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1、1 31 ? 1 9 2007 c 9 ? p U ? n ? ? ? n HIGHENERGYPHYSICSANDNUCLEARPHYSICS Vol. 31, No. 9 Sep., 2007 Probing Noncommutative Space-Time Scale Using Z at ILC HE Xiao-Gang1,2LI Xue-Qian2 1 (Department of Physics and Center for Theoretical Sciences, Taiwan University, Taipei, China) 2 (Department of Physi
2、cs, Nankai University, Tianjin 300071, China) AbstractIn this talk we report our work on testing Noncommutative Space-Time Scale Using Z at ILC. In ordinary space-time theory, decay of a spin-1 particle into two photons is strictly forbidden due to the Yangs Theorem. With noncommutative space-time t
3、his process can occur. This process thus provides an important probe for noncommutative space-time. The collision mode at the ILC provides an ideal place to carry out such a study. Assuming an integrated luminosity of 500fb1, we show that the constraint which can be achieved on (Z) is three to four
4、orders of magnitude better than the current bound of 5.2105GeV. The noncommutative scale can be probed up to a few TeVs. Key wordsnoncommutative space-time, Yangs theorem, photon collider In this talk we report our work1on testing Non- commutative Space-Time Scale Using Z at ILC. In ordinary space-t
5、ime fi eld theory, decay of a spin- 1 particle into two photons is strictly forbidden due to the Yangs Theorem2.Therefore Z can- not occur in the Standard Model (SM). With non- commutative space-time this process can occur. This process thus provides an important probe for non- commutative space-tim
6、e. The collision at the ILC by laser backscattering of the electron and positron beams provides an ideal place to carry out such a study. To start with, let us briefl y review why Z cannot occur in ordinary space-time fi eld theory by constructing Z- interaction from Z, F. The Lagrangian must be sym
7、metric in the two photons F1and F2due to the Bose-Einstein statis- tics. Using F= 0, the independent terms with even parity that can be constructed are Z(F 1 F 2+F 2 F 1), Z(F 1 F 2+F 2 F 1), Z(F 1 F 2+F 2 F 1). In momentum space, the fi rst term is given by (k1+k2)?Z(k1?2k2?1k1k2?1?2), which is zer
8、o for on-shell Z. Similarly, one can show that the other terms are also zero when particles are on-shell. Anothertypeoftermsinvolves F= (i/2)?Fwhich has odd parity. Using F=0, we fi nd the independent terms to be given by Z(F 1 F 2+ F 2 F 1), Z( F 1 F 2+ F 2 F 1), ?Z(F 1 F 2+F 2 F 1), ?Z(F 1 F 2+ F
9、2 F 1). The fi rst term in momentum space is given by ?Zk 1k 2(? 1? k1? 2?1 k2)(k 1k 2)? 1? 2k1 k2. (1) In this frame the momenta and polarizations of the prticles are given by Pz=(mz,0,0), Z=(0,?Z), k1=(kz,0,kz), k2=(kz,0,kz), ?L 1(kz,0,kz), ? L 2(kz,0,kz), ?T 1 =(0,a,0), ?T 2 =(0,b,0). Received 30
10、 March 2007 844 848 19?f?3ILC?ZLu?I?845 Inserting the above into Eq. (1), one can easily check that the contribution is zero. Similarly, one can show that the other terms are also zero for on-shell parti- cles. Z and Z are forbidden. In noncommutative (NC) space-time3, the pro- cesses Z and Z are no
11、t forbidden. We now describe how this can happen by using a sim- ple and commonly studied noncommutative quantum fi eld theory based on the following commutation re- lation of space-time4, x, x=i,(2) as an example. In the above expression, xis the non- commutative space-time coordinates. is a con- s
12、tant, real, anti-symmetric matrix, and has mass2 dimension. The size of 1/p| represents the non- commutative scale NC. There have been extensive studies on related phenomenology5. Quantum fi eld theory based on the commutation relation in Eq. (2) can be easily studied using the Weyl-Moyal correspond
13、ence replacing the product of two fi elds A( x) and B( x) with NC coordinates by the star “*” product6 A( x)B( x) A(x) B(x)= exp ? i1 2 x y ? A(x)B(y)|x=y.(3) Here the fi elds with and without hat indicate the fi elds in the noncommutative space-time and the or- dinary space-time, respectively. The
14、promotion of the usual space-time coordi- nates xto the noncommutative space-time coordi- nates xhas very interesting consequences7.We denote the noncommutative gauge fi eld to be A= Aa T a of a group with generators normalized as Tr(TaTb) = ab/2.In noncommutative space-time two consecutive local ga
15、uge transformations and of a gauge fi eld Aof the type =i on matter fi eld , transforming as a fundamental representa- tion of the gauge group, is given by () = ( ). This commutation relation is consis- tent with U(N) Lie algebra, but not consistent with SU(N) Lie algebra since it cannot be reduced
16、to the matrix commutator of the SU(N) generators. Also note that even with U(1) group the above consecutive transformation does not commute implying that the charge for a U(1) gauge theory is fi xed to only three possible charges which can be normalized to 1, 0, 1. The above properties pose diffi cu
17、lties in construct- ing noncommutative standard model for the strong and electroweak interactions because the standard gauge group contains SU(3)Cand SU(2)Lwhich can- not be naively gauged with noncommutative space- time. Also the charges of U(1)Yare not just 1, 0, 1, some of them are fractionally c
18、harged after nor- malizing the right-handed electron to have 1 hy- percharge, such as, 1/6, 1/2, 2/3, 1/3 for left- handed quarks, left-handed leptons, right-handed up and down quarks, respectively. This is the so called charge quantization problem. However, all these dif- fi culties can be overcome
19、 with the use of the Seiberg- Witten (SW)6map which maps noncommutative gauge fi eld to ordinary commutative gauge fi eld. A consistent noncommutative SU(N) gauge theory can be constructed by expanding to powers of with = +(1) ab : TaTb: +.+(n1) a1.an: T a1.Tan : . to form a closed envelop algebra.
20、Here : T a1.Tan : is totally symmetric in exchanging ai. Detailed descrip- tion of the method can be found in Ref. 8. One can then expand gauge and mater fi elds in powers of to have a consistent SU(N) gauge theory order by order in . To the fi rst order in , one has for the gauge fi eld8 A=A 1 4g N
21、 A ,A+F. (4) Using the above gauge fi eld new terms in the interac- tion Lagrangian compared with the ordinary SU(N) gauge theory will be generated. For example the term (1/2)Tr(FF) in the Lagrangian for a SU(N) gauge fi eld will become, to the fi rst order in 8, L = 1 2TrF F + gN 1 4TrF FF 4F FF .
22、(5) The SW map can also cure the charge quantization problem by associating a gauge fi eld A(n) for the a matter fi eld (n)with U(1) charge gQ(n), A(n) = A(gQ(n)/4)A,A+F, where Ais the gauge fi eld of U(1) in ordinary space-time. With the help of SW map specifi c method to construct NCSM 846p U ? n
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27、nzburg I et al. Nucl. Instrum. Methods Phys. Res., 1983, 202: 57 14Particle Data Group. Phys. Lett., 2004, B592: 1 3ILC?ZLu?I? ?f 1,2 od 2 1 (?nX?n?%?) 2 (Hm?nXU9300071) ?u3ILCgamma gamma?ZLu?UI(?u3hep-ph/0604115). 3? ?f|?, d?n?g?1?f?UPC?1f. ?3?dL#N ?. d?LU?u?. ILC?1fE?“Uy?L. XJo?U? 500fb1, ?yGamma (Z to gamma gamma)?yk?(5.2105GeV)34? ?. ?UI?u?p?A?TeV. ?c?n1fE 2007 03 30 v