Constraints on the Parity-Violating Couplings of a New Gauge Boson

Physics Letters B 608 (2005) 87–94 www.elsevier.com/locate/physletb

Constraints on the parity-violating couplings of a new gauge boson

C. Bouchiat, P. Fayet

Laboratoire de Physique Théorique de l’ENS (UMR 8549 CNRS), 24 rue Lhomond, 75231 Paris cedex 05, France Received 19 October 2004; accepted 15 December 2004 Available online 7 January 2005 Editor: G.F. Giudice

Abstract High-energy particle physics experiments allow for the possible existence of a new light, very weakly coupled, neutral gauge boson (the U boson). This one permits for light (spin- 1 or spin-0) particles to be acceptable Dark Matter candidates, by inducing 2 + − sufﬁcient (stronger than weak) annihilation cross sections into e e . They could be responsible for the bright 511 keV γ ray line observed by INTEGRAL from the galactic bulge. Such a new interaction may have important consequences, especially at lower energies. Parity-violation atomic-physics experiments provide strong constraints on such a U boson, if its couplings to quarks and electrons violate parity. With the constraints coming from an unobserved axionlike behaviour of this particle, they favor a pure vector coupling of the U boson to quarks and leptons, unless the corresponding symmetry is broken sufﬁciently above the electroweak scale.  2004 Elsevier B.V. All rights reserved.

1. Introduction This only applies directly, in fact, to heavy neutral gauge bosons. For light gauge bosons having small The SU(3) × SU(2) × U(1) standard model gives a couplings to Standard Model particles, the discussion very good description of strong and electroweak phe- is different [1]. When the mass mU of the exchanged nomena, so that the possible existence, next to the boson is small (compared to the momentum transfer gluons, photon, W ± and Z, of an additional neutral |q2| ), propagator effects are important. U-induced gauge boson, called here the U boson, is severely con- cross sections then generally decrease with energy,as strained. The U contributions to neutral-current ampli- for electromagnetic ones, as soon as |q2| gets larger ≈ 2 tudes should be sufficiently small, as well as its mixing than mU (as for Z-exchanges, above the Z mass), with the Z. According to the usual belief, any such and may be sufficiently small, if the U couplings are new interaction must be weaker than ordinary weak small enough. In particular, the existence of a new interactions, or it would have been seen already. light gauge boson U having couplings f to matter particles such that f 2 g2 + g 2 E-mail addresses: [email protected] (C. Bouchiat), ∼ (or ∼ G ), (1) [email protected] (P. Fayet). 2 2 F mU mZ 0370-2693/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.12.065 88 C. Bouchiat, P. Fayet / Physics Letters B 608 (2005) 87–94 for example, is not necessarily excluded by high- U-induced Dark Matter annihilation cross section into + − 2 energy scattering experiments. Experiments performed e e (σannvrel/c) also includes, naturally, a vdm low- at lower energies, such as those measuring parity- energy suppression factor (as desirable to avoid exces- violation effects in atomic physics (as we shall discuss sive γ rays from residual light Dark Matter annihila- here), or neutrino scattering cross sections at lower tions [9]). This requirement is satisfied, in the case of a | 2| 1 q , are particularly relevant to search for such a par- spin- 2 Dark Matter particle axially coupled to the U, ticle, and constrain its properties [1–3]. if this one is vectorially coupled to electrons [8]. Let us recall, however, that even if it is very weakly A new interaction stronger than weak interactions coupled, a light spin-1 U boson could still have de- could seem, naively, to be ruled out experimentally. tectable interactions. And this, even in the limit in In fact, however, the U-mediated Dark-Matter/Matter which its couplings f to quarks and leptons would interactions should be stronger than ordinary weak almost vanish, a very surprizing result indeed (appar- interactions but only at lower energies, when weak in- ently)! teractions are really very weak. But weaker at higher In fact a very light spin-1 U boson behaves in this energies, at which they are damped by U propagator → → | 2| 2 case (f very small, mU very small) very much effects (for s or q >mU ), when weak-interaction as a quasimassless spin-0 axionlike particle, if the cur- cross sections, still growing with energy like s,be- rent to which it is coupled includes a (non-conserved) come important. The smallness of the U couplings to axial part.1 This axionlike behavior then restricts ordinary matter (f ), as compared to e, by several or- rather strongly its possible existence and properties, ders of magnitude, and of the resulting U amplitudes implying that the corresponding gauge symmetry be compared to electromagnetic ones, can then account broken at a scale at least somewhat above the elec- for the fact that these particles have not been observed troweak scale; or even at a very high scale, according yet. The U boson, in addition, may well have dominant to the “invisible U-boson” mechanism [1,5]. invisible decay modes into unobserved Dark Matter It is also possible that the new current to which the particles. U couples is purely vectorial, involving a linear com- We indicated in may 2003 that a gamma ray sig- bination of the conserved B, L and electromagnetic nature from the galactic centre at low energy could currents, as in a class of models discussed in Refs. [5, be due to the existence of a light new gauge boson, 6]. In this case there is no such axionlike behavior inducing annihilations of Light Dark Matter particles of the U boson. No significant extra contribution to into e+e− [7]. The observation, a few months later, by parity-violation effects is then to be expected. the satellite INTEGRAL of a bright 511 keV γ ray We now turn to the recent suggestion of Light Dark line from the galactic bulge [10], requiring a rather Matter particles. Contrasting with the heavy WIMPs, large number of annihilating positrons, may then be such as the neutralinos of supersymmetry, light (an- viewed as originating from Light Dark Matter anni- 1 1 nihilating) spin- 2 or spin-0 particles can also be ac- hilations [11]. Indeed spin-0, or as well spin- 2 parti- ceptable Dark Matter candidates. This requires, how- cles, could be responsible for this bright 511 keV line, ever, that they annihilate very efficiently, necessitating which does not seem to have an easy interpretation in new interactions, as induced by a light U boson [7,8]. terms of known astrophysical processes [8,12].One The required annihilation cross sections (≈ 4–10 pb, should, however, also keep in mind that Light Dark depending on whether Dark Matter particles are self- Matter particles may still exist, even if they are not re- conjugate or not), must be significantly larger than sponsible for this line. (And that a light U boson may weak-interaction cross sections (for this energy), oth- be present, even if Light Dark Matter particles do not erwise the relic abundance would be too large! The exist at all.2)

1 This is very similar to what happens in supersymme- 3 2 try/supergravity theories, in which a very light spin- 2 gravitino does In addition, a U that would be both extremely light and ex- not decouple in the κ → 0 limit, but interacts (proportionately to tremely weakly coupled would lead to a new long-range force, 2 1 κ/m3/2 or 1/Λss) like the massless spin- 2 goldstino of global su- and to the possibility of (apparent) violations of the Equivalence persymmetry (a feature largely used in “GMSB” models) [4]. Principe. C. Bouchiat, P. Fayet / Physics Letters B 608 (2005) 87–94 89

Returning to Standard Model particles, a new inter- We now proceed with the phenomenological analy- action that would be stronger than weak interactions sis, expressing the relevant couplings in the La- at lower energies (at least when dealing with Light grangian density as follows: Dark Matter particle annihilations) could have impor- µ tant implications on ordinary physics, especially at L =−eAµJ em lower energies or momentum transfer,evenifithas − 2 + 2 µ − 2 µ no significant influence on high-energy neutral current Zµ g g J3 sin θJem processes. − U fγ¯ µ(f − γ f )f. (2) As we shall see, parity-violation atomic-physics ex- µ Vf 5 Af f =l,q periments [13] provide new strong constraints on such − a gauge boson – whether light or heavy – if its cou- P = 1 γ5 The left- and right-handed projectors are L 2 , plings to quarks and electrons violate parity, then re- 1+γ5 PR = (so that a left-handed U-current would cor- quiring that the corresponding symmetry be broken 2 = = 0 1 significantly above the electroweak scale. respond to fV fA), with γ5 1 0 ,andthemetric (+−−−). The relevant terms in the Z weak neutral current are given by

2. The effective weak charge of a nucleus µ µ J = J − sin2 θJµ Z 3 em 1 1 Such models, in which the standard gauge group = eγ¯ µγ e + − + s2 eγ¯ µe 4 5 4 is extended to include SU(3) × SU(2) × U(1) × extra-U(1) at least, have been discussed in detail. They 1 µ 1 2 2 µ − uγ¯ γ5u + − s uγ¯ u involve an additional neutral gauge boson U (which 4 4 3 may also be called Z or Z), initially associated, be- 1 ¯ µ 1 1 2 ¯ µ U( ) + dγ γ5d + − + s dγ d, (3) fore gauge symmetry breaking, with the extra- 1 4 4 3 generator. Mixing effects with the Z, however, in general play an important role, as far as the U couplings with s2 = sin2 θ = g 2/(g2 + g 2), θ being the elec- are concerned [1,5,6]. When the extra-U(1) gauge troweak mixing angle.3 The vector part of the Z weak coupling constant (g ) is small compared to g and neutral current is associated with the Z (vectorial) g, the mixing angle (∝ g/ g2 + g 2) turns out to be weak charge, which reads, as far as the quark contri- small also, and the modification to the Z weak neu- bution is concerned,4,5 µ − 2 µ tral current, still given by (J3 sin θJem) up to very small corrections (∝ g 2/(g2 + g 2)), is in fact negli- 3 We disregard here, as explained earlier, the very small influence gible. of Z–U mixing effects on the Z current. The current to which the U boson couples, how- 4 This may also be obtained from the quark vector couplings to ever, is significantly affected by the mixing, acquiring, the Z as in addition to the initial extra-U(1) term, a new contri- 1 2 1 1 µ − 2 µ Q = (2Z + N) − s2 + (Z + 2N) − + s2 bution proportional to (J3 sin θJem). The resulting Z 4 3 4 3 U-current includes in general a vector part which ap- 1 1 ≡ Z 1 − 4s2 − N = Q (Z, N) . pears as a linear combination of the conserved B, L 4 4 W SM and electromagnetic currents, as well as an axial part (which may, however, not be present at all, depend- 5 = 1 − 2 More generally, QZ 2 T3(L+R) sin θQmay be rewritten, ing on the models considered). In particular, in a class = − 1 − using T3(L+R) Q 2 (B L), as the conserved charge of simple one-Higgs-doublet models the quark-and- lepton contribution to the U current turns out to be 1 1 2 QZ =− (B − L) + − sin θ Q, purely vectorial [5,6] – which also provides the desired 4 2 2 1 vdm factor in the annihilation cross section of spin- 2 leading to a Standard Model “weak charge” −(B − L) + (2 − light Dark Matter particles [8]. 4sin2 θ)Q, identical to (5) in the case of a nucleus. 90 C. Bouchiat, P. Fayet / Physics Letters B 608 (2005) 87–94

T + T = − 2 = 3L 3R − 2 H = e†(r)γ e(r) QZ (T3L)V sin θQ sin θQ eff 5 2 2 + 2 − g g 2 Z N 2 1 × Z 1 − 4s − N = − sin θZ = Qweak(Z, N). (4) 2 4 4 16mZ eff The quantity fAefVq − 3(Z + N) δ(r) 2 2 mU QW (Z, N) = Z 1 − 4sin θ − N ≈−N (5) = † G√F eff e (r)γ5e(r) QW (Z, N)δ(r). (8) to which it is proportional, is usually referred to as 2 2 the “weak charge” of a nucleus of Z protons and N In the non-relativistic limit (with small components neutrons, and governs the parity-violation effects in expressed as σ·p/(2m ) acting on the electron atomic physics we are interested in [14,15]. e wave-function), this turns into the parity-violating The corresponding effective Lagrangian density in- Hamiltonian for an atomic electron, volves the products of the (Z and U) axial currents G σ·pδ(r) + δ(r) σ·p of the electron by the vector neutral currents of the = √F eff Heff QW (Z, N), (9) quarks (i.e., ultimately the vector currents associated 2 2 2me with protons and neutrons). It may be written, in the 2 p being the electron momentum operator. local limit approximation (assuming mU somewhat This Hamiltonian is expressed in terms of an “ef- | 2| larger than the relevant q , cf. Section 4)as: fective weak-charge” of the nucleus, which includes, −L in addition to the standard contribution QW (Z, N)SM eff (given by Eqs. (4), (5), plus radiative correction 2 + 2 g g 1 terms), an additional U contribution, in the case of = eγ¯ µγ5e m2 4 a parity-violating U current: Z 1 2 1 1 eff × − 2 ¯ µ + − + 2 ¯ µ Q (Z, N) = QW (Z, N)SM s uγ u s dγ d W √ 4 3 4 3 eff 2 2 fAefVq fAe µ ¯ µ − 3(Z + N). (10) − eγ¯ µγ5e fVuuγ¯ u + fVddγ d . (6) 2 2 GF mU mU This applies even if the vector coupling of the U dif- The quark (or proton and neutron) contribution to fers for the u and d quarks, the effective quark vector Q the charge U associated with the vector part of the coupling f eff being defined from Eq. (7) by U current reads Vq fVu(2Z + N)+ fVd(Z + 2N) = + + + f eff = . (11) QU (2fVu fVd)Z (fVu 2fVd)N Vq 3(Z + N) = eff + = eff 3fVq(Z N) 3fVqA. (7) This proportionality to the total number of nucleons 3. Expression in terms of the symmetry-breaking A holds only, strictly speaking, when the U has equal scale vector couplings to the u and d quarks, fVu = fVd.If not, we can still use Eq. (7) as defining the average ef- Eq. (10), namely, fective vector coupling f eff of the U boson to a quark, √ Vq eff within the nucleus considered. 2 2 fAefVq Qeff(Z, N) =− 3(Z + N), (12) The effective Lagrangian density (6) responsible W 2 GF mU for atomic parity-violation effects leads to the parity- may be identified with the one given in [2], violating Hamiltonian density for the electron field, in 6 2 the vicinity of the nucleus: QW = r cϕ3(Z + N), (13)

6 Finite-size effects of the nucleus may be taken into account (assuming here for simplicity that the p and n densities have the by replacing δ(r) by the nuclear density ρn(r) , normalized to unity same radial behaviour). C. Bouchiat, P. Fayet / Physics Letters B 608 (2005) 87–94 91 in a simple situation with a universal axial contribution a large scale” through this (large) extra singlet v.e.v.8 to the U current. The axial and vector couplings are This can make the physical effects of the U boson es- then parametrized, in terms of the extra-U(1) gauge sentially invisible in particle physics, very much as for coupling g,7 as an axion, according to the “invisible U boson” (or sim-  ilar “invisible axion”) mechanism [1,5]. − 1/2 Reexpressing Q as in (13) allows us to com-  −f = g = 2 3/4G m r W  A 4 F U pare directly the extra amount of parity-violation due  −6 2 × 10 mU (MeV)r, (14) to the U boson to the Z contribution, in terms of r 1.  g −3/4 1/2  fV = cϕ = 2 G mU rcϕ The U contribution (for parity-violating couplings to  4 F −6 quarks and electrons) would be roughly of the same 2 × 10 mU (MeV)rcϕ. order as the standard one (and then excessively large), if the extra-U(1) were broken at a scale comparable to r 1 (here simply defined by g = g r)isa mU mW the electroweak scale. dimensionless parameter related to the extra-U(1) symmetry-breaking scale, and cϕ (initially denoted cos ϕ, although not necessarily smaller than 1 in mod- 4. Propagator effects for a very light U boson ulus) measures the magnitude of the quark vector coupling relatively to the axial one. The parity-violating In addition, if the U is light enough (i.e., as com- U-exchange amplitudes are proportional to the Fermi- | 2| like constant pared to the typical q in the experiment considered), one can no longer use for its propagator the local 2 limit approximation. One writes instead, fAfV g GF − = c = √ r2c , (15) 2 2 ϕ ϕ eff eff 2 m 16m 2 2 fAefVq fAefVq m U U = U , (16) m2 − q2 m2 m2 − q2 allowing the identification of expressions (12) and (13) U U U of QW . which leads to a corrective factor The parameter r 1 represents more generally, 2 2 m m 0form2 q2, in such models, the scale at which the extra-U(1) U = U U (17) 2 − 2 2 +2 1form2 q2, symmetry gets spontaneously broken (to which it is, mU q mU q U roughly, inversely proportional) [1,5]. In a class of as compared to a calculation that would have been per- models r = 1 would correspond to an extra U(1) bro- formed in the local limit approximation. ken “at the electroweak√ scale” by two√ Higgs doublets Expression (16) is associated with a Yukawa- ◦ = ◦ = only ( ϕd v1/ 2and ϕu v2/ 2 ); this, how- like (or Coulomb-like, if the U is massless) parity- ever, is excluded experimentally as the light U would violating potential, i.e., then behave very much as a standard axion (with pre- eff sumably, in the present case, invisible decay modes fAefVq into Light Dark Matter particles dominating over the m2 − q2 + − U visible ones into e e ). −m |r| eff 2 −m |r| e U fAefVq m e U r is smaller than one, if an extra Higgs singlet pro- ←→ f f eff = U , (18) Ae Vq || 2 || vides an additional contribution to the U mass, r<1 4π r mU 4π r measuring the amount by which the extra-U(1) sym- where metry gets broken “above the electroweak scale” or “at − || 2 mU r →∞ mU e mU = “δm ”(r) −→ δ(r), (19) 4π|r| U 7 The couplings to the left-handed and right-handed fermion − g − g + fields were expressed as (1 cϕ), (1 cϕ), respectively, 8 4 4 In particular, if we define v /v = 1/x = tan β,theaxial g 2 1 which corresponds to a vector coupling fV = cϕ , and an axial coupling f is given, after Z–U mixing effects, by −f = 4 Ae Ae =−g −3/4 1/2 coupling fA 4 . 2 GF mU r/x. 92 C. Bouchiat, P. Fayet / Physics Letters B 608 (2005) 87–94 in the case of a sufficiently “heavy” U (typically mU order as obtained from a local limit approximation, for 2 2 100 MeV/c as we shall see), leading to the parity- mU afewMeV/c , as can be seen from Table 1. violating Hamiltonian (9) (with δ(r) replaced by the normalized nuclear density ρn(r) , if the nucleus is not taken as pointlike). 5. Experimental limits on fAefVq For a too light U boson, however, the local limit approximation is not valid, and the new contribution From the present comparison between experimen- eff QW as given by (12) should be multiplied by a cor- tal results on QW (Z, N) for cesium [13],whichrely rection factor K(mU ) obtained by replacing δ(r) in (8) on atomic physics calculations [15], and theoretical or (9) by the appropriate Yukawa distribution δm (r) predictions from standard model estimates [16], U of Eq.(19), which extends over a range ≈ h/(m¯ U c). QW exp =−72.74 (29)exp(36)theor, The normalized nuclear density ρn(rn) (which may (22) QW SM =−73.19 ± 0.13, be approximated by a δ(rn) distribution although it 1/3 extends over a radius Rn r0A 6Fermifor one gets Cs) gets replaced by its convolution product with the = − = ± − QW QW exp QW SM 0.45 0.48, (23) δmU (r rn) Yukawa distribution, corresponding to the exchange of a very light U between an electron at r which corresponds to an uncertainty of less than 1%, and the nucleus at rn. i.e. (working conservatively at a pseudo “2σ ”level) This can be expressed through a corrective factor −0.51 <QW < 1.41. (24) √ Heff(mU ) K(mU ) = For cesium (A = 133), using GF /(2 23A) 1.03 × Heff(m →∞) −14 −2 U 10 MeV and expression (12) of QW (for e†(r)γ e(r) ρ (r )δ (r −r )d3rd 3r m 100 MeV/c2), we get the constraint 5 n n mU n n U † 3 e (r)γ 5e(r) ρn(r)d r −f f (20) − × −14 −2 Ae Vq 0.53 10 MeV < 2 mU evaluated in [3], and given numerically in Table 1. − − 2 × 14 2 The mass mU 2.4MeV/c ,forwhichK(mU ) = <1.46 10 MeV , 1/2, defines the typical momentum transfer associated (25) with cesium parity-violation experiments. The corre- or, approximately, ¯ sponding h/(mU c) is 80 Fermi: the electron in- −f f − × −3 Ae Vq × −3 volved in the parity-violating transition of the cesium 0.5 10 GF < 2 < 1.3 10 GF . (26) atom “feels” in fact the new U-mediated interaction, mU even if it is relatively long-ranged, essentially in the For a lighter U the limits get divided by the corrective vicinity of the nucleus, where the screening of the factor K(mU ) of Table 1 (i.e., approximately doubled, 2 Coulomb potential of the nucleus by the core electrons for mU afewMeV/c ’s). can be neglected. One has therefore, ultimately, This analysis applies to heavy as well as to light √ U’s. In the first case it can constrain a U (unmixed eff 2 2 fAefVq with the Z, with couplings ∼ g, g or e) to be heavier Qeff(Z, N) =− 3(Z + N)K(m ). W G 2 U than several hundred GeV/c2 or even more, depend- F mU (21) ing on its couplings. As a toy-model illustration, an 2 For mU < 100 MeV/c the presence of the factor K extra Z or U boson that would have the same cou- weakens the expressions of the limits that would other- plings as the Z would lead directly to a negative contri- 2 wise be obtained from (12), especially in the case of a bution QW QW SM (mZ/mextraZ) [17]. Assum- very light gauge boson.9 Still they remain of the same ing for simplicity that no other contribution has to

9 ∝ 2 Furthermore, in the limit of a very light U (i.e., for mU K(mU ) mU (as one can see from (17)), and the limits may then 2 ¯ ¯ × −8 | eff| ··· | eff | 2 ··· meα 4keV/c so that h/(mU c) h/(mecα) 0.5 10 cm), be expressed as fAefVq < , instead of fAefVq /mU < . C. Bouchiat, P. Fayet / Physics Letters B 608 (2005) 87–94 93

Table 1 Atomic factor K(MU ) giving the correction to the weak charge QW (mU ) for the cesium atom mU 0.10.37 0.51 2.4 5 10 20 50 100 (MeV/c2) Corr. factor 0.025 0.15 0.20 0.33 0.50.63 0.74 0.83 0.93 0.98 K(mU ) be considered, it would have to verify, approximately, requiring typically |Q | < 0.55. The new gauge boson should then be W 2 at least 11.5 times heavier than the Z, i.e.: fAq 1 < GF , (29) 2 10 2 mU mextra Z > 1.05 TeV/c , (27) from ψ or Υ → γ + U decays, or even which is above present direct collider bounds [18]. For a light U on the other hand, the constraint (cf. f 2 1 As G , (30) (26)) adds to those already obtained from low-energy 2 F mU 300 ν–e scattering cross sections, e.g., for m larger than U + + afewMeV/c2’s, from K → π U decays [1,5,8,20]. If such an axial contribution is actually present, the extra-U(1) sym- |f f | Vν Ve G (28) metry should then be broken sufﬁciently above the 2 F mU electroweak scale – a conclusion reenforced here, in and anomalous magnetic moments of charged lep- the case of a U boson inducing atomic physics parity- tons [1,2,7,8]. The latter constraints, however, should violation effects, constrained to be very small. This il- be considered with appropriate care, especially in the lustrates, also, how parity-violation atomic physics ex- case of parity-violating couplings, due to the possibil- periments can give very valuable informations, com- ity of cancellations between (positive) vector contribu- plementing those obtained from particle physics. tions and (negative) axial ones.10,11 More signiﬁcant in fact are the limits from the non- observation of an axionlike particle, which severely 6. Conclusion constrain an axial contribution in the quark U current, This analysis of parity-violation effects in atomic physics (which also applies to heavy bosons), com- 10 While the g − 2 constraints on the vector and axial couplings bined, in the case of a light U, with earlier constraints to the electron, and vector coupling to the muon, are in general on a possible axionlike behavior of this particle, favors not so restrictive (e.g., for a U somewhat heavier than e but lighter −4 −4 a situation in which the quark and lepton contribu- than µ, fVe 2 × 10 mU (MeV), fAe 0.6 × 10 mU (MeV), −4 fVµ 6 × 10 ), the one for an axial coupling to the muons tiontotheU current is purely vectorial, as in a class × −6 2 2 (fAµ 3 10 mU (MeV), i.e., fAµ/mU

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