On the QCD of a Massive Vector Field in the Adjoint Representation Alfonso R. Zerwekh UTFSM December 9, 2012
Outlook 1 Motivation 2 A Gauge Theory for a Massive Vector Field Local Symmetry 3 Quantum Theory: BRST Symmetry 4 Unitarity 5 Conclusions Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 2 / 18
Motivation Models Beyond the Standard Model Many New Physics models predict new spin-1 color octet elds: Model Top Color One-Family Technicolor Extra-dimensions Chiral Color Field Coloron Color-Octet Technirho Kaluza-Klein excitations of the gluon Axigluon Table: Spin-1 Color-Octet Fields predicted by some New Physics models Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 3 / 18
Motivation How can we describe them? Extending the gauge symmetry (Deconstruction Theory) SU(3) c U U U SU(3) SU(3) SU(3) SU(3) Coloron as a matter eld using Vector-Meson-Dominance (VMD)-like Lagrangians (for example: Technicolor) Kroll-Lee-Zumino-like Lagrangians (based on kinetic mixing) Sakurai-like Lagrangian (based on mass mixing) OBS: these two versions of VMD are completely equivalent This description quickly violate unitarity and is not renormalizable All of them are eective descriptions Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 4 / 18
Motivation Why not a renormalizable theory? We know how to construct a renormalizable QCD with scalars and spin- 1 elds as matter elds. 2 Is it possible to construct a consistent and renormalizable theory with massive spin-1 elds without introducing additional structures (Higgses, extended local symmetries, etc)? Common lore says it's not possible because: 1 Scattering of longitudinally polarized massive vector violates unitarity 2 The propagator of a massive spin-1 eld has a bad UV behavior and spoil the power counting in loops diagrams µν = i ( q 2 M 2 g µν q ) µq ν M 2 The aim of this talk is to give an example of construction of a reasonable candidate. Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 5 / 18
A Gauge Theory for a Massive Vector Field The very beginning: Proca Lagrangian Let's consider a generalization of Proca theory for a non-abelian global symmetry: L = 1 4 F a µν F aµν + 1 2 M2 V a µ V aµ V a µ Jaµ (+L int ) with Fµν a = µ Vν a ν Vµ a and V µ a global symmetry (V µ U V µ U). The equation of motion is transforms homogeneously under the ρ F aρν + M 2 V aν = J aν The anti-symmetry of F a µν implies the Lorentz µv aµ = 0, assuming that J aµ is a conserved current. Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 6 / 18
A Gauge Theory for a Massive Vector Field Propagator From the Proca Lagrangian we can obtain the propagator and we get: µν = ( i q 2 M 2 g µν q ) µq ν M 2 We have already comment that this propagator has a bad UV behavior and spoil renormalizability The origin of this propagator and all its problems is the anti-symmetric structure of the F a µν That means that µ V ν µ V v and µ V ν ν V µ enter in the Lagrangian with the same weight Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 7 / 18
A Gauge Theory for a Massive Vector Field Local Symmetry Recipe for a local gauge theory A direct way to turn the Proca-Stueckelberg Lagrangian into a gauge theory is: replace all the derivatives by covariant derivatives ( µ D µ ) Include in L int all the gauge invariant and renormalizable terms we can form with V µ, D µ V ν and G µν (strength eld of gluons) with arbitrary coecients. You may include a kinetic mixing term G µν F µν (KLZ-like Lagrangian) but this term can be removed by a simple eld redenition. L = 1 2 Tr {G µνg µν } Tr {D µ V ν D µ V ν } + (1 + a) Tr {D µ V ν D ν V µ } +a 11 Tr {D µ V ν V µ V ν } + a 12 Tr {D µ V ν V ν V µ } +a 21 Tr {V µ V ν V µ V ν } + a 22 Tr {V µ V ν V µ V ν } +a 3 Tr {G µν [V µ, V ν ]} + M 2 Tr{V ν V ν } Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 8 / 18
A Gauge Theory for a Massive Vector Field Local Symmetry Yang-Mills Theory with Two Connections Consider two color-octet vector elds, A a 1µ and Aa 2µ, which transform like connections of the same group SU(3): A a 1µ UA a 1µ U + 1 g 1 ( µ U) U A a 2µ UA a 2µ U + 1 g 2 ( µ U) U Key observation: V a µ = ga a 1µ g 2A a 2µ transforms homogeneously (it is covariant under gauge transformations). Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 9 / 18
A Gauge Theory for a Massive Vector Field Local Symmetry Yang-Mills Theory with Two Connections Now we can write the following Lagrangian: L 2CYM = 1 2 Tr [F 1µνF µν ] 1 1 2 Tr [F 2µνF µν ] 2 ] + M2 g1 2 + g 2 2 Tr [(g 1 A 1µ g 2 A 2µ ) 2 + a g1 2 + g 2 2 Tr [(g 1 D µ A 1ν g 2 D µ A 2ν ) (g 1 D ν A µ 1 g 2D ν A µ )] 2 +L NM where L NM contains all the gauge invariant and renormalizable terms we can form with V µ, D µ V ν, F 1µν and F 2µν, with arbitrary coecients. Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 10 / 18
A Gauge Theory for a Massive Vector Field Local Symmetry General Lagrangian The physical elds (mass eigenstates) are: G µ = V µ = g 2 g 2 1 + g 2 2 g 1 g 2 1 + g 2 2 A 1µ + A 1µ g 1 g 2 1 + g 2 2 g 2 g 2 1 + g 2 2 A 2µ A 2µ In terms of them, the previous Lagrangian can be written as L = 1 2 Tr {G µνg µν } Tr {D µ V ν D µ V ν } + (1 + a) Tr {D µ V ν D ν V µ } +a 11 Tr {D µ V ν V µ V ν } + a 12 Tr {D µ V ν V ν V µ } +a 21 Tr {V µ V ν V µ V ν } + a 22 Tr {V µ V ν V µ V ν } +a 3 Tr {G µν [V µ, V ν ]} + M 2 Tr{V ν V ν } Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 11 / 18
Quantum Theory: BRST Symmetry BRST Symmetry Because we have two connections (A 1µ and A 2µ ) it is natural to write the gauge-xing and ghost sector as: L BRST = 1 2 ξ 1B a 1 Ba 1 B a 1 µ A a 1µ + c a 1 µ D ab 1µ cb 1 2 ξ 2B a 2 Ba 2 B a 2 µ A a 2µ + c a 2 µ D ab 2µ cb where Diµ ab = δab µ g i f abc A c iµ. The whole Lagrangian is invariant under: BRST transformations δ B A a iµ = 1 g i D ab iµ cb δ B c a = 1 2 f abc c b c c δ B c i a = Bi a δ B Bi a = 0 In order to avoid kinetic mixing terms, we impose ξ 1 = ξ 2 = ξ Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 12 / 18
Quantum Theory: BRST Symmetry BRST Symmetry In terms of the physical elds, the gauge xing + ghost Lagrangian reads: L BRST = 1 2ξ ( µ G a µ ) 2 1 2ξ ( µ V a µ ) 2 + + c a µ D ab µ cb +αf abc ( µ c a ) V c µ cb + βf abc ( µ η a ) V c µ cb where c = c 1 + c 2 and η = c 2 c 1. Notice that η doesn't have a kinetic term. Indeed its equation of motion produce the constrain: f abc µ ( V c µ cb ) = 0 Putting it back to the Lagrangian, we get: L BRST = 1 2ξ ( µ G a µ ) 2 1 2ξ ( µ V a µ ) 2 + c a µ D ab µ cb Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 13 / 18
Quantum Theory: BRST Symmetry Propagator Again With these gauge-xing terms the propagators of the gluon and the massive vector boson are G = (g iδab µν + (ξ 1) qµ q ν ) q 2 q 2 ( iδ ab V = q 2 M 2 g µν q µ q ν ) + (ξ + ξa 1) (1 ξa)q 2 ξm 2 Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 14 / 18
Unitarity V L V L V L V L In order to study the constraints on the theory imposed by the unitarity of the S-matrix, we will use a simpler version of the previous Lagrangian. The commutator where introduced in order to cancel terms proportional to d abc L = 1 2 Tr {G µνg µν } Tr {D µ V ν D µ V ν } + (1 + a) Tr {D µ V ν D ν V µ } +a 1 Tr {(D µ V ν D ν V µ ) [V µ, V ν ]} +a 2 Tr {[V µ, V ν ] [V µ, V ν ]} +a 3 Tr {G µν [V µ, V ν ]} + M 2 Tr{V ν V ν } +L BRST The results for V L V L V L V L scattering are too long to be shown (they were computed with REDUCE), but in order to cancel the unitarity violating terms, it should happen that: a = 0, a 1 = 0 a 2 = g 2 Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 15 / 18
Unitarity GG V L V L In order to cancel the unitarity violating terms in GG V L V L, we need a 3 = g. So, the Lagrangian now reads: L = 1 2 Tr {G µνg µν } Tr {D µ V ν D µ V ν } + Tr {D µ V ν D ν V µ } g 2 Tr {[V µ, V ν ] [V µ, V ν ]} gtr {G µν [V µ, V ν ]} + M 2 Tr{V ν V ν } +L BRST Which is exactly the Lagrangian obtained from the minimal Yang-Mills theory with two connections with g 1 = g 2 = 2g L m2cym = 1 2 Tr [F 1µνF µν ] 1 [ ] 1 2 Tr [F 2µνF µν ] + M2 2 2 Tr (A 1µ A 2µ ) 2 +L BRST Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 16 / 18
Unitarity Consequences The a = a 1 = 0, a 2 = g 2 and a 3 = g assignment makes the Lagrangian invariant under the Z 2 transformation G G and V V This correspond to a symmetry of L m2cym under A 1 A 2 interchange This symmetry implies that V doesn't couple to quarks V can not decay in two gluon. The Z 2 symmetry makes V stable V conveniently dressed by gluons should originate a new neutral and stable hadron which may be a dark matter candidate The Z 2 symmetry makes the observation of a V very dicult. Now we have a theory BRST invariant, with a good propagator and unitarity protected by a Z 2 symmetry. I expect the theory be renormalizable Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 17 / 18
Conclusions Conclusions We have analyzed the construction of the QCD of a massive spin-1 color-octet particle In the general case, the GVV coupling is modied by the Stueckelberg term and the expectation that a massive spin-1 color-octet particle be produced with QCD intensity may not be true in general Unitarity put constraints on the couplings and implies: A new discrete symmetry A dark matter candidate Probably a renormalizable theory without introducing higher structures We gave a meaning to a Yang-Mills theory with 2 connections and we have shown that it is consistent and maybe renormalizable. Alfonso R. Zerwekh UTFSM () On the QCD of a Massive Vector Field in the Adjoint December Representation 9, 2012 18 / 18