Search for contact interactions, large extra dimensions and finite quark radius in ep collisions at HERA

A search for physics beyond the Standard Model has been performed with high-Q^2 neutral current deep inelastic scattering events recorded with the ZEUS detector at HERA. Two data sets, e^+ p \to e^+ X and e^- p \to e^- X, with respective integrated luminosities of 112 pb^-1 and 16 pb^-1, were analyzed. The data reach Q^2 values as high as 40000 GeV^2. No significant deviations from Standard Model predictions were observed. Limits were derived on the effective mass scale in eeqq contact interactions, the ratio of leptoquark mass to the Yukawa coupling for heavy leptoquark models and the mass scale parameter in models with large extra dimensions. The limit on the quark charge radius, in the classical form factor approximation, is 0.85 10^-16 cm.


Introduction
The HERA ep collider has extended the kinematic range of deep inelastic scattering (DIS) measurements by two orders of magnitude in Q 2 , the negative square of the fourmomentum transfer, compared to fixed-target experiments. At values of Q 2 of about 4 × 10 4 GeV 2 , the eq interaction, where q is a constituent quark of the proton, is probed at distances of ∼ 10 −16 cm. Measurements in this domain allow searches for new physics processes with characteristic mass scales in the TeV range. New interactions between e and q involving mass scales above the center-of-mass energy can modify the cross section at high Q 2 via virtual effects, resulting in observable deviations from the Standard Model (SM) predictions. Many such interactions, such as processes mediated by heavy leptoquarks, can be modelled as four-fermion contact interactions. The SM predictions for ep scattering in the Q 2 domain of this study result from the evolution of accurate measurements of the proton structure functions made at lower Q 2 . In this paper, a common method is applied to search for four-fermion interactions, for graviton exchange in models with large extra dimensions, and for a finite charge radius of the quark.
In an analysis of 1994-97 e + p data [1], the ZEUS Collaboration set limits on the effective mass scale for several parity-conserving compositeness models. Results presented here are based on approximately 130 pb −1 of e + p and e − p data collected by ZEUS in the years 1994-2000. Since this publication also includes the early ZEUS data, the results presented here supersede those of the earlier publication [1].

Standard Model cross section
The differential SM cross section for neutral current (NC) ep scattering, e ± p → e ± X, can be expressed in terms of the kinematic variables Q 2 , x and y, which are defined by the four-momenta of the incoming electron 1 (k), the incoming proton (P ), and the scattered electron (k ′ ) as Q 2 = −q 2 = −(k − k ′ ) 2 , x = Q 2 /(2q · P ), and y = (q · P )/(k · P ). For unpolarized beams, the leading-order electroweak cross sections can be expressed as where α is the electromagnetic coupling constant. The contribution of the longitudinal structure function, F L (x, Q 2 ), is negligible at high Q 2 and is not taken into account in this analysis. At leading order (LO) in QCD, the structure functions F NC 2 and xF NC 3 are 1 Unless otherwise specified, 'electron' refers to both positron and electron.
given by where q(x, Q 2 ) and q(x, Q 2 ) are the parton densities for quarks and antiquarks. The functions A q and B q are defined as where the coefficient functions V L,R q and A L,R q are given by: In Eq. (2), the superscript i denotes the left (L) or right (R) helicity projection of the lepton field; the plus (minus) sign in the definitions of V i q and A i q is appropriate for i = L(R). The coefficients v f and a f are the SM vector and axial-vector coupling constants of an electron (f = e) or quark (f = q); Q f and T 3 f denote the fermion charge and third component of the weak isospin; M Z and θ W are the mass of the Z 0 and the electroweak mixing angle, respectively.

General contact interactions
Four-fermion contact interactions (CI) represent an effective theory, which describes lowenergy effects due to physics at much higher energy scales. Such models would describe the effects of heavy leptoquarks, additional heavy weak bosons, and electron or quark compositeness. The CI approach is not renormalizable and is only valid in the lowenergy limit. As strong limits have already been placed on scalar and tensor contact interactions [2], only vector currents are considered here. They can be represented by additional terms in the Standard Model Lagrangian, viz: where the sum runs over electron and quark helicities and quark flavors. The couplings η eq ij describe the helicity and flavor structure of contact interactions. The CI Lagrangian (Eq. (3)) results in the following modification of the functions V i q and A i q of Eq. (2): It was assumed that all up-type quarks have the same contact-interaction couplings, and a similar assumption was made for down-type quarks 2 : leading to eight independent couplings, η eq ij , with q = u, d. Due to the impracticality of setting limits in an eight-dimensional space, a set of representative scenarios was analyzed. Each scenario is defined by a set of eight coefficients, ǫ eq ij , each of which may take the values ±1 or zero, and the compositeness scale Λ. The couplings are then defined by Note that models that differ in the overall sign of the coefficients ǫ eq ij are distinct because of the interference with the SM.
In this paper, different chiral structures of CI are considered, as listed in Table 1. Models listed in the lower part of the table were previously considered in the published analysis of 1994-97 e + p data [1]. They fulfill the relation η eq LL + η eq LR − η eq RL − η eq RR = 0 , which was imposed to conserve parity, and thereby complement strong limits from atomic parity violation (APV) results [3,4]. Since a later APV analysis [5] indicated possible deviations from SM predictions, models that violate parity, listed in the upper part of Table 1, have also been incorporated in the analysis. The reported 2.3σ deviation [5] from the SM was later reduced to around 1σ, after reevaluation of some of the theoretical corrections [6,7].

Leptoquarks
Leptoquarks (LQ) appear in certain extensions of the SM that connect leptons and quarks; they carry both lepton and baryon numbers and have spin 0 or 1. According to the general classification proposed by Buchmüller, Rückl and Wyler [8], there are 14 possible LQ states: seven scalar and seven vector 3 . In the limit of heavy LQs (M LQ ≫ √ s), the effect of s-and t-channel LQ exchange is equivalent to a vector-type eeqq contact interaction 4 . The effective contact-interaction couplings, η eq ij , are proportional to the square of the ratio of the leptoquark Yukawa coupling, λ LQ , to the leptoquark mass, M LQ : where the coefficients a eq ij depend on the LQ species [11] and are twice as large for vector as for scalar leptoquarks. Only first-generation leptoquarks are considered in this analysis, q = u, d. The coupling structure for different leptoquark species is shown in Table 2. Leptoquark models S L 0 andS L 1/2 correspond to the squark statesd R andũ L , in minimal supersymmetric theories with broken R-parity.

Large extra dimensions
Arkani-Hamed, Dimopoulos and Dvali [12][13][14] have proposed a model to solve the hierarchy problem, assuming that space-time has 4 + n dimensions. Particles, including strong and electroweak bosons, are confined to four dimensions, but gravity can propagate into the extra dimensions. The extra n spatial dimensions are compactified with a radius R. The Planck scale, M P ∼ 10 19 GeV, in 4 dimensions is an effective scale arising from the fundamental Planck scale M D in D = 4 + n dimensions. The two scales are related by: For extra dimensions with R ∼ 1 mm for n = 2, the scale M D can be of the order of TeV. At high energies, the strengths of the gravitational and electroweak interactions can then become comparable. After summing the effects of graviton excitations in the extra dimensions, the graviton-exchange contribution to eq → eq scattering can be described as a contact interaction with an effective coupling strength of [15,16] where M S is an ultraviolet cutoff scale, expected to be of the order of M D , and the coupling λ is of order unity. Since the sign of λ is not known a priori, both values λ = ±1 are considered in this analysis. However, due to additional energy-scale dependence, reflecting the number of accessible graviton excitations, these contact interactions are not equivalent to the vector contact interactions of Eq. (3). To describe the effects of graviton exchange, terms arising from pure graviton exchange (G), graviton-photon interference (γG) and graviton-Z (ZG) interference have to be added to the SM eq → eq scattering cross section [17]: whereŝ,t andû, witht = −Q 2 , are the Mandelstam variables, while the other coefficients are given in Eq.
(2). The corresponding cross sections for e ±q scattering are obtained by changing the sign of Q q and v q parameters.
Graviton exchange also contributes to electron-gluon scattering, eg → eg, which is not present at leading order in the SM: For a given point in the (x, Q 2 ) plane, the e ± p cross section is then given by where q(x, Q 2 ),q(x, Q 2 ) and g(x, Q 2 ) are the quark, anti-quark and gluon densities in the proton, respectively.

Quark form factor
Quark substructure can be detected by measuring the spatial distribution of the quark charge. If Q 2 ≪ 1/R 2 e and Q 2 ≪ 1/R 2 q , the SM predictions for the cross sections are modified, approximately, to: where R e and R q are the root-mean-square radii of the electroweak charge of the electron and the quark, respectively.

Data samples
The data used in this analysis were collected with the ZEUS detector at HERA and correspond to an integrated luminosity of 48 pb −1 and 63 pb −1 for e + p collisions collected in 1994-97 and 1999-2000 respectively, and 16 pb −1 for e − p collisions collected in 1998-99. The 1994-97 data set was collected at √ s = 300 GeV and the 1998-2000 data sets were taken with √ s = 318 GeV.
The analysis is based upon the final event samples used in previously published cross section measurements [18][19][20]. Only events with Q 2 > 1000 GeV 2 are considered. The SM predictions were taken from the simulated event samples used in the cross section measurements, where selection cuts and event reconstruction are identical to those applied to the data. Neutral current DIS events were simulated using the Heracles [21] program with Djangoh [22,23] for electroweak radiative corrections and higher-order matrix elements, and the color-dipole model of Ariadne [24] for the QCD cascade and hadronization. The ZEUS detector was simulated using a program based on Geant 3.13 [25]. The details of the data selection and reconstruction, and the simulation used can be found elsewhere [18][19][20].
The distributions of NC DIS events in Q 2 , measured separately for each of the three data sets, are in good agreement with SM predictions calculated using the CTEQ5D parameterization [26,27] of the parton distribution functions (PDFs) of the proton. The CTEQ5D parameterization is based on a global QCD analysis of the data on high energy lepton-hadron and hadron-hadron interactions, including high-Q 2 H1 and ZEUS results based on the 1994 e + p data. The ZEUS data used in the CTEQ analysis amount to less than 3% of the sample considered in this analysis. In general, SM predictions in the Q 2 range considered here are dominantly determined by fixed-target data at Q 2 < 100 GeV 2 and x > 0.01 [28].
5 Analysis method

Monte Carlo reweighting
The contact interactions analysis was based on a comparison of the measured Q 2 distributions with the predictions of the MC simulation. The effects of each CI scenario are taken into account by reweighting each MC event of the type ep → eX with the weight The weight w was calculated as the ratio of the leading-order 5 cross sections, Eq. (1), evaluated at the true values of x and Q 2 as determined from the four-momenta of the exchanged boson and the incident particles. In simulated events where a photon with energy E γ is radiated by the incoming electron (initial-state radiation), the electron energy is reduced by E γ . This approach guarantees that possible differences between the SM and the CI model in event-selection efficiency and migration corrections are properly taken into account. Under the assumption that the difference between the SM predictions and those of the model including contact interactions is small, higher-order QCD and electroweak corrections, including radiative corrections, are also accounted for.

Limit-setting procedure
For each of the models of new physics described above, it is possible to characterize the strength of the interaction by a single parameter: 4π/Λ 2 for contact interactions; (λ LQ /M LQ ) 2 for leptoquarks; λ/M 4 S for models with large extra dimensions; and R 2 q for the quark form factor. In the following, this parameter is denoted by η. For contact interactions, models with large extra dimensions and the quark form factor model, scenarios with positive and negative η values were considered separately. For a given model, the likelihood was calculated as where the product runs over all Q 2 bins, n i is the number of events observed in Q 2 bin i and µ i (η) is the expected number of events in that bin for a coupling strength η. The likelihood for the complete e ± p data set was obtained by multiplying the likelihoods for each of the three running periods.
The value of η for which L(η) is maximized is denoted as η • . First η data • , the value of η that best describes the observed Q 2 spectra was determined. Using ensembles of Monte Carlo experiments (MCE), the expected distribution of η • was then determined as a function of η M C , the coupling value used as the input to the simulation. The 95% C.L. limit on η was defined as the value of η M C for which the probability that |η • | > |η data • | was 0.95.
For each value of η M C , the nominal number of events expected in each Q 2 bin i, denoted µ i (η M C ) was calculated by reweighting the SM MC prediction according to Eq. (4). Theoretical and experimental systematic uncertainties were taken into account by treating each uncertain quantity as a random variable. For each uncertainty, 100% correlation between systematic variations in different bins was assumed. For each individual MCE, an independent random variable, δ j , with zero mean, was generated for each systematic uncertainty j. The expected number of events in each Q 2 bin i was then given by the product of the nominal expectation,μ i , and N sys random factors which account for the uncertainties in the estimation of µ i as follows: The coefficent c ij is the fractional change in the expected number of events in bin i for a unit change in δ j . This definition of µ i reduces to a linear dependence of µ i on each δ j when δ j is small, while avoiding the possibility of µ i becoming negative which would arise if µ i was defined as a linear function of the δ j 's. For most of the systematic uncertainties, δ j follows a Gaussian distribution, except for a few where it follows a uniform distribution, as noted in the next section. For a Gaussian δ j distribution, the definition of µ i corresponds to a Gaussian distribution in log µ i . About one million MCEs were generated for each model, so that the statistical error was negligible.

Systematic uncertainties
Uncertainties in the SM cross sections considered in this study were estimated using the Epdflib program [29] based on Qcdnum [30]. Fractional variations estimated from Epdflib were used to rescale the nominal SM expectations calculated with CTEQ5D. The following uncertainties were included: • statistical and systematic uncertainties of the data used as an input to the NLO QCD fit. These errors were the largest uncertainty in the SM expectations. At high Q 2 , the uncertainty is up to about 4.5% (3%) for e + p (e − p) data; • uncertainty in the value of α S (M 2 Z ) used in the NLO QCD fit. The resulting uncertainties of NC DIS cross sections at high Q 2 , estimated assuming an error on α S (M 2 Z ) of ±0.002 [31], is about 1.6%; • uncertainties in the nuclear corrections applied to the deuteron data (K D ) and to the data from neutrino scattering on iron (K F e ) used in Qcdnum. As suggested in Epdflib, variations by up to 100% for K D and 50% for K F e were applied, treating the corrections as uniformly distributed random variables. The corresponding uncertainties of NC DIS cross sections at high Q 2 , are up to about 1.7% (0.8%) for K D and up to about 3% (0.7%) for K F e , for e + p (e − p) data.
The PDF uncertainties calculated using Epdflib are similar to those obtained from a ZEUS NLO QCD fit [28], when high-Q 2 HERA data were excluded from the fit.
In addition to the uncertainty in the SM prediction, the following experimental uncertainties were taken into account: • the scale uncertainty on the energy of the scattered electron of ±(1-3)% depending on the topology of the event [32]. The resulting uncertainty of NC DIS cross section at high Q 2 is about 0.6% (1.3%), for e + p (e − p) data; • the uncertainty in the hadronic energy scale of ±(1-2)% depending on the topology of the event [33]. The resulting cross section uncertainty at high Q 2 is about 1%, for both e + p and e − p data; • uncertainties on the luminosity measurement of 1.6% for the 1994-97 e + p data, 1.8% for the 1998-99 e − p data and 2.5% for the 1999-2000 e + p data. Correlations between luminosity uncertainties for different data-taking periods are small and were neglected in the analysis.
As the double-angle method used to reconstruct the kinematics of the events [18][19][20] is relatively insensitive to uncertainties in the absolute energy scale of the calorimeter, the largest experimental uncertainty in the numbers of NC DIS events expected at high Q 2 is due to the luminosity measurement.

Results
No significant deviation of the ZEUS data from the SM prediction using the CTEQ5D parameterization of the proton PDF was observed. For all models considered, the best description of the data was obtained for very small values of |η data • |, i.e. close to the SM. The probability of obtaining larger best-fit coupling from the SM, i.e. the probability that an experiment would produce a value of |η • | greater than that obtained from the data, |η • | > |η data • |, calculated with MCEs assuming the SM cross section, was above 25% in all cases. Therefore, limits on the strength parameters of the models described in Sec. 3 are presented in this paper.
The measured Q 2 spectra for e + p and e − p data, normalized to the SM predictions are shown in Fig. 1. Also shown are curves, for VV and AA contact-interaction models (Section 3.1), which correspond to the 95% C.L. exclusion limits on Λ. The 95% C.L. limits on the compositeness scale Λ, for different CI models, are compared in Fig. 2 and Table 1 The 95% C.L. lower limits on the compositeness scale Λ are compared in Table 1 with limits from the H1 collaboration [34], the Tevatron [35,36] and the LEP [37][38][39][40] experiments (where only the results from e + e − → qq channel are quoted). In Table 1 the relations between CI couplings for the compositeness models considered are also included. The results on the compositeness scale Λ presented here are comparable to those obtained by other experiments, where they exist. For many models, this analysis sets the only existing limits.
The leptoquark analysis takes into account LQs that couple to the electron and the firstgeneration quarks (u, d) only (Section 3.2). Deviations in the Q 2 distribution of e + p and e − p NC DIS events, corresponding to the 95% C.L. exclusion limits for selected scalar and vector leptoquark models, are compared with ZEUS data in Fig. 3. The 95% C.L. limits on the ratio of the leptoquark mass to the Yukawa coupling, M LQ /λ LQ , are summarized in Table 2 together with the coefficients a eq ij describing the CI coupling structure. The limits range from 0.27 TeV forS R • model to 1.23 TeV for V L 1 model. Table 2 also shows the LQ limits obtained by the H1 collaboration [34] and by the LEP experiments [37,39]. In general, comparable limits are obtained. For the S L 1 , V R 1/2 andṼ L 1/2 leptoquarks, the ZEUS analysis provides the most stringent limits.
When only the NC DIS event sample is considered, the leptoquark limits obtained in the contact-interaction approximation are similar to, or better than, the high-mass limits from the ZEUS resonance-search analysis [10]. However, for S L 0 , S L 1 and V L 0 models these previously published limits are more stringent, as the possible leptoquark contribution to charged current DIS was also taken into account. were obtained. In Fig. 4, effects of graviton exchange on the Q 2 distribution, corresponding to these limits, are compared with ZEUS e + p (Fig. 4a) and e − p (Fig. 4b) data. The limits on M S obtained in this analysis are similar to those obtained by the H1 collaboration [34] and stronger than limits from qq production at LEP [41]. However, if all final states are considered, the limits derived from e + e − collisions exceed 1 TeV [41]. Limits above 1 TeV are also obtained in pp from the measurement of e − e + and γγ production [42].
Assuming the electron to be point-like (R e = 0), the 95% C.L. upper limit on the effective quark-charge radius (Section 3.4) of R q < 0.85 · 10 −16 cm was obtained. The present result improves the limits set in ep scattering by the H1 collaboration [34] (R q < 1.0 · 10 −16 cm) and is similar to the limit set by the CDF collaboration in pp collisions using the Drell-Yan production of e + e − and µ + µ − pairs [35] (R q < 0.79·10 −16 cm). 7 The L3 collaboration has presented a stronger limit (R q < 0.42·10 −16 cm, assuming R e = 0), based on quark-pair production measurement at LEP2 [39] and assuming the same effective charge radius for all produced quark flavors.
If the charge distribution in the quark changes sign as a function of the radius, negative values can also be considered for R 2 q . For such a model, the ZEUS 95% C.L. upper limit on the effective quark-charge radius squared can be written as: Cross section deviations corresponding to the 95% C.L. exclusion limits for the effective radius, R q , of the electroweak charge of the quark are compared with the ZEUS data in Fig. 4c. The limits derived in this analysis are comparable to the limits obtained by the H1 collaboration and by the LEP and Tevatron experiments. For many models the analysis presented here provides the most stringent limits to date.

Conclusions
ZEUS 1994-2000 e ± p 95% C.L. ( TeV)  Table 1: Coupling structure [ǫ LL , ǫ LR , ǫ RL , ǫ RR ] of the compositeness models and the 95% C.L. limits on the compositeness scale, Λ, resulting from the ZEUS analysis of 1994-2000 e ± p data. Each row of the table represents two scenarios corresponding to η > 0 (Λ + ) and η < 0 (Λ − ). The same coupling structure applies to d and u quarks, except for the models U1 to U6, for which the couplings for the d quarks are zero. Also shown are results obtained by the H1 collaboration, the pp collider experiments DØ and CDF, and the LEP experiments ALEPH, L3 and OPAL. For the LEP experiments, limits derived from the channel e + e − → qq are quoted.