Further tests of special interactions of massive particles from the Z polarization rate in e + e − →Zt t ¯ and in e + e − →Z W + W −
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Further tests of special interactions of massiveparticles from the Z polarization rate in e + e − → Z t ¯ t and in e + e − → Z W + W − . F.M. Renard
Laboratoire Univers et Particules de Montpellier, UMR 5299Universit´e de Montpellier, Place Eug`ene Bataillon CC072F-34095 Montpellier Cedex 5, France.
Abstract
We propose further tests of the occurence of scale dependent heavy particle masses(Z,W,t) and of strong final state interactions by comparing Z longitudinal polarizationrates in different kinematical distributions of the e + e − → Zt ¯ t and in e + e − → ZW + W − processes. INTRODUCTION
In previous papers [1, 2, 3, 4] we have shown that the rate of Z L polarization in several Zt ¯ t and ZW W production processes is directly sensitive to the occurence of scale dependentmasses (see [5, 6]) and of final state interactions between heavy particles (for example dueto a substructure [7, 8, 9, 10, 11] like in the hadronic case or to a dark matter environment[12, 13, 14]).We now want to improve these tests by looking at different kinematical distributions ofthe Z L rate and by comparing the effects in the Zt ¯ t and ZW W production processes inorder to identify the origin of the effects, pure t , pure Z, W or both.We will concentrate on the e + e − → ZW W − and e + e − → Zt ¯ t processes and illustrate theeffects on the distributions of the Z L rates versus different final 2-body invariant energies.The SM properties (from the respective Born diagrams) have been recalled in [1, 4] andillustrated for the p Z distribution. The sensitivity of the Z L rate to the concerned massesis natural in SM due to the Goldstone equivalence [15].In this paper we will first compute the corresponding s W W , s ZW + , s ZW − and s t ¯ t , s Zt , s Z ¯ t distributions of the Z L rate. Like in the previous papers we will then introduce thetwo different types of modifications, scale dependent top quark, Z , W masses and 2-bodypossible final state interactions. Illustrations will be made with simple kinematical depen-dences but one can easily imagine what would give more elaborated forms like resonanceswith Breit-Wigner forms.In Section 2 we consider the e + e − → ZW W − process with scale dependent Z, W masses(keeping c W at its SM value) and W W , ZW + , ZW − final state interactions.In Section 3 we consider the e + e − → Zt ¯ t − process with scale dependent masses for the Z or the t or both and t ¯ t , Zt , Z ¯ t final state interactions.In Section 4 we will conclude by summarizing the informations that may be obtained fromthe comparison of the two processes, in particular about the simultaneous occurence ornot of the scale dependence of the top quark mass and of the Z, W masses. e + e − → Z W W − The Born SM diagrams have been given in [4] with illustration of the p Z distribution forthe Z L rate R L = σ ( Z L W W ) σ ( Z T W W ) + σ ( Z L W W ) (1)The scale dependence of the
Z, W masses has been studied (assuming that the m W /m Z ratio (i.e. c W ) is fixed) with the test form m W ( s ) = m W ( m th + m )( s + m ) (2)2ffects of final state interactions were illustrated by multiplying the amplitudes by the(1 + C ( s ZW + ))(1 + C ( s ZW − ))(1 + C ( s W + W − )) ”test factor” with C ( x ) = 1 + m Z m ln − x ( m Z + m W ) , (3)In Fig.1 (up) we plot the s W W distribution of the Z L rate for √ s = 5 TeV and θ = π/ p Z distribution shown in [4] as s W W = s + m Z − E Z √ s .In Fig.1 (down) we plot the s ZW + distribution for the same kinematical conditions; we donot show the s ZW − distribution which is very similar.In both cases we can see the basic SM contributions and the effect of a modification ofthe Z, W masses according to eq.(2). The shapes of the distributions and of their modi-fications are typically different in the s W W and in the s ZW ± cases.In Fig.2 we then show, with the same conditions, the effects of final state interactionsaccording to eq.(3) and as in [4] from the addition of the Z and of the G intermediatecontributions. We can also see the differences between the shapes of these distributionsand between the ones due to scale dependent masses or final interactions.With other types of ”test forms” the differences could even be stronger and specific of theorigin of these new interactions (for example with resonance contributions). e + e − → Z t ¯ t The behaviour of the Z L rate R L = σ ( Z L t ¯ t ) σ ( Z T t ¯ t ) + σ ( Z L t ¯ t ) (4)in this process has been studied in [1] where one can find the SM diagrams and thecorresponding p Z distributions.In addition to the scale dependence of the Z, W masses one may now have a scaledependence of the top quark mass that we will similarly study with the test form m t ( s ) = m t ( m th + m )( s + m ) (5)Final state interactions may now appear differently between ( Zt ) or ( Z ¯ t ) and ( t ¯ t ).So we will separately study their effects with the test factors affecting the amplitudesrespectively:(1 + C ( s Zt )), (1 + C ( s Z ¯ t )), and (1 + C ( s t ¯ t ))with C ( x ) = 1 + m t m ln − x ( m Z + m t ) , (6)3esults of scale dependent masses and of final state interactions are respectively illus-trated in Fig.3 and 4.As expected from the expression of the Z L polarization vector, a decrease of the Z mass leads to an increase of the corresponding amplitudes. On another hand a decreaseof the top quark mass leads to a decrease of the longitudinal amplitudes; this is expected,by Goldstone equivalence ([15]), from the couplings of the Goldstone boson to the topquark which is proportional to the top quark mass.Consequently the presence of both Z and t scale dependent masses may cancel and leadto almost no visible effect if the forms of the dependences are similar. This is illustratedin Fig.3 for both s t ¯ t and s Zt (and similarly s Z ¯ t ). This is the remarkable feature of thisprocess.For comparison we then show, in Fig.4, the effects of specific final state interactions onthe s t ¯ t and s Zt distributions. We separately illustrate the effects of s t ¯ t interactions (label t ), of s Zt and s Z ¯ t interactions (label Z ), and of all of them (label Zt ) giving progressivelystronger effects and again specific shapes as compared to the above ones. In this paper we have made a comparative study of the longitudinal Z polarization rate inthe e + e − → Zt ¯ t and e + e − → ZW W − processes; this has shown its remarkable richness.In ZW W production this rate is directly controlled by the W and Z masses; the W massdependence occurs in the ZGW couplings and both the W and Z masses in the respectivepolarization vector. We assumed that the SM structure is maintained ( m W /m Z = c W )even with scale dependent masses. This leads to an increase of the Z L rate as shown inFig.1.In Zt ¯ t production the rate is controlled by both Z and t masses. Contrarily to the ZW W case there is no obvious relation between them in SM. The Z mass controls the Z polar-ization vector and the t mass the Gtt couplings (with Z L − G equivalence). Their effectsare opposite and almost cancel in the total Z L rate (Fig.3).In addition we have shown that the shapes of the s W W , s ZW ± , s t ¯ t and s Zt,Z ¯ t are kinemat-ically different and differently affected by masses and by specific final state interactions(Fig.2,4).The illustrations were made with arbitrary choices of parameters controlling the scaledependence of the masses and the sizes and energy dependences of the final state inter-actions. Our figures just show that one may indeed suspect the presence of BSM effectsand guess their type from the behaviours of the Z L rates, for example those originatingfrom substructures or from special interactions with a dark matter environment.For experimental possibilities relative to these processes see [16].As already mentioned in [4] other production processes may be interesting for confirmingpossible indications coming from the present proposal, for example γ − γ , see [17], or4luon-gluon in hadronic collisions; for LHC possibilities see [18, 19]. References [1] F.M. Renard, arXiv: 1803.10466.[2] F.M. Renard, arXiv: 1805.06379.[3] F.M. Renard, arXiv: 1807.00621.[4] F.M. Renard, arXiv: 1807.08938.[5] G.J. Gounaris and F.M. Renard, arXiv: 1611.02426.[6] F.M. Renard, arXiv: 1708.01111.[7] H. Terazawa, Y. Chikashige and K. Akama, Phys. Rev.
D15 , 480 (1977); for otherreferences see H. Terazawa and M. Yasue, Nonlin.Phenom.Complex Syst. ,1(2016);J. Mod. Phys. , 205 (2014).[8] D.B. Kaplan and H. Georgi, Phys. Lett. , 183 (1984).[9] K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B719 , 165 (2005); hep/ph0412089.[10] G. Panico and A. Wulzer, Lect.Notes Phys. ,1(2016).[11] R. Contino, T. Kramer, M. Son and R. Sundrum, J. High Energy Physics (2007)074.[12] B. Penning, arXiv: 1712.01391. We also thank Mike Cavedon for interesting infor-mations about this subject.[13] F.M. Renard, arXiv: 1712.05352.[14] F.M. Renard, arXiv: 1801.10369.[15] J.M.Cornwall, D.N.Levim and G.Tiktopoulos, Phys. Rev.D10(1974)1145 ; D11(1975)972E; C.E.Vayonakis, Lett. Nuovo Cimento17(1976) 383; B.W.Lee, C.Quiggand H.Thacker, Phys. Rev.D16(1977) 1519 ; M.S.Chanowitz and M.K.Gaillard,Nucl. Phys.B261(1985) 379; M.S.Chanowitz, Ann.Rev.Nucl.Part.Sci.38(1988)323;G.J.Gounaris, R.Koegerler and H.Neufeld, Phys. Rev.D34(1986) 3257.[16] G. Moortgat-Pick et al, Eur. Phys. J.C75, 371 (2015), arXiv: 1504.01726.[17] V.I. Telnov, Nucl.Part.Phys.Proc. (2016)219.518] R. Contino et al, arXiv: 1606.09408.[19] F. Richard, arXiv: 1703.05046. 6
10 15 200.20.40.60.81.0 ( m m SM ) R L ( s W W ) ( θ Z = π/ s WW ( m m SM ) R L ( s ZW + ) ( θ Z = π/ s ZW + Figure 1: e + e − → Z L W W ratio in SM and with an effective Z mass with parameter m = 20 or 40 in eq.(2); invariant distributions for s W W (up) and s ZW (down).7
10 15 200.10.20.30.40.50.60.7 ( ZG )( Z )( SM ) R L ( s W W ) ( θ Z = π/ s WW ( ZG )( Z )( SM ) R L ( s ZW + ) ( θ Z = π/ s ZW + Figure 2: e + e − → Z L W W ratio in SM and in the cases of an effective final (
W W and ZW ) interaction (Z) and of an additional Goldstone contribution ( ZG ) contribution.8
10 15 200.00.20.40.60.8 ( Z )( SM )( Zt )( t ) R L ( s t ¯ t ) ( θ Z = π/ s t ¯ t ( Z )( Zt )( SM )( t ) R L ( s Zt ) ( θ Z = π/ s Zt Figure 3: e + e − → Z L t ¯ t ratio in SM and in the cases of an effective top mass (t), of aneffective Z mass (Z) and of both (Zt); invariant distributions for s t ¯ t (up) and s Zt (down).9