Wess-Zumino-Witten term in QCD-like theories

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1 Wess-Zumino-Witten term in QCD-like theories Helena Kolešová University of Stavanger Joint work with Tomáš Brauner Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 1

2 Outline Motivation: QCD-like theories in strong magnetic field Stuck in mathematics: WZW term and theory of cohomology Wess-Zumino-Witten Helena Koles ova : Wess-Zumino-Witten term in QCD-like theories 2

3 Outline Motivation: QCD-like theories in strong magnetic field Stuck in mathematics: WZW term and theory of cohomology Results Other applications of WZW term I will ask a question at the end!!! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 2

4 Motivation - Chiral Soliton Lattice phase in QCD [Son,Stephanov(2008)][Brauner,Yamamoto(2017)] B [GeV 2 ] Dense QCD matter in strong B (remember also [Ferrer,Incera(Mon)]) Method: low-energy effective field theory (expansion in p 4πf π ) Coupling of neutral pions to electromagnetic field due to chiral anomaly! low-energy effective field theory: chiral anomaly described by the so-called Wess-Zumino-Witten term in the chiral Lagrangian! Groud state for Bµ 16πm πfπ 2 : inhomegeneous condensate of neutral pions carrying baryon charge and magnetic moment n B (z) = B µ zφ(z), m(z) = 4π2 4π zφ(z) 2 for Bµ 2 16π 4 fπ 4 BEC of charged pions [Brauner,Yamamoto(2017)] zφ/mπ µ [MeV] z/l B [GeV 2 ] Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 3 nb/nb,sat µ=800 GeV µ=400 GeV

5 QCD-like theories in strong magnetic filds? Is there similar phase also in case of two-color QCD? Two-color QCD accessible for lattice simulations (no sign problem)! Analytic calculation using EFT in progress! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 4

6 QCD-like theories in strong magnetic filds? Is there similar phase also in case of two-color QCD? Two-color QCD accessible for lattice simulations (no sign problem)! Analytic calculation using EFT in progress! QCD: SU(N f ) L SU(N f ) R SU(N f ) V quarks in pseudo-real representation of the gauge group: SU(2N f ) Sp(2N f ) EFT with different coset space different shape of the WZW term! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 4

7 QCD-like theories in strong magnetic filds? Is there similar phase also in case of two-color QCD? Two-color QCD accessible for lattice simulations (no sign problem)! Analytic calculation using EFT in progress! QCD: SU(N f ) L SU(N f ) R SU(N f ) V quarks in pseudo-real representation of the gauge group: SU(2N f ) Sp(2N f ) EFT with different coset space different shape of the WZW term! Gauged WZW term for general G/H? Results exist, but... [Hull,Spense(1991)][D Hoker(1995)] Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 4

8 Outline Motivation: QCD-like theories in strong magnetic field Stuck in mathematics: WZW term and theory of cohomology Wess-Zumino-Witten Helena Koles ova : Wess-Zumino-Witten term in QCD-like theories 5

9 Chiral anomaly and Wess-Zumino-Witten term T A T C ( d ABC Tr T A {T B, T C } ) T B Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 6

10 Chiral anomaly and Wess-Zumino-Witten term Q γ 5 τ 3 L eff 1 32π 2 π 0 ε µνρσ F µν F ρσ f π Q Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 6

11 Chiral anomaly and Wess-Zumino-Witten term Q γ 5 τ 3 L eff 1 32π 2 π 0 ε µνρσ F µν F ρσ f π Q coset construction [Callan,Coleman,Wess,Zumino(1969)] L χpt f 2 π 4 Tr [ µ U µ U +... ] +??? ) (U e iπa T a fπ Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 6

12 Chiral anomaly and Wess-Zumino-Witten term Q γ 5 τ 3 L eff 1 32π 2 π 0 ε µνρσ F µν F ρσ f π Q coset construction [Callan,Coleman,Wess,Zumino(1969)] L χpt f 2 π 4 Tr [ µ U µ U +... ] +??? ) (U e iπa T a fπ [Witten(1983)] L LO χpt invariant under naive parity : x x, t t, U U L LO χpt invariant under U U 1 (i.e. π a π a ) but QCD invariant only under the combination of these two symmetries! Remember K + K π + π 0 π! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 6

13 Wess-Zumino-Witten term in QCD includes K + K π + π 0 π and chiral anomaly in the chiral Lagrangian first identified in [Wess,Zumino(1971)] [Witten(1983)] gives it nice geometrical meaning: Γ = ω 5 Q (spacetime compactified to 4-sphere M; U : M SU(3), the 4-sphere in SU(3) defined by U(x) = boundary of 5-dimensional disc Q; ω 5 : closed SU(3)-invariant 5-form) ω 5 = Q+Q S ω 5 = 2πn Topological term! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 7

14 Wess-Zumino-Witten term in QCD includes K + K π + π 0 π and chiral anomaly in the chiral Lagrangian first identified in [Wess,Zumino(1971)] [Witten(1983)] gives it nice geometrical meaning: Γ = ω 5 Q (spacetime compactified to 4-sphere M; U : M SU(3), the 4-sphere in SU(3) defined by U(x) = boundary of 5-dimensional disc Q; ω 5 : closed SU(3)-invariant 5-form) But what does the Lagrangian density look like? Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 7

15 Wess-Zumino-Witten term in QCD Γ = Q ω 5 = Q Stokes dω 4 = ω 4 = ω 4 = U ω 4 Q U(M) M (spacetime compactified to 4-sphere M; U : M SU(3), the 4-sphere in SU(3) defined by U(x) = boundary of 5-dimensional disc Q; ω 5 : closed SU(3)-invariant 5-form) if ω 4 SU(3)-invariant, invariant Lagrangian density is obtained if ω 4 not SU(3)-invariant, the Lagrangian density is invariant up to a total derivative - WZW term! ω 4 not SU(3)-invariant ω 5 is closed but not exact 5th de Rham cohomology group of SU(3) Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 8

16 Wess-Zumino-Witten term in QCD Γ = Q ω 5 = Q Stokes dω 4 = ω 4 = ω 4 = U ω 4 Q U(M) M (spacetime compactified to 4-sphere M; U : M SU(3), the 4-sphere in SU(3) defined by U(x) = boundary of 5-dimensional disc Q; ω 5 : closed SU(3)-invariant 5-form) if ω 4 SU(3)-invariant, invariant Lagrangian density is obtained if ω 4 not SU(3)-invariant, the Lagrangian density is invariant up to a total derivative - WZW term! ω 4 not SU(3)-invariant ω 5 is closed but not exact 5th de Rham cohomology group of SU(3) Looking for WZW term for general G/H = looking for 5th de Rham cohomology group of G/H Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 8

17 WZW term for general G/H: Existing literature explicit results for gauged WZW terms for general G L G R /G V [Alvarez-Gaumé,Ginsparg(1985)][Mañes(1985)] explicit results for ungauged WZW terms for general G/H [D Hoker(1995)] not so explicit results for gauged WZW terms for general G/H [Hull,Spense(1991)][D Hoker(1995)] Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 9

18 WZW term for general G/H: Existing literature explicit results for gauged WZW terms for general G L G R /G V [Alvarez-Gaumé,Ginsparg(1985)][Mañes(1985)] explicit results for ungauged WZW terms for general G/H [D Hoker(1995)] not so explicit results for gauged WZW terms for general G/H [Hull,Spense(1991)][D Hoker(1995)] Refresh you knowledge of differential geometry + understand the notation + redo all the calculations = O(1) months + O(100) pages of paper Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 9

19 WZW term for general G/H: Existing literature explicit results for gauged WZW terms for general G L G R /G V [Alvarez-Gaumé,Ginsparg(1985)][Mañes(1985)] explicit results for ungauged WZW terms for general G/H [D Hoker(1995)] not so explicit results for gauged WZW terms for general G/H [Hull,Spense(1991)][D Hoker(1995)] Refresh you knowledge of differential geometry + understand the notation + redo all the calculations = O(1) months + O(100) pages of paper Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 9

20 WZW term for general G/H: Results [ 1 ω 5 = Tr 10 φ ( W + F ) φ 3 + ( W 2 + F 2 ) φ + 1 ] 2 ( W F + F W ) φ Notation: Goldstone boson field: U gauged Maurer-Cartan form (Lie-algebra valued 1-form): θ = U 1 (d + A)U = θ A T A = θ a T a + θ α T α φ + V (a/α - broken/unbroken generators) W = d V + V 2, F = d θ + θ 2 Tr ( φ5 ) = φ a φ b φ c φ d φ e 1 8 Tr ( {[T a, T b ], [T c, T d ]}T c) = φ a φ b φ c φ d φ e f abg f cdh 1 8 Tr ( {T G, T H }T e) dω 5 = 0 if only unbroken generators gauged! ( ) dω 5 = Tr {T A, T B }T C F A F B F C [Tr ( {T α, T β }T γ) = 0 so that ω 5 is well defined] If also broken generators gauged, a closed form can be constructed: ˆω 5 = ω 5 ω U=1 5, but ˆω 5 is not gauge invariant! The gauge non-invariance proportional to Tr ( {T A, T B }T C )! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 10

21 WZW term for general G/H: Results [ 1 ω 5 = Tr 10 φ ( W + F ) φ 3 + ( W 2 + F 2 ) φ + 1 ] 2 ( W F + F W ) φ ω 5 = ω A=0 5 + dω 4 { ω 4 = Tr 12 φ 3 (Ā + Ā ) φā φ(ā + Ā ) φ2 [Ā, Ā ] Ā Ā3 1 2 Ā Ā3 4 1 Ā Ā Ā Ā + φ( 2 1 Ā Ā2 Ā Ā Ā Ā2 Ā Ā Ā Ā Ā Ā2 + Ā3 ) + 1 F [Ā Ā, φ] ( W + W )[ 2 1 Ā + Ā, φ] + ( 1 F W } W )[Ā, Ā ] Notation: Goldstone boson field: U gauged Maurer-Cartan form (Lie-algebra valued 1-form): θ = U 1 (d + A)U = θ A T A = θ a T a + θ α T α φ + V (a/α - broken/unbroken generators) W = d V + V 2, F = d θ + θ 2 {θ, φ, V } = { θ, φ, V } A=0, W = dv + V 2 Ā = U 1 AU = Āa T a + Āα T α Ā + Ā Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 10

22 WZW term for symmetric coset G/H U transforms non-linearly under G: U guh 1 (g G, h(u, g) H) but if there is an automorphism: R : T a T a, T α T α then U 2 Σ gσr(g) 1 [ ω 4 = Tr dσdσ 1 dσσ 1 A dσ 1 dσdσ 1 ΣA dσσ 1 AdΣΣ 1 A 64 5 dσ 1 ΣAdΣ 1 ΣA dσadσ 1 A 1 4 dσdσ 1 AΣAΣ dσ 1 dσaσ 1 AΣ 5 16 ΣAΣ 1 A Σ 1 AΣA Σ 1 AΣAΣ 1 AΣA dσσ 1 A dσ 1 ΣA dσaσ 1 A dσ 1 AΣA dσa2 Σ 1 A 32 3 dσ 1 A 2 ΣA 32 9 dσσ 1 AΣAΣ 1 A dσ 1 ΣAΣ 1 AΣA ( dσ 1 ( Σ 16 7 dσσ 1) {F, A} + 1 dσ 1 AΣF + dσ 1 F ΣA dσaσ 1 F dσf Σ 1 ) A 8 ] ΣAΣ 1 {F, A} 8 1 Σ 1 AΣ{F, A} Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 11

23 WZW term for symmetric coset G/H U transforms non-linearly under G: U guh 1 (g G, h(u, g) H) but if there is an automorphism: R : T a T a, T α T α then U 2 Σ gσr(g) 1 In case of G L G R /G V only terms odd in Σ Σ 1 survive ( ω G L G R 4 = Tr [ 1 2 dσdσ 1 dσσ 1 A dσσ 1 AdΣΣ 1 A 1 2 dσdσ 1 AΣAΣ ΣAΣ 1 A dσσ 1 A dσaσ 1 A dσa2 Σ 1 A 1 2 dσσ 1 AΣAΣ 1 A dσ 1 Σ{F, A} ( + 1 dσ 1 AΣF + dσ 1 ) F ΣA ΣAΣ 1 {F, A} ] Σ Σ 1) 1 4 Tr[ Σ 1 AΣAΣ 1 AΣA ] Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 11

24 WZW term for symmetric coset G/H U transforms non-linearly under G: U guh 1 (g G, h(u, g) H) but if there is an automorphism: R : T a T a, T α T α then U 2 Σ gσr(g) 1 In case of G L G R /G V only terms odd in Σ Σ 1 survive If only single generator gauged all terms with more than one A disappear [Son,Stephanov(2008)] ( ω G L G R 4 = Tr [ 1 2 dσdσ 1 dσσ 1 A dσ 1 Σ{F, A} ( dσ 1 AΣF + dσ 1 ) ] 1 ) F ΣA Σ Σ ToDo: ω G L G R 4 with dσ µσ dx µ, A A µdx µ Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 11

25 WZW term for symmetric coset G/H Decomposable generator of 5th cohomology group (ω 2,3 generators of lower cohomology groups): ω 5 = ω 2 ω 3 If H contains U(1) X subgroup: ω 2 = F X = da X applicable to U(1) B in QCD: ω 5 = F B ω 3 = d(a B ω 3 ) ω 3 : Goldstone-Wilczeck current! [Goldstone,Wilczek(1981)] Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 12

26 WZW term for symmetric coset G/H Decomposable generator of 5th cohomology group (ω 2,3 generators of lower cohomology groups): ω 5 = ω 2 ω 3 If H contains U(1) X subgroup: ω 2 = F X = da X applicable to U(1) B in QCD: ω 5 = F B ω 3 = d(a B ω 3 ) ω 3 : Goldstone-Wilczeck current! [Goldstone,Wilczek(1981)] Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 12

27 Other applications of the WZW-like terms Lower cohomology groups d-dim spacetime: WZW term d + 1st cohomology group of G/H Γ d Stokes = ω d+1 = dω d = ω d = ω d = U ω d Q d+1 Q d+1 Q d+1 U(M d ) M d We have formulas for ω 2, ω 3, ω 4! [D'Hoker(1995)] [4]=[Witten(1983)] [12]=[Goldtone,Wilczek(1981)] [8]=[Chern:Complex Manifolds without Potential Theory(1979)] [13]=[Jackiw,in:Current Algebra and Anomalies(1985)] Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 13

28 Other applications of the WZW term Composite dark matter [Hochberg,Kuflik(,Murayama,Volansky,Wacker)(2014,15,16)] Strongly Interacting Massive Particles (SIMP) as dark matter Relic abundance set by 3 2 self-interactions SIMPle realization: pion of strongly coupled QCD-like hidden sector 3 2 interaction generated by the WZW term for given G/H! (Remember K + K π + π 0 π ) Gauging U(1) D H dark photon mixing with hypercharge gauge boson mediator enabling kinetic equilibrium between dark and visible sectors SIMPlest scenario: SU(4)/Sp(4) symmetry breaking! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 14

29 Other applications of the WZW term Composite dark matter [Hochberg,Kuflik(,Murayama,Volansky,Wacker)(2014,15,16)] Strongly Interacting Massive Particles (SIMP) as dark matter Relic abundance set by 3 2 self-interactions SIMPle realization: pion of strongly coupled QCD-like hidden sector 3 2 interaction generated by the WZW term for given G/H! (Remember K + K π + π 0 π ) Gauging U(1) D H dark photon mixing with hypercharge gauge boson mediator enabling kinetic equilibrium between dark and visible sectors SIMPlest scenario: SU(4)/Sp(4) symmetry breaking! Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 14

30 Conclusions Physics problem: QCD-like theories in strong magnetic field Construction of WZW term for general G/H necessary The existing results translated to usable form Other applications of WZW terms mentioned Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 15

31 Conclusions Physics problem: QCD-like theories in strong magnetic field Construction of WZW term for general G/H necessary The existing results translated to usable form Other applications of WZW terms mentioned Are the results on WZW term for general G/H present somewhere in the literature? Yes No Do you know about some other applications of WZW term? What form of the results would be useful in that particular case? Helena Kolešová: Wess-Zumino-Witten term in QCD-like theories 15