Constraints on Electron-quark Contact Interactions and Implications to models of leptoquarks and Extra Z Bosons

We update the global constraint on four-fermion $ee q q$ contact interactions. In this update, we included the published data of H1 and ZEUS for the 94--96 run in the $e^+ p$ mode and the newly published data of H1 for the 1999 run in the $e^- p$ mode. Other major changes are the new LEPII data on hadronic cross sections above 189 GeV, and the atomic parity violation measurement on Cesium because of a new and improved atomic calculation, which drives the data within $1\sigma$ of the standard model value. The global data do not show any evidence for contact interactions, and we obtain 95% C.L. limits on the compositeness scale. A limit of $\Lambda^{eu}_{LL+(-)}>23 (12.5)$ TeV is obtained. Implications to models of leptoquarks and extra Z bosons are examined.

The purpose of this note is to update the analysis that examines the NC data sets from current accelerator experiments to see if there is any sign of contact interactions. If so it is a signal of new physics; if not we put limits on the compositeness scale Λ.
In the next section, we describe the formalism and followed by the descriptions of various data sets in Sec. III. We present the fits and limits in Sec. IV. In Sec. V and VI, we extend the analysis to models of leptoquarks and extra Z bosons, respectively. We conclude in Sec. VII.

II. PARAMETRIZATION
The conventional effective Lagrangian of eeqq contact interactions has the form [13] L N C = q η eq LL (e L γ µ e L ) (q L γ µ q L ) + η eq RR (e R γ µ e R ) (q R γ µ q R ) +η eq LR (e L γ µ e L ) (q R γ µ q R ) + η eq RL (e R γ µ e R ) (q L γ µ q L ) , where eight independent coefficients η eu αβ and η ed αβ have dimension (TeV) −2 and are conventionally expressed as η eq αβ = ǫg 2 /Λ 2 eq , with a fixed g 2 = 4π. The sign factor ǫ = ±1 allows for either constructive or destructive interference with the SM γ and Z exchange amplitudes and Λ eq represents the mass scale of the exchanged new particles, with coupling strength g 2 /4π = 1. A coupling of this order is expected in substructure models and Λ eq is often called the "compositeness scale".
In models with SU(2) L symmetry, we expect some relations among the contact interaction coefficients. The particle content has the left-handed leptons and quarks in SU(2) doublets L = (ν L , e L ) and Q = (u L , d L ), while the righthanded electrons and quarks in singlets. The most general SU (2) L × U (1) invariant contact term Lagrangian is given by L SU(2) = η 1 Lγ µ L Qγ µ Q + η 2 Lγ µ T a L Qγ µ T a Q +η 3 Lγ µ L u R γ µ u R + η 4 Lγ µ L d R γ µ d R +η 5 e R γ µ e R Qγ µ Q + η 6 e R γ µ e R u R γ µ u R +η 7 e R γ µ e R d R γ µ d R . ( By expanding the η 5 term we have In addition, the four neutrino and the lepton couplings are also related by SU (2): In our analysis, the relations of Eqs. (3) and (4) are only used when neutrino scattering data are included in the analysis. We shall state clearly when these SU(2) relations are used or not. This is because in some combinations of η's, at least one of the SU(2) relations cannot be held, then we are forced not to use the SU(2) symmetry. Even though we expect that SU (2) L × U (1) will be a symmetry of the renormalizable interactions which ultimately manifest themselves as the contact terms of Eq. (1), electroweak symmetry breaking may break the mass degeneracy of SU(2) multiplets of the heavy quanta that give rise to (1). This would result in a violation of the relations of Eqs. (3) and (4).
Because of severe experimental constraints on intergenerational transitions like K → µe we restrict our discussions to first generation contact terms. Only where required by particular data (e.g. the muon sample of Drell-yan production at the Tevatron) shall we assume universality of contact terms between e and µ.
Let us start with the scattering process qq → ℓ + ℓ − (ℓ = e, µ). The amplitude squared for qq → ℓ + ℓ − or ℓ + ℓ − → qq (without averaging initial spins or colors) is given by where M ℓq αβ (s) = where s, t, u are the usual Mandelstam variables. In the above equations, g f L = T 3f − Q f sin 2 θ w , g f R = −Q f sin 2 θ w , Q f is the electric charge of the fermion f in units of proton charge. The SM amplitude can be recovered by setting η's to zero.
For a large class of new interactions the new physics contributions η eq αβ vary slowly with q 2 , effectively being constant at energies accessible to present experiments, e.g., if the mass of the exchange quanta is much heavier than the energy scale of the experiments. In this case the η f f ′ αβ correspond to constant four-fermion contact interactions, and Eq. (6) relates the sensitivity to new physics of all experiments probing a given combination of external quarks and leptons, such as ep → eX, pp → e + e − X, e + e − → hadrons and atomic physics parity violation experiments. Based on the formula in Eq. (6) the amplitude squared for the deep-inelastic scattering at HERA can be obtained by a simple interchange of the Mandelstam variables.

III. GLOBAL DATA
The global data used in this analysis have been described in Ref. [1]. Here we only describe those that have been updated since then. We have used the most recent CTEQ (v.5) parton distribution functions [14] wherever they are needed.
A. HERA data ZEUS [7] and H1 [8] have published their results on the NC deep-inelastic scattering (DIS) at e + p collision with √ s ≈ 300 GeV. The data sets of H1 and ZEUS are based on accumulated luminosities of 35.6 and 47.7 pb −1 , respectively. H1 [8] also published NC data for the most recent run of e − p collision at √ s ≈ 320 GeV with an integrated luminosity of 16.4 pb −1 . We used the double differential cross section d 2 σ/dxdQ 2 given by the H1 [8] data and the single differential cross section dσ/dQ 2 given by ZEUS [7] data in our fits. At e + p collision, the double differential cross section for NC DIS, including the effect of η's, is given by where Q 2 = sxy is the square of the momentum-transfer and f q/q (x) are parton distribution functions. The reduced amplitudes M eq αβ are given by Eq. (6). The single differential cross section dσ/dQ 2 is obtained by integrating over x. The corresponding formulas for e − p collision can be obtained from the above equation by interchanging (LL ↔ LR, RR ↔ RL).
We normalize the tree-level SM cross section to the low Q 2 part of the data set by a scale factor C (C is very close to 1.) The cross section σ th used in the minimization procedure is then given by where σ interf is the interference cross section between the SM and the contact interactions and σ cont is the cross section due to contact interactions.

B. Drell-yan Production
Both CDF [15] and DØ [16] measured the differential cross section dσ/dM ℓℓ for Drell-Yan production, where M ℓℓ is the invariant mass of the lepton pair. While CDF analyzed data from both electron and muon samples, DØ analyzed only the electron sample.
The differential cross section, including the contributions of contact interactions, is given by where M eq αβ is given by Eq. (6),ŝ = M 2 ℓℓ , √ s is the center-of-mass energy of the pp collision, M ℓℓ and y are, respectively, the invariant mass and the rapidity of the lepton pair, and x 1,2 = M ℓℓ √ s e ±y , and y is numerically integrated. The QCD K-factor is given by K = 1 + αs(ŝ) . We scale our tree-level SM cross section by normalizing to the Z-peak cross section data. The cross section used in the minimization procedure is then given similarly by Eq. (8).

C. LEP
The LEP Electroweak Working Group (LEPEW) combined the data on qq production from the four LEP collaborations [9] for energies between 130 and 202 GeV. In our previous fits, we have data upto 183 GeV only. In the LEPEW report, they also noted that the hadronic cross section, on average, is about 2.5σ above the SM prediction. In fact, we see this effect in our fits.
In the report, both the experimental cross sections and predictions from the next-leading-order (NLO) cross sections are given [9]. Since the NLO calculation for contact interactions is not available, we do the calculation by first normalizing our tree-level results to the NLO cross sections given in the report and then multiplying this scale factor to the new cross sections that include the SM and the contact interactions.
At leading order in the electroweak interactions, the total hadronic cross section for e + e − → qq, summed over all flavors q = u, d, s, c, b, is given by where K = 1 + α s /π + 1.409(α s /π) 2 − 12.77(α s /π) 3 is the QCD K factor. We found that some of the fits are dominated by these e + e − → qq hadronic cross sections. If data at even higher energies > 202 GeV are available, the limits will increase. In our fits, we assumed a more conservative scenario that contact interactions only appear in eu and ed channels. Have we assumed the universalities of eu = ec and ed = es = eb, the limits obtained would have been significantly higher.

D. Atomic Parity Violation
The APV is measured in terms of weak charge Q W . The updated experimental value with an improved atomic calculation [10,11] is about 1.0σ larger than the SM prediction [12], namely, ∆Q W ≡ Q W (Cs)−Q SM W (Cs) = 0.44±0.44. The contribution to ∆Q W from the contact parameters is given by [17,1] Note that the η's come in special combinations. If for some specific combinations: e.g., vector-vector There are also electron-nucleon scattering data, which have not been updated since our previous fits. The contributions to the asymmetries that were measured in these experiments are automatically zero for similar combinations of η's.

E. Charged-current (CC) Universality
The difference η ed LL − η eu LL = η 2 /2 measures the exchange of isospin triplet quanta between left-handed leptons and quarks, as indicated by the presence of the SU (2) generators T a = σ a /2 in the η 2 term. This term also provides an eνud contact term in CC processes. Such contributions, however, are severely restricted by lepton-hadron universality of weak charged currents [18] within the experimental verification of unitarity of the CKM matrix. The experimental values [19] |V exp ud | = 0.9735 ± 0.0008 , |V exp us | = 0.2196 ± 0.0023 , |V exp ub | = 0.0036 ± 0.0010 , lead to the constraint when flavor universality of the contact interaction is assumed. As a result η 2 must be small, though not necessarily negligible, Other data we used in our fits include low-energy electron-nucleon scattering experiments [20] and neutrino-nucleon scattering experiments [21]. When considering constraints from neutrino-nucleon scattering experiments, we invoke the SU (2) relations and e-µ universality in order to restrict the number of free parameters. In addition to using the relations of Eqs. (3) and (4), we will also impose the CC constraint on η 2 when neutrino data are included in the fits.

IV. FITS AND LIMITS
The fits of contact parameters are obtained by minimizing the χ 2 of the data sets. In order to see how each data set affects the fit, we first show the fits with each data set added one at a time, as shown in Table I. We observe the following: (i) the SM model fits the data well with χ 2 SM /d.o.f. < ∼ 1 for all five columns in Table I. (ii) The contact interaction fits the data slightly better than the SM. In the last column of Table I for APV in all cases are zero, which means that the minimization procedure prefers the APV data to be satisfied. In other words, other choices of η's would give a too large χ 2 if APV data is violated to a large extent.
In view of these, we conclude that the global data do not show any sign of contact interactions. Thus, we can derive 95% C.L. limits on the compositeness scale, below which the contact interaction is ruled out. The 95% C.L. one-sided limits η 95 ± are defined, respectively, as where P (η) is the fit likelihood given by P (η) = exp(−(χ 2 (η) − χ 2 min )/2). The 95% C.L. limits on Λ ± = ± 4π The limits on Λ ± are summarized in Tables II-IV. In Table II, for each chirality coupling considered the others are put to zero. The limits on Λ obtained range from 10-26 TeV, which improve significantly from each individual experiment. We also calculate the limits on the compositeness scale when some symmetries on contact terms are considered, as shown in Table III: V V stands for vector-vector: η LL = η LR = η RL = η RR = η V V , while AA stands for axial-vector-axial-vector: η LL = −η LR = −η RL = η RR = η AA . These limits, in general, are not as strong as those in the previous table because the additional symmetry automatically satisfies the parity violation experiments: APV and e-N. Finally, we show the limits that can be obtained from each set of data by looking at the results of LL and V V cases. The former is constrained severely by the APV and CC data, while the latter is free from the APV data. For the LL case the most dominant constraint is the CC universality, followed closely by the APV data (as indicated by the error of the best fit values.) The stringencies of the CC universality is understood because the LL interaction affects the V − A structure. The CC universality will not constrain chirality combinations other than LL. On the other hand, parity-violating experiments will not be able to constrain the V V case, and the strongest constraint then comes from the LEP hadronic cross sections.

V. IMPLICATIONS TO LEPTOQUARK MODELS
The interaction Lagrangians for the F = 0 and F = −2 (F is the fermion number) scalar leptoquarks are [22] where q L , ℓ L denote the left-handed quark and lepton doublets, u R , d R , e R denote the right-handed up-type quark, down-type quark, and lepton singlet, and q (c) R denote the charge-conjugated fields. The subscript on leptoquark fields denotes the weak-isospin of the leptoquark, while the superscript (L, R) denotes the handedness of the lepton that the leptoquark couples to. The color indices of the quarks and leptoquarks are suppressed. Note that the above Lagrangians have the SU(2) L symmetry and thus obey the SU(2) relations in Eqs. (3) and (4).
It is convenient to express the effects of leptoquarks in terms of the contact interaction coefficients η's. This is made possible when the mass of the leptoquark is much larger than the momentum transfer in the process. We classify the effects as follows. (i) S L,R 1/2 : (ii)S L 1/2 : (iv)S R 0 : (v) S L 0 : Once we expressed the effects in terms of η's, we can directly analyze the combinations of η's in the global fit. The resulting limits are given in terms of λ 2 /2M 2 LQ . Conventionally, the coupling constants λ L,R or g L,R,3L are assumed the electromagnetic strength, i.e., λ L,R = e = g L,R,3L and thus we can obtain the lower limits on M LQ . These results are summarized in Table V. Roughly, the leptoquark masses are required to be larger than 1 TeV, in order to satisfy all the constraints (except that S L,R 0 has to be heavier than 1.7 TeV and S L,R 1/2 can be as light as 0.67 TeV) when the coupling constants are assumed an electromagnetic strength e. The strongest constraint comes from APV and CC universality, the latter of which constrains the LL chirality severely.
This is an interesting result in view of the recent measurement of muon anomalous magnetic moment [23] with respect to a couple of leptoquark solutions [24]. The most favorable leptoquark solution to the muon anomaly is S L,R 1/2 that has both left-and right-handed couplings with the allowed mass range in 0.8 TeV < M S 1/2 < 2.2 TeV. This solution is in total consistency with the global NC constraint (as shown in (i) of Table V.)

VI. IMPLICATIONS TO Z ′ MODELS
We can write down the Lagrangian of a generic Z ′ model coupling to fermions as where P L,R = (1 ∓ γ 5 )/2, g E = 5λ g /3 e/ cos θ w and λ g is typically in the range 2/3 − 1 and for grand-unified theories breaking directly into SU(3)×SU(2)×U(1)×U(1)' λ g = 1, and ǫ L,R (f ) are the left-and right-handed chiral couplings to the Z ′ . Here in this simple analysis, we assume that the Z ′ does not mix with the SM Z boson such that the Z ′ is not constrained by the electroweak precision data [25]. The contribution of the Z ′ to the reduced amplitudes is given by In other words, if M 2 Z ′ ≫ q 2 the effects of Z ′ can be expressed in terms of the contact interaction parameters η's as Once we expressed the effects of Z ′ in terms of contact interaction parameters, we can easily analyze the Z ′ models in our global fit. We shall analyze the following Z ′ models (λ g = 1) [26]: (i) Sequential Z model: (ii) Left-right Z LR model: (iii) Z χ model: (iv) Z ψ model: (v) Z η model: Note that the Z ψ gives an axial-vector-axial-vector interaction, which evades strong constraints of APV and CC universality. In fact, Z χ and Z η are also not constrained by CC universality. The resulting best estimates of g 2 for each model are shown in Table VI, with the corresponding lower limits on Z ′ masses. The results shown in the Table are not satisfactory. First, the lower mass limits on Z ψ and Z η are rather low, 0.16 and 0.43 TeV respectively. We have verified that Z ψ gives only AA-type interactions and the result of Z ψ is consistent with η eq AA of Table III. Such low mass values invalidate the assumption of Eq. (25), which means that we cannot apply the simple contact interaction analysis to these Z ′ models. In this case, more sophisticated q 2 -dependent analysis is necessary to get an accurate result, which is beyond the scope of the present paper 2 . Nevertheless, it should be reasonably applicable to sequential Z model, Z LR and Z χ .

VII. CONCLUSIONS
In conclusion, we have examined the NC eeqq data and found that the data do not support the existence of eeqq contact interactions with the compositeness scale upto 6-26 TeV, depending on the chiralities. We have also demonstrated that the low-energy data (APV and CC universality) dominate the fit for the LL chirality. In the case of parity-conserving contact interactions, the LEP hadronic cross section dominates the fit.
The above analysis has also been applied in a straight-forward fashion to other new physics such as leptoquark and Z ′ models. For leptoquark models we found the 95% C.L. lower mass limits range from 0.67 to 1.7 TeV. Especially, the leptoquark S L,R 1/2 , which couples to both left-and right-handed charged leptons, has a mass limit of 0.67 TeV, which is consistent with the best leptoquark solution to the muon anomalous magnetic moment anomaly. For Z ′ models we found that our analysis is applicable to the sequential Z, Z LR , and Z χ models, with mass limits ranging from 0.68 to 1.5 TeV. We found that our analysis is not applicable to Z ψ and Z η models.
I would like to thank Vernon Barger, Karou Hagiwara, and Dieter Zeppenfeld for previous collaborations, which lead to the present work. This research was supported in part by the National Center for Theoretical Science under a grant from the National Science Council of Taiwan R.O.C.