Probing CP Violating Two Higgs Doublet Model through Interplay between LHC and ILC

a r X i v :0803.2889v 2 [h e p -p h ] 14 J u l 2008Mapping Out SU (5)GUTs with Non-Abelian Discrete Flavor SymmetriesFlorian Plentinger ∗and Gerhart Seidl †Institut f¨u r Physik und Astrophysik,Universit¨a t W¨u rzburg,Am Hubland,D 97074W¨u rzburg,Germany(Dated:December 25,2013)We construct a class of supersymmetric SU (5)GUT models that produce nearly tribimaximal lepton mixing,the observed quark mixing matrix,and the quark and lepton masses,from discrete non-Abelian flavor symmetries.The SU (5)GUTs are formulated on five-dimensional throats in the flat limit and the neutrino masses become small due to the type-I seesaw mechanism.The discrete non-Abelian flavor symmetries are given by semi-direct products of cyclic groups that are broken at the infrared branes at the tip of the throats.As a result,we obtain SU (5)GUTs that provide a combined description of non-Abelian flavor symmetries and quark-lepton complementarity.PACS numbers:12.15.Ff,11.30.Hv,12.10.Dm,One possibility to explore the physics of grand unified theories (GUTs)[1,2]at low energies is to analyze the neutrino sector.This is due to the explanation of small neutrino masses via the seesaw mechanism [3,4],which is naturally incorporated in GUTs.In fact,from the perspective of quark-lepton unification,it is interesting to study in GUTs the drastic differences between the masses and mixings of quarks and leptons as revealed by current neutrino oscillation data.In recent years,there have been many attempts to re-produce a tribimaximal mixing form [5]for the leptonic Pontecorvo-Maki-Nakagawa-Sakata (PMNS)[6]mixing matrix U PMNS using non-Abelian discrete flavor symme-tries such as the tetrahedral [7]and double (or binary)tetrahedral [8]groupA 4≃Z 3⋉(Z 2×Z 2)and T ′≃Z 2⋉Q,(1)where Q is the quaternion group of order eight,or [9]∆(27)≃Z 3⋉(Z 3×Z 3),(2)which is a subgroup of SU (3)(for reviews see, e.g.,Ref.[10]).Existing models,however,have generally dif-ficulties to predict also the observed fermion mass hierar-chies as well as the Cabibbo-Kobayashi-Maskawa (CKM)quark mixing matrix V CKM [11],which applies especially to GUTs (for very recent examples,see Ref.[12]).An-other approach,on the other hand,is offered by the idea of quark-lepton complementarity (QLC),where the so-lar neutrino angle is a combination of maximal mixing and the Cabibbo angle θC [13].Subsequently,this has,in an interpretation of QLC [14,15],led to a machine-aided survey of several thousand lepton flavor models for nearly tribimaximal lepton mixing [16].Here,we investigate the embedding of the models found in Ref.[16]into five-dimensional (5D)supersym-metric (SUSY)SU (5)GUTs.The hierarchical pattern of quark and lepton masses,V CKM ,and nearly tribi-maximal lepton mixing,arise from the local breaking of non-Abelian discrete flavor symmetries in the extra-dimensional geometry.This has the advantage that theFIG.1:SUSY SU (5)GUT on two 5D intervals or throats.The zero modes of the matter fields 10i ,5H,24H ,and the gauge supermul-tiplet,propagate freely in the two throats.scalar sector of these models is extremely simple without the need for a vacuum alignment mechanism,while of-fering an intuitive geometrical interpretation of the non-Abelian flavor symmetries.As a consequence,we obtain,for the first time,a realization of non-Abelian flavor sym-metries and QLC in SU (5)GUTs.We will describe our models by considering a specific minimal realization as an example.The main features of this example model,however,should be viewed as generic and representative for a large class of possible realiza-tions.Our model is given by a SUSY SU (5)GUT in 5D flat space,which is defined on two 5D intervals that have been glued together at a common endpoint.The geom-etry and the location of the 5D hypermultiplets in the model is depicted in FIG.1.The two intervals consti-tute a simple example for a two-throat setup in the flat limit (see,e.g.,Refs.[17,18]),where the two 5D inter-vals,or throats,have the lengths πR 1and πR 2,and the coordinates y 1∈[0,πR 1]and y 2∈[0,πR 2].The point at y 1=y 2=0is called ultraviolet (UV)brane,whereas the two endpoints at y 1=πR 1and y 2=πR 2will be referred to as infrared (IR)branes.The throats are supposed to be GUT-scale sized,i.e.1/R 1,2 M GUT ≃1016GeV,and the SU (5)gauge supermultiplet and the Higgs hy-permultiplets 5H and2neously broken to G SM by a 24H bulk Higgs hypermulti-plet propagating in the two throats that acquires a vac-uum expectation value pointing in the hypercharge direc-tion 24H ∝diag(−12,13,15i ,where i =1,2,3is the generation index.Toobtainsmall neutrino masses via the type-I seesaw mechanism [3],we introduce three right-handed SU (5)singlet neutrino superfields 1i .The 5D Lagrangian for the Yukawa couplings of the zero mode fermions then readsL 5D =d 2θ δ(y 1−πR 1) ˜Y uij,R 110i 10j 5H +˜Y d ij,R 110i 5H +˜Y νij,R 15j5i 1j 5H +M R ˜Y R ij,R 21i 1j+h.c. ,(3)where ˜Y x ij,R 1and ˜Y x ij,R 2(x =u,d,ν,R )are Yukawa cou-pling matrices (with mass dimension −1/2)and M R ≃1014GeV is the B −L breaking scale.In the four-dimensional (4D)low energy effective theory,L 5D gives rise to the 4D Yukawa couplingsL 4D =d 2θ Y u ij 10i 10j 5H +Y dij10i 5H +Y νij5i ∼(q i 1,q i 2,...,q i m ),(5)1i ∼(r i 1,r i 2,...,r im ),where the j th entry in each row vector denotes the Z n jcharge of the representation.In the 5D theory,we sup-pose that the group G A is spontaneously broken by singly charged flavon fields located at the IR branes.The Yukawa coupling matrices of quarks and leptons are then generated by the Froggatt-Nielsen mechanism [21].Applying a straightforward generalization of the flavor group space scan in Ref.[16]to the SU (5)×G A represen-tations in Eq.(5),we find a large number of about 4×102flavor models that produce the hierarchies of quark and lepton masses and yield the CKM and PMNS mixing angles in perfect agreement with current data.A distri-bution of these models as a function of the group G A for increasing group order is shown in FIG.2.The selection criteria for the flavor models are as follows:First,all models have to be consistent with the quark and charged3 lepton mass ratiosm u:m c:m t=ǫ6:ǫ4:1,m d:m s:m b=ǫ4:ǫ2:1,(6)m e:mµ:mτ=ǫ4:ǫ2:1,and a normal hierarchical neutrino mass spectrumm1:m2:m3=ǫ2:ǫ:1,(7)whereǫ≃θC≃0.2is of the order of the Cabibbo angle.Second,each model has to reproduce the CKM anglesV us∼ǫ,V cb∼ǫ2,V ub∼ǫ3,(8)as well as nearly tribimaximal lepton mixing at3σCLwith an extremely small reactor angle 1◦.In perform-ing the group space scan,we have restricted ourselves togroups G A with orders roughly up to 102and FIG.2shows only groups admitting more than three valid mod-els.In FIG.2,we can observe the general trend thatwith increasing group order the number of valid modelsper group generally increases too.This rough observa-tion,however,is modified by a large“periodic”fluctu-ation of the number of models,which possibly singlesout certain groups G A as particularly interesting.Thehighly populated groups would deserve further system-atic investigation,which is,however,beyond the scopeof this paper.From this large set of models,let us choose the groupG A=Z3×Z8×Z9and,in the notation of Eq.(5),thecharge assignment101∼(1,1,6),102∼(0,3,1),103∼(0,0,0),52∼(0,7,0),52↔4FIG.3:Effect of the non-Abelian flavor symmetry on θ23for a 10%variation of all Yukawa couplings.Shown is θ23as a function of ǫfor the flavor group G A (left)and G A ⋉G B (right).The right plot illustrates the exact prediction of the zeroth order term π/4in the expansion θ23=π/4+ǫ/√2and the relation θ13≃ǫ2.The important point is that in the expression for θ23,the leading order term π/4is exactly predicted by thenon-Abelian flavor symmetry G F =G A ⋉G B (see FIG.3),while θ13≃θ2C is extremely small due to a suppression by the square of the Cabibbo angle.We thus predict a devi-ation ∼ǫ/√2,which is the well-known QLC relation for the solar angle.There have been attempts in the literature to reproduce QLC in quark-lepton unified models [26],however,the model presented here is the first realization of QLC in an SU (5)GUT.Although our analysis has been carried out for the CP conserving case,a simple numerical study shows that CP violating phases (cf.Ref.[27])relevant for neutri-noless double beta decay and leptogenesis can be easily included as well.Concerning proton decay,note that since SU (5)is bro-ken by a bulk Higgs field,the broken gauge boson masses are ≃M GUT .Therefore,all fermion zero modes can be localized at the IR branes of the throats without intro-ducing rapid proton decay through d =6operators.To achieve doublet-triplet splitting and suppress d =5pro-ton decay,we may then,e.g.,resort to suitable extensions of the Higgs sector [28].Moreover,although the flavor symmetry G F is global,quantum gravity effects might require G F to be gauged [29].Anomalies can then be canceled by Chern-Simons terms in the 5D bulk.We emphasize that the above discussion is focussed on a specific minimal example realization of the model.Many SU (5)GUTs with non-Abelian flavor symmetries,however,can be constructed along the same lines by varying the flavor charge assignment,choosing different groups G F ,or by modifying the throat geometry.A de-tailed analysis of these models and variations thereof will be presented in a future publication [30].To summarize,we have discussed the construction of 5D SUSY SU (5)GUTs that yield nearly tribimaximal lepton mixing,as well as the observed CKM mixing matrix,together with the hierarchy of quark and lepton masses.Small neutrino masses are generated only by the type-I seesaw mechanism.The fermion masses and mixings arise from the local breaking of non-Abelian flavor symmetries at the IR branes of a flat multi-throat geometry.For an example realization,we have shown that the non-Abelian flavor symmetries can exactly predict the leading order term π/4in the sum rule for the atmospheric mixing angle,while strongly suppress-ing the reactor 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a r X i v :h e p -p h /9703459v 1 31 M a r 1997DPNU-97-18March 1997Possible Signals of D 0-¯D0Mixing and CP Violation ∗Zhi-zhong Xing†Department of Physics,Nagoya University,Chikusa-ku,Nagoya 464-01,JapanAbstractIn view of the discovery potential associated with the future experiments of high-luminosity fixed target facilities,B -meson factories and τ-charm factories,we highlightsome typical signals of D 0-¯D0mixing and CP violation which are likely to show up in neutral D -meson decays to the semileptonic final states,the hadronic CP eigenstates,the hadronic non-CP eigenstates and the CP -forbidden states.Both time-dependent andtime-integrated measurements are discussed,and particular interest is paid to D 0/¯D0→K S,L +π0and D 0/¯D0→K ±π∓transitions.1IntroductionThe study of D 0-¯D0mixing and CP violation in neutral D -meson decays is not only comple-mentary to our knowledge obtained and to be obtained from K 0-¯K0and B 0-¯B 0systems,but also important for exploring underlying new physics that is out of reach of the standard model predictions.A large discovery potential associated with this topic is expected to exist in the future delicate experiments of high-luminosity fixed target facilities,B -meson factories,and τ-charm factories [1].WithoutCPTviolation,the mass eigenstates of D 0and ¯D0mesons can be written as |D L =p |D 0 +q |¯D0 ,|D H =p |D 0 −q |¯D0 ,(1.1)where p and q are complex parameters determined by off-diagonal elements of the D 0-¯D0mixing Hamiltonian.The rate of D 0-¯D0mixing is commonly measured by two well-defined dimensionless quantities,x D ≡∆m2Γ,(1.2)which correspond to the mass and width differences of D H and D L (i.e.,∆m ≡m H −m L and ∆Γ≡ΓL −ΓH ).The latest result from the Fermilab experiment E791has set an upper boundon the rate of D 0-¯D0mixing [2]:r D ≡x 2D +y 2D|p |4+|q |4.(1.4)It is expected that the magnitude of ∆D should be at most of the order 10−3in the standard model.However,a reliable estimation of ∆D suffers from large long-distance uncertainties.(b)CP violation in direct decay .For a decay mode D 0→f and its CP -conjugate process¯D0→¯f ,this implies | ¯f|H eff|¯D 0 |≡nA n e i(δn −φn )=| f |H eff|D 0 |≡nA n e i(δn +φn ),(1.5)where a parametrization of the decay amplitudes with the weak(φn)and strong(δn)phases is also given.We see that n≥2,φm−φn=0orπandδm−δn=0orπare necessary conditions for the above direct CP violation.(c)CP violation from the interplay of decay and mixing.Let us define two rephasing-invariant quantitiesλf≡qf|H eff|D0,¯λ¯f≡p¯f|H eff|¯D0,(1.6)where the hadronic states f and¯f are common to the decay of D0(or¯D0).Even in the assumption of|q/p|=1,indirect CP violation can appear ifImλf−Im¯λ¯f=0.(1.7) Provided f is a CP eigenstate(i.e.,|¯f =±|f )and the decay is dominated by a single weak phase,then we have¯λ¯f=λ∗f.CP violation at the percent level has not been observed in experiments[7].But signals of O(10−3)are expected in some neutral D decays within the standard model,and those of O(10−2)cannot be ruled out in some channels beyond the standard model.Subsequently we shall highlight some typical signals of D0-¯D0mixing and CP violation which are likely to show up in weak decays of neutral D mesons.A systematic and compre-hensive study of this topic can be found in Ref.[6]and references therein.2Typical signals of D0-¯D0mixingFor simplicity and instruction,we assume∆D=0in the discussion of D0-¯D0mixing effects. This assumption should be safe both within and beyond the standard model,and it can be tested by detecting CP violation in the semileptonic decays of D0and¯D0mesons.A.Time-integrated measurementsForfixed target experiments or e+e−collisions at theΥ(4S)resonance,the produced D0 and¯D0mesons are incoherent.Knowledge of D0-¯D0mixing is expected to come from ratios of the wrong-sign to right-sign events of semileptonic D decays:R(D0phys→K+l−¯νl)R(¯D0phys→K+l−¯νl)≈r D(2.1)in the assumption made above.The Fermilab experiment E791gives r D<0.5%at the90% confidence level[2],the best model-independent limit on D0-¯D0mixing today.For aτ-charm factory running on theψ(3.77)resonance,coherent D0¯D0events with odd C-parity can be produced.If e+e−collisions take place at theψ(4.16)resonance,coherent D0¯D0events will be produced throughψ(4.16)→γ(D0¯D0)C−even orψ(4.16)→π0(D0¯D0)C−odd. Three types of joint decay modes are interesting for measuring D0-¯D0mixing:(D0phys¯D0phys)C−→(l±X∓)D(l±X∓)¯D,−→(K±π∓)D(l∓X±)¯D,−→(K±π∓)D(K±π∓)¯D,(2.2) where we have used the notations X+≡K+¯νl and X−≡K−νl.Note that D0→K+π−is a doubly Cabibbo-suppressed decay(DCSD).This effect is usually measured by the following ratio of decay rates:R DCSD≡ K+π−|H eff|D0R DCSD[y D cos(δKπ−φD)−x D sin(δKπ−φD)],T−int=R(l±X∓;l±X∓)CR(K±π∓;l∓X±)CR(K±π∓;K±π∓)CR[D0(t)→K−π+]R[¯D0(t)→K−π+]R[D0(t)→K−π+]=R DCSD+T+int(Γt)+r DR[¯D0(t)→K+π−]=R DCSD+T−int(Γt)+r D¯K0π0|H eff|D02,(2.6)0.60.81.01.21.41.61.8012345ΓtR [¯D0(t )→K L π0]R [D 0(t )→K S π0]Figure 2:Illustrative plot for changes of two D 0-¯D0mixing observables with proper time t ,where (a)x D =y D =0.05,φD =π/4(corresponding to the solid curves)and (b)x D =0.05,y D =0,φD =π/2(corresponding to the dark solid curves)have been taken.analogous to R DCSD defined above.Obviously R ′DCSD and R DCSD are comparable in magnitude.To a good degree of accuracy,the time-dependent decay rates of D 0(t )→K S,L +π0and ¯D 0(t )→K S,L +π0are independent of R ′DCSD [6]:R [¯D0(t )→K S π0]2+2(y D cos φD +x D sin φD )(Γt )+y 2D (Γt )2,R [¯D0(t )→K L π0]2−2(y D cos φD +x D sin φD )(Γt )+y 2D (Γt )2,(2.7)where a tiny CKM phase from the direct transition amplitude has been neglected.One cansee that the deviation of each of the above two observables from unity measures nonvanishing x D sin φD ,while the difference between them measures the magnitude of y D cos φD −x D sin φD .Thus some useful information about x D ,y D and φD should be achievable from the experimental study of those decay modes.We typically take (a)φD =π/4,x D =y D =0.05and (b)φD =π/2,x D =0.05,y D =0,to plot changes of R [¯D0(t )→K S,L +π0]/R [D 0(t )→K S,L +π0]with proper time t in Fig.2for the purpose of illustration.Finally it is worth pointing out that a comparison of the interference terms in (2.7)with T ±int in (2.4)can provide a model-independent constraint on the strong phase shift δKπ.3Typical signals of CP violationIt is expected that CP violation induced by D 0-¯D0mixing (i.e.,∆D )can manifest itself,ap-parently or indirectly,in all neutral D -meson decay modes.But this effect should be negligiblysmall in most cases.A constraint on∆D is possible through measuring the semileptonic decays of either incoherent or coherent D0and¯D0mesons.For example,R(¯D0phys→K−l+νl)−R(D0phys→K+l−¯νl)R(K−l+νl;K−l+νl)C+R(K+l−¯νl;K+l−¯νl)C=∆D(3.2) for both C-odd and C-even cases.In the following we shall pay main attention to the direct and indirect CP asymmetries in some nonleptonic D transitions,where∆D=0will be assumed.A.Time-integrated measurementsFor neutral D mesons decaying to a hadronic CP eigenstate f,the observables of direct CP violation in the decay amplitude and indirect CP violation from the interplay of decay and D0-¯D0mixing are expressed asA dir≡1−|ρf|21+|ρf|2,(3.3)whereρf≡ f|H eff|¯D0 / f|H eff|D0 .In the time-integrated measurements,the following CP asymmetries can be used to probe A dir and A ind:(a)For incoherent decays of D0and¯D0mesons,we haveR(D0phys→f)−R(¯D0phys→f)R(l−X+;f)C+R(l+X−;f)C ≈ A dir(C−odd)A dir+2x D A ind(C−even).(3.5)Clearly it is possible to distinguish between direct and indirect CP-violating signals,if the magnitude of A dir is comparable with that of x D A ind.(c)At theψ(4.16)resonance there may be a type of CP violation arising from the CP-forbidden decay channels.For example,R(f;f)C−oddR(K Sπ0;K Lπ0)≈R(K Lπ0;K Lπ0)If r D were close to its experimental upper bound andφD were enhanced by new physics,this signal could be measured at aτ-charm factory‡.Now we turn our attention to CP violation in neutral D decays to non-CP eigenstates. Again D0/¯D0→K±π∓can be taken as a good example for illustration.It is expected that only indirect CP violation appears in these four decay modes.We denote the signals as follows:A Kπ≡ R DCSD sinφD(y D sinδKπ+x D cosδKπ),(3.8)whereφD andδKπhave been defined before.If|y D|≪|x D|,as anticipated in some non-standard models[4,10],we arrive at A′Kπ≈−A Kπ.For incoherent D-meson decays,A Kπcan be measured from the decay-rate asymmetryR(¯D0phys→K+π−)−R(D0phys→K−π+)R(l+X−;K+π−)C−R(l−X+;K−π+)CR(l−X+;K+π−)CR(l+X−;K+π−)C4A′KπR(K−π+;K+π−)C −R(K−π+;K−π+)C‡In contrast with(3.7),there may be a similar CP-violating signal for B d decays to K S X c and K L X c on theΥ(4S)resonance,where X c=J/ψ,ψ′,ηc,η′c,etc[11].This signal is expected to be of O(10%)due to the large B0d-¯B0d mixing rate(x B≈0.7)and significant CP-violating phase(φB∼19◦−70◦)in the standard model,thus it should be detectable at the forthcoming B-meson factories.-0.8%-0.6%-0.4%-0.2%0.0%0.2%12345ΓtD 0(t )vs ¯D0(t )→K S π0D 0(t )vs ¯D0(t )→K L π0R [D 0(t )→f ]+R [¯D0(t )→f ]≈A dir +x D A ind (Γt ),(3.10)where A dir and A ind have been defined before.Taking D 0/¯D0→K S,L +π0for example,we obtain R [D 0(t )→K S π0]−R [¯D0(t )→K S π0]R [D 0(t )→K L π0]+R [¯D0(t )→K L π0]≈−2Re ǫK −x D sin φD (Γt )(3.11)in the assumption of ∆D =0.Here Re ǫK ≈1.6×10−3,signifying the CP asymmetry inducedby K 0-¯K 0mixing,cannot be neglected [12].Even if the x Dsin φD term is vanishingly small,the effect of Re ǫK is still detectable from the above decay modes.For the purpose of illustration,we take x D =0.01and φD =0.1to plot changes of the CP asymmetries (3.11)with proper time t in Fig.3.Indirect CP violation in neutral D -meson decays to hadronic non-CP eigenstates can beillustrated by taking D 0/¯D0→K ±π∓for example.Assuming ∆D =0,we have R [¯D0(t )→K +π−]−R [D 0(t )→K −π+]-0.10.10.30.50.7012345ΓtD 0(t )→K +π−vs¯D0(t )→K −π+¯D0(t )→K +π−vsD 0(t )→K −π+R [D 0(t )→K +π−]+R [¯D0(t )→K −π+]≈A ′Kπ2(Γt )+r DI would like to thank S.Pakvasa for inviting me to participate in this nice conference.I am grateful to A.I.Sanda for his encouragement and support,which make my participation realizable.This work was supported by the Japan Society for the Promotion of Science.References[1]See,e.g.,Proceedings of the Workshop on the Future of High Sensitivity Charm Experi-ments,Batavia,Illinois,1994,edited by D.M.Kaplan and S.Kwan(Fermilab Report No.94/190,Batavia,1994);Proceedings of the Workshop on the Tau-Charm Factory in the Era of B-Factory and CESR,Stanford,California,1994,edited by L.V.Beers and M.L.Perl(SLAC Report No.451,Stanford,1994);T.Liu,Ph.D.Thesis,Harvard University,Report No.HUHEPL-20(1995).[2]E791Collaboration,E.M.Aitala et al.,Phys.Rev.Lett.77(1996)2384;M.Purohit,in these proceedings.[3]E.Golowich,Report No.hep-ph/9701225(invited talk given at the4th KEK TopicalConference on Flavor Physics,29-31October,1996),to be published in Nucl.Phys.B (Proc.Suppl.).[4]E.Golowich,in these proceedings.[5]I.I.Bigi and A.I.Sanda,Phys.Lett.B171(1986)320;I.I.Bigi,in Proceedings of the Tau-Charm Factory Workshop,Standford,California,1989,edited by L.V.Beers(SLAC Report No.343,Stanford,1989),p.169.[6]Z.Z.Xing,Phys.Rev.D55(1997)196.[7]CLEO Collaboration,J.Bartelt et al.,Phys.Rev.D52(1995)4860;E687Collaboration,P.L.Frabetti et al.,Phys.Rev.D50(1994)R2953.[8]CLEO Collaboration,D.Cinabro et al.,Phys.Rev.Lett.72(1994)406;E791Collaboration,E.M.Aitala et al.,Report No.hep-ph/9608018(1996).[9]Z.Z.Xing,Phys.Lett.B372(1996)317.[10]T.E.Browder and S.Pakvasa,Phys.Lett.B383(1996)475;L.Wolfenstein,Phys.Rev.Lett.75(1995)2460;G.Blaylock,A.Seiden,and Y.Nir,Phys.Lett.B355(1995)555.[11]Z.Z.Xing,Report No.DPNU-97-16(invited talk given at the1997Shizuoka Workshopon Masses and Mixings of Quarks and Leptons,Shizuoka,March19-21,1997),to be published in the workshop proceedings;Z.Z.Xing,Phys.Rev.D53(1996)204.[12]Z.Z.Xing,Phys.Lett.B353(1995)313.。

a r X i v :h e p -p h /9701419v 2 4 A p r 1997DTP/97/12January 1997(revised March 1997)The rapidity gap Higgs signal at LHC V.A.Khoze a,b ,A.D.Martin a and M.G.Ryskin a,b a Department of Physics,University of Durham,Durham,DH13LE,UK.b Institute for Nuclear Physics,St.Petersburg,SU-188350Gatchina,Russia.Abstract We compare the W W and pomeron-pomeron fusion mechanisms for the double-diffractive production of a Higgs boson.We determine the suppression of the ‘rapid-ity gap’pomeron-pomeron fusion events due to QCD radiative effects.In particular we use leading log techniques to estimate the cross sections for both exclusive and inclusivedouble-diffractive Higgs production at LHC energies.The same approach can be applied to the double-diffractive central production of large E T dijets.These two processes pro-vide one of the most justified applications of various aspects of leading logarithm QCD techniques.The biggest challenge facing the experiments at the forthcoming very high energy proton-proton collider(LHC)is the search for possible Higgs bosons.The present estimates based on the Standard Model and its minimal supersymmetric extension favour the existence of an intermediate mass Higgs boson(M H<∼2M W)[1].In this case the best signals,which are based on the decay modes H→b1For each reaction the short-distance process is gg→H,where we regard g in the case of I P I P fusion as agluonic constituent of the pomeron.Let us start from the ordinary gg→H fusion process.A relevant Feynman diagram for ‘rapidity gap’production is shown in Fig.1,where the additional t channel gluon is needed to screen the colour.(The reason for the dashed and dotted gluon lines will be explained below.) Within this two-gluon exchange picture of the pomeron it is clear that the most optimistic scenario isfirst to assume that the gluon,which screens the colour,does not couple to the Higgs, and second,to assume that it has small virtuality Q2T to enhance the probability of screening via a large value ofαS.This idea was invoked in an attempt to describe the diffractive events in small x deep inelastic scattering[12].The simplest and most extreme prediction is given in ref.[13].It was assumed that the‘screening’gluon is so soft that there is no suppression, apart from a factor of1/N2c which is the probability of forming a colour singlet gg t-channel state.The perturbative realisation of the soft screening approach has been studied for Higgs production[9]and for dijet production[14].An important question,which has not yet been addressed in the literature,concerns the probability of relatively hard gluon emission coming from distance scalesλ>∼1/M H shorter than the characteristic transverse size(∼1/Q T)of the pomeron at which the colourflow is screened.Such gluons couldfill up the rapidity gaps.The goal of the present paper is to estimate the suppression of the rapidity gap events due to these effects.We will show that the typical values of Q T of the‘screening’gluon are indeed much smaller than M H,but nevertheless are sufficiently large for perturbative QCD to be applicable.Of course,there is also a suppression of rapidity gap events due to parton-parton rescattering and to the possibility of multiple(or‘pile-up’)interactions at high luminosities[3,4,6,15]. For example,a rough estimate of the former suppression is[16][1−2(σel+σSD)/σtot]2∼1−10%depending on the value of the cross section,σSD,for single diffraction.These suppressions are common effects for any Higgs production model,including I P I P and W W fusion,as well as for the background processes.Such effects will not be discussed further.We calculate the rate of both exclusive and inclusive Higgs production.In the exclusive process,pp→ppH,only the Higgs and the recoil protons occur in thefinal state.Due to the presence of the proton form factors,the Higgs is produced with small transverse momentum q T.Wefind that the production cross section is negligibly small.On the other hand in the inclusive process,pp→X+gap+H+gap+X′,the initial protons are destroyed.The phase space available for Higgs production,and hence q T,are large.The cross section is found to be comparable to that for W W fusion.We will work in the double logarithmic approximation(DLA)and even for the Born am-plitude we will use the leading power of all logarithms to simplify the calculations.Due to the large value of the Higgs mass M H,this approach is rather well justified.Indeed double-diffractive Higgs production provides probably one of the most justified applications of various aspects of leading log techniques to date.Exclusive productionWe start with the calculation of the(double-diffractive)exclusive process which,at the quark level,is shown in Fig.1.The Born amplitude for the process(shown by the solid lines in thefigure)is of the formM(qq→qHq)=2Q2k21k224α2S(Q2)(k1.k2),(3)where22G Fα2S(M2H)N2/9π2(4) where G F is the Fermi coupling,and where N≈1provided that we are away from the threshold M H=2m t.Note that in the forward direction[where t i=(q i−q′i)2→0for i=1,2andk1T=k2T=Q T]the integral over the gluon loop reduces to d2Q T/Q4.Hence,as mentioned above,small values of Q T of the screening gluon are favoured.In order to make the(Born)calculation more realistic wefirst have to include the ladder ‘evolution’gluons(shown by the dashed lines in Fig.1)and to consider the process pp→pHp at the proton,rather than the quark,level.This is achieved by the replacements[17,18]4αS(Q2)∂ln Q2(5)where x=x1or x2for the upper or lower ladders in Fig.1respectively,and where f(x,Q2)is the unintegrated gluon density2of the proton.The second correction to the Born formula,(3),is the inclusion of the Sudakov form factor F S,which is shown schematically by the dotted curved line in Fig.1.F S is the probability not to emit bremsstrahlung gluons(one of which is shown by p T)in the interval Q T<∼p T<∼M H/2. Clearly the upper bound of the interval is p T≃M H/2.The lower bound,Q T<∼p T,occurs because there is destructive interference of the amplitude in which the bremsstrahlung gluon is emitted from a‘hard’gluon k i with that in which it is emitted from the soft‘screening’gluon Q.That is there is no emission whenλ≃1/p T is larger than the separation,∆ρ∼1/Q T, of the two t-channel gluons in the transverse plane since then they act as a single coherent colour-singlet system.The Sudakov form factor for the above interval of p T isF S=exp −S(Q2T,M2H) (6)where S is the mean multiplicity of bremsstrahlung gluonsS (Q 2T ,M 2H )= M 2H /4Q 2T C A αS (p 2T )E dp 2T4πln 2 M 2HQ 4e −S (Q 2T ,M 2H )f (x 1,Q 2T )f (x 2,Q 2T )(8)in the leading log approximation.The integral has a saddle point given byln(M 2H /4Q 2)=(2π/N c αS (Q 2))(1−2γ)(9)where γis the anomalous dimension of the gluon,g (x,Q 2)∝(Q 2)γ.Suppose that we were to assume a constant γ=0.15.Then for M H =100(200)GeV the saddle point would occur at Q 2=7.3(14)GeV 2,well into the perturbative region,and the Sudakov suppression of the cross section would be (F S )2=0.04(0.025).However,a more realistic evaluation using,say,the MRS(R2)set of partons[19]shows that the integrand reaches its maximum at Q 2∼2GeV 2,where the suppression is (F S )2=0.003(0.0004).If γwere frozen in the region Q 2≤4GeV 2then the cross section would be decreased by a further factor of 2—a factor which is typical of the uncertainty.Table 1shows the values of the exclusive cross section,dσ162π3b 2,(10)calculated from (8).We have integrated over the dt i assuming form factors exp(−bt i /2)at the proton-pomeron vertices,with b =5.5GeV −2.We find that the cross section is more than a factor of 105smaller than the inclusive pp →gg →H cross section,without rapidity gaps;and even a factor 10less than the γγ→H cross section [20].Exclusive double-diffractive Higgs production is thus only of academic interest.Inclusive productionWe find that the cross section for inclusive double-diffractive Higgs production is much larger.Here the initial protons may be destroyed and the transverse momentum of the Higgs is no longer limited by the proton form factor,and so the Sudakov suppression is weaker.The process is shown in Fig.2in the form of the amplitude multiplied by its complex conjugate.The partonic quasielastic subprocess is ab →a ′+gap +H +gap +b ′.If the partons a,b are quarks then the Born amplitude for the subprocess is given by (3).However,the form factor suppressions are more complicated than for the exclusive process.As the momenta transferred,t i =(Q −k i )2,are large we can no longer express the upper and lower ‘blocks’in terms of the gluon structure function,but instead they are given by BFKL non-forward amplitudes.We begin with the expression for the Born cross section for the subprocess gg →g +H +gdσ28πI (11)withI =1Q 2dQ ′2k 21k ′21d 2k 2T2k 21T k 22T ≃1π∆ηi ln k 2iT2 dQ 2Q ′2dt 1t 2exp −n 12−n 22−S −S ′ (16)where the exponential factor represents the total form factor suppression in order to maintainthe rapidity gaps ∆η1and ∆η2in Fig.2.The Sudakov form factor,exp(−S (k 2T ,M 2H )),arisesfrom the insistence that there is no gluon emission in the interval k T <p T <13The resummation corresponds to the Reggeization of the t -channel gluons.4Here ∆ηi (or Y i )plays the role of ln(1/x )in the BFKL evolution.In the region of interest at LHC energies,Y i <∼0.3,it is sufficient to include only the O (Y i )term,which gives Φ≃1+Y i Q 2T /k 2iT ≃1.1±0.1[21].At ourlevelof accuracy we may neglect the enhancement due to Φ,and hence we obtain (16),which is valid in the double logapproximation.To evaluate I of (16)we first perform the Q 2and Q ′2integrations and obtain (Y 1+Y 2)−2.Then we integrate over ln(t 1/t 2)which gives 14Y 1Y 2(Y 1+Y 2) 1t exp(−2S (t,M 2H )).(18)For fixed αS the final (dt )integration gives π(2N c αS )−1s for central Higgs production.However,at large k iT the rapidities of the a ′,b ′jets are small in the Higgs rest frame;ηa ′=ln(x a ′√5The K factor enhancement (analogous to that in Drell-Yan production)is expected to be 1.6–2[22],and there could be a factor of up to 2from the single log BFKL enhancement term Φ4.6For W W fusion one half of the cross section comes from events with k jT <∼M W ,while for IP I P fusion one half comes from k T <13(25)GeV for M H =100(300)GeV if ∆η=2.Indeed we could reduce W W fusion incomparison with I P I P fusion by about a factor (k 2jT /M 2W )2by selecting events with both ‘border’jets satisfyingthe cut-offk 2jT <k 20,with k 20taken to be much less than M 2W .For example,for M H =100GeV,∆η=2if we take k 0=20GeV then we find dσin (I P I P →H )/dy =200fb as compared to dσin (W W →H )/dy =4fb.implicitly assumed that there is no interference between the I P I P→H and W W→H ampli-tudes.This is a good approximation since the two amplitudes(i)are essentially out of phase, (ii)produce Higgs with different q T distributions,and(iii)involve different partons(namely a=g for I P I P fusion and a=q for W W fusion.)DiscussionRecall that our DLA approach to I P I P→H is only justified for the asymmetric configuration of the t channel gluons,Q2T≪k2iT.We must check that this is in fact the case.We have seen above that typically k iT∼20GeV at LHC energies.Now,takingαS=0.2,we have Y≃0.1∆η≃0.25.Thus,using(15),wefind ln(k2iT/Q2T)≃1/2Y≃2.So indeed Q2T<0.15 k2iT.Since k iT is rather large,the suppression due to the Sudakov form factor is not so strong for inclusive production,(F S)2≃0.5.We conclude that for relatively small Y,say Y<0.3,the approach is self-consistent and we may use the DLA expressions,exp(−Y i ln(k2iT/Q2T)),for the BFKL non-forward amplitudes,see(15)and(16).Moreover we have seen that the suppression has a clear physical interpretation.At large Y,say∆η>5,the situation is different.As∆ηincreases we enter the symmetric BFKL gluon configuration,Q2T∼k2iT.We no longer have double logs(and moreover we lose three logs from the Q2,Q′2and d(t1−t2)integrations in I of(16)).Instead,at large Y and t=0, we have the familiar exponential growth of the BFKL amplitude arising from the resummation of the(single)longitudinal logs7.We obtainΦ(Y)∼exp(λ∆η)/(∆η)32λ−1>∼5(if we takeλ∼0.3from the rise of F2observed at HERA with decreasing x).This rapidity gap configuration is beyond the LHC energy range and is not discussed further here,although it could become important at very high energies.Our conclusion is that the interesting proposal“that the Higgs signal could be improved by studying production in the double-diffractive configuration”does not look so optimistic as it first seemed.Exclusive production is negligibly small and even the inclusive cross section is of the same order as the cross section for the more familiar W W→H process.The problem is that QCD radiation has a large probability tofill the parton-level rapidity gaps.Finally we note that our approach may be used to estimate the cross section for the central production of a pair of high E T jets with a rapidity gap either side of the pair.We simply need to replace the gg→H cross section by that for gg→dijet.Since the latter cross section is much larger,and since we have an extra parameter E T,such dijet production(even at Fermilabenergies)offers an excellent opportunity to study QCD(double and single)leading log tech-niques.Moreover,estimates of this dijet production will be important to determine the level of the background to the H→bs=14TeV via I P I P or W W fusion.Here I P denotes the‘hard’QCD pomeron,andσex,in refer to exclusive,inclusive production respectively.The MRS(R2) set of partons[19]is used.We take runningαS in the evaluation of the Sudakov form factors, withαS(M2Z)=0.118,butfixedαS=0.2in the BFKL amplitude.M H(GeV)σin(I P I P)∆η=2(3)18×10−2220(60)20085(9)4.4×10−4140(40)References[1]For a recent review seeM.Carena,P.M.Zerwas,E.Accomando et al.,Higgs physics,in Physics at LEP2,CERN 96-01,vol.1,p.351and references 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J.C.Collins,Nucl.Phys.B474(1996)183.[15]E.Gotsman,E.M.Levin and U.Maor,Phys.Lett.B353(1995)526.[16]A.Rostovtsev and M.G.Ryskin,Phys.Lett.B390(1997)375.[17]M.G.Ryskin,Z.Phys.C57(1993)89.[18]M.G.Ryskin,R.G.Roberts,A.D.Martin and E.M.Levin,Durham preprint DTP/95/96,Z.Phys.(in press).[19]A.D.Martin,R.G.Roberts and W.J.Stirling,Phys.Lett.B387(1996)419.[20]E.Papageorgiu,Phys.Lett.B352(1995)394.[21]J.R.Forshaw and M.G.Ryskin,Z.Phys.C68(1995)137.[22]M.Kr¨a mer,enen and M.Spira,CERN preprint,CERN-TH/96-231.Figure CaptionsFig.1The Born amplitude for the exclusive double-diffractive production of a Higgs boson of transverse momentum q T,shown together with the QCD radiative corrections arising from‘evolution’gluons(dashed lines)and the Sudakov form factor(curved dotted lines).The‘soft screening’gluon has four-momentum Q.Fig.2The amplitude multiplied by its complex conjugate for the inclusive central production of a Higgs boson with rapidity gaps∆η1and∆η2on either side.The suppression due to QCD radiative effects comes from the double log resummations exp(−n i/2)in the BFKL non-forward amplitudes and from the Sudakov form factors exp(−S)shown by the dotted curves;see eq.(16).10q 1q 1′q 2q2′(Higgs) Fig.1Fig.2。