Weak Interacting Holographic QCD Doron Gazit Institute for
Weak Interacting Holographic QCD Doron Gazit Institute for Nuclear Theory University of Washington & Ho-Ung Yee ICTP, Trieste Phys. Lett. B (2008), doi:10.1016/j.physletb.2008.10.045, arXiv: 0807.0607 Large N QCD has a dual classical theory in 5-D?! Large N factorization of gauge invariant operators: 1 O1 ( x1 )O2 ( x 2 )L On ( x n ) = O1 ( x ) O2 ( x ) L On ( x ) + O 2 N Implies a classical theory for gauge invariant operators (AKA master fields). RG running survives the large N limit, thus the master field is a function of the energy scale: O( x ) ( ) The RG equations constrain flow in this scale Holographic QCD is a gravitational theory of gauge dimensions. invariant fields in 5 5th dimension corresponds roughly to the energy scale. 2 PANIC08 - Weak interacting holographic QCD - Gazit & Yee Things that we know AdS/CFT Duality proposal N=4 Super Yang-Mills theory in (3+1)D for Nc, 2 =gYM NC gYM0 and fixed but large
is equivalent to Type IIB Supergravity inAdS5S5 with size Mandalcena 1998 3 PANIC08 - Weak interacting holographic QCD - Gazit & Yee We thus expect the dual theory of QCD In the UV regime: highly nonlocal, corresponding to asymptotic freedom. In the IR regime: local, corresponding to the strongly correlated QCD. Thus, current models of Holographic QCD model the gravitational dual as a local theory. Properties of existing models of Holographic QCD: Chiral symmetry. Confinement. Explain experimental observables to 20%. 4 PANIC08 - Weak interacting holographic QCD - Gazit & Yee Motivation for Weak Interacting Holographic QCD Many important processes include weak interaction and low-energy QCD. Numerical calculation of observables from first principles QCD possible only using lattice calculations. Weak interacting Holographic QCD will provide an additional tool for estimating Hadronic processes. 5 PANIC08 - Weak interacting holographic QCD - Gazit & Yee Low-energy Weak interaction
q q Z 0 ,W q q g + 2 MB Tfi ~ J 2 J 2 q + MB 2 4GF 2 HW = J J + cos J + 0 W( Z ) ]= 2 [ W W 2 4GF 1 2 2 2 3 2 = J + J + J
sin J ( L) ( L) ( L W Q) ] 2 [ SU(2)L 6 J La = f a f f EM How to perturb the QCD Lagrangian? Gauge Gravity Perturbation to the Lagrangian. Single trace operator O. Deforming boundary conditions of field near UV boundary. tor
ra e p o A 5D field, such that: e tr O ( x ,zs c1l(e x ) z ac+ c 2 ( x )z + ) i~ng T a z O N s i ion t a b r u t er p A Lagrangian k a e ,w r e v e w Boundary pertutbation:
c1 ( x )conditions: = f (x) Ho 4 L = d xf ( x )O ( x ) 7 c2 ( x) = O ( x) PANIC08 - Weak interacting holographic QCD - Gazit & Yee For a general functional perturbation of a single trace operator L = d 4 xF [O ( x )] F [O ] c1 ( x ) = O O c c2 ( x) = O ( x) Witten, Yee, Marolf-Ross 8 PANIC08 - Weak interacting holographic QCD - Gazit & Yee 2 ( x) Implementation The idea is general enough to implement in any Holographic Model. We demonstrated on two models: Top Down Model: Sakai-Sugimoto Model [Sakai and Sugimoto, Witten]
Bottom Up Model: Hard/Soft Wall Model [Erlich-Katz-SonStephanov, Da Rold -Pomarol]. Both models lead to essentially the same results, as the 5D gauge field includes pions and spin-1 massive bosons when we expand it in terms of 4D modes. 9 PANIC08 - Weak interacting holographic QCD - Gazit & Yee Sakai-Sugimoto Model Set-up of D8 and D8 branes in UV, joined at IR boundary of NC D4 branes. Geometrical realization of U(Nf)RU(Nf)LU(Nf)I SD 8 = = 1/ 3 1 d 4 x dzTr (1+ z 2 ) F2 + (1+ z 2 ) FZ2 2 2 f 4 z ( z 2 z A ) = 0
The dynamics resides in (1) A () L /R the U(Nf) gauge field A ( z) ~ A z A(x,z). N e a r U V z b 2 (1) Values near UV couple to om2 lim J = 2A = z Fz ) ( ) ( L /R z the QCD Nther currents.
u n d a r 10 PANIC08 - Weak interacting holographic QCD y - Gazit & Yee Hard/Soft Wall Model S5D = A phenomenological model, with an artificial IR cutoff, embedded in the dilaton field. is The chiral symmetry due to an expectation value of the bifundamental field broken hidden symmetries. ds 2 = d x 4 1 2 2
2 dz Ge Tr - 2 ( FL,R ) + DX + 3 X 4g 5 1 X ( z) = ( M q z + z 3 ) 2 1 2 dz + dx dx ( ) z2 Mesons come from the 5D gauge fields, as well as the field X, upon KK mode expansion to 4D. N e a r U V z 0 b o u Including non-vanishing
quark-mass natural. PANIC08 - Weak interacting holographic QCD n 11 - Gazit & Yee d Implementing the prescription: 1 For quarks with U(1) charge: JQ = J L3 + J R3 + ( JUL (1) + JUR (1) ) 6 Treating the lepton current as an external a current. U (1) a A=A The current is decomposed: SakaiSugimoto 2 +A 122 Hard/Soft wall model 12 PANIC08 - Weak interacting holographic QCD - Gazit & Yee Charged Pion Decay
+ . el od m l l L,lepton a Treating the lepton current asJan = lLtw L f external source. + o S / d r Thus: A + (+) = 4GF J L+,lepton Ha e there will be no 5th h We define the pion field so that t ( x) 1 Az ( x,z) = inin the kinetic energy: dimension mixing term 1+ z 2 d
e 1 v SS: S = d x idzTr 1+ z F 1+ z A + Tr f Tr A e ( ) ( ) ( ) ( ) ( ) 2 [ ] h
c a As a result s L = 2G f ( )( l ) i lt u s result as old current algebra, and without Same e rfree parameters (due to the chiral symmetry)! e m a S14 PANIC08 - Weak interacting holographic QCD - Gazit & Yee 2 1/ 3 4 2 z l F 2
2 L L 2 How to calculate different reactions? Write equation of motion for the global gauge field (i.e. the U(NF) current). Solve it with the prescribed boundary conditions. If youd like pions to be involved, do it in the previous gauge fixing for Az. For reactions that include nucleons, choose a model for baryons, and calculate baryon-pion coupling from the kinetic term, and from magnetic type of couplings: 15 PANIC08 - Weak interacting holographic QCD - Gazit & Yee Nucleons Sakai-Sugimoto: Nucleons are instantons in 5D. We take only lowest excitation, such that spin connections can be cancelled
S = i d 4 x d [B M (M iAM )B - mB ( ) + CB MN FMNB + ...] Hard/Soft Wall model: Nucleons are spin doublets. 16 Weak interacting holographic QCD - Gazit & Yee ong, Rho, Yee, Yi, Phys. Rev.PANIC08 D 77,- 014030 (2008). Neutron -decay Sakai-Sugimoto Hard/Soft wall model Ln pe = 2GF [ n p + e q q +gA 2 q n 5 p Ln pe = 2GF [ n p + e qq +gA 2 q i(0.84) n q p] ( L eL )
With: i(0.48) Dn q p] ( L eL ) gA = 1.3 n 5 p With: gA = 0.33 + 1.02D The axial constant is measured using the neutron decay beta 17 gAexp = 1.2695(29) PANIC08 - Weak interacting holographic QCD - Gazit & Yee Parity non-conserving pionnucleon coupling First example without an external source. We are interested in parity violating couplings of mesons to the nucleons. To this end, we consider only charged pionnucleon coupling. In both models, the result in the zero q limit is identical to the current algebra result: weak N L
+ = 2GF f ( p n )( ) Still, a lot to be done! 18 PANIC08 - Weak interacting holographic QCD - Gazit & Yee Conclusions We have outlined a prescription to include weak interactions in the framework of holographic QCD. Applicable up to energies of a few GeV, when strong coupling is still valid. We have shown its strength by using Sakai-Sugimoto and Hard/Soft wall models to calculate few exemplar reactions. The current approach, contrary to other approaches (such as PT), gives not only the operator structure, but the numerical coefficients, to about 20%, and valid for energies above the chiral limit. 19 PANIC08 - Weak interacting holographic QCD - Gazit & Yee
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