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Pseudo-Chern-Simons Terms in the Standard Model (With Applications)

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Pseudo-Chern-Simons Terms in the Standard Model (With Applications) Pseudo-Chern-Simons Terms in the Standard Model (with Applications) Rutgers, Dec. 5, 2007 Based on Baryon-Number-Induced Chern-Simons Couplings of Vector and Axial-Vector Mesons in Holographic QCD, PRL 99,14 (2007); arXiv:0704.1604 w/ Sophia Domokos Anomaly mediated neutrino-photon interactions at finite baryon density; arXiv:0708.1281 w/ Chris Hill and Richard Hill. Pseudo-Chern-Simons Terms in the Standard Model; arXiv:07mm.xxxx w/ Chris Hill and Richard Hill Work in preparation “Pseudo-Chern-Simons Term” is an awkward phrase for interaction terms of the form S = A A dA = d4x !µνλρA A ∂ A pCS 1 ∧ 2 ∧ 3 1µ 2ν λ 3ρ ! ! with the A i vector fields. This talk will be about either the origin of such terms or their applications, but I want to first make a few general comments: If A 1 is a background vector field (e.g. the baryon 0 ! current) then in its rest frame A 1 = ( A 1 , 0) we have S A0!ijkA ∂ A pCS ∼ 1 2i j 3k ! CS term in 3 dimensions In a Parity invariant theo! ry"# an o$dd number of the A’s must be axial-vector rather than vector fields. S pC S is not gauge invariant, so for fundamental gauge fields A 1 and A 2 must be massive (Stuckelberg) gauge fields or bkgnd vector fields. Main Points Such terms arise in the low-energy effective theory of QCD coupled to the electroweak theory and are part of the Standard Model. These terms are linked to anomalies, familiar from π0 γ + γ → One such term may explain the MiniBoone excess at low-energies and should have astrophysical implications. Other pCS terms make new predictions for vector meson couplings. Outline 1. pCS terms in a toy model 2. pCS terms in the Standard Model 3. Sanity check: f1 ρ + γ → 4. Application to the MiniBoone excess 5. pCS terms in AdS/QCD (time permitting) 6. Conclusions A Toy Model with pCS Terms BL BR LL LR “quarks” qL 1 0 0 0 qR 0 1 0 0 ! 0 0 1 0 “leptons” L !R 0 0 0 1 We gauge U (1) U (1) with generators L × R Q = B L L L − L Q = B L R R − R The action is 4 S = d x q¯L(i∂/ + A/ L)qL + q¯R(i∂/ + A/ R)qR +"¯ (i∂/ A/ )" + "¯ (i∂/ A/ )" ! L − L L R − R R 1 (F )2 1 (F )2 − 4 L − 4 R The quark sector by itself has triangle anomalies 1 δSeff = # dA dA + # dA dA q 24π2 − L L L R R R ! " # But this anomaly is cancelled by the leptons so that we can consistently gauge U(1)L U(1)R × We now generalize this model in two ways: We add a quark mass term and integrate out the quarks. We add in a background field coupling to the (anomalous) vector baryon current. To add a mass term consistent with gauge invariance we add a Higgs field with a non- zero vev Φ = v e iφ /f coupled to the quarks leading to iφ/f 0 mqe q¯LqR + h.c. φ/f π /fπ in QCD ∼ U(1) U(1) The L × R acts as δLqL = i"LqL, δLAL = d"L, δLφ/f = "L δ q = i" q , δ A = d" , δ φ/f = " R R R R R R R R − R Now consider the low-energy theory after integrating out the massive quarks. This is a function of φ , A L , A R and since the anomalies must still cancel, it must have an anomalous eff variation equal to that of S q (D’Hoker and Farhi). This “WZW” contribution is 1 Γ = A A dA + A A dA W ZW − 24π2 L R L L R R ! " φ + dA dA + dA dA + dA dA f L L R R L R % # $ We now add a new ingredient: a background field coupled to the baryon current (like the omega meson in QCD). S d4x q¯ (i∂/ A/ ω/ )q + q¯ (i∂/ A / ω/ )q q → L − L − L R − R − R ! This leads to new terms in the variation of the WZW term proportional to omega: 1 δΓ = # 2dA dω + dωdω W ZW −24π2 L L ! # " 2dA dω + dωdω# − R R " # Since omega doesn’t couple to leptons, the anomaly no longer cancels. But, as one might expect, this can be fixed by adding a local counterterm whose variation cancels the omega dependent terms in δ Γ W Z W : 1 Γ = 2ωA dA ωA dω (R L) c 24π2 R R − R − ↔ ! " # The effective action then has the anomaly related terms Γeff = ΓW ZW (φ, AL + ω, AR + ω) + Γc = ΓW ZW (φ, AL, AR) + ΓpCS Variation cancelled by leptons ! "# $ where in terms of AL = Z + A, AR = A 1 Γ = ω[2dA + dZ] + ωdω Z dφ/f pCS 8π2 − ! " gauge invariant g#"auge invari#ant massless field ma!ssiv"#e vec$tor strength This term correctly generates the anomaly in the baryon current by varying δω = d#B One can apply the same procedure to the SM starting with Γ W Z W ( U , A L , A R ) with A L , A R gauging S U (2) U (1) and then adding a L × Y background of QCD vector and axial-vector mesons ρ , ω , a 1 , f 1 . This gives us Γ W Z W ( U , A L , B L , A R , B R ) where for two flavors, ρ0 + ω √2ρ+ BL + BR = 0 √2ρ− ρ + ω ! − " 0 + a1 + f1 √2a1 BL BR = 0 − √2a− a + f ! 1 − 1 1" This leads to a large number of pCS terms: 2 2 sW 0 3 1 sW 0 3sW + + sW ΓpCS = dZZ c2 ρ + 2c2 3 ω 2c2 f + dAZ c ρ c ω + dZ [W −ρ + W ρ−] c C W W − − W − W − W W " # + $ + % " + % + 3 1 ! +dA [W −ρ + W ρ−] ( s ) + (DW W − + DW −W ) ω f , − W − 2 − 2 2 2 0 3 sW 0 3 3 0 & 3 ' 0 sW + Z dρ ω a + + 3cW f + dω ρ + + 3cW a f C − 2cW − cW − 2cW − 2cW − 2cMinW iBo−ocWne ( ) # $ * ) # $ * ! 2 2 0 sW 0 3 1 3 0 sW 1 0 +da ρ + 3cW ω f + df 3cW ρ + ω a cW 2cW − − 2cW 2cW − cW Ex− 2cWcess ) # $ * ) # $ *+ 2 0 0 0 0 sW + + +dA sW ρ a + 3sW ρ f + 3sW ωa + sW ωf + dZ (ρ a− + ρ−a ) − cW ( + ( + + + +dA sW (ρ a− + ρ−a ) ( + 3 + + 3 + + + + [W Dρ− + W −Dρ )] ( ω + f) + [W ( ρ− + a−) + W −( ρ + a )] dω 2 − 2 − − 1 + + 1 + + + + [W Da− + W −Da ] (f31ω f) + ρ[W+( 3γρ− a−) + W −( 3ρ a )] df , 2 − −→ 2 − − − − + + + 0 0 0 + + 0 0 + 2 (ρ−f + ωa−) Dρ + (ωa + ρ f) Dρ− + ωa + ρ f Dρ + ρ a− + ρ−a + ωf + ρ a dω , C ) * ! , - , - + + 1 + i W W − 3cW Z ω + W W − cW + Z f , C 2cW ( ) * ) * + ! # $ + 3 0 0 1 0 0 + i W W − (ρ + a )ω (ρ a )f C 2 − 2 − ( ) * ! + 3cW 3cW cW 3cW 1 +W Z ρ−f ρ−ω a−f + ωa− ρ−f 2 − 2 − 2 2 − cW ) * 3cW + 3cW + cW + 3cW + 1 + +W −Z ρ f + ρ ω + a f ωa + ρ f , − 2 2 2 − 2 cW ) *+ + 0 0 0 0 + i W ρ−ρ (ω 2f) ρ−ωa + ρ ωa− + ωa−a C − − ( ) * ! + 0 + 0 0 + + 0 +W − ρ ρ ( ω + 2f) + ρ ωa ρ ωa ωa a − − − ) * + 1 2 + 1 +Z ρ ρ− ω + 4cW + f + ρ ωa− 2cW + cW − cW − cW ) # # $ $ # $ + 1 + 1 +ρ−ωa 2cW + ωa a− . − cW cW # $ # $ *+ What do we do with such couplings? Integrate out the massive W ± , Z to get couplings for light fields (e.g. γ , ν , ν¯ ) in the presence of background fields (e.g. baryon number) Treat the QCD mesons as fundamental fields in the spirit of Vector Meson Dominance. The first is more clearly justified, the second involves an approximation which is not under good control, but often works reasonably well, and receives some justification from AdS/QCD. The decay f 1 ρ + γ provides a useful sanity → check of this analysis. It is observed with a 5% branching ratio Γ(f1 ρ + γ) = 1.32MeV → Our coupling leads to λ2 (m2 m2)3 1 1 Γ = f − ρ + 96π m2 m2 m2 f ! f ρ " 2 with λ = 3 g ρ / 4 π 9 we find reasonable ! agreement with the overall rate. More importantly, we get the observed helicity structure. There are two amplitudes: A(γ) = (J (f) = 0) (J (ρ) = 1) + (J (γ) = +1) 0 z → z − z A(γ) = (J (f) = +1) (J (ρ) = 0) + (J (γ) = +1) 1 z → z z (γ) We predict A 1 = 0 (transverse rho) in contrast to an earlier quark model analysis of Babcock and Rosner. We also predict the rate f ω + γ for 1 → down by a factor of 9 from the above and not seen in the PDG. I now want to focus on the term Nc egωg2 µ ν ρσ 2 #µνρσω Z F 48π cos θW which gives rise to a “321 widget” which links the strong, weak, and EM interactions: This interaction may be relevant to the results of the MiniBooNE experiment looking for ν µ ν e oscillations. → MiniBoone sees an excess at low energies: 300 < Eν < 475MeV 96 17 20events ± ± excess 3.7σ (MiniBoone presentation at Lepton-Photon ‘07) MiniBooNE distinguishes electrons from muons, but cannot discriminate between final state photons and electrons: Electron Muon Two Photons Thanks to Fermilab for graphics.
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