Constraining flavoured contact interactions at the LHC
aa r X i v : . [ h e p - ph ] N ov Constraining flavoured contact interactions at the LHC
Sacha Davidson , ∗ and S´ebastien Descotes-Genon , † IPNL, Universit´e de Lyon, Universit´e Lyon 1, CNRS/IN2P3, 4 rue E. Fermi 69622, Villeurbanne cedex, France Laboratoire de Physique Th´eorique, CNRS/Univ. Paris-Sud 11 (UM8627), 91405 Orsay Cedex.
Abstract
Contact interactions are the low-energy footprints of New Physics, so ideally, constraints upon them should be asgeneric and model independent as possible. Hadron colliders search for four-quark contact interactions with incidentvalence quarks, and the LHC currently sets limits on a flavour sum (over uu, dd and ud ) of selected interactions. Weapproximately translate these bounds to a more complete (and larger) set of dimension-six interactions of specifiedflavours. These estimates are obtained at the parton level, are mostly analytic and are less restrictive than theexperimental bounds on flavour-summed interactions. The estimates may scale in a simple way to higher energyand luminosity. Contact interactions inevitably arise as the low-energy remnants of high-energy theories. Should the LHC not find(additional) new particles, it can nonetheless be sensitive to their traces in contact interactions. This paper focuseson four-quark contact interactions, with two incoming valence quarks, as is most probable at the LHC. New Physicsfrom beyond the LHC energy can induce various operators [1, 2], with different Lorentz and gauge structures as wellas flavour indices. Bounds on contact interactions involving specific flavours are appropriate for constraining NewPhysics models. The great variety of high-energy models have different low-energy footprints, such as different flavourstructures for contact interactions. For instance, the contact interactions involving singlet (i.e., right-handed) u quarksmight differ from those involving d quarks. Furthermore, the flavour structure of contact interactions induced by NewPhysics is explored by precision flavour physics (mostly for flavour off-diagonal operators), so the collider boundsshould also be for specific flavour indices, to allow comparison and combination with low-energy observations.However, the current LHC bounds are given for a subset of flavour-summed operators [3, 4, 5]. The aim of thispaper is to make an approximate translation of the collider constraints onto individual, flavoured operators and toillustrate the limits of such a translation. Two issues arise in attempting to apply the published contact interactionsbounds to a different operator: the Lorentz and gauge structure of the operator affects the partonic cross section, andthe flavour indices affect possible interferences with QCD, as well as controlling the probability of finding the initialstate quarks in the proton. Let us mention that four-quark contact interactions have been searched for at the Tevatronin q ¯ q → dijets [6], and at the LHC in qq → dijets [3, 4, 5]. In both cases, the initial state q or ¯ q are first generationvalence quarks, whereas the final state can be of any flavour, so the Tevatron and LHC can constrain different flavourstructures. In this paper, we focus on the qq → dijets process at the LHC, recently used to constrain models withcompositeness and/or heavy coloured octets [7, 8].Section 2 reviews the kinematical variables used by the experimental collaborations [3, 4, 6] to constrain contactinteractions from the rapidity distribution of high energy dijets. These variables allow an approximate “factorisation”of the pp → dijets cross section into an integral over parton distribution functions (pdfs), multiplying a partonic crosssection. In section 3.1, a basis of Standard Model gauge-invariant, dimension-six effective operators are listed, which,in the presence of electroweak symmetry breaking, induce the effective interactions listed in section 3.2. To obtainbounds as widely applicable as possible, we will constrain the coefficients of these effective interactions (this is discussedin section 3). Issues regarding flavour and interferences are discussed in section 4. Section 5 uses this approximatefactorisation to estimate bounds on individual operators based on the analysis of ref. [4]. The Appendix collects thepartonic cross sections for various contact interactions. Its aim is to allow an interested reader to estimate boundsfrom future data on their selection of contact interactions, following the mostly analytic recipe given in section 5. ∗ [email protected] † [email protected] i p mp nk j k i p np mk j k i p mp nk j k i p np mk j Figure 1: Possible QCD and contact interaction diagrams for q i q j → q m q n . QED diagrams with the gluon replacedby a photon are also possible. At the grey blob representing the contact interaction, the quark lines may have chiralprojectors and/or a colour matrix. Which gauge diagrams interfere with a given contact interaction will depend on itsflavour indices and operator structure; for instance, for V − A operators with i = j = n = m , all four gauge diagramscould interfere. pp → dijets At the LHC, the cross section for pp → dijets contains contributions from QCD, electroweak bosons, and possiblyfrom four-parton contact interactions. The purely QCD (or QED) contribution falls off as 1 / ˆ s , where ˆ s = M dijet is thefour-momentum-squared of the pair of jets, and grows in the forward/backward directions. On the other hand, thecontact interaction contribution grows with ˆ s and is fairly central. So bounds on contact interactions can be obtainedfrom the meagre population of central high energy dijets. With a clever choice of variables, the distribution in rapidityand invariant-mass squared of the two jets can be approximated as the partonic cross section, multiplying an integralof pdfs. We review here this approximation.The parton-level diagrams for quark-quark scattering q i ( k ) q j ( k ) → q m ( p ) q n ( p ) are given in Figure 1, where i, j, m, n are flavour indices, and { k, p } are four-momenta. The resulting cross sections, which can be parametrised withthe partonic mandelstam variables ˆ s, ˆ t , and ˆ u are listed in the Appendix. To obtain an observable, the partons mustbe embedded in the incident protons, here taken to have four-momenta P ± . We denote f j ( x ) the probability densitythat the parton j carries a fraction x of the four-momentum of an incident proton, so that ˆ s = x x ( P + + P − ) . Thetotal cross section can be written σ ( pp → dijets ) = X i,j,m,n Z dx Z dx f i ( x ) f j ( x ) σ ( i ( x P + ) j ( x P − ) → mn ) (1)where the sum runs over all possible incident partons for each partonic process, and over all the partonic processeswhich contribute ( e.g. , in the case of dijets, qq → qq, q ¯ q → q ¯ q, gg → q ¯ q, gg → gg , etc). In our estimates, we onlyinclude uu → uu, dd → dd , ud → ud and ug → ug in the partonic QCD cross section. These should be the maincontributions to the dijet cross section. The partonic cross sections increase by a factor ∼ / σ ( qq → qq ) : σ ( qg → qg ) : σ ( gg → gg ) ∼ : 1 : . However, the density of gluons in the proton, at the large values of x which are relevant here, is at least a factor of 1/10 (1/3) below that of valence u ( d ) quarks, and the density of seaquarks, is two orders of magnitude below the u density, which justifies our approximation.It can be convenient to introduce the pseudo-rapidities y = 12 ln E + p z E − p z (2)of the individual jets, and their combined mass squared M dijet = ˆ s = x x s . (3)We interchangeably refer to M dijet or ˆ s throughout the paper, using the convention that partonic variables, such asthe Mandelstam ˆ s, ˆ t, ˆ u (defined in eq. (35)), wear hats.The CMS search [4] for contact interactions in the angular distribution of dijets at high M dijet uses as variablesthe dijet mass squared eq. (3), the pseudo-rapidity of the partonic centre of mass frame y + = ( y + y ) / , (4)and χ = exp | y − y | = 1 + | cos θ ∗ | − | cos θ ∗ | , (5)where θ ∗ is the centre-of-mass scattering angle away from the beam axis (see the Appendix).2t the parton level, the QCD contributions to the differential cross section d ˆ σ/d ˆ t have a 1 / ˆ t divergence, leadingto a large rate for small-angle scatterings along the beam pipe. On the other hand, the dijets produced by contactinteractions have a more isotropic distribution. The most sensitive place to look for contact interactions is thereforein large-angle scatterings, producing dijets in the central part of the detector. Expressed as a function of χ , the QCDcontribution to the dijet cross section is approximately flat, whereas the contact interaction contribution peaks atsmall χ . This is illustrated in figs. 1 and 2 of ref. [4], where the main effect of contact interaction occurs for 1 ≤ χ ≤ o ≤ θ ∗ ≤ o .With these variables, the differential dijet cross section is dσdy + dχ dM dijet = f i ( x ) f j ( x ) s d ˆ σd ˆ t ˆ t ˆ s . (6)For fixed dijet mass, the pdfs depend on y + but not χ (since x = e y + M dijet / √ s, x = e − y + M dijet / √ s ), and thepartonic differential cross section depends on χ but not y + , so the expected number of events can be factorised as aintegral-of-pdfs, multiplied by a partonic cross section. Therefore, in an ideal world with contact interactions, the dijetdistribution in mass and χ could allow one to determine the actual partonic cross section, providing the necessaryinformation to identify the operator(s) that induced it. New Physics from a scale
M > m W can be described, at scales ≪ M , by an effective Lagrangian containing therenormalisable Standard Model (SM) interactions, and various SU (3) × SU (2) × U (1) Y -invariant operators of dimension >
4, with coefficients determined by the New Physics model. There can be relations among the operators, arisingfrom symmetries and equations of motion. This section gives a basis of dimension six, four-quark operators, takenfrom the Buchmuller-Wyler [1] list as pruned by ref. [2] .In the following list of four quark operators, Q are electroweak doublets, U and D are singlets, λ A are the generatorsof SU(3), ~τ / i, j, m, n are generationindices which all run from 1. . . 3 (so there is a factor 1/2 in front of operators made of same current twice, to obtainthe Feynman rule described below in eq. (23)). O , QQ = 12 ( Q m γ µ Q i )( Q n γ µ Q j ) (7) O , QQ = 12 ( Q m γ µ ~τ Q i )( Q n γ µ ~τ Q j ) (8) O , UU = 12 ( U m γ µ U i )( U n γ µ U j ) (9) O , DD = 12 ( D m γ µ D i )( D n γ µ D j ) (10)There are also operators contracting currents of singlet quarks of different charge: O , UD = ( U m γ µ U i )( D n γ µ D j ) O , UD = ( U m γ µ λ A U i )( D n γ µ λ A D j ) (11)and operators contracting doublet and singlet currents: O , QD = ( Q m γ µ Q i )( D n γ µ D j ) O , QD = ( Q m γ µ λ A Q i )( D n γ µ λ A D j ) (12) O , QU = ( Q m γ µ Q i )( U n γ µ U j ) O , QU = ( Q m γ µ λ A Q i )( U n γ µ λ A U j ) (13)and finally there is a scalar operator, with antisymmetric SU(2) index contraction across the parentheses: O S, , Q ¯ Q = ( Q m U i )( Q n D j ) O S, , Q ¯ Q = ( Q m λ A U i )( Q n λ A D j ) . (14) We restrict our analysis to dimension-6 operators. This is a reasonable perturbative approximation when the next-order terms, relativelysuppressed by ∼ ˆ s/ Λ , v / Λ can be neglected — which appears to be barely the case here. .2 Effective interactions Once electroweak symmetry is broken, a particular operator induces one or several effective interactions among masseigenstates. We aim at constraining these effective interactions, one at a time, through the LHC searches on contactinteractions.This leads us to make a distinction between (gauge-invariant) operators, and effective interactions (having distinctexternal legs). The aim of this distinction is to address a general problem with setting bounds on the coefficients ofgauge-invariant operators [10]: such bounds may not transfer, in a simple way, from one operator basis to another.The selection of SM gauge-invariant operators made in section 3.1 is not unique, as expected for a choice of basis. Adifferent list of operators might be more suited to describing some models because they capture the symmetries of themodel in a more economical way. Ideally, the constraints on four-fermion operator coefficients should be transferablefrom one operator basis to another. However, this is not possible if these constraints are obtained by turning on oneoperator at a time, as will be done here. One difficulty is that an operator, such as the SU(2) triplet of eq. (9), inducesseveral four-fermion interactions, so quoting a bound on the operator loses the correlation carrying the informationabout which interaction the bound arose from. In addition, if the operators can interfere among themselves (forinstance, the singlet and triple operators of eqs. (7) and (9) can interfere), then the bound on a sum of operators couldbe less restrictive than the bounds on single operators.To circumvent the problem of “basis-dependent” bounds , we follow ref. [9], and set limits on the coefficientsof “effective four-quark interactions”, These effective interactions should be distinct, so that they do not interfereamong each other, and should include all the interactions induced by the effective operators. Unfortunately, these tworequirements are not quite compatible; there are two interactions which can interfere, but we neglect this effect.We obtain the list by considering the U (1) × SU (3) invariant operators generated after electroweak symmetry break-ing by the previous basis of SM gauge invariant operators (decomposing the SU(2) doublets Q into their components).These interactions almost never interfere among themselves (the exceptions are eqs. (18) and (22)), which ensures thatthe prediction of a sum of interactions will be the sum of the predictions of the interactions taken separately. Thepossibilities are the following: • the “neutral-current” left-left or right-right interactions, X = L or R : O ,XXu m u i u n u j = 12 ( u m γ µ P X u i )( u n γ µ P X u j ) (15) O ,XXd m d i d n d j = 12 ( d m γ µ P X d i )( d n γ µ P X d j ) (16) O ,XXu m u i d n d j = ( u m γ µ P X u i )( d n γ µ P X d j ) O ,RRu m u i d n d j = ( u m γ µ P R λ A u i )( d n γ µ P R λ A d j ) . (17)Notice that, with the previous basis of gauge-invariant operators, the octet O ,RRuudd only arises for singlet currents. • the “charged-current” interactions O ,CC = ( u m γ µ P L d i )( d n γ µ P L u j ) O ,CC = ( u m γ µ P L λ A d i )( d n γ µ P L λ A u j ) , (18)the second of which can be rearranged to a linear combination of O ,CCu m d i d n u j and O ,LLu m u j d n d i . We therefore do notinclude octet charged-current interactions. We include the singlet O ,CCu m d i d n u j , which unfortunately can interferewith O ,LLu m u j d n d i (see eq. (48)). • the “neutral-current” left-right operators, O ,XYu m u i u n u j = 12 ( u m γ µ P X u i )( u n γ µ P Y u j ) O ,XYu m u i u n u j = 12 ( u m γ µ P X λ A u i )( u n γ µ P Y λ A u j ) (19) O ,XYd m d i d n d j = 12 ( d m γ µ P X d i )( d n γ µ P Y d j ) O ,XYd m d i d n d j = 12 ( d m γ µ P X λ A d i )( d n γ µ P Y λ A d j ) (20) O ,XYu m u i d n d j = ( u m γ µ P X u i )( d n γ µ P Y d j ) O ,XYu m u i d n d j = ( u m γ µ P X λ A u i )( d n γ µ P Y λ A d j ) (21)where P X , P Y ∈ { P L , P R } , X = Y. • the scalar operators stemming from eq. (14), which each give two interactions: O S q m u i q n d j = ( d m P R u i )( u n P R d j ) − ( u m P R u i )( d n P R d j ) O S q m u i q n d j = ( d m P R λ A u i )( u n P R λ A d j ) − ( u m P R λ A u i )( d n P R λ A d j ) . (22) Ref. [11] shows some parameter choices where interference among operators could reduce sensitivity. O S and O S given above,because the interactions in the sums interfere between each other. Following the convention of collider constraints on contact interactions, we suppose that the coupling in the four-quarkFeynman rule is iη π Λ , with η = ± . (23)The operators/interactions, of the previous sections are normalised such that their coefficient should be η π/ Λ . Foreach flavoured contact interaction, there will be a different lower bound on Λ. If the contact interaction is generatedperturbatively by the exchange of a particle of mass M with coupling g ′ , one would expect a contact interaction like g ′ /M , leading to the scale Λ ∼ M/ √ α ′ .In the matrix element squared, the contact interaction may interfere with QCD (depending on its colour, flavourand chiral structure) as well as with itself. The pp → dijet cross section, in the central part of the detector, is thereforeof the form dσdχ ∼ C QCD α s ˆ s + C α s Λ + C ˆ s Λ , (24)where the C x are O (1) constants that depend on the specific contact interaction. C , related to the contributionof contact interaction alone, is always positive and will dominate for very large ˆ s , whereas C , stemming from theinterference between QCD and contact interactions, can be positive or negative depending on the value of η , and canthus induce either a deficit or an excess of events at intermediate values of ˆ s . Requiring that the contact interactionsinduce a deviation of ǫ < ∼ sα s ∼ Λ . (25)suggesting that for √ ˆ s = 3 TeV at the 7-8 TeV LHC [4], the limits on Λ should be O (10) TeV. In the following, wewill impose a tighter constraint on eq. (24) for specific contact interactions, providing different bounds depending ontheir flavour structure. Each of the interactions given in the section 3.2 exists in a plethora of flavoured combinations, only some of which canbe constrained by colliders. The flavour indices have two effects on the collider bounds: first, some flavours are moreplentiful in the proton than others, and second, the cross section can involve interferences with QCD or QED, whichdepend on the flavours.First, at the LHC, bounds can be set on contact interactions with two incoming valence quarks ( uu, dd, ud ). Thisis because the density of sea quarks and antiquarks is very suppressed. The incident partons should both carry asignificant fraction of the proton momentum, to produce a pair of jets of large combined mass (the last bin in dijetmass of the CMS analysis [4] is M dijet > x x ≥ / x , the density of all flavours of sea quark and anti-quark is of similar size and about two orders of magnitude belowthe density of u quarks. It is therefore doubtful, with current data, to set a bound on contact interactions with aninitial sea quark or anti-quark, because a huge cross section would be required to compensate the relative suppressionof sea pdfs.The second effect of flavour indices is on the partonic cross section. For instance, the cross section for ud → ud ,mediated by QCD + contact interactions, is different from uu → uu (see eqs. (38)-(43)). In particular, the singletinteractions mediating uu → uu interfere with QCD, whereas there is not interference between QCD and singletinteractions mediating ud → ud . Flavour-summed contact interactions constrained by the experimental collaborationshave thus a reduced sensitivity to destructive interferences as they get contributions from several flavour operators,most of them being positive contributions coming from contact interactions alone that increase the cross sections.In the Appendix, we attempt to classify the possible flavour index combinations, and provide their partonic crosssections.Another comment is in order at this stage. The experimental final state considered here is a pair of jets, so thefinal state quarks can be of any flavour other than top. This means the LHC is sensitive to curious ∆ F = 2 and5 η LL , η RR , η LR ) operatorsΛ ± LL ( ± , , ± [ O ,LLuuuu + 2 O ,LLuudd + O ,LLdddd ]Λ ± RR (0 , ± , ± [ O ,RRuuuu + 2 O ,RRuudd + O ,RRdddd ]Λ ± V V ( ± , ± , ± ± [ P m,n = u,d ( O ,LLmmnn + O ,LRmmnn + O ,RLmmnn + O ,RRmmnn )]Λ ± AA ( ± , ± , ∓ ± [ P m,n = u,d ( O ,LLmmnn − O ,LRmmnn − O ,RLmmnn + O ,RRmmnn )]Λ ± V − A (0 , , ± ± [ O ,LR = RLuuuu + O ,LRuudd + O ,LRdduu + O ,LR = RLdddd ]Table 1: Sums of operators contributing (at the LHC) to some commonly studied contact interactions (see eq. (26)).On the left, the sub/super-scripts for Λ indicate the choice of η coefficients. On the right are given the correspondinginteractions from section 3.2. In principle, contact interactions studied in collider experiments have a sum over allflavours; however, the LHC is principally sensitive to contact interactions with two incoming valence quarks, so theflavour sum is over u, d .∆ F = 1 flavour-changing operators mediating processes like uu → cc or ud → ub . Flavour physics ( e.g. meson mixingand B decays) can impose more stringent bounds on some of them. In the following, we will give the LHC bounds onthe various flavour-changing operators. Traditionally [12], collider searches for contact interactions quote bounds on the mass scale Λ appearing in an inter-action of the form: L P ythia = 4 π Λ X i,j = u,d,s,c,b h η LL q i γ µ P L q i )( q j γ µ P L q j ) + η RR q i γ µ P R q i )( q j γ µ P R q j ) + η LR ( q i γ µ P R q i )( q j γ µ P L q j ) i . (26)Specifically, this is the contact interaction coded into pythia η coefficients can be chosen to be ± η coefficients are given in table 1.With the aim of obtaining useful and conservative bounds, our constraints will differ in three ways:1. We include the octet operators O of section 3.2. They generally give smaller modifications to the dijet rate, sothe bound on Λ is lower, as can be seen from the tables of section 5.2. We constrain each combination of flavour indices separately.3. We only consider O ,LR (or O ,RL ), and O ,LL (or O ,RR ), but not the various linear combinations available intable 1.Turning on one effective interaction of given flavours at a time (as done here) gives conservative bounds if twoconditions are satisfied. First, the interactions should not interfere among themselves, so that the contribution of thesum is the sum of the contributions. This is almost the case for the interactions of section 3.2. Second, each boundshould arise from requiring that the operator not induce an excess of events (as opposed to a deficit of events). Thiswill be true for the bounds that we will derive.Suppose now that one wishes to set a bound on a specific New Physics model, which induces a sum of low-energycontact interactions — for instance the V V combination. It is simple and conservative to take the strongest of thebounds obtained one-at-a-time for the operators in the sum. However, the true limit should be better. It is notstraightforward to obtain the bound on Λ + V V given the limits on Λ + XX ( X = R or L ) and Λ + LR . Despite that theexcess events induced by O ,V V are the sum of the excesses due to O ,XY and O ,XX (for X = Y ∈ { L, R } ), thereare two hurdles to obtaining a bound on Λ + V V : first, one must calculate the partonic cross section for the
V V operatorcombination, then one must know how it is constrained by the data. To address the first hurdle, we collect in theAppendix the partonic cross sections for a variety of contact interactions. The second hurdle is a problem, becauseit is clear that the experimental collaborations cannot constrain all possible combinations of all contact interactions.However, contact interactions induce excess high-mass dijets in the central part of the detector, so observables whichmeasure this, such as F χ ( M dijet ) [3] of the ATLAS collaboration, should be translatable to limits on generic contactinteractions. 6 Estimating bounds on flavoured contact interactions
Suppose that an effective interaction has been selected, with incident flavours of the first generation. The recipe toguess a bound on Λ, from the dijet distribution in M dijet and χ , is simple:1. look up the (flavoured) partonic cross section for the selected contact interaction plus QCD, and evaluate at the χ corresponding to the bin.2. integrate the pdfs of the incident partons over the y + values which are consistent with the experimental cuts.3. multiply 1) by 2), and require that it agree with the QCD expectation for the bin.We want to illustrate in detail the procedure in the case of the CMS analysis [4]. The CMS collaboration measuredthe distribution of dijets in χ from 1 →
16, and M dijet from 0.4 to ≥ M dijet > − of data), for which CMS plots the normalised differential cross sections ( ≡ σ dσdχ ),corresponding to the data, to the QCD expectation, and to the predictions of QCD plus contact interactions (denotedQCD+CI). As indicated in section 2, the highest sensitivity to contact interactions is obtained for the 1 ≤ χ ≤ σ QCD + CI dσ QCD + CI dχ (cid:20) σ QCD dσ QCD dχ (cid:21) − , (27)where all the cross sections are for pp → dijets (partonic cross sections will wear hats): dσ QCD + CI dχ = X i,j,m,n Z dy + f i f j (cid:18) d ˆ σ QCD dχ ( ij → mn ) + d ˆ σ QCD ∗ CI dχ ( ij → mn ) + d ˆ σ CI dχ ( ij → mn ) (cid:19) , (28)and are summed over the χ and ˆ s ranges of the bin (we will return to these sums later).Clearly, the prediction of contact interactions should be compared with the data, not with the QCD prediction.However, we notice that the data [3, 4] agree with the QCD prediction (for M dijet > χ bins and above for high- χ bins, in the case of ref. [4]). So we “normalise” our (incomplete)leading order parton-level QCD cross section, to the QCD expectation obtained by ATLAS and CMS at next-to-leadingorder (NLO) with hadronisation and detector effects. Then we estimate bounds on contact interactions by requiringthat they add < ∼ . σ to the QCD contribution, where σ is the experimental statistical and systematic uncertaintiesfor the relevant bin, added in quadrature. On the basis of the results of ref. [4], we estimate that this allows contactinteractions to contribute from 1/3 to 1/2 of the QCD contribution either positively or negatively. In other words,even though we base our analysis on the most sensitive χ -bin (from 1 to 3) at maximal M dijet , which has an observedvalue slightly below the QCD prediction, we consider that the spread of data with respect to QCD in the other binsprevents us from interpreting the deficit in the 1-3 bin as a negative contribution from contact interactions. We thustake the more conservative approach to set a bound on contact interactions as a fraction (positive or negative) fromQCD (we will come back to this point in section 5.3).The ratio (27) can be related, in a series of steps, to a ratio of partonic cross sections. • The first step consists in canceling σ QCD + CI ≃ σ QCD in the ratio. This is a self-consistent approximation,because contact interactions only contribute in the low χ bins, where they are bounded to be a fraction of QCD. • The second step amounts to writing the ratio dσ QCD + CI dχ (cid:20) dσ QCD dχ (cid:21) − = 1 + ǫ (29)where ǫ = P m,n R d ˆ s R dy + f i f j δ d ˆ σdχ ( q i q j → q m q n ) P i,j,m,n R d ˆ s R dy + f i f j d ˆ σ QCD dχ ( q i q j → q m q n ) , (30)where the partonic differential cross section δ d ˆ σdχ is the modification to ( q i q j → q m q n ) induced by contactinteractions (alone or through interference with QCD, corresponding to the last two terms inside the parentheses At this stage, we suppress the M dijet dependence; dσdχ means R dy + d σdy + dM dijet dχ , and σ = R dχdy + d σdy + dM dijet dχ . ij r ij = integrated density(normalised to uu ) u u d d u d u g r ij = I − ij /I − uu , which allow translating the experimental bound on σ ( pp → dijets) to a bound on partonic cross sections. r ud and r ug are multiplied by 2 because the u valence quark could comefrom either incoming proton.of eq. (28)). The denominator amounts to the QCD contribution which is summed over the possible initialflavour combinations (limited to uu , dd , ud and ug ). The integral over the ˆ s = M dijet range of the bin has beenreinstated. • The third step corresponds to factorising the integrals in the ratio 1 + ǫ . As discussed in section 2, the partoniccross sections depend on M dijet and χ , and the pdfs depend on M dijet and y + . We now want to factor the partoniccross section out of the integral across the M dijet > x , so contribute most of the integral in a narrow range of M dijet ∼ / ˆ s , sothe cross sections evaluated at ˆ s min = (3 TeV) can be factored out of the integrals and replaced by ˆ s min / ˆ s asfar as the s -dependence is concerned (see Appendix). We are left to integrate for N = − I Nij = Z maxmin d ˆ s (cid:18) ˆ s ˆ s min (cid:19) N Z dy + f i ( x ) f j ( x ) , x = e y + r ˆ ss , x = e − y + r ˆ ss , (31)over the y + region consistent with the CMS cuts ( y + < .
1) and the value of M dijet (corresponding to 2 y + < ln s/ ˆ s ), and over the range of energy from ˆ s min = (3 TeV) to ˆ s max = (6 TeV) . We use CTEQ10 [14] pdfs (atNLO) at a scale of 3 TeV. The results, normalised to I − uu , are given in table 2. The ratios in the table changeby only a few percent when we change ˆ s max to 4.5 TeV.The second step is to take δ d ˆ σdχ , evaluated at ˆ s = (3 TeV) , out of the integral in the numerator of eq. (30).We have then to evaluate I Nij for N = 1 ,
0, which corresponds to the ˆ s dependence of the | CI | and interferencecontributions respectively. For ij = uu, dd or ud , these integrals can be up to 20% larger than for N = −
1. Weconservatively neglect this effect, and use the r ij given in table 2.In the end, as desired, we have obtained an analytic formulation of the experimental bound on contact interactions.The data gives ǫ < ∼ / ij , ǫ = r ij δ d ˆ σdχ ( q i q j → q m q n )(1 + r dd ) d ˆ σ QCD dχ ( uu → uu ) + r ud d ˆ σ QCD dχ ( ud → ud ) + r ug d ˆ σ QCD dχ ( ug → ug ) (32)where r ij is from table 2, δ d ˆ σdχ is the parton-level excess with respect to QCD induced by contact interactions (givenby the Appendix), whereas the cross sections in the denominator are induced by QCD (see eqs. (38), (37) and (43)). We are now in a position to translate the CMS results in terms of flavoured operators. The recipe to guess a boundon Λ, as given in the previous section, can now be reformulated analytically:1. look up in the Appendix the contribution to the partonic cross section of the selected contact interaction(s) plusinterference with QCD, and evaluate it at the χ corresponding to the bin of interest . The same must be donefor the QCD cross sections (38), (37) and (43). For instance, 1 ≤ χ ≤ − ˆ t, − ˆ u ) between ( ˆ s, ˆ s ) and ( ˆ s, ˆ s ). minj ˆ s/ Λ Λ < + O XY uuuu .64 .72 3 . − . −O XY uuuu .20 .28 6 . − . O XX uuuu .19 .21 6 . − . −O XX uuuu .06 .09 12 . −
10 TeV+ O XY uuuu .19 .23 6 . − . −O XY uuuu .15 .19 7 . − . O XX dddd .43 .52 4 . − . −O XX dddd .31 .39 5 . − . O XY dddd .60 .74 3 . − . −O XY dddd .56 .70 4 . − . O pythiaXX .15 .17 7 . − . . −O pythiaXX .06 .07 12 . − . . O pythiaXY .17 .20 7 . − . . −O pythiaXY .13 .17 8 . − . . O pythia is the flavour-summed operator of eq. (26) for comparison. The flavourindices of the second column are in the order of the fields in the operator, and correspond to ij → mn . The boundsare for α s ( M dijet ) = 0 .
09. In the third column are given the bounds on ˆ s/ Λ from requiring that the relative excessof dijets induced by contact interactions be | ǫ | < / | ǫ | < /
2. The bound in the last column is obtained withˆ s = M dijet = (3 TeV) , for the two values of ǫ . The bounds in parentheses on O pythia are those of CMS [4].Operator flavours minj ˆ s/ Λ Λ < + O XX dduu .59 .67 3 . − . −O XX dduu .21 .28 6 . − . −O XY dduu .50 .66 4 . − . O S dduu ..60 .74 3 . − . O XX dduu .17 .22 7 . − . −O XX dduu .16 .20 7 . − . O CC uddu .27 .31 5 . − . −O CC uddu .10 .14 9 . − . O XY dduu .33 .41 5 . − . −O XY dduu .31 .39 5 . − . O S dduu .39 .48 4 . − . ij → mn = ud → ud flavour structure. In the third column are given the bounds on ˆ s/ Λ from requiring | ǫ | < / | ǫ | < /
2. The last column is obtained with ˆ s = M dijet = (3 TeV) .9perator flavours minj ˆ s/ Λ Λ < O XY cucu .35 .44 5 . − . O XX cucu .11 .13 9 . − . O XY cucu .17 .21 7 . − . O XX sdsd, bdbd .37 .45 4 . − . O XY sdsd, bdbd .59 .71 3 . − . O XX sdbd .30 .37 5 . − . O XY sdbd .59 .71 3 . − . ij → mn flavour structure, ij = uu or dd , and ∆ F = 2. In the third column are given the bounds onˆ s/ Λ from requiring | ǫ | < / | ǫ | < /
2. The bound in the last column is obtained with ˆ s = M dijet = (3 TeV) .Operator flavours minj ˆ s/ Λ Λ < O XY cuuu, uucu .35 .44 5 . − . O XX cuuu .09 .11 10 − . O XY cuuu, uucu .16 .21 7 . − . O XX qddd .30 .37 5 . − . O XY qddd, ddqd .59 .71 3 . − . ij → mn , ij = uu or dd , and ∆ F = 1. In this table, q = b, s . In the third column are given thebounds on ˆ s/ Λ from requiring | ǫ | < / | ǫ | < /
2. The bound in the last column is obtained with ˆ s = M dijet = (3TeV) . Operator flavours minj ˆ s/ Λ Λ < O XX cuqd .36 .43 5 . − . O XY cuqd, qdcu .67 .84 3 . − . O XX , O CC cuqd .20 .25 6 . − . O XY cuqd, qdcu .32 .40 5 . − . O S − cuqd + qucd .56 .67 4 . − . ij → mn , ij = ud , and ∆ F = 2. In this table, q = b, s . In the third column are given the bounds onˆ s/ Λ from requiring | ǫ | < / | ǫ | < /
2. The bound in the last column is obtained with ˆ s = M dijet = (3 TeV) .Operator flavours minj ˆ s/ Λ Λ < O XX cudd, uuqd .36 .43 5 . − . O XY cudd, ddcuuuqd, qduu .66 .84 3 . − . O XX , O CC uuqd , cudd .20 .25 6 . − . O XY cudd, ddcuuuqd, qduu .32 .40 5 . − . O S − cudd + ducd − uuqd + quud .56 .67 4 . − . ij → mn , ij = ud , and ∆ F = 1. q = b or s . In the third column are given the bounds on ˆ s/ Λ fromrequiring ǫ < / ǫ < /
2. The bound in the last column is obtained with ˆ s = M dijet = (3 TeV) .10. weight the various contact interactions by the appropriate r ij factor from table 2, and the QCD cross sectionsas given in the denominator of eq. (32).3. impose that the ratio of eq. (32), ǫ ≤ / s/ Λ whose root givesthe estimated bound on Λ for ˆ s taken at the lower end of the range allowed in the highest dijet mass bin.In practice, we integrate the partonic cross sections over 1 ≤ χ ≤
3, and plot 1 + ǫ in figures 2, 3 and 4. Imposing ǫ ≤ / / M dijet / Λ , by taking M dijet = 3TeV [4].The bound that we obtain on Λ is the solution of a quadratic polynomial in ˆ s/ ( α s Λ ) (see eq. (24)), so dependson the numerical value of α s . We take α s (3TeV) ≃ .
09, in agreement with the leading order running, since we do aleading order calculation (this is analogous to using α s ( m Z ) = .
139 in pythia [15]). If instead, we take the ParticleData Group value [16] α s ( m Z ) ≃ .
12, then α s (3TeV) ≃ .
08. If the scale of evaluation of α s is changed by a factorof two, α s varies by about 0.005. We conclude that varying α s between 0.08 and 0.09 gives some notion of the NLOuncertainties, and take the larger value, since this yields the more conservative limit.The study performed here neglects several effects, such as hadronisation (partons are not jets), and NLO corrections(calculated for O ,LL in ref. [11]). We also do not consider dimension 8 operators: as pointed out by ref. [17], matrixelements to which QCD contributes could also have an O (1 / Λ ) term from QCD interference with a dimension 8operator. This interference is in principle suppressed with respect to dimension 6-contributions by a factor α s .We do include interference between QED and contact interactions, when there is no interference with QCD, butwe neglect effects of weak interactions. There are (almost) no gluon contributions in this analysis: we include the gq → gq contribution to the QCD cross section, but neglect gg → gg (because we assume f g < ∼ f d / < ∼ f u / gg ¯ qq contact interactions [18] in a later analysis. The reach of the future LHC for the usual flavour-summed contact interactions has been studied in ref. [19], who findexpected limits Λ > ∼
20 TeV. If the LHC with more energy and luminosity still does not find contact interactions, howwould our flavoured estimates scale ? The bounds obtained here are on the dimensionless variable ˆ s/ Λ , and theydepend on the experimental uncertainty via ǫ , as well as the scale at which the pdfs were evaluated. There are twouseful approximations/assumptions:1. suppose that the ratios of integrated pdfs, r ij , given in table 2, will not change significantly in going from the 8to 14 TeV LHC.2. suppose the last bin in dijet mass will always have an experimental uncertainty of 20%-30% and remain com-patible with the QCD prediction. This may be reasonable, because the bound on contact interactions profitsmore from going to a larger ˆ s than from reducing the statistical uncertainties to the size of the systematics. So ǫ would remain approximately 1/3 (or 1/2).Then the estimated bounds we quote on ˆ s/ Λ remain valid, and the bound on Λ will be multiplied by a factor M ( new ) dijet /(3 TeV), where M ( new ) dijet is the lower bound on the dijet mass of the highest bin of a future analysis.In relation with this issue, we come back to the interpretation of the current result of CMS in the lowest- χ bin for M dijet > ∼
30% deficit in the 1 ≤ χ ≤ O ,XXuuuu , O ,CCuddu , O ,XYuuuu and O ,XXuddu (with a scale of around 11 TeV for the singlet operatorsand a scale around 7 TeV for the octet operators). These operators show a similar behaviour once extrapolated tohigher M dijet . They yield a deficit of events around 10% in the bin 2.4-3 TeV (where CMS data are in very goodagreement with QCD), and of around 7% in the bin 1.9-2.4 TeV. The effect is within QCD uncertainties at lower M dijet .At higher M dijet , of interest in the context of the LHC upgraded to 14 TeV, the contribution from contactinteractions-squared becomes larger and starts dominating over the interference with QCD: for 4 TeV, the totalcontribution from contact interactions approximately vanishes, and becomes positive at higher dijet mass, with anexcess of 30% at 5 TeV, and more than 100% at 6 TeV. In this particular scenario, the slight deficit currently observedin the M dijet > Λ /s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ∈ + uu->uu ² Λ /s0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ∈ + dd->dd Figure 2: Ratio 1 + ǫ (see eq. (32)) as a function of ˆ s/ Λ for uu → uu (up) and dd → dd (down) contact interactions.Continuous (dashed) lines correspond to interactions with the same (opposite) chiral projector in the two currents,and thick (thin) lines are for a positive (negative) coefficient in the Lagrangian. Red lines are for O , and blue for O . Black are the flavour-summed O pythia for comparison. The bounds derived on Λ are obtained by requesting thatˆ s/ Λ should be small enough to give 1 + ǫ < ∼ . → . ∼ Λ /s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ∈ + ud->ud Figure 3: Ratio 1 + ǫ (see eq. (32)) as a function of ˆ s/ Λ for ud → ud contact interactions. Continuous (dashed) linescorrespond to interactions with the same (opposite) chiral projector in the two currents, and thick (thin) lines are fora positive (negative) coefficient in the Lagrangian. Red lines are for O , blue for O , green for O CC , yellow for O S ,pink for O S .massive states, whereas the former, linked with a scale Λ ∼
11 TeV, could still remain a contact interaction if studiedby a LHC running at 14 TeV.
Contact interactions can be induced by the exchange of new resonances which are not resolved as mass eigenstates,because they are either broad or heavier than the exchanged four-momentum. The tree-level exchange of an off-shellboson of mass M new > ∼ M dijet , interacting with the quarks via a coupling g ′ , would give 4 π/ Λ ∼ g ′ /M . However,the order of magnitude of collider bounds, estimated in eq. (25), is 1 / Λ ∼ α s / ˆ s , so M new > ∼ M dijet if α ′ > ∼ α . Thecontact interaction approximation is appropriate at a collider to describe the exchange of particles with O (1) couplings.So if the 14 TeV LHC reaches a sensitivity of Λ ∼ −
30 TeV, this excludes strongly coupled particles, without aconserved parity, up to masses ∼
10 TeV.Heavy particles interacting with quarks can evade contributing to contact interactions in various ways. If newparticles have a conserved parity (as is convenient to obtain dark matter), and couplings g ′ < ∼ g s , they can generatecontact interactions via a closed loop of heavy new particles. This implies either a contact interaction coefficient4 π/ Λ ∼ g ′ / (16 π M new ) , (33)which is small enough to be allowed for M new ∼ M dijet (it gives Λ > M new /α ′ ), or the absence of a contact interactionfor g ′ / (16 π M new ) ∼ g s /M dijet , (34)since the new particles could then be produced in pairs. Heavy new bosons that are less strongly coupled to quarks canalso evade contact interaction bounds. For instance, a Z ′ with Standard Model couplings induces various four-quarkcontact interactions with a coefficient 4 π/ Λ ∼ g / (8 c W M Z ′ ).Despite that a contact interaction which is absent at 3 TeV might be present at lower energies, it is interesting tocompare the collider bounds on four-quark contact interactions to limits from precision flavour physics. The first step13 Λ /s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ∈ + uu->cc ² Λ /s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ∈ + ud->cs Figure 4: Ratio 1 + ǫ (see eq. (32)) as a function of ˆ s/ Λ for flavour changing contact interactions uu → cc (up) or ud → cs (down). Continuous (dashed) lines correspond to interactions with the same (opposite) chiral projector inthe two currents. There is no interference with QCD, so the sign of the contact interactions does not matter. Redlines are for O , blue for O , yellow for O S , pink for O S . Black are the flavour-summed O pythia for comparison.should be to evolve the operator coefficients between the TeV scale and low energy ( e.g. m b ):4 π Λ (cid:12)(cid:12)(cid:12)(cid:12) m b ≃ c π Λ (cid:12)(cid:12)(cid:12)(cid:12) ∼ (cid:18) α s (3 TeV) α s ( m b ) (cid:19) γ/ β π Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β = (11 N c − N f ) / π . For a few cases where we know the anomalous dimension γ of the contact interaction,0 . < c < . π/ Λ obtained at 3 TeV. Dueto the importance of Fermi interaction in low-energy precision physics, it is convenient to define the parameter β as4 π Λ = β G F √ ⇒ β = 4 πv Λ = (cid:18) . (cid:19) so the collider bounds discussed previously, ranging from 3 to 11 TeV, imply β < ∼ × − → × − . Let usadd that flavour physics and collider searches do not have the same scope in probing the flavour structure of contactinteractions. As discussed above, proton-proton collider searches have the potential to search four-quark contactinteractions involving at least two quarks of the first generation, but the two other quark lines are left free (andperhaps could be identified through jet tagging). In flavour physics, neutral-meson mixing is a sensitive probe offlavour off-diagonal (∆ F = 2) contact interactions [21, 22], with particularly stringent bounds on Λ for operatorsinducing kaon mixing (assuming, as we have done here, that the coefficients η are O (1)). Bounds on other four-quarkoperators could in principle come from processes where a meson decays into two lighter mesons non-leptonically –however, such processes are very challenging from the theoretical point of view, and can hardly be considered as usefulto set constraints on contact interactions.If a signal for contact interactions was observed at the LHC, it could indicate strongly coupled New Physics at ascale just beyond the reach of the LHC, or the (perturbative) exchange of a (broad) resonance in t or s -channel. Ineither case, it would be interesting to identify the flavour of the final state jets — not only to distinguish gluons andheavy flavours ( b, c ) from light quarks, but even to distinguish u from d jets using jet charge. This would allow oneto predict the expected rate for the crossed process. For instance, if a t -channel Z ′ mediates uu → uu , then it couldalso be searched for as a bump in u ¯ u → u ¯ u whereas an s -channel diquark inducing the contact interactions ud → cs is not worth searching for in t channel. Many models with new particles at high energy have contact interactions as their low-energy footprints. As such,contact interactions are a parametrisation of New Physics, so it is important that constraints on upon them be aswidely applicable as possible. Current collider bounds are calculated for a palette of colour-singlet interactions (nocolour matrices in the vertex), summed on flavour. In this paper, we estimate bounds on an almost complete basis offour-quark contact interactions, with specified flavour indices mediating q i q j → q m q n , where q i , q j are first-generationquarks.We start from a basis of dimension six, Standard Model gauge-invariant, four-quark operators. After electroweaksymmetry breaking, each operator induces one or several four-quark interactions. We constrain the coefficients ofthese effective interactions by turning them on one at a time. We prefer to constrain the coefficients of the effectiveinteractions, rather than the gauge-invariant operators, because this allows us to apply the bounds to a differentoperator basis. The effective interactions (almost) do not interfere among themselves, so that the bounds obtainedby turning them on one at a time are conservative. A more stringent bound could apply in the presence of severalcontact interactions.The bounds profit from a convenient choice of variables made by the experimental collaborations, which allows oneto approximate the pp → dijets cross section as an integral over parton distribution functions multiplying a partoniccross section. We estimate the expected limit on contact interactions by comparing their partonic cross sections to theleading order QCD prediction. The data agree with QCD, so we require that the contact interactions contribute lessthan 1/3 or 1/2 of the QCD expectation. Our bounds are listed in tables 3-8. Our flavoured estimates are genericallylower than the limits of ATLAS and CMS, who constrain a flavour-sum of contact interactions given in eq. (26). Ourestimate for this flavour-summed operator is comparable to the experimental bounds.The original aim of this paper was to identify “classes” among the large collection of contact interactions, such thata bound obtained on a representative of the class could be translated by some simple rule to the others. The situationis however more complicated: the obstacle seems to be the interference between QCD and the contact interaction,which precludes any simple scaling of the limit on ˆ s/ Λ from one interaction to another . An interesting feature ofour analytic recipe is that it constrains the dimensionless ratio M dijet / Λ for each contact interaction. The futureLHC, with more energy and luminosity, will be able to probe a higher dijet mass than the M dijet = 3 TeV [4] usedhere to extract bounds on Λ. The estimated bounds on Λ from tables 3-8 should therefore scale as M dijet ( new ) / (3TeV), under assumptions given in section 5.3. It is mildly surprising that a limit on contact interactions at a hadron collider can be estimated analytically; however, in practice, itmay be just as simple to use Madgraph5 [23]. ≤ χ ≤ M dijet > We thank M. Gouzevitch, A. Hinzmann, G. Salam, and P. Skands for useful conversations. This project was performedin the context of the Lyon Institute of Origins, ANR-10-LABX-66.
A Kinematics and cross sections
This Appendix gives the contact interaction correction, in sections labeled by flavour structure, for a larger set ofcontact interactions than the basis of section 3.2 (in particular, we give cross sections for octet interactions involvingfour u - or d -type quarks,although these are not in the list of section 3.2). The formulae are labeled to the right bythe contact interaction to which they apply. The partonic cross sections mediated by contact interactions are given inseveral references [17, 24] , and are collected below for convenience. A.1 Definitions
For momenta as given in the first diagram of figure 1, with time running left to right, the Mandelstam variables aredefined as ˆ s = ( k + k ) = 2 k · k = 2 p · p = 4 E ∗ ˆ t = ( k − p ) = − k · p = − p · k = − E ∗ (1 − cos θ ∗ )ˆ u = ( k − p ) = − k · p = − p · k = − E ∗ (1 + cos θ ∗ ) (35)where the last expressions are in the parton centre-of mass frame. If time runs upwards in the same diagram, itdescribes q ¯ q annihilations, and the Mandelstam variables exchange their definitions:ˆ s scat = ˆ u ann , ˆ t scat = ˆ s ann , ˆ u scat = ˆ t ann (36)In the following, we give various partonic differential cross sections, d ˆ σd ˆ t = |M| π ˆ s −→ d ˆ σdχ = ˆ t ˆ s (cid:18) d ˆ σd ˆ t (ˆ s, ˆ t, ˆ u ) + d ˆ σd ˆ t (ˆ s, ˆ u, ˆ t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ ∗ <π/ where |M| is averaged over incoming spins and colours, and contains a factor 1/2 when the final state particles areidentical. Since the change of variables to χ is simple for cos θ ∗ >
0, we restrict 0 ≤ θ ∗ ≤ π/
2, and add d ˆ σ/d ˆ t withˆ t ↔ ˆ u .We include the QCD contribution, the contact-interaction interference with QCD (and sometimes QED), and thecontact interaction squared. We neglect the pure QED contribution (subdominant with respect to QCD), but includeQED-contact interference when there is no QCD-contact interference. Indeed the interference term ∝ α em ˆ s/ Λ canreduce the cross section and have a minor effect on the bound. A.2 q i g → q i g — QCD only dσd ˆ t = 4 πα s s (cid:20) − ˆ u ˆ s − ˆ s ˆ u + 94 ˆ s + ˆ u ˆ t (cid:21) (37) These formulae do not always agree with ref. [25], and disagree on the sign of some interferences with respect to ref. [26] .3 q i q j → q m q n A.3.1 qq → qq , ( i = j = m = n ) d ˆ σd ˆ t ( qq → qq ) = 2 πα s s (cid:20) ˆ s + ˆ u ˆ t + ˆ t + ˆ s ˆ u −
23 ˆ s ˆ t ˆ u (cid:21) ≡ QCD (38) d ˆ σd ˆ t ( qq → qq ) = QCD − η ,XX Λ πα s
27 ˆ s ˆ u ˆ t + | η ,XX | Λ π O ,RR , O ,LL (39)= QCD + η ,XY Λ πα s s ˆ u + ˆ t ˆ t ˆ u + | η ,XY | Λ π u + ˆ t )ˆ s O ,RL = O ,RL (40)= QCD − η ,XX Λ πα s s ˆ u ˆ t + | η ,XX | Λ π O ,LL , O ,RR (41)= QCD + η ,XY πα em Λ (cid:18) ˆ u ˆ t + ˆ t ˆ u (cid:19) s + | η ,XY | Λ π (ˆ t + ˆ u )ˆ s O ,RL = O ,RL (42)A factor of 1/2 for identical fermions in the final state is included in eqs. (38, 41, and 39). When the final state quarkshave opposite chirality, they are not identical, but the operators O RL = O ,RL are, so its the same to include oneoperator for distinct fermions, or two operators for identical fermions.Interference with QED is included for the singlet operators involving quarks of different chirality: O ,RL + O ,RL .The interference with QCD is absent because the spinor traces vanish: a non-zero trace must contain an even numberof colour matrices, so both the QCD vertices must appear in the trace. Therefore there is only one trace, in whichappear both contact vertices, with conflicting chiral projection operators. However, QED has no colour matrices, sothere is an interference term with two spinor traces. A.3.2 qq ′ → qq ′ , i = m = j = n . d ˆ σd ˆ t ( ud → ud ) = 4 πα s t ˆ s + ˆ u ˆ s ≡ QCD ′ (43)= QCD ′ + η ,XX Λ πα s t + | η ,XX | Λ π O ,XX (44)= QCD ′ + η ,XY Λ πα s u ˆ t ˆ s + | η ,XY | Λ π u ˆ s O ,XY (45)(46)= QCD ′ + 2 πη ,XX α em Λ t + π | η ,XX | Λ O ,RR (47)= QCD ′ + 2 πη ,LL α em Λ t + 8 π η ,CC α Λ t + π (cid:18) | η ,LL | Λ + 23 η ,CC η ,LL Λ Λ + | η ,CC | Λ (cid:19) O ,LL + O ,CC (48)= QCD ′ + 2 πη ,XY α em Λ ˆ u ˆ t ˆ s + π | η ,XY | Λ ˆ u ˆ s O ,XY (49)= QCD ′ + π u + 4ˆ t − ˆ s ˆ s O S (50)= QCD ′ + π u + 2ˆ t + ˆ s ˆ s O S (51)Notice that for ud → ud , O ,LR is different from O ,RL . so their contributions are not summed in the above formulae.Interference with QED is included when there is no interference with QCD, and the interference between O ,LL and O ,CC is given, although we constrain the two operators separately. A.3.3 q ′ q ′ → qq If the quark flavour changes at the contact interaction, there is no interference with QCD. However, there are twocontact interaction diagrams, and an interference term when the initial and final states contain identical quarks. At17he LHC, this can describe uu → cc , dd → ss , and dd → bb . d ˆ σd ˆ t = | η ,XX | Λ π O ,XX (52)= | η ,XY | Λ π t + ˆ u )ˆ s O ,XY = O ,Y X (53)= | η ,XX | Λ π O ,XX (54)= | η ,XY | Λ π (ˆ t + ˆ u )ˆ s O ,XY = O ,Y X (55)A factor of 1/2 for identical fermions in the final state is included. In practice, this list is just last terms from qq → qq . A.3.4 q ′ q ′′ → qq , or qq → q ′ q ′′ In the case where there are identical fermions in either the initial or final states, but not both (at the LHC: uu → uc , dd → ds , dd → db , dd → sb ), there are still two diagrams, but no interference term: d ˆ σd ˆ t ( dd → sb ) = | η ,XX | Λ π O ,XX (56)= | η ,XY | Λ π t + ˆ u )ˆ s O ,XY (57)= | η ,XX | Λ π O ,XX (58)= | η ,XY | Λ π ˆ t + ˆ u ˆ s O ,XY (59)and in the case where the identical fermions are in the final state, the given formulae should be multiplied by 1/2. A.3.5 q ′′ q → qq ′ and any vertex with more than three different flavours At the LHC this can describe ud → cs , ud → cb , and also ud → us , ud → ub , ud → cd : d ˆ σd ˆ t = | η ,XX | Λ π , O ,XX (60)= | η ,XY | Λ π u ˆ s , O ,XY (61)= | η ,XX | Λ π , O ,XX , O ,CC (62)= | η ,XY | Λ π ˆ u ˆ s , O ,XY (63)= π u + 4ˆ t − ˆ s ˆ s O S (64)= π u + 2ˆ t + ˆ s ˆ s O S (65) A.4 q i ¯ q m → ¯ q j q n For contact interactions with two incident first generation quarks, the best bounds arise from qq → qq . However, a“flavour diagonal” interaction involving a quark and anti-quark of the first generation, going to a quark and anti-quarkof a higher generation, is better constrained by the Tevatron, who had valence q q in the initial state. The crosssections for contact interactions in quark-anti-quark collisions, can be obtained by crossing (36), the previous formulae,and removing, if neccessary, the factor of 1/2 for identical fermions in the final state. References [1] W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions and Flavor Conservation,”Nucl. Phys. B (1986) 621. 182] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, “Dimension-Six Terms in the Standard ModelLagrangian,” JHEP (2010) 085 [arXiv:1008.4884 [hep-ph]].[3] G. Aad et al. [ATLAS Collaboration], “ATLAS search for new phenomena in dijet mass and angular distri-butions using pp collisions at √ s = 7 TeV,” JHEP (2013) 029 [arXiv:1210.1718 [hep-ex]].[4] S. Chatrchyan et al. [CMS Collaboration], “Search for quark compositeness in dijet angular distributionsfrom pp collisions at √ s = 7 TeV,” JHEP (2012) 055 [arXiv:1202.5535 [hep-ex]].[5] S. Chatrchyan et al. [CMS Collaboration], “Search for contact interactions using the inclusive jet p T spectrumin p p collisions at √ s = 7 TeV,” Phys. Rev. D (2013) 052017 [arXiv:1301.5023 [hep-ex]].[6] V. M. Abazov et al. [D0 Collaboration], “Measurement of dijet angular distributions at s**(1/2) = 1.96-TeVand searches for quark compositeness and extra spatial dimensions,” Phys. Rev. Lett. (2009) 191803[arXiv:0906.4819 [hep-ex]].[7] M. Redi, V. Sanz, M. de Vries and A. Weiler, “Strong Signatures of Right-Handed Compositeness,” JHEP (2013) 008 [arXiv:1305.3818 [hep-ph]].[8] R. S. Chivukula, E. H. Simmons and N. Vignaroli, “A Flavorful Top-Coloron Model,” Phys. Rev. D (2013) 075002 [arXiv:1302.1069 [hep-ph]].[9] M. Carpentier and S. Davidson, “Constraints on two-lepton, two quark operators,” Eur. Phys. J. C (2010)1071 [arXiv:1008.0280 [hep-ph]].[10] A. E. Nelson, “Contact terms, compositeness, and atomic parity violation,” Phys. Rev. Lett. (1997) 4159[hep-ph/9703379].[11] J. Gao, C. S. Li, J. Wang, H. X. Zhu and C. -P. Yuan, “Next-to-leading QCD effect to the quark compositenesssearch at the LHC,” Phys. Rev. Lett. (2011) 142001 [arXiv:1101.4611 [hep-ph]].[12] E. Eichten, K. D. Lane and M. E. Peskin, “New Tests for Quark and Lepton Substructure,” Phys. Rev. Lett. (1983) 811.[13] T. Sjostrand, S. Mrenna and P. Z. Skands, “A Brief Introduction to PYTHIA 8.1,” Comput. Phys. Commun. (2008) 852 [arXiv:0710.3820 [hep-ph]].T. Sjostrand, S. Mrenna and P. Z. Skands, “PYTHIA 6.4 Physics and Manual,” JHEP (2006) 026[hep-ph/0603175].[14] H. -L. Lai, M. Guzzi, J. Huston, Z. Li, P. M. Nadolsky, J. Pumplin and C. -P. Yuan, “New parton distributionsfor collider physics,” Phys. Rev. D (2010) 074024 [arXiv:1007.2241 [hep-ph]].[15] L. Hartgring, E. Laenen and P. Skands, “Antenna Showers with One-Loop Matrix Elements,” arXiv:1303.4974[hep-ph].P. Skands, Talk on generator tuning, CERN, oct 18, 2013, http://skands.web.cern.ch/skands/slides/[16] J. Beringer et al. (Particle Data Group), Phys. Rev. D , (2012) 010001[17] R. S. Chivukula, A. G. Cohen and E. H. Simmons, “New strong interactions at the Tevatron?,” Phys. Lett.B (1996) 92 [hep-ph/9603311].[18] H. Potter and G. Valencia, “Probing lepton gluonic couplings at the LHC,” Phys. Lett. B (2012) 95[arXiv:1202.1780 [hep-ph]].[19] L. Apanasevich, S. Upadhyay, N. Varelas, D. Whiteson and F. Yu, “Sensitivity of potential future pp collidersto quark compositeness,” arXiv:1307.7149 [hep-ex].[20] A. J. Buras, “Weak Hamiltonian, CP violation and rare decays,” hep-ph/9806471.[21] M. Bona et al. [UTfit Collaboration], “Model-independent constraints on ∆ F=2 operators and the scale ofnew physics,” JHEP (2008) 049 [arXiv:0707.0636 [hep-ph]].1922] G. Isidori, Y. Nir and G. Perez, “Flavor Physics Constraints for Physics Beyond the Standard Model,” Ann.Rev. Nucl. Part. Sci. (2010) 355 [arXiv:1002.0900 [hep-ph]].J. Charles, S. Descotes-Genon, Z. Ligeti, S. Monteil, M. Papucci and K. Trabelsi, “Future sensitivity to newphysics in B d , B s and K mixings,” arXiv:1309.2293 [hep-ph].[23] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, “MadGraph 5 : Going Beyond,” JHEP (2011) 128 [arXiv:1106.0522 [hep-ph]].[24] E. Argyres, U. Baur, P. Chiappetta, C. Papadopoulos, M. Perrottet, M. Spira, S. Vlassopulos and P. M. Zer-was, “Compositeness,” CERN-TH-5980-91D.[25] E. Eichten, I. Hinchliffe, K. D. Lane and C. Quigg, “Super Collider Physics,” Rev. Mod. Phys. (1984)579 [Addendum-ibid.58