January 2005 PM/05-1 hep-ph/0501046 corrected version
arXiv:hep-ph/0501046v2 4 Mar 2005
About helicity conservation in gauge boson scattering at high energy?.
Gounarisa and F.M. Renardb
Department of Theoretical Physics, Aristotle University of Thessaloniki, Gr-54124, Thessaloniki, Greece.
Laboratoire de Physique Th?orique et Astroparticules, UMR 5207 e Universit? Montpellier II, F-34095 Montpellier Cedex 5. e
Abstract We remark that the high energy gauge boson scattering processes involving two-body initial and ?nal states, satisfy certain selection rules described as helicity conservation of the gauge boson amplitudes (GBHC). These rules are valid at Born level, as well as at the level of the leading and sub-leading 1-loop logarithmic corrections, in both the Standard Model (SM) and the Minimal Supersymmetric Standard Model (MSSM). A ”fermionic equivalence” theorem is also proved, which suggests that GBHC is valid at all orders in MSSM at su?ciently high energies, where the mass suppressed contributions are neglected. PACS numbers: 11.30.-j, 11.30.Pb, 12.15.-y, 12.15.Lk
Work supported by the European Union under contract HPRN-CT-2000-00149.
Many people may have noticed that at high energy where masses are neglected, twobody processes involving transverse gauge bosons (V = gluon, photon, Z, W ±) satisfy certain selection rules implying asymptotic helicity conservation in the s-channel. This can easily be seen at Born level in either the Standard Model (SM) or its renormalizable SUSY extensions; e.g. the Minimal Supersymmetric Standard Model (MSSM). For example, ′ considering the processes VλV + VλV ′ → AλA + A′λA′ , and computing the diagrams of Fig.1 corresponding to A, A′ beings scalars, one observes that the high energy helicity amplitudes FλV λV ′ λA λA′ vanish for λV = λV ′ ; while for the fermion production case of Fig.2, the vanishing of the high energy amplitudes is guaranteed whenever either of the relations λV = λV ′ or λA = λA′ is satis?ed. Correspondingly, the amplitudes for the ′ crossed process VλV + AλA → VλV ′ + A′λ′ vanish when λV = ?λV ′ for the case of Fig.1; or A when either of the relations λV = ?λV ′ or λA = ?λA′ is satis?ed for the fermion case of Fig.2. Similar asymptotic rules also exist for the purely gauge helicity amplitudes Fλ1 λ2 λ3 λ4 of the processes V1λ1 V2λ2 → V3λ3 V4λ4 involving four gauge bosons. Thus, it has been observed in  that these amplitudes satisfy asymptotically F+++? = F++?+ = F+?++ = F?+++ = F???+ = F??+? = F?+?? = F+??? = F++?? = F??++ = 0
at the Born level, in either SM or MSSM. Consequently, only the helicity amplitudes satisfying λ1 + λ2 = λ3 + λ4 can survive asymptotically, at this level. These properties of gauge boson helicity conservation (GBHC), are a priori di?erent and complementary to the well-known fermion helicity conservation in processes involving external fermions. The later is an essentially kinematical consequence of the fermionic vertices in SM or MSSM, valid at a diagram by diagram basis, provided that the energy is su?cient high, so that all masses can be neglected1 . GBHC though, referring speci?cally to the external gauge boson helicities, is more subtle. Contrary to the fermionic case, detail cancellation among the contributions of various diagrams must take place, before GBHC is established. This can be seen from the Born processes described by Figs.1 or 2, where the asymptotic vanishing of the helicity amplitudes for λV = λV ′ is established through the occurrence of ”large gauge cancellations” among the V f f and V V V vertices; or among the V ss, V V V and V V ss vertices, with s describing generic scalar particles. It should also be emphasized that such cancellations are only realized when the minimal gauge couplings, characterizing the renormalizable gauge theories, are used. They would be violated if e.g. higher dimensional operators are inserted the theory, even though SU(3) × SU(2) × U(1) gauge symmetry is still respected . Renormalizability of the theory is therefore crucial, for these rules to be valid2 .
See below the discussion of the e?ects of Yukawa couplings. The simplest illustration is the scalar coupling of the type φF ?ν F?ν . It is perfectly gauge invariant, but if used in a scalar exchange diagram, it violates the above rules. Another simple example is the ”anomalous” quadruple coupling [2, 1]. The complete list of such anomalous gauge invariant couplings can be found in .
Sofar we have only considered tree diagrams, and one may wonder whether these high energy helicity conservation properties remain true beyond the Born approximation. Indeed, for processes receiving a Born contribution, one can immediately check that these properties remain true at the level of the 1-loop leading ln2 s and subleading ln s logarithmic corrections, according to the theory developed in [4, 5, 6]. This we have also checked explicitly for e? e+ → γγ , ZZ , γZ using the complete 1-loop results of , and for e+ e? → W + W ? using . We have also looked at the process γγ → γγ , γγ → ZZ  and γγ → γZ , where there is no Born term and the high energy 1-loop behavior is known. The validity of GBHC for the leading and sub-leading logarithmic terms is again observed in both SM and MSSM. However, at the level of the sub-sub-leading (constant) 1-loop contributions, GBHC is generally violated within SM, but it is still preserved in MSSM. Motivated by this observation and the surprising analogy between the fermionic helicity conservation and GBHC, we have looked at its justi?cation, on the basis of supersymmetric invariance and renormalizability. The aim of the present paper is to release this justi?cation. We work in the framework of the exact supersymmetric limit of MSSM, assuming in addition that the Higgs-bilinear ?-term of the superpotential is also vanishing. In such a theory, all particles are massless, and the electroweak gauge symmetry is not broken. We denote the leptons and quarks by the chiral spin=1/2 ?elds (ψL , ψR ), the sleptons and ? ? squarks by the corresponding scalar ?elds (ψL , ψR ), the gauge bosons by Vj? , their gaugino ? partners by χj = χjL + χjR , the higgsino doublets by H(1,2)L , and the corresponding Higgs doublets by H(1,2)L . The later include also the Goldstone bosons. In fact, since all particles are massless in this theory, the notation of the fermionic ?elds may be further simpli?ed by denoting them as (ψλ , χλ ), with λ being the helicity of the particle the ?eld absorbs. The corresponding scalar ?elds may also be de?ned by ? this helicity and written as ψλ ; in fact it is advantageous to think of this scalar ?eld as carrying a ”formal helicity” 2λ. The same de?nition applies also to higgsino and Higgs ?elds. In this massless theory, all purely scalar self interactions consist of 4-leg-vertices arising either from the F-terms generated by the superpotential, or from D-terms. In each of these vertices the total ”formal helicity” de?ned above is conserved. The sum of fermion helicity and ”formal helicity” of the scalar ?elds, is also conserved in all gaugino-fermion-sfermion and gaugino-higgsino-Higgs MSSM vertices. Thus, e.g. a massless quark of a de?nite helicity can be transformed to an opposite gaugino helicity, emitting at the same time a scalar ?eld, that remembers it; so that the sum of the fermion-helicity and the ”formal helicity” is conserved at each vertex separately. The fermion helicity in each of the gauge-fermion vertices, is of course also conserved, for all kinds of fermions, including gauginos and higgsinos. In this respect, we think of the massless gauge bosons of our theory as carrying vanishing ”formal helicity”, and claim that all gauge-fermion vertices also conserve the sum of fermion and formal helicities. It might be useful to think of this conservation of the sum of fermion and formal helicities, as a new global U(1) symmetry respected by all vertices in our framework, 3
except the fermion vertices induced by the Yukava terms in the superpotential. However, if we restrict to processes determined by diagrams in which the Yukawa terms can only appear in hermitian conjugate pairs, then this overall generalized helicity conservation rule will not be a?ected. Since we only consider two-body scattering amplitudes, this is achieved e.g. by restricting to processes involving an even number of external transverse gauge bosons, and/or an even number of external gauginos. In such amplitudes, the number of external Higgs ?elds, as well as the number of external Higgsinos, are also always even. These are in fact the processes which constitute our main interest. With these de?nitions, it is straightforward to check helicity conservation for any 2-fermion to 2-fermion process at high energy, when all masses are neglected. More explicitly, in any allowed such process, the helicities of the incoming and outgoing particles in an amplitude which is not forced to vanish asymptotically, should satisfy
′ ′ F (fλ fλ′ → f? f?′ )
λ + λ′ = ? + ? ′ ,
to all orders in our framework. We emphasize that this result is valid separately for each contributing diagram, independently of the nature of the fermions involved; i.e. whether some or all of them are quarks or leptons or their antiparticles, or gauginos, or higgsinos. The same result (2) remains true, if two of the fermions (irrespective of whether they are in- or out-going) are replaced by scalars. In this case of course, the helicities for the scalar particles actually refer to their ”formal helicities” de?ned above. Since these are ±1 though, while the fermionic ones are half-integers, it is immediately seen that the only relevant amplitudes which may be asymptotically non-vanishing, have the structure
′ F (fλ s → fλ s′ )
′ F (fλ f?λ → ss′ ) ,
where (s, s′ ) denote any kind of scalars3 , and (f, f ′ ) are fermions with their helicities indicated as indices in (3). It is important to realize that (2, 3) imply conservation of physical helicities at asymptotic energies, for any processes involving only external fermions and/or scalars. The physical helicities of all scalars are, of course, vanishing. For proving GBHC for the physical helicities of the transverse gauge bosons, we just rely upon the validity of (2, 3), and the supersymmetric transformation properties of the external ?elds4 . For simplicity we start from the 2-fermion to 2-fermion amplitudes in (2), for the case where all incoming and outgoing fermions describe gauginos. We then remark that the supersymmetric transformation for the gaugino ?elds is 1 j δχj = σ ?ν F?ν γ5 ? ? D j ? , 2
Including of course also the Goldstone bosons. The notion of ”formal helicity” is not needed for this.
j where j is the gaugino group index, F?ν and D j are the corresponding gauge-strength and auxiliary ?elds, and ? is the usual SUSY Majorana constant . This implies that a massless incoming gaugino state of helicity ? and momentum p along the z -axis, trans? forms completely into a massless gauge state with helicity λ and the same momentum and gauge quantum numbers. The explicit result is5
δχ? = δ
(1 + 2?γ5 ) j χ 2
(1 + 2?λ) ip √ (1 + λγ5 )(iλσ 23 + σ 13 )? . 2 2
The crucial term in (5) is the factor (1 + 2?λ) on the r.h.s, which guarantees that the helicities of the transverse gauge bosons generated under a SUSY transformation, will always have the same signs as those of the initial gauginos6 . Thus, any asymptotic helicity structure of the 2-gaugino to 2-gaugino process, will be transformed into a 2-gauge to 2gauge process having the same structure. Starting therefore from (2) applied to gauginos, we conclude that the physical helicities of the asymptotically non-vanishing 2-transverse gauge to 2-transverse gauge amplitudes, satisfy
′ ′ F (Vλ Vλ′ → V? V?′ )
λ + λ′ = ? + ? ′ ,
to all orders in our framework. This procedure can be straightforwardly extended to amplitudes involving any even number of gauginos. Thus, the only asymptotically non-vanishing amplitudes involving two transverse gauge bosons should have the helicity structure
′ ′ ′ ′ F (Vλf? → Vλ′ f? ) , F (VλV?λ → f? f?? ) , F (Vλ s → Vλ′ s′ ) , F (Vλ V?λ → ss′ ) ,
with (f, f ′ ) and (s, s′ ) being fermions and scalars respectively; with the appropriate quantum numbers of course, so that the process is allowed. In the above study we have proved the ”physical helicity” conservation rules (2, 3, 6, 7), in an exactly supersymmetric theory, where all particles are massless and electroweak symmetry (EW) is not broken. Longitudinal gauge bosons do not exist in this theory, but the Goldstone boson (Higgs) ?elds do appear, among the scalar external states of (3, 7). After EW breaking and masses are generated, (2, 3, 6, 7) will of course remain asymptotically true for transverse gauge bosons. At the same time, the equivalence theorem, guarantees that the external Goldstone bosons may readily be replaced by longitudinal gauge bosons in these amplitudes . Thus, if e.g. the scalars in the last of the amplitudes (7) are Goldstone bosons, then the equivalence theorem guarantees also the existence of the asymptotic amplitudes
′ F (VλV?λ → V0′′ V0′′′ ) .
5 The derivation of this relation only involves the standard algebra for the massless fermionic and gauge states, for the aforementioned momenta and helicities. 6 The D-term in (4), being always a product of 4 ?elds in an unbroken SUSY theory, gives no contribution to the single particle projection in (5).
Doing such replacements, in all possible ways, it is easy to see that the complete set of the asymptotically allowed gauge-involving amplitudes is again described by7 (6, 7), with the vector bosons helicities now allowed to acquire vanishing values, while (s, s′) are now interpreted as sfermions or physical Higgs particles only8 . Eqs.(2, 3) will of course also remain true, under this interpretation. The above proof of ”fermionic equivalence” assumes that SUSY is indeed realized in Nature at a moderate scale, such that the corresponding selection rules can be observed at high energy. In such a case in fact, eqs.(2, 3, 6, 7) can be extended to any two-body process which is not determined by diagrams of odd order in the Yukava couplings. Thus, the asymptotically non-vanishing amplitudes should satisfy F (aλ1 bλ2 → cλ3 dλ4 ) ? λ1 + λ2 = λ3 + λ4 , (8)
to all orders in α, for any kind of particles (a, b, c, d) with physical helicities (λ1 , λ2 , λ3 , λ4 ), provided the process is of even order on the Yukawa couplings, and it is of course allowed. As already mentioned, a su?cient condition for this is that the process involves an even number of transverse gauge and an even number of gaugino states. If both initial particles have spin 1/2, and the ?nal are gauge or scalar bosons, (or vice versa), the helicity constraint in (8) is further restricted as λ1 + λ2 = λ3 + λ4 = 0; while if one of the particles in each of the initial and ?nal state has spin 1/2, and the other is boson, helicity is conserved separately for the fermions and the bosons of the process; compare (7). In case SUSY would not be realized at a moderate scale, or not realized at all, then SM will provide the appropriate framework. In this framework, GBHC would remain valid only at the Born approximation, including the leading and sub-leading 1-loop logarithmic corrections9 . Depending on the process, it may be broken at the sub-sub-leading (constant) level, though. We have already mentioned that this is the case in 2-gauge boson to 2-gauge boson processes. Speci?c studies of other processes should be done in order to see if this is a general feature, i.e. if indeed there is a residual GBHC-violating term in SM, which is only cancelled when the supersymmetric partner contributions are added. A priori, there could also be cases in which the sub-sub-leading terms cancel separately in SM and in SUSY contributions. Incidently one should also mention that the cancellation of the GBH-violating amplitudes leads to a remarkable simpli?cation of the actual theoretical description of the processes; about half of the helicity amplitudes disappear and the expressions of the remaining ones are noticeably simpli?ed.
The purely Goldstone four-body amplitude will, of course also be needed here. In principle, it could even be possible to have asymptotically non vanishing amplitudes of the form V0 s → s′ s′′ , where the vector boson is longitudinal. Conservation of other quantum numbers like e.g. CP, forbids the appearance of such terms in MSSM. 9 Occasionally it may be possible to extend this rule to non asymptotic energies also. As an example ? we mention the tree level observation in , that the projections of the t and t spins along the ”o? ? diagonal axis” in the e? e+ → tt c.m. frame, must be equal for any energy. This ”o?-diagonal” axis ? coincides asymptotically with the t ? t helicity axis.
Theoretically, GBHC looks like an appealing simple rule. Experimentally, it may be possible to check it at LHC or ILC, by looking at processes involving gluons, photons, Z or W ’s in processes like q q → gg , gγ , gZ , gW , γγ , γZ , ZZ , W + W ? , γW , ZW , ? gq → gq , γq , Zq , W q , gg → gg , q q , ? + ? e e → γγ , γZ , ZZ , W + W ? , γe → γe , Ze , W ν , ? γγ → f f , γγ , γZ , ZZ , W + W ? , as well as processes involving external supersymmetric particles, like e.g. gg → g g , q? ?? ?q ? ? ?f , χχ , H + H ? , H 0 H ′ 0 . These checks can be done either through a direct ? and γγ → f measurement of the polarization of the initial or the ?nal states, whenever possible; or by looking at the agreement between the di?erential cross section measured experimentally and the theoretical predictions based on the leading helicity conserving amplitudes.
G.J.G. gratefully acknowledges also the support by the European Union RTN contract MRTN-CT-2004-503369.
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V′ V ′′
V (b) V′
Figure 1: Born diagrams for V V ′ → ss′ , with V V ′ being gauge bosons and ss′ being scalar particles.
V′ V ′′
? ? Figure 2: Born diagrams for V V ′ → f f ′ with V V ′ being gauge bosons and f f ′ being fermions.