered by HERA and dedicated instanton searches will allow to test the absolute prediction of the cross-section in the near future.