Supersymmetric QCD Parity Nonconservation in Top Quark Pairs at the Tevatron

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Can the Supersymmetric $mu$ parameter be generated dynamically without a light Singlet

a r X i v :h e p -p h /9404257v 1 12 A p r 1994DESY 94-061ISSN 0418-9833April 1994hep-ph/9404257Can the Supersymmetric µparameter be generated dynamically without a light Singlet?Ralf Hempfling Deutsches Elektronen-Synchrotron,Notkestraße 85,D-22603Hamburg,Germany ABSTRACTIt is generally assumed that the dynamical generation of the Higgs mass param-eter of the superpotential,µ,implies the existence of a light singlet at or below the supersymmetry breaking scale,M SUSY .We present a counter-example in which the singlet ﬁeld can receive an arbitrarily heavy mass (e.g.,of the order of the Planck scale,M P ≈1019GeV).In this example,a non-zero value of µis generatedthrough soft supersymmetry breaking parameters and is thus naturally of the order of M SUSY .The cancellation of quadratic divergences in the unrenormalized Green func-tions is one of the main motivations of supersymmetry(SUSY).It stabilizes any mass scale under radiative corrections and thus allows the existence of diﬀerent mass scales such as the electroweak scale,given by the Z boson mass,m z,and the Planck scale,M P.The minimal supersymmetric standard model(MSSM)is the most popular model of this kind due to its minimal particle content[1].In this model,the SU(2)L⊗U(1)Y symmetry breaking is driven by soft SUSY break-ing parameters.Thus,the SUSY breaking scale,M SUSY,has to be at or slightly above m z.For this mechanism to work it is also necessary that the SUSY Higgs mass parameter,|µ|<∼M SUSY.This parameter also determines the chargino and neutralino mass spectrum.From here one can deduce a experimental lower bound from LEP experiments of|µ|>∼m z/4independent of tanβ[2].The fact that in the MSSM theµ-parameter,which is a priori arbitrary,has to lie within the narrow range1will be evaded[5].We will demonstrate in the following that it is also possible to make N1heavy [say m N1=O(M P)]while keeping N1 =O(M SUSY)withoutﬁne-tuning.In this limit we recover the predictive Higgs sector of the MSSM[6]with its well deﬁned upper limit of the lightest Higgs boson mass[eq.(2)].First we need to extend the symmetry group of our Lagrangian in order to forbid the explicit Higgs mass term of the superpotential,W H=µH8(ξ+2N∗1N1−2N∗2N2−N∗3N3)2.(4)Here the inclusion of a Fayet-Iliopoulos term[7],ξ,is the easiest way of breaking the U(1)Y′gauge symmetry but one can envisage other alternatives[8].The VEVs are denoted byn1= N1 =0,n2= N2 =1λ+ λ2+4ξ , n3= N3 = λ.(5)The CP-even and CP-odd components of the scalarﬁeld N1are mass-degenerate mass-eigenstates with m N1=(m2+λ2n23)1/2.The gauge boson,g′,acquires a massm g′=g′(n22+n23/4)1/2via the Higgs mechanism.The masses of the remaining CP-,m g′(m N1,0;the zero mass eigenvalue corresponds even(CP-odd)scalars are m N1to the Goldstone boson which is absorbed to give mass to the gauge boson).Theand±m g′as required if mass eigenvalues of the fermionic components are±m N1SUSY is unbroken.Note that in addition to the gauge and the SUSY transformations the La-grangian is invariant under the global U(1)R-symmetry[9]which does not com-mute with SUSY.This symmetry transformsΦ→exp(inΦα)Φ,where nΦ= 2,0,0,0for the bosons and n Φ=1,−1,−1,1for the fermions(Φ=N1,N2,N3,g′). We now break SUSY explicitly in the standard fashion by including soft SUSY breaking terms[10]V soft=BmN1N2−AλN1N23+h.c.,(6) where A,B=O(M SUSY)are the soft SUSY breaking parameters.With these terms the R-symmetry is broken down to a discrete Z2symmetry(α=±π).If we minimize the full potential,V=V SUSY+V soft,weﬁndN1 ≈(A−B)mn2ξ.However,the condition N1 =0is protected by R-symmetry to all orders in perturbation theory and is only broken by adding soft SUSY breaking terms[eq.(6)].We now include in our model the full particle content of the MSSM.The Z2symmetry is equivalent to the usual R-parity that prevents baryon and lepton number violating interactions. The full superpotential can then be written asW=W N+W H+W Y,(8) where W H=κN1Hof W in eq.(8)and by requiring the absence of anomalies.These constraints can be satisﬁed by introducing additional pairs of SUSY multiplets T∼(n c,n w,Y,Y′1) and T c∼(¯n c,n w,−Y,Y′2).These representations have been included in pairs such√that below the U(1)Y′breaking scale,Useful conversations with W.Buchm¨u ller are gratefully ac-knowledgedREFERENCES1.H.P.Nilles,Phys.Rep.110,1(1984);H.E.Haber and G.L.Kane,Phys.Rep.117,75(1985);R.Barbieri,Riv.Nuovo Cimento11,1(1988).2.J.-F.Grivaz,in Proceedings of the Workshop on e+e−Collisions at500GeV:The Physics Potential,Munich,Annecy,Hamburg,DESY report DESY92-123B(1992).3.G.F.Guidice and A.Masiero,Phys.Lett.B206,480(1988);J.E.Kimand H.P.Nilles,Phys.Lett.B263,79(1991);J.A.Casas and C.Mu˜n oz, Phys.Lett.B306,288(1993).4.E.Witten,Phys.Lett.B105,267(1981);L.Ib´a˜n ez and G.G.Ross,Phys.Lett.B110,215(1982);P.V.Nanopoulos and K.Tamvakis,Phys.Lett.B113,151(1982).5.see,e.g.,J.Ellis,J.F.Gunion,H.E.Haber,L.Roszkowski and F.Zwirner,Phys.Rev.D39,844(1989).6.see,e.g.,J.F.Gunion,H.E.Haber,G.L.Kane,and S.Dawson,The HiggsHunter’s Guide,(Addison-Wesley,Redwood City,CA,1990).7.P.Fayet and J.Iliopoulos,Phys.Lett.B51,461(1974).8.e.g.,L.O’Raifeartaigh,Nucl.Phys.B96,331(1975).9.P.Fayet,Nucl.Phys.B90,104(1975);A.Salam and J.Strathdee,Nucl.Phys.B87,85(1975).10.L.Girardello and M.T.Grisaru,Nucl.Phys.B194,65(1982).。

Dynamical supersymmetry breaking with a large internal dimension

Abstract Supersymmetry breaking in string perturbation theory predicts the existence of a new dimension at the TeV scale. The simplest realization of the minimal supersymmetric Standard Model in the context of this mechanism has two important consequences: (i) A natural solution to the µ-problem; (ii) The absence of quadratic divergences in the cosmological constant, which leads to a dynamical determination of the supersymmetry breaking and electroweak scale. We present an explicit example in which the whole particle spectrum is given as a function of the top quark mass. A generic prediction of this mechanism is the existence of Kaluza-Klein excitations for gauge bosons and higgses. In particular the ﬁrst excitation of the photon could be accessible to future accelerators and give a clear signal of the proposed mechanism.

Hypermultiplets, domain walls and supersymmetric attractors

Abstract We establish general properties of supersymmetric ﬂow equations and of the superpotential of ﬁve-dimensional N = 2 gauged supergravity coupled to vector multiplets and hypermultiplets. We provide necessary and suﬃcient conditions for BPS domain walls and ﬁnd a set of algebraic attractor equations for N = 2 critical points. As an example we describe in detail the gauging of the universal hypermultiplet and a vector multiplet. We study a two-parameter family of superpotentials with supersymmetric AdS critical points and we ﬁnd, in particular, an N = 2 embedding for the UV–IR solution of Freedman, Gubser, Pilch and Warner of the N = 8 theory. We comment on the relevance of these results for brane world constructions.
Anna Ceresole⋆, Gianguido Dall’Agata♯, Renata Kallosh♮ and Antoine Van Proeyen♭

Aspects of Split Supersymmetry

1
Jeﬀerson Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
2
Physics Department, Stanford University, Stanford, California 94305, USA CERN, Theory Division, CH-1211 Geneva 23, Switzerland
1
Introduction
Our concept of Naturalness – the principle that Nature abhors ﬁne-tunings – is based on theories with a few vacua. It has led to the proposal of low-scale supersymmetry, in order to avoid tuning short-distance parameters to 30-decimals. But this principle faces a serious challenge from the cosmological constant problem, where we see no new physics at the scale ∼ 10−3 eV required by naturalness. Recent developments in string theory [1] suggest the existence of an enormous “landscape” of long-lived metastable vacua. These can have a signiﬁcant impact on the criterion of naturalness, and may provide the key to resolving the cosmological constant problem along the lines of ref. [2]. They may also have a bearing on the solution to the hierarchy problem. For example, if the number of vacua that break supersymmetry at high-scale is more than 1030 times larger than those with low-scale supersymmetry, then the breaking of supersymmetry at high scales is favored [3]. The 30-decimal tuning is compensated by the enormous “entropy factor” favoring high-scale breaking of supersymmetry. In this case, the simplest possibility would be that the Standard Model (SM) is preferred to the usual low-energy supersymmetric SM [4]. This would then account for why we have not seen any evidence for supersymmetry – either in the spectrum or in rare process, such as FCNCs, CP violation, proton decay, etc. – at the expense of giving up two major successes of the supersymmetric SM: gauge coupling uniﬁcation [5] and natural dark-matter candidate [6]. A more interesting possibility that preserves these successes is that approximate chiral symmetries protect the fermions of the supersymmetric SM down to the TeV scale [7, 8]. So, the sparticle spectrum of these theories is “split” in two: (i) the scalars (squarks and sleptons) that get a mass at the high-scale of supersymmetry breaking m ˜ , which can be as large as the GUT scale, and (ii) the fermions (gauginos and higgsinos) which remain near the electroweak scale and can account for both gauge-coupling uniﬁcation and DM. The only light scalar in this theory is a ﬁnely-tuned Higgs. So, rather than the dull prediction that the LHC will discover just the Higgs, these theories – coined Split Supersymmetry – predict gauginos and higgsinos at a TeV, maintain the successes of the supersymmetric SM, and account for the absence of problematic ﬂavor and CP-violation, of fast proton decay, and of an excessively light Higgs, caused by the presence of light squarks and sleptons in the supersymmetric SM. In this paper we address some novel theoretical, cosmological and phenomenological aspects of Split Supersymmetry1 . An important theoretical issue is how the spectrum of Split Supersymmetry can naturally emerge from a high-energy theory. In sects. 2 and 3 we

Supersymmetric QCD corrections to the W-boson width

a r X i v :h e p -p h /9610284v 1 8 O c t 1996DTP/96/88hep-ph/9610284October 1996Supersymmetric QCD corrections to the W -boson width hanas a and V.C.Spanos b a University of Athens,Physics Department,Nuclear and Particle Physics Section,GR–15771Athens,Greece b University of Durham,Department of Physics,Durham DH13LE,England Abstract We calculate the one-loop supersymmetric QCD corrections to the width of the W -boson.We ﬁnd that these are of order ∼αs 20M 2W 20these are at least two ordersof magnitude smaller than the standard QCD corrections ∼αs E-mail:a alahanas@atlas.uoa.gr,b V.C.Spanos@The second phase of the LEP(LEP2)collider has already started,and theﬁrst e+e−→W+W−events have been collected.Studying for the veryﬁrst time directly this process,one will have the opportunity to test the non-abelian character of the Standard Model(SM),through the precise measurements of the trilinear gauge boson couplings.In addition,it will be possible to measure precisely the mass and width of the W-boson[1].Speciﬁcally the measurement of the W-width is of special interest,as it is used as an input parameter in many other processes.(It is understood that all the W production events are detected through the hadronic and/or(semi)leptonic decays of the W-boson.)So it is very essential,both for theoretical and experimental reasons,to know as precise as possible the theoretical prediction for this parameter.The one-loop corrections to the W-width in the context of the Standard Model(SM) are already known[2,3],and there has been also a calculation in the context of a two Higgs doublet model[4].The possible existence of new physics of characteristic scale M new may aﬀect the theoretical predictions for the W-boson decay width.The magnitude of these eﬀects is not a priori known without knowledge of the underlying theory1and thus manifestation of new physics from a direct measurement of the W-boson width is not possible.The corrections to the W-boson observables which are induced by new physics are expected to be small,possibly smaller than the experimental precision of LEP2which will be in the percent region.With increasing experimental accuracy in the future,the W-boson observables may provide a laboratory for testing new physics and Supersymmetry is a prominent candidate.It is known that strong interaction eﬀects yield the largest contribution,O(4%),to the W-width at the one-loop order.With the SM being promoted to a supersymmetric theory,the QCD sector is also supersymmetrized(SQCD)and new species which interact strongly aﬀect the QCD predictions.Therefore it seems natural to calculate the SQCD corrections to the hadronic width of the W.The size of these corrections depends on the supersymmetry breaking scale M S and is obviously negligible as M S becomes large. However the existing experimental lower bounds on sparticle masses does not exclude values of M S in the vicinity of the electroweak scale M S≃O(few M W),in which case these eﬀects may not be suppressed.In this Letter we undertake this problem and calculate the supersymmetric QCD corrections to the W-boson width.We perform our calculations using the on-shell renor-(a)g(b)Figure 1:Graphs which contribute to the one-loop supersymmetric QCD correc-tions to the W -width.There are corrections to the W u ¯dvertex (a)and corrections to the external quark propagators (b).malization scheme [6,7]which has been extensively used in the SM calculations (see for instance Ref.[8,9]).In order to study the SQCD corrections to the W -boson hadronicwidth we need to calculate the corrections to W u ¯dvertex as well as the wave function renormalizations to the external fermion propagators (see Fig.1(a)and (b)respectively).In order to simplify our discussion we shall neglect mixings of the up ˜u L ,˜u c L and down ˜d L ,˜d c Lleft handed squarks of the ﬁrst two generations since these mixings are propor-tional to the corresponding fermion masses and hence small.Therefore the above squark states are mass eigenstates in this approximation.By using the well known Passarino–Veltman functions [7,10]B 0,B 1,B ′1,C 0,C 1,C ijetc.,through which the two and three point functions are usually expressed 2,we ﬁnd forthe W u ¯dvertex correction of Fig.1(a)A 1=i (αs √(4π),where g s is the strong coupling constant.M u,d are themasses of the u,d external quarks and the factor c F =4/3is the value of the quadraticCasimir operator of the fundamental representation of the SU(3)symmetry group.The arguments of the C ij functions appearing above are deﬁned as follows:C ij=C ij(p1,−p1−p2,M2˜g,m2˜u L,m2˜d L).In this expression p1(−p2)is the momentum carried by the outgoing(incoming)u(d) quark.The ultraviolet inﬁnity of the vertex correction is contained within the factor C24of the amplitude M0.This inﬁnity is canceled by the vertex counterterm in the Lagrangian[9],∆L CT=(δZ L+δZ W1−δZ W2)g2W+µ¯u Lγµd L,(2)δZ L≡Z L−1,where Z L is the wave function renormalization constant of the left handed doublet(u L,d L).There are no strong interaction contributions to the diﬀerence δZ W1−δZ W2so that onlyδZ L needs be considered.For the down quark the on-shell renormalization condition is−→(/P−M d)−1,(3)S down(P)/P→M dwhere S down(P)denotes the down quark propagator.Thisﬁxes the wave function renor-malization constant of both left and right handed components of the down quark.By a straightforward calculation of the graph shown in Fig.1(b),and using Eq.(3),weﬁnd forδZ L,which is needed for our calculation,αsδZ L=(√Note that since left handed up (I 3=1/2)and down (I 3=−1/2)components belong to the same multiplet and Z L has already been ﬁxed by Eq.(3)we cannot have z u =1.The residue z u is ﬁnite and is given byz u ≡1+δz u =1+(αs√2)=i c F gαs 2(Πu (M 2u )−Πd (M 2d ))M 0.(7)This completes our calculation of the SQCD corrections to the amplitude for W +→u ¯d.We now proceed to discussing the corrections to the hadronic width of the W -boson.The one-loop hadronic width Γ(1)can be written asΓ(1)=Γ(0)u ¯d (1+δ),(8)where Γ(0)u ¯d is the tree level hadronic width for one family.In the limit of vanishing quark masses this is given by Γ(0)u ¯d =αw M W /4.The SQCD corrections to δcan be found from the amplitudes A 1,A 1′,A 2we have just calculated.It is found thatδSQCD =αs2]+ (9)In order to avoid confusion we should say that δSQCD accounts for only the supersymmet-ric corrections,that is those due to the exchange of gluinos and squarks.The functions C 24,B 1,Πu,d are as they appear in the deﬁnitions of the amplitudes A 1,A ′1,A 2,while the ellipses denote terms proportional to the external quark masses.In the limit of vanishing quark masses 3,δSQCD can be cast in the following integral formδSQCD =2αs(M 2W x 2y (y −1)+(m 2˜d L−m 2˜u L )xy +(m 2˜u L −M 2˜g )x +M 2˜g )2}.(10)00.010.020.030.040.0500.20.40.60.81F(a)F'(a)a Figure 2:The function F (a )and its derivative as described in the text.¿From the form above we can easily get ﬁrst estimates of the magnitude of the SQCD corrections as will be seen in the sequel.In order to simplify the discussion let us assume that m ˜u L ≈m ˜dL ≈M ˜g =M S ,where M S sets the order of the supersymmetry breaking scale.In this case the expression above for δSQCD is simpliﬁed,to becomeδSQCD =4αsM 2W x 2y (y −1)+M 2S }≡4αs 24+a 23360+...(12)Keeping the ﬁrst term of the expansion results in δSQCD =αs18M 2W3The O (M u,d )terms give a negligible contribution and hence it is permissible to omit them.fact the function F(a)is almost linear in the interval0<a<1with almost constant derivative,justifying the linear approximation to F which led to the result above.Note the appearance of an extra suppression factor1/18in addition to the expected M2WπM2WπM2Wπ,which is the contribution of gluons toδ,we see that the appearance of gluinos and squarks has a negligible eﬀect≤O(10−2)αsπ.Therefore supersymmetric QCD corrections to the W-boson width are at best of the order of the two-loop electroweak corrections and not of relevance to current experiments.AcknowledgementsWe want to thank James Stirling for a careful reading of this manuscript.This work was supported by the EU Human Capital and Mobility Programme,CHRX–CT93–0319.References[1]‘Determination of the Mass of the W Boson’,Z.Kunszt and W.J.Stirling et al.,in Physics at LEP2,eds.G.Altarelli,T.Sj¨o strand and F.Zwirner,CERN Report 96-01(1996),vol.1,p.141;‘WW cross-sections and distributions’,W.Beenakker and F.A.Berends et al.,in Physics at LEP2,eds.G.Altarelli,T.Sj¨o strand and F.Zwirner,CERN Report 96-01(1996),vol.1,p.81.[2]T.H.Chang,K.J.F.Gaemers and W.L.van Neerven,Nucl.Phys.B202(1982)407;K.Inoue,A.Kakuto,H.Komatsu and S.Takeshita,Prog.Theor.Phys.64(1980) 1008;D.Bardin,S.Riemann and T.Riemann,Z.Phys.C32(1986)121;J.W.Jun and C.Jue,Mod.Phys.Lett.A6(1991)2767.[3]A.Denner and T.Sack,Z.Phys.C46(1990)653.[4]D.-S.Shin,Nucl.Phys.B449(1995)69.[5]J.Rosner,M.Worah and T.Takeuchi,Phys.Rev.D49(1994)1363.[6]D.A.Ross and J.C.Taylor,Nucl.Phys.B51(1973)125.[7]G.Passarino and M.Veltman,Nucl.Phys.B160(1979)151.[8]M.B¨o hm,H.Spiesberger and W.Hollik,Fortschr.Phys.34(1986)687.[9]“Renormalization of the Standard Model”,W.Hollik,appears in Precision tests ofthe Standard Model,Advanced series on directions in high-energy physics,ed.Paul Langacker,World Scientiﬁc,1993.[10]G.’t Hooft and M.Veltman,Nucl.Phys.B153(1979)369.。

Hidden Symmetries of Large N QCD

2
1
Introduction
Recently [1], some progress was achieved in a long-standing problem of string representation ˆ µ of the endpoint of the QCD string of large N QCD. We introduced the position operator X and applied the momentum loop equation [8, 9] (MLE) to get some relations between the planar connected moments of vacuum expectation values of this operator. The non-commutative probability theory [6, 5, 7] was then used to build the Fock space ˆ µ in powers of the creation operator a† and operator expansion of X µ which obeys the Cuntz † † † † algebra aµ aν = δµν . The Fock space consists of words aµ1 aµ2 . . . aµn |0 , where the vacuum |0 is annihilated by aµ . There are several problems with this approach. One problem is that some coeﬃcients of this operator expansion ˆ µ = aµ + Aa† + Ba† a† a† + C a† a† a† + a† a† a† + . . . . X µ ν ν νtract

Manifestly N=3 supersymmetric Euler-Heisenberg action in light-cone superspace

a r X i v :h e p -t h /0002134v 1 16 F eb 2000ITP–UH–02/00February 2000NBI–HE–00–05hep-th/0002134Manifestly N=3Supersymmetric Euler-Heisenberg Action in Light-Cone SuperspaceThomas B¨o ttner,Sergei V.Ketov 1and Thomas Lau Institut f¨u r Theoretische Physik,Universit¨a t HannoverAppelstraße 2,Hannover 30167,Germany boettner,ketov,lau@itp.uni-hannover.deAbstractWe ﬁnd a manifestly N=3supersymmetric generalization of the four-dimensional Euler-Heisenberg (four-derivative,or F 4)part of the Born-Infeld action in light-cone gauge,by using N=3light-cone superspace.1IntroductionThe Born-Infeld (BI)action in ﬂat spacetime,2S BI =1−det(ηµν+bF µν),(1.1)is the particular non-linear generalization of Maxwell theory,F µν=∂µA ν−∂νA µ.The action (1.1)was initially introduced to regularize both the electric ﬁeld and the self-energy of a point-like charge in electrodynamics [1].Much later,the BI action was recognized as the leading contribution to the eﬀective action of open strings in an abelian background with constant ﬁeld strength F [2],and as the essential part of the D3-brane action as well [3],with b =2πα′.The action (1.1)has many remarkable properties,e.g.,causal propagation and electric-magnetic duality [4,5].The BI Lagrangian can be rewritten to the formL =−1b 21−4p µνp µν,Q =i2εµνρσp ρσ,(1.4)have been introduced.Eliminating p µνfrom eq.(1.2)results in the equivalent La-grangian L =11+b 216(F ˜F )2,(1.5)where we have deﬁned F 2=F µνF µν,˜Fµν=12We use ηµν=diag(+,−,−,−)and ¯h =c =1.supersymmetric generalizations of the four-dimensional bosonic BI action(1.1)are not known in any form.Supersymmetry apparently prefers the parametrization of the BI action in terms of the Maxwell term L2=−132 (F2)2+(F˜F)2 =12 Fµν±i˜Fµν .(1.6) This term is known as the Euler-Heisenberg(EH)Lagrangian[11].The EH action alsoappears as the bosonic part of the one-loop eﬀective action in N=1supersymmetric√scalar electrodynamics with the parameter b−1=2b2 1−Lorentz invariance.The light-cone formulation is,therefore,very suitable for an oﬀ-shell formulation of N-extended supersymmetric gaugeﬁeld theories with manifest supersymmetry in N-extended light-cone superspace[17,18,19].We deﬁne light-cone coordinates in Minkowski spacetime asx+=12 x0+x3,x−=12 x0−x3 ,x=12 x1+ix2,¯x=12 x1−ix2 ,(2.1)and similarly for the gauge vectorﬁeld,Aµ→(A+,A−,A,¯A).The real coordinate x+is going to be considered as‘light-cone time’.The linear transformation(2.1)of spacetime coordinates is obviously non-singular(with the Jacobian equal to i),while it does not preserve the Minkowski metric(i.e.it is not a Lorentz-transformation).The light-cone gauge readsA+=0.(2.2) In this(physical)gauge the A−component of the gaugeﬁeld Aµis supposed to be eliminated via its(non-dynamical)equation of motion,whereas the transverse components(A,¯A)are supposed to represent the physical propagatingﬁelds.It is easy to solve the equation of motion for A−in the Maxwell theory,where it takes the form of a linear equation in the light-cone gauge(cf.refs.[17,18,19]). It becomes,however,a highly non-trivial problem in the BI or EH theory,where it takes the form of a non-linear partial diﬀerential equation.The equations of motion amount to the conservation law for the p-tensor,∂µpµν=0,(2.3) while the pµνin the BI theory is given bypµν=b2Fµν−b4 2F2−b42∂µFµ−F2+14Fµ−∂µF2+b4 −116∂µFµ−(F˜F)2+132Fµ−∂µ(F˜F)2 .(2.5)We use a perturbative Ansatz,in powers of the small parameter b2,for a solution to eq.(2.5),A−(x)=∞ n=0b2n A−(2n)(x).(2.6) As regards the leading and sub-leading terms,weﬁndA−(0)=1(∂+)2 −14˜Fµ−∂µ(F˜F)+14F2+b22∂+A−∂+¯A∂22δm n P+,m,n=1,2,3,(3.1)where the supersymmetry charges Q r transform in the fundamental representation of SU(3).A natural representation of the algebra(3.1)in N=3light-cone superspace Z=(xµ,θm,¯θn)is given byQ m=−∂√∂θn−i2¯θn∂+.(3.2)The covariant derivatives in N=3light-cone superspace areD m=−∂√∂θn +i2¯θn∂+.(3.3)They anticommute with the supersymmetry charges(3.2)and obey the same alge-bra(3.1).The irreducible oﬀ-shell representations of N=3light-cone supersymmetry are easily obtained by imposing the covariant chirality condition on N=3light-cone superﬁeldsφ(Z),D mφ(Z)=0.(3.4)A solution to eq.(3.4)in components is just given by an arbitrary complex function φ(x+,x−+i2θm¯θm,x,¯x;θn)≡φ(y;θ).Its expansion in the chiral superspace readsφ(y;θ)=1∂+θm¯χm(y)+i3!εmnpθmθnθpψ(y).(3.5)The light-cone N=3supersymmetry transformation laws for the components areδA=iεn¯χn,δ¯χm=√2εpqr¯εq¯χr−iεpψ,δψ=−√where (D )3=εmnp D m D n D p and similarly for (¯D)3.After some trials and errors,we ﬁnd36(−i √∂+¯φ+2b 2 1∂+3∂+2φ¯∂2φ ∂+2¯φ∂2¯φ+14∂+3 ∂+2φ2φ ∂+2¯φ2¯φ−12∂+3 ∂+2φ2φ¯∂2φ ∂+2¯φ−12∂+3¯∂∂+φ¯∂∂+φ2φ ∂+2¯φ−12∂+A −∂+¯A ∂22C p 2¯Cp −2b 222∂+2∂+2C p ¯∂2A +¯∂2C p ∂+2A∂+2¯Cp ∂2¯A +∂2¯C p ∂+2¯A+14Cp2∂∂+¯Cp ∂∂+¯A 2¯A +2¯C p ∂∂+¯A∂∂+¯A(3.9)−14∂+2∂+2C p2A ¯∂2A +¯∂2C p ∂+2A 2A +2C p ¯∂2A∂+2A∂+2¯Cp +h .c ..One of the obvious features of both eqs.(3.8)and (3.9)is the apparent presence ofhigher derivatives,as may have been expected from the experience with the manifestly N=2supersymmetric generalization of the BI action in the covariant N=2superspace [9].The expected correspondence to the component D3-brane eﬀective action having non-linearly realized extended supersymmetry and no higher derivatives implies the existence of a ﬁeld redeﬁnition that would eliminate the higher-derivative terms in our action and make its N=3supersymmetry to be non-linearly realised (i.e non-manifest)[6].We also note the absence of quartic (C 4)scalar terms and the on-shell (2A =2C =0)invariance of our action under constant shifts,C p (x )→C p (x )+c p ,which are supposed to be related to the possible interpretation of the C p ﬁelds as the Goldstone scalars associated with spontaneoulsy broken translations in the full N=3BI action.4ConclusionOur main results are given by eqs.(2.8),(3.8)and(3.9).Our initial motivation was to construct an N=4supersymmetric generalization of the EH action in the light-cone gauge.The N=4light-cone supersymmetry algebra is given by eq.(3.1),where the indices now take four values.Equations(3.2),(3.3)and(3.4)are still valid in N=4 light-cone superspace,where they have to supplemented by an extra(generalized reality)condition[17],D m D n¯φ=148∂+2εmnpq¯D m¯D n¯D p¯D qφ.(4.1)The restricted chiral N=4light-cone superﬁeldφis equivalent to the chiral N=3 superﬁeld in eq.(3.5).Our eﬀorts to construct an N=4generalization of eq.(2.8) along the similar lines(sect.3)unexpectedly failed,while eq.(4.1)was the main obstruction.We conclude that even a manifestly N=4supersymmetric generalization of the EH action in the light-cone gauge seems to be highly non-trivial,if any,not to mention an even more ambitious(manifest)N=4supersymmetrization of the BI action.AcknowledgementsWe are grateful to Norbert Dragon,Gordon Chalmers,Olaf Lechtenfeld and Daniela Zanon for useful discussions.References[1]M.Born,Proc.Roy.Soc.A143(1934)410;M.Born and L.Infeld,Proc.Roy.Soc.A144(1934)425;M.Born,Ann.Inst.Poincar´e7(1939)155[2]E.S.Fradkin and A.A.Tseytlin,Phys.Lett.163B(1985)123;A.Abouelsaood,C.G.Callan,C.R.Nappi and S.A.Yost,Nucl.Phys.B280(1987)599[3]C.G.Callan,and J.Maldacena,Nucl.Phys.B513(1998)198;G.W.Gibbons,Nucl.Phys.B514(1998)603;R.G.Leigh,Mod.Phys.Lett.A4(1989)2767[4]J.Plebanski.Lectures on Non-Linear Electrodynamics,Nordita(1968);S.Deser,J.McCarthy and O.Sarioglu,Class.and Quantum Grav.16(1999) 841[5]M.K.Gaillard and B.Zumino,Nucl.Phys.B193(1981)221;G.W.Gibbons and D.A.Rasheed,Nucl.Phys.B454(1995)185[6]A.A.Tseytlin,Born-Infeld Action,Supersymmetry and String Theory,in“YuriGol’fand Memorial Volume”,edited by M.Shifman,World Scientiﬁc,2000;hep-th/9908105[7]S.Cecotti and S.Ferrara,Phys.Lett.187B(1987)335[8]S.Deser and R.Puzalowski,J.Phys.A13(1980)2501[9]S.V.Ketov,Mod.Phys.Lett.A14(1999)501;Nucl.Phys.B553(1999)250[10]S.M.Kuzenko and S.Theisen,Supersymmetric Duality Rotations,M¨u nchenpreprint;hep-th/0001068[11]W.Heisenberg and H.Euler,Z.Phys.98(1936)714[12]M.Roˇc ek and A.A.Tseytlin,Phys.Rev.D59(1999)106001[13]A.De Giovanni, A.Santambrogio and D.Zanon,(α′)4Corrections to theN=2Supersymmetric Born-Infeld Action,Milano and Leuven preprint,hep-th/9907214[14]H.Liu and A.A.Tseytlin,JHEP9910(1999)003[15]F.Gonzalez-Rey and M.Roˇc ek,Phys.Lett.434B(1998)303[16]G.G.Harwell and P.S.Howe,Class.and Quantum Grav.12(1995)1823;P.S.Howe and P.C.West,Phys.Lett.389B(1996)273[17]L.Brink,O.Lindgren and B.E.W.Nilsson,Phys.Lett.123B(1983)323andNucl.Phys.B212(1983)401[18]S.Mandelstam,Nucl.Phys.B213(1983)149[19]S.V.Ketov,Theor.Math.Phys.63(1985)470.。

BPS Equations, BPS States, and Central Charge of N=2 Supersymmetric Gauge Theories

1
exact solution of the low–energy eﬀective Lagrangian and BPS spectrum of the Coulomb phase of N =2 SU(2) Yang-Mills theory, and subsequent work, has provided us with solvable examples of gauge theories exhibiting many of the strong–coupling phenomena associated with QCD [1, 2]. BPS states appear at several key steps in Seiberg and Witten’s derivation, which leads to a formula for their exact mass spectrum. A crucial role is played by the central charge Z of the N =2 algebra in the presence of a charged state, which appears in the BPS mass inequality √ M ≥ 2| Z | The masses of BPS states saturate this inequality, and are therefore determined by the central charge, which in turn depends on the vevs and masses that parameterize the moduli space of Coulomb vacua. As given by Seiberg and Witten for pure SU(2), the formula for the central charge is Z = a∞ ne + aD∞ nm (1.1)

Parity non-conservation in nuclear excitation by circularly polarized photon beam

a r X i v :n u c l -t h /0605083v 2 9 J u n 2006Parity non-conservation in nuclear excitationby circularly polarized photon beamA.I.Titov 1,2,M.Fujiwara 1,3and K.Kawase 31Kansai Photon Science Institute,Japan Atomic Energy Agency,Kizu,Kyoto619-0215,Japan2Bogoliubov Laboratory of Theoretical Physics,JINR,Dubna 141980,Russia3ResearchCenter for Nuclear Physics,Osaka University,Ibaraki,Osaka 567-0047,JapanParity non-conservation (PNC)is well known after the discovery of the mirror symmetry violation in β-decays by Wu [1],following the suggestion by Lee and Yang [2].The origin of this mirror-asymmetry is now clearly understoodPreprint submitted to Elsevier Science9February 2008as a manifestation of the exchange processes of weak bosons,W±,which are mediators ofβ-decay.Observations of the PNC eﬀect in the nucleon-nucleon interaction are not new. The trial to observe the PNC eﬀect began with theﬁrst report by Tanner in 1957[3],followed by the famous work of Feynman and Gell-Mann[4]for the universal current-current theory of weak interaction.Wilkinson[5]also triggered the enthusiastic long studies ofﬁnding the tiny PNC eﬀect in nuclear excitation processes.The process contributing to the PNC eﬀect is due to the non-trivial quark interactions with weak Z0and W±bosons in the eﬀective nucleon-nucleon-meson vertices.The details of the PNC studies were reviewed in Refs.[6,7]. However,the current problem is essentially focused on the fact that the weak meson-nucleon coupling constants(and in particular,the pion-nucleon cou-pling constant)deduced from various experiments are not consistent.It is concluded by Haxton et al.,[8]that the experimental PNC studies are still not satisfactory,and more experimental as well as theoretical studies are needed.One of experimental PNC studies is to measure the parity mixing between the parity-doublet states.On the basis of theﬁrst order perturbation theory,the wave functions of two closely-located states,|φπ and|φ−π are mixed by the PNC interaction V P NC as|˜φπ =|φπ + φ−π|V PNC|φπThen,the PNC eﬀect appears in the asymmetry of emitted circularly polarized photons[6].Pγ∼2RA a RL=σa R −σaL2−(E x =109.9keV)state in19F.This example is very transparent and canbe easily extended to other parity doublets with higher spins.Therefore,we use it as a starting point of our consideration,leaving discussion of practical utilization of this particular transition,which may be not so easy at once because of the ﬁnite life time of the radiative 19F.It is assumed that theground state with J π=12−are the paritydoublet|12+−α|12−≃|12+,(4)withα=12+/∆E,(5)and ∆E =E 12+.The amplitude of the process γi +A →A ∗→γf +A(A =19F)may be expressed as a product of absorption (T a )and decay (T d )-amplitudes4Tλiλf =T a m∗;λi,m i·T dλf,m f;m∗,(6)where m i,m∗,m f,andλi,λf are the spin projections of the nucleus A in the initial,excited,and theﬁnal states,and the photon helicities in the initial and theﬁnal states,respectively.Here,we assume that the spin projection of the excited state is conserved during its short decay lifetime.Fig.1.Reaction scheme ofγi+19F(12−)→γf+19F(12π(2L+1)× J i m i Lλ|J f m f2J f+1[F EL+λF ML],(7) where J i,f and m i,f are the spin and spin projection of the initial and theﬁnalstates,λis the photon helicity,F E/M L= f||T LE/M||i is the reduced matrix element of the multipole operators.In the case of J i=J f=12δm i,−λ2πF E1and M1≡√2δm i,−λiwithµ±=√2±||T1M||12δm f,−λf2,¯m′ =d12,m ,(12) where d j mm′(θ)is the Wigner function,andθis the angle between the beam direction and the direction ofﬂight of the emitted photon.This relation leads toTλf,m f;λf 2λi2(θ)δm f,−λf2λi2(θ))2=1, λfλf(d12λfand neglecting the terms proportional toα2,we get the PNC-asymmetry in the following formA RL(θ)=(1+cosθ)<A RL>,(15) with<A RL>=2αRe µ+)→19F(12+)has“1+cosθ”-dependence.It is enhanced(suppressed)atθ∼02(θ∼π).This idea can be extended to the other nuclei.In case of18F,there are parity doublet states with Jπ=0+and0−at the energies E x=1.042and1.081 MeV,respectively.Although the1+ground state of18F is unstable and thus the experimental feasibility is very low,we show the result of corresponding calculation as a prediction for any1+→0−(0)+transitions.In case of the transitionγ+(1+)→(0−)[1081keV]→γ+(1+)in18F,the asymmetry is isotropic,because the excited state with J=0loses information about the spin-helicity in the initial stateA RL(θ)=<A RL>=2 0−|V PNC|0+ E1 .(17) Here E1and M1are the amplitudes of the1+→0−and1+→0+transitions, respectively.−)[2789keV]→γ+(3 The PNC asymmetry for the transitionγ+(32A RL(θ)≃(1+12+|V PNC|1E12−Re M12+→12+→14in Eq.(18)reﬂects the fact that the spin projectionm i of the ground state at theﬁxed photon helicityλi may be−12λi. Instead of the reactions with polarized photons and unpolarized target,one can analyze the reactions with polarized target and unpolarized beam.The spin asymmetry is deﬁned asA S=σ+−σ−2cosθ)<A RL>.(23)8In Table1,we show other possible examples for studying the PNC-transitions in light nuclei by NRF.For completeness,we also show the corresponding angular correlations for the photon asymmetries for transitions with the spin in initial andﬁnal states are J i and J f,respectively.Transitions0→1A RL(θ)= 1+cosθ1−3cos2θ+4cos4θ <A RL>.(25) Transitions1→1A RL(θ)= 1+2cosθ73+21cos2θ <A RL>.(27) In the equations described above,the average value of the asymmetry is deﬁned as a the product of the PNC-matrix element and the nuclear ampliﬁer factor |<A i RL>|=2|R i NTable 1Possible candidates for studying the PNC asymmetry in the light nuclei.The energy levels and the ampliﬁer factors |R N /∆E |are given in keV and MeV −1,respectively.A Ztransition(J πi ;I i )[E i ]→(J πf ;I f)[E f ]admixture(J −πf )[E ′f]|R N /∆E |14N (1+,0)→(1+,0)[6203][5691]7.0±2.0(1+,0)→(0+,1)[8624][8776]40±5(1+,0)→(2−,1)[9509][9172]45±52−,12−,116O(0+,0)→(2−,0)[8872][6917]18±2[11520]9.5±0.720Ne (0+,0)→(1−,0)[11270][11262]670±7000parity doublet levels was excited via a nuclear reaction,and the admixture of the conﬁguration of the opposite parity was manifested as the asymmetry A γof γ-rays emitted from the excited states with a polarization,or as the cir-cular polarization P γof γ-rays emitted from unpolarized excited states.The isospin-structure of the corresponding transitions in each of these cases are diﬀerent.This results in diﬀerent structure of the transition matrix elements,and therefore it is possible to get independent information on the elementary parity-violated meson-nucleon coupling constants.10At present,there is no data available to measure the PNC eﬀect with circular polarized photons although there are theoretical estimations for the PNC eﬀect in the deuteron photodisintegration[14,15]for which the A RL asymmetry are expected to be very small as the10−7level.In case of the PNC measurement for the transition from the1/2+ground state to the109.9keV1/2−state in19F,for example,a high intensity photon source from the synchrotron radiation facilities at SPring-8is useful.The intensity of photons from a elliptical multipole wiggler system at SPring-8[16]reaches at around1013photons/second even at Eγ=109.9keV with an energy width (∆E)of100eV.The expected yield rate R of theγA→A∗→γA reaction reads [17]ΓR=π2λ2One method to overcome this problem is to use a multi-segmented detector in order to greatly reduce the counting rate of each detector and obtain the nec-essary total counts of N∼1010as the NRF events.The use of newly developed lutetium oxyorthosilicate(Lu2SiO5,LSO)and lutetium-yttrium oxyorthosili-cate(Lu2(1−x)Y2x SiO5,LYSO)crystals[19,20]with a decay constant of about 40ns and an energy resolution of7-10%is also promising for the NRF mea-surement with a high-counting rate.Another way is to obtain a photon beam with an ultra high resolution of∆E/E∼10−5−10−6.In this case,the back-ground photons due to Compton scattering are greatly reduced,and theγ-ray events due to the NRF process are relatively enhanced to get a high counting rate necessary for performing a high-statics PNC measurement.AcknowledgementsWe thank H.Akimune,H.Ejiri,S.Dat’e,M.Itoh,Y.Ohashi,H.Ohkuma, Y.Sakurai,S.Suzuki,K.Tamura,H.Toki,and H.Toyokawa for fruitful discussions.One of the authors(A.I.T.)thanks T.Tajima for his hospitality to stay at SPring-8.This work was strongly stimulated by a new project to produce a high-intensity MeVγ-rays at SPring-8.References[1]Wu C S et al1957Phys.Rev.1051413[2]Lee T D and Yang C N1956Phys.Rev.104254[3]Tanner N1957Phys.Rev.1071203[4]Feynman R P and Gell-Mann M1958Phys.Rev.109193[5]Wilkinson D H1958Phys.Rev.1091603[6]Adelberger E G and Haxton W C1985Ann.Rev.Nucl.Sci.35501[7]Desplanques B1998Phys.Rep.2971[8]Haxton W C,Liu C-P and Ramsey-Musolf M J2002Phys.Rev.C65045502[9]Earle E D,McDonald A B,Adelberger E G,Snover K A,Swanson H E,vonLintig R,Mak H B and Barnes C A,1983Nucl.Phys.A396221c[10]Holstein B.R“Weak Interactions in Nuclei”,(Princeton University Press,1989)[11]“Symmetries and Fundamental Interaction in Nuclei”,ed.by W.C.Haxtonand E.M.Henley.World Scientiﬁc Publishing Co.Pte.Ltd.1995,p.17 [12]Adelberger E G,Hindi M M,Hoyle C D,Swanson H E,Von Lintig R D andHaxton W C1983Phys.Rev.C272833[13]De Forest T and Walecka J D1966Adv.Phys.151[14]Fujiwara M and Titov A I2004Phys.Rev.C69065503[15]Liu C P,Hyun C H and Desplanques B2004Phys.Rev.C69065502[16]Ma´r echal X.-M,Hara T,Tanabe T,Tanaka T and Kitamura H1998J.Synchroton Rad.5431[17]Skorka S.J.“The electromagnetic interaction in nuclear spectroscopy”ed.byW.D.Hamilton.North-Holland1975,p.283[18]Ajzenberg-Selove F,1972Nucl.Phys.A1901,and see Table19.10[19]Qin L,Pei Yu,Lu S,Li H,Yin Z,and Ren G2005Nucl.Instr.and Meths inPhy.Res.A545273,and references therein[20]Chen J,Zhang L and Zhu R2004Nuclear Science Symposium ConferenceRecord IEEE1117。

Lepton number and lepton flavor violations in seesaw models

(3)
and from solar neutrinos (LMA solution)
2 −5 |m2 2 − m1 | ∼ 10 eV,
(4)
or less favoured (LOW solution)
2 −7 |m2 2 − m1 | ∼ 10 eV.
(5)
3
Other two experimental informations come from single beta decay [10],
II. NEUTRINO MASSES AND MIXINGS
where ǫ 0.1. The name bimaximal comes from the fact that both Uµ3 and Ue2 have the √ value 1/ 2, while Ue3 is very small. However, the best ﬁt is closer to 1 2 √ √ ǫ 6 3 1 1 . 1 (2) U ≃ √ √ √ − 6 3 2 1 1 1 √ √ −√ 6 3 2 The element Uµ3 is determined by atmospheric oscillations, with a mixing angle nearly maximal. The element Ue2 by solar oscillations, with a mixing angle large but not maximal. The smallness of element Ue3 is obtained from reactor neutrinos [9]. These studies also provide values for square mass diﬀerences of left-handed neutrinos. From atmospheric neutrinos we get

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a r X i v :h e p -p h /9709275v 2 12 F eb 1998PKU-TP-97-20MSUHEP-70825THU-TP-97-08hep-ph/9709275Supersymmetric QCD Parity Nonconservation inTop Quark Pairs at the TevatronChong Sheng Li (a ),C.–P.Yuan (b ),Hong-Yi Zhou (c )(a )Department of Physics,Peking University,Beijing 100871,China (b )Department of Physics and Astronomy,Michigan State University,East Lansing,Michigan 48824,USA(c )Institute of Modern Physics and Department of Physics,Tsinghua University,Beijing 100084,ChinaABSTRACTIn the supersymmetry (SUSY)models,because of the mass diﬀerence between the left-and right-top squarks,the supersymmetric QCD in-teractions can generate parity violating eﬀects in the production of t ¯t pairs.We show that SUSY QCD radiative corrections to the parity vi-olating asymmetry in the production rates of the left-and right-handedtop quarks via the q ¯q →t ¯tprocess can reach about 3%at the Fermilab Tevatron with√1IntroductionIn a recent paper[1],we studied the parity violating asymmetry induced from the supersymmetric electroweak(SUSY EW)and Yukawa(SUSY Yukawa)corrections at the one loop level.Two classes of supersymmetry(SUSY)models were considered:the mini-mal supergravity(mSUGRA)models[2]and the minimal supersymmetric models(MSSM) with scenarios motivated by current data[3,4].After sampling a range of values of SUSY parameters in the region that might give large contributions to the parity-violating asym-metry A,and which are also consistent with either of the above two classes of models, we found that the asymmetry A due to the one-loop SUSY EW(αm2t/m2W)and SUSY Yukawa corrections for the production process q¯q→g→t¯t at the upgraded Tevatron is generally small,less than a few percent.However,the sign can be either positive or negative depending on the values of the SUSY parameters.(The eﬀect from the Standard Model(SM)weak corrections to this asymmetry is typically less than a fraction of percent [5,6].)In the supersymmetric Standard Model,some superparticles experience not only the electroweak interaction but also the strong interaction.Although the SM QCD interaction respects the discrete symmetries of charge conjugation(C)and parity(P),the SUSY QCD interactions for superparticles,in their mass eigenstates,need not be C and P invariant. (Needless to say,in the strong interaction eigenstates,the SUSY QCD interaction is C-and P-invariant.)For either the mSUGRA or the MSSM models,the masses of the left-stop(the supersymmetric partner of the left-handed top quark)and the right-stop can be noticeably diﬀerent due to the large mass of the top quark.This is a general feature of the supersymmetry models in which the electroweak symmetry is broken spontaneously via radiative corrections.Since both the left-stop and the right-stop contribute to the loop corrections for the t¯t pair production process q¯q,gg→t¯t,the diﬀerent masses of the top-squarks will induce a parity violating asymmetry.It is this eﬀect that we shall study in this paper.Because the t¯t pairs are produced predominantly via the QCD process q¯q→t¯t√at the Tevatron(a p¯p collider with CM energyThis amounts to a signal at∼90%c.l.(conﬁdence level)with2fb−1,or99%c.l.with10 fb−1.Thus,a study of A at the Tevatron could yield information about the allowed range of SUSY model parameter space.2SUSY QCD Corrections and Parity ViolationI.Squark mixingsIn the MSSM the mass eigenstates˜q1and˜q2of the squarks are related to the(strong) current eigenstates˜q L and˜q R via the mixing angleθ˜q by˜q1=˜q L cosθ˜q+˜q R sinθ˜q,˜q2=−˜q L sinθ˜q+˜q R cosθ˜q.(1) For the top squarks,the mixing angleθ˜t and the masses m˜t1,2can be calculated by diago-nalizing the following mass matrix[3],M2˜t = M2˜t L m t m LRm t m LR M2˜t R,M2˜t L =m2˜t L+m2t+(13sin2θW)cos(2β)m2Z,M2˜t R =m2˜t R+m2t+2N R+N L=σR−σLSome of the one loop scattering amplitudes of q ¯q →t ¯twere already presented in Refs.[10,11]for calculating the total production rates of t ¯tpairs.To calculate the parity violating asymmetry A in the t ¯tsystem,additional renormalized amplitudes are needed.In terms of the tree-level amplitude,M 0,and the next-to-leading order SUSY QCD corrections,δM ,the renormalized amplitudes at the one-loop level can be writtenas M =M 0+δM .Denote the momenta of the initial and the ﬁnal state particles asq l (p 4)¯q m (p 3)→t i (p 2)¯tj (p 1),and the Dirac four-spinor as u i ≡u (p i )(v i ≡v (p i ))for particle (anti-particle)i .Then,M 0=ig 2s (T c ji T clm )J 1·J 2/ˆs ,where J µ1=¯v (p 3)γµu (p 4)and J µ2=¯u (p 2)γµv (p 1);ˆs is the invariant mass of the t ¯t pair;g s and T c ij are the gauge coupling andthe generator of the group SU (3)c ,respectively.To calculate the parity violating asymmetry induced by the SUSY QCD eﬀects,we fol-low the method presented in Ref.[12],in which the asymmetry was calculated numerically using the helicity amplitude method.To obtain the renormalized scattering amplitudes,we adopt the dimensional regularization scheme to regulate the ultraviolet divergences and the on-mass-shell renormalization scheme [13]to deﬁne the input parameters.The SUSY QCD corrections to the scattering amplitudes arise from the vertex diagram,the gluon self-energy and the box diagrams,as well as the crossed-box diagrams.The renormalized amplitudes can be written asδM =δM v 1+δM v 2+δM s +δM DB +δM CB ,(4)where δM v 1and δM v 2are vertex corrections,δM s is the self-energy correction,and δM DB and δM CB are the contributions from the box diagrams and crossed-box diagrams,respec-tively.The results for these separate contributions are,δM v 1=ig 2s(T c ji T c lm )¯u (p 2)[F v 10·J 1+F v 11/J 1+/J 1/F v 13+/F v 14/J 1+/F v 16·J 1+(F Av 11/J 1+/J 1/F Av 13+/F Av 14/J 1+/F Av 16·J 1)γ5]v (p 1)/ˆs ,(5)δM v 2=ig 2s (T c ji T c lm )¯v (p 3)(F v 21/J 2+/F v 26·J 2)u (p 4)/ˆs ,(6)δM s =F s0M 0,(7)δM DB =ig 2s7δM CB=ig2s1ˆs=M t¯t.1As discussed in the previous section,the SUSY parameters relevant to our study arem˜t1,m˜t2,θ˜t(or m˜tL,m˜tR,m LR),m˜b R,m˜qL,R,and m˜g.To simplify our discussion,we assumem˜qL,R =m˜b R=m˜tL,so that there are only four SUSY parameters to be considered,m˜t1,m˜t2,θ˜t and m˜g.(The SU(2)L gauge symmetry requires that m2˜b L=m2˜t L.)The mSUGRA models predict radiative breaking of the electroweak gauge symmetryinduced by the large top quark mass.Consequently,it is possible to have large splitting in the masses of the left-stop and the right-stop,while the masses of all the other(left-or right-)squarks are about the same[17].For the MSSM models with scenarios motivated by current data[4],a light˜t1is likely to be the right-stop(˜t R),with a mass at the order of m W;the other squarks are heavier than˜t1.Since heavy superparticles decouple in loopcontributions,we expect that a lighter˜t1would induce a larger asymmetry.Because theparity-violating eﬀects from the SUSY QCD interactions arise from the mass diﬀerencebetween˜t1and˜t2,it is obvious from Eq.(1)that the largest parity violating eﬀect occurswhenθ˜t is±π/2for m˜tR≤m˜t L.Whenθ˜t=±π/4,the parity asymmetry should be zero. This is evident from the results shown in the Appendix,which indicate that the amplitudesthat contribute to A are all proportional to Z i=∓cos(2θ˜t).In either the mSUGRA or the MSSM models,the gluinos are usually as heavy as thelight squarks,on the order of a few hundred GeV.However,Farrar has argued[18]that lightgluinos are still a possibility.If gluinos are light,then a heavy top quark can decay into astop and a light gluino for m˜t1<(m t−m˜g)such that the branching ratio of t→bW+could show a large diﬀerence from that(∼100%)predicted by the SM.The CDF collaborationhas measured the branching ratio of t→bW+to be0.87+0.13−0.30+0.13−0.11[19].At the1σlevel,this implies that a50(90)GeV˜t1requires the mass of the gluino to be larger than about 120(80)GeV forθ˜t=±π/2.However,at the2σlevel(i.e.95%c.l.),there is no useful limit on the mass of the gluino.2To represent diﬀerent classes of SUSY models in which the parity-violating asymmetry induced by the SUSY QCD interactions can be large,we show in Table1four represen-tative sets of models.They are labeled by the set of parameters(m˜t1,m˜t2,θ˜t),which areequal to(50,1033,−1.38),(90,1033,−1.38),(50,558,−1.25)and(90,558,−1.25),respec-tively.(All the masses are in units of GeV.)Based upon Eq.(2),one can also label thesemodels by(m˜tL ,m˜tR,m LR),which are(1000,90,1100),(1000,118,1100),(500,40,520)and(500,88,520),respectively,forβ=π/4.It is interesting to note that for all the models listed in Table1,the asymmetry A is negative(i.e.σR<σL)for m˜g<200GeV,and its magnitude can be as large as 3%for models with light˜t1.The maximal|A|occurs when m˜g is about equal to(m t−m˜t1)because of the mass threshold enhancement.For m˜g>200GeV,the asymmetry A becomes positive,with a few percent in magnitude,and monotonically decreases as m˜g paring these results with those induced by the SUSY EW and SUSY Yukawa corrections[1],it is clear that SUSY QCD interactions can generate a relatively larger parity-violating asymmetry.The diﬀerential asymmetry A(M t¯t)also exhibits an interesting behaviour as a function of the t¯t invariant mass M t¯t.This is illustrated in Table2for theﬁrst SUSY model inTable1((m˜t1,m˜t2,θ˜t)=(50,1033,-1.38)).As shown,|A(M t¯t)|increases as M t¯t increasesfor m˜g<200GeV,which is similar to the eﬀects from the SUSY EW and SUSY YukawaTable1:Parity violating asymmetry A in p¯p→t¯t+X,as a function of m˜g,for four sets of SUSYmodels labeled by(m˜t1,m˜t2,θ˜t).m˜g(GeV)(90,1033,−1.38)(90,558,−1.25)-1.10%-0.98%-1.53%-1.40%-2.34%-2.21%-2.86%-2.89%-3.16%-3.43%-2.58%-2.80%-1.18%-1.30%0.99%0.82%1.60% 1.40%1.53% 1.35%1.27% 1.16%1.04%0.95%3Without the cuts in(10),the values of A for theﬁrst model in Table1are−1.0%,−2.65%,and +0.94%for m˜g=2,120,200GeV,respectively.4These apparent problems in Ref.[22]were also pointed out in Ref.[21].6Table2:The diﬀerential asymmetry A(M t¯t)and cross section dσ/d M t¯t(in unit of fb/GeV)as a function of M t¯t for theﬁrst SUSY model in Table1with various m˜g values.M t¯t(GeV)m˜g=120GeV358-0.73%16.3-0.42%36.60.95%31.4 378-1.63%29.0-0.67%38.7 2.12%34.2 398-2.17%27.2-0.82%35.0 3.67%32.4 425-2.64%22.8-1.13%24.1 1.05%20.0 475-3.40%13.8-1.47%13.7-0.62%10.8 525-3.81%7.9-1.76%7.7-1.71% 5.9 575-4.34% 4.4-3.26%0.032-4.66%0.024Table3:The SUSY QCD corrections(∆σ)to the q¯q→t¯t production rates at the Tevatron with √2501001201351501752002252502753001.170.26-0.04-0.18-0.87-0.49-0.020.330.300.240.190.16m˜g=200GeV and m˜t=m˜q=75GeV,we obtain a39%,in contrast to33%,correction in the total cross section without cuts.Including cuts in(10)only slightly increases the correction to40%.For completeness,in Table3we show the SUSY QCD corrections∆σto the q¯q→t¯t√production rates at the Tevatron withUp to now,we have only considered the one loop SUSY QCD eﬀects on the parity violating asymmetry A in t¯t pair production.Amusingly,the parity-violating asymmetry induced by the SUSY QCD interactions can also occur at the Born level.If gluinos are very light,of the order of1GeV,this asymmetry can be generated by the tree level process ˜g˜g→t¯t.Unfortunately,its production rate is smaller than the gg→t¯t rate,which is only about one tenth of the q¯q→t¯t rate at the Tevatron.Hence,it cannot be measured at the Tevatron.However,at the CERN Large Hadron Collider(LHC),the production rate of˜g˜g→t¯t is large enough to allow the measurement of the parity-violating asymmetry induced by the SUSY QCD interactions.The asymmetry in the production rates of t L¯t and t R¯t,generated by the˜g˜g fusion process alone,can reach about10%for M t¯t larger than about500GeV.We shall present its details and include the eﬀect from the gg and q¯q fusion processes in a future publication[23].This work is supported in part by the National Natural Science Foundation of China, and by the U.S.NSF grant PHY-9507683.AppendixWe give here the form factors for the matrix elements appearing in Eqs.(8)-(12). 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