Pseudo-Chern-Simons Terms in the (with Applications)

Rutgers, Dec. 5, 2007 Based on

Baryon-Number-Induced Chern-Simons Couplings of Vector and Axial-Vector in Holographic QCD, PRL 99,14 (2007); arXiv:0704.1604 w/ Sophia Domokos

Anomaly mediated - interactions at finite 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 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 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 “” qL 1 0 0 0 qR 0 1 0 0 ! 0 0 1 0 “” 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 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 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 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 -Photon ‘07) MiniBooNE distinguishes from , but cannot discriminate between final state and electrons:

Electron

Muon

Two Photons

Thanks to Fermilab for graphics. Michel

Two Photons from pi0 decay

Thanks to Fermilab for graphics. Thus the Z ω γ vertex gives a background − − to the charged current events ( ν + N e − + N " ) MiniBooNE is looking for: e → The most naive estimate ignores recoil, form factors, effects (Fermi motion, Pauli blocking), and replaces the neutrino beam by a mono-energetic beam at the peak energy of 700 MeV. This gives 4 1 2 gω 6 σ 6 GF α 4 Eν ! 480π mω Which for every 2x10^5 CCQE events gives g 140 ( ω )4 ∼ 10 events from the anomaly-induced neutrino- photon interaction. We are working on a more detailed comparison. Including some of the simpler effects leads to reasonable fits to data. Including nuclear recoil and a simple choice of form factor, but using a mono-energetic beam and scattering off of rather than nuclei gives, up to normalization. # E v e n t s Note that MiniBoone plots the number of events vs. the reconstructed neutrino energy, assuming a two body final state (electron + ).

The previous graphs plots the number of events vs. the (visible) photon energy, which is shared roughly equally with the final state neutrino in a three body final state (neutrino + nucleon + photon). A neutrino beam with a distribution of neutrino energies peaked at 700 MeV is shifted to a distribution of events with final state photons with the photon energy peaked at ~ 350 MeV. pCS terms in AdS/QCD

2 1 2 µ 5D AdS with IR cutoff: ds = dz + dx dxµ z2 − 0

Fields: Aa,µ ja,µ = q¯ taγµq L ∼ L L L Aa,µ ja,µ = q¯ taγµq R ∼ R R R Xα,β q¯αqβ ∼ We will gauge U ( N ) U ( N ) and focus on f L × f R Nf = 2 1 S = d4xdz g T r DX 2 + 3 X 2 (F 2 + F 2 ) | | | | | | − 4g2 L R # 5 $ ! " 2 2 with g5 = 12π /Nc One finds the pi, rho, a1 etc. by expanding around the solution 1 1 X (z) = m 1z + σ1z3 0 2 q 2

Coeff. of q¯ q in SQCD Exp. value q¯q ! "

The model is defined by three parameters: zm, mq, σ Add Chern-Simons terms

In QCD there are anomalies:

jL jR

The AdS dual involves terms which are gauge invariant in bulk, but vary on the bndy in the same way that QCD would if coupled to fictitious flavor gauge fields. N S = c ω (A ) ω (A ) CS 24π2 5 L − 5 R ! 3 with dω5 = T rF 1 δω5 = dω4 This can also be understood in the S-S model where such couplings arise from a term on the D8-branes S = C ch(F ) = C T rF 3 + anom ∧ 3 ∧ · · · !Σp !Σ9 which in the D4-background with G4 = 2πNc S4 gives the same couplings ! Restrict to lightest isoscalar modes:

a a π (x, z) = π (x)ψπ(z)

Vµ(x, z) = g5 ωµ(x)ψω(z)

Aµ(x, z) = g5 fµ(x)ψf (z) one finds a pCS term (among others)

S g d4x!µνλρf ω ∂ ω fωω ∼ fωω µ ν λ ρ ! with g f ω ω determined in terms of integrals over z and the parameters of the model, leading to g f ω ω 9 . ∼ This might be visible in polarized photon- scattering experiments planned for Hall D at JLab γ + p f + p with γ ω → 1 → using VMD and measurement of the f 1 polarization used to distinguish omega exchange from other effects. Other Applications:

Neutron Star Cooling, Supernova dynamics?

Detection of coherent neutrino scattering off of nuclei. Conclusions

There are a set of SM couplings which involve both the SM gauge fields and the vector fields of QCD which are distinguished by their violation of “natural parity.”

The couplings give a reasonable prediction for certain vector meson decays and may account for the MiniBoone excess at low-energies.

These couplings should have a variety of other applications in astrophysics and nuclear physics/ QCD. The cure is well known. The theory also contains a WZW term: 1 Γ = d4x "µναβ Tr[π∂ π∂ π∂ π∂ π] + W ZW 80π2 µ ν α β ! " 2 e 0 2 π Fµν Fαβ + 16π fπ · · · This term is related to anomalies. One cannot# consistently gauge the full global symmetry and the coupling of the π 0 to reflects an anomaly in the axial current.

In the SM we want to gauge S U (2) L U (1) Y and 0 × we will have couplings to γ , W ± , Z . There are anomalies that are cancelled by the leptons.