1 / 46

The Strong CP Problem and Axions

The Strong CP Problem and Axions. Joint ILIAS-CAST-CERN Axion Training CERN November 30, 2005. R. D. Peccei UCLA. The Strong CP Problem and Axions. The U(1) A Problem of QCD The QCD Vacuum and the Strong CP Problem Approaches to the Strong CP Problem U(1) PQ and Axions Axion Dynamics

ryu
Télécharger la présentation

The Strong CP Problem and Axions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Strong CP Problem and Axions Joint ILIAS-CAST-CERN Axion Training CERN November 30, 2005 R. D. Peccei UCLA

  2. The Strong CP Problem and Axions • The U(1)A Problem of QCD • The QCD Vacuum and the Strong CP Problem • Approaches to the Strong CP Problem • U(1)PQ and Axions • Axion Dynamics • Invisible Axion Models • Concluding Remarks

  3. The U(1)A Problem of QCD • In the 1970’s the strong interactions had a puzzling problem, which became particularly clear with the development of QCD. • The QCD Lagrangian for N flavors LQCD = -1/4Fa Fa- Σfqf (-iD + mf) qf in the limit mf -> 0 has a large global symmetry: U(N)Vx U(N)A qf -> [e iaTa/2]ff’qf’ ; qf -> [e iaTa5/2]ff’qf’ Vector Axial

  4. Since mu, md << ΛQCD, for these quarks mf -> 0 limit is sensible. Thus expect strong interactions to be approximately U(2)Vx U(2)A invariant. Indeed, experimentally know that U(2)V = SU(2)V x U(1)V≡ Isospin x Baryon # is a good approximate symmetry of nature  (p, n) and (, °) multiplets in spectrum For axial symmetries, however, things are different. Dynamically, quark condensates break SU(2)A down spontaneously and no mixed parity multiplets

  5. However, because U(2)A is spontaneously broken symmetry, expect appearance in the spectrum of approximate Nambu-Goldstone bosons, with m  0 [ m0 as mu, md 0 ] For U(2)Awould expect 4 such bosons (, ). Although pions are light, m  0, see no sign of another light state in the hadronic spectrum, since m2>> m2 . Weinberg dubbed this the U(1)A problem and suggested that, somehow, there was no U(1)A symmetry in the strong interactions

  6. The QCD Vacuum and the Strong CP Problem • The resolution of the U(1)Aproblem, came through the realization that the QCD vacuum is more complicated [‘t Hooft]. • This complexity, in effect, is what makes U(1)Anot a symmetry of QCD, even though it is an apparent symmetry of LQCD in the limit mf -> 0 • However, this more complicated vacuum gives rise to the strongCP problem. In essence, as we shall see, the question becomes why in QCD is CP not very badly broken?

  7. A possible resolution of the U(1)Aproblem seems to be provided by the chiral anomaly for axial currents[Adler Bell Jackiw] The divergence of axial currents, get quantum corrections from the triangle graph Aa J5 Ab with fermions going around the loop

  8. This anomaly gives a non-zero divergence where , even in symmetry limit Hence, in the mf -> 0limit, although formally QCD is invariant under a U(1)A transformation qf ->ei/25qf the chiral anomaly affects the action However, matters are not that simple!

  9. This is because the pseudoscalar density entering in the anomaly is, in fact, a total divergence [Bardeen]: where K= Aa [Fa -g/3 fabc Ab Ac] This makes W a pure surface integral  W= g2N/322 dK Hence, using the naïve boundary condition Aa=0 at   dK = 0 U(1)A appears to be a symmetry again!

  10. What ‘t Hooft showed, however, is that the correct boundary condition to use is that Aa be a pure gauge at  i.e. either Aa=0 or gauge transformation of0 It turns out that, with these B. C., there are gauge configurations for which dK  0 and thus U(1)A is not asymmetry of QCD This is most easily understood for SU(2) QCD and in Aoa=0 gauge [Callan Dashen Gross]. In this case one has only spatial gauge fields Aia

  11. Under a gauge transformation the Aia gauge fieldstransform as: ½aAia≡Ai Ai -1 + i/g i -1 Thus vacuum configurations are either 0 or have the form i/g i -1 In the Aoa=0 gauge can further classify vacuum configurations by how  goes to unity as r  n  e i2n as r   [n=0, 1, 2,…] The winding numbern is related to the Jacobian of an S3 S3 map and is given by

  12. This expression is closely related to the Bardeen current K. Indeed, in the Aoa=0 gauge only K0≠0 and one finds for pure gauge fields: K0=-g/3ijkabc Aia Ajb Akc =4/3ig ijkTr AiAj Ak True vacuum is superposition of these, so-called, n-vacua and is called the -vacuum: |> =  e -in |n> Easy to see that in vacuum to vacuum transitions there are transitions with dK  0 n|t= + - n|t= - = g2/322dK |t=+ t= -

  13. Pictorially, one has _-3 _--2 _ -1 _ 0 _1 _ 2 _3 _4 _ t =+ g2/322 dK =0 g2/322dK =2 _-4 _-3 _--2 _ -1 _ 0 _1 _ 2 _3 _4 _ t = - In detail one can write for the vacuum to vacuum transition amplitude +<|>- =eim e -in+<m|n>- =  ei n+<n+|n>-

  14. Here the difference in winding numbers is given by Using the usual path integral representation for +<|>- one sees that which allows to re-interpret  term as addition to usual QCD action

  15. Resolution of U(1)A problem, by recognizing complicated nature of QCD’s vacuum, effectively adds and extra term to LQCD This term violatesP and T, but conserves C. Strong bound on the neutron electric dipole momentdn<1.1 x 10-26 ecm requires the angle to be very small [dne mq/MN2  < 10-9 -10-10] Why should this be so is the strong CP problem Problem actually worse if one considers the effect of chiral transformations on -vacuum

  16. Chiral transformations, because of the anomaly, change the -vacuum [Jackiw Rebbi ]: eiQ5 |  > = |  +  > (see Appendix) If one includes weak interactions, the quark mass matrix is in general complex LMass = -qiR Mij qjL + h. c. To diagonalize it one must, among other things, perform a chiral transformation which changes  into total =  + Arg det M Strong CP Problem: Why is this angle, coming from the strong and weak interactions, so small?

  17. Approaches to the Strong CP Problem • There are three possible “solutions”to Strong CP Problem: • Unconventional dynamics • Spontaneously broken CP • An additional chiral symmetry However, in my opinion, only iii. is viable solution • It is, of course, also possible that, as a result of some anthropic reasons total =  + Arg det M just turns out to be of O(10-10), but I doubt it!

  18. Approaches to i.unconventional dynamics are also not very believable: They either suggest that B.C. which gave rise to -vacuum is an artifact [but then, what is solution to U(1)A problem?], or use periodicity of vacuum energy E() ~ cos  to deduce that  vanishes [but, why E/=0?] The second possibility, iiSpontaneously broken CP, is more interesting If CP is a symmetry of nature, which is spontaneously broken, then can set =0 at the Lagrangian level

  19. However,  gets induced back at the loop-level, and to get  < 10-9 one needs, in general, also to insure that 1-loop=0 Although models exist where this is so, theories with spontaneously broken CP need complex Higgs VEVs, leading to FCNC and domain walls, and introduce recondite physics [Barr Nelson] to avoid these problems In my view, however, the biggest drawback for this “solution” to the strong CP problem is that experimental data is in excellent agreement with the CKM Model– a model where CP is explicitly not spontaneously broken

  20. Introducing iii anadditional chiral symmetry is a very natural solution to strong CP problem since it, effectively, rotates  -vacua away e-iQ5 |  > = | 0 > Two suggestions for this chiral symmetry: The u-quark has no mass, mu = 0 SM has an additional global U(1) chiral symmetry [Peccei Quinn] mu = 0 is disfavored by current algebra analysis [ Leutwyler]. Further, it is difficult to understand why Arg det M = 0 What is the origin of this chiral symmetry?

  21. U(1)PQ and Axions • Introducing a global U(1)PQ symmetry, which is necessarily spontaneously broken, replaces: total =  + Arg det M a(x) / fa Static CP viol. Angle Dynamical CP cons. Axion field • The axion is the Goldstone boson of the broken [Weinberg Wilczek] U(1)PQ symmetry and fa is scale of the breaking. Hence under U(1)PQ a(x)  a(x)   fa

  22. Formally, for U(1)PQ invariance the Lagrangian of SM is augmented by axion interactions: Last term needed to give chiral anomaly of JPQ and acts as an effective potential for axion field Minimum of potential occurs at <a>=-fa/ total

  23. Easy to understand the physics of PQ solution. If one neglects the effects of QCD then U(1)PQ symmetry allows any value for <a>: 0≤ <a> ≤ 2 Including the effects of the QCD anomaly generates a potential for the axion field which is periodic in the effective vacuum angle Veff ~ cos[total + <a>/fa ] Minimizing this potential with respect to <a> gives the PQ solution <a>=-fa/ total

  24. Hence theory written in terms of aphys= a- <a> has no longer a -term [ this is the PQ solution] Furthermore, expanding Veff at minimum gives the axion a mass [anomaly gives NG a mass] Calculation of axion mass first done explicitly by current algebra techniques [Bardeen Tye] Here will give an effective Lagrangian derivation [Bardeen Peccei Yanagida], as it also gives readily axion couplings

  25. Axion Dynamics In the originalPeccei Quinnmodel, the U(1)PQ symmetry breakdown coincided with that of electroweak breaking fa = vF, with vF 250 GeV. However, this is not necessary. If fa >> vF then axion is very light, very weakly coupled and very long lived [invisible axion models] Useful to derive first properties of weak-scale axions and then generalize the discussion to invisible axions To make SM U(1)PQ invariant must introduce 2 Higgs fields to absorb independent chiral transformations of u- and d-quarks (and leptons)

  26. Yukawa interactions in SM involve Higgs Defining x=v2/v1 and vF= √(v12 + v22), the axion is the common phase field in 1 and 2 which is orthogonal to the weak hypercharge See that LYukawais invariant under the U(1)PQtransformation aavF ;uRi e-ixuRi;dRi lRi e-i/xdRi lRi

  27. Let us focus on the quark pieces. The current JPQ=-vF ∂ a+ x Σi uiR  uiR+ 1/xΣi diR  diR identifies the strong anomaly coefficient as: =N/2(x +1/x)=Ng(x +1/x) To compute the axion mass and mixings from an effective chiral Lagrangian we need to separate out lightu- and d-quarks from rest. For these purposes one introduces a 2x2 matrix of NG fields Σ = exp[ i(. +)/f ] and the U(2)VxU(2)A invariant eff. Lagrangian Lchiral =-f2/4 Tr∂ † ∂ 

  28. To Lchiralmust add U(2)VxU(2)A breaking terms which mimic the U(1)PQ invariant Yukawa interactions of the u- and d-quarks. This is accomplished by adding Lmass=½(f mo )2Tr[ΣAM+(ΣAM)†] where and under PQ-transformations

  29. However, Lmassonly gives part of the physics. Indeed, the quadratic terms in Lmass involving neutral fields L2mass=-½mo2{mu/(mu+md)[+-xf/vFa]2 +md/(mu+md)[--f/xvFa]2} give the wrong ratio for m2/ m2 m2/ m2= md/mu 1.6[ the U(1)A problem!] and the axion is still massless To account for the effect of the anomaly in both U(1)Aand U(1)PQone must add a further effective mass term which gives the the right mass and produces a mass for the axion

  30. It is easy to see that such a term has the form [Bardeen Peccei Yanagida] Lanomaly=-½mo2 [+ {[f/vF] [(Ng-1)(x +1/x)/2]}a]2 where mo2  m2>>m2 Coefficient in front of a in Lanomaly details the relative strength of the couplings of  and a to .Naively, one would imagine {} =f/vF/2= f/vFNg/2(x +1/x) However, only the contribution of heavy quarks to the PQ anomaly should be included (hence Ng(Ng-1)) since light quark interactions of axions are included already in Lmass

  31. Diagonalization of the quadratic terms in LmassandLanomaly gives both the axion mass andthe parameters for a- and a – mixingfor thePQ model. Convenient to define mast = mf /vF [mumd/(mu+md)]  25 KeV Then can characterize all axion models by 4 parameters { m; 3;0;K a } of O(1). To wit: ma=mmast[vF / fa] a=3 [f/ fa] ;a = 0 [f/ fa]

  32. A simple calculation for weak-scale axions, where fa=vF, gives: m=Ng(x +1/x) 3=½[(x -1/x)-Ng(x +1/x)(md-mu)/(mu+md)] 0 =½(1-Ng)(x+1/x) To compute the coupling K aone must considerthe em anomaly of the PQ current Leptons also contribute to  and one finds =Ng{[3(2/3)2]x+[3(-1/3)2+(-1)2]1/x} =4/3Ng(x +1/x)

  33. As before, we must separate out the light quark contributions in the anomaly, since they are counted by the coupling of the °and  to 2. Adding the lepton and heavy quark contributions of the axion coupling to the em anomaly eff=4/3Ng(x +1/x)-4/3x -1/3x to that coming from the  and  mixing 3 + 5/3 0 gives, finally, K a=Ng(x +1/x)[mu/(mu+md)]

  34. Invisible Axion Models • Original PQ model, where fa=vF, was long ago ruled out by experiment. • For example, one can estimate the branching ratio [Bardeen Peccei Yanagida] BR(K+ + +a)  3 x 10 -50 2  3 x 10 -5(x+1/x)2 which is well above the KEK bound BR(K+ + +nothing) <3.8 x 10 -8 • However, invisible axion models, where fa>>vF, are still viable

  35. These invisible axion models introduce fields which carry PQ charge but are SU(2)XU(1) singlets Two types of models have been proposed i) KSVZ [Kim;Shifman Vainshtein Zakharov] Only a scalar field with fa= <> >> vFand a superheavy quark Q with MQ~facarry PQ charge ii)DFSZ [Dine Fischler Srednicki; Zhitnisky] Adds to PQ model a scalar fieldwhich carriesPQ chargeandfa= <>>> vF

  36. For these models, one can repeat the calculations we just did to get the axion mass and couplings Will do this for the KSVZ model because it is simple and illustrates well what we just did The KSVZaxion does notinteract withleptons and only interacts with light quarks as the result of the strong and em anomalies The superheavy quark Q induces the following couplings [eQ is the em charge of Q]

  37. Since in the KSVZ model the ordinary Higgs do not carry PQ charge, the only interactions of the axion come from the effective anomaly mass term which here is given by Lanomaly=-½mo2 [+ {[f/fa] [1/2]}a]2 To the above one must add the standard quadratic term coming from the light quarks L2=-½mo2{mu/(mu+md)[+]2+md/(mu+md)[-]2} Diagonalizing Lanomaly plus L2 gives the axion parameters: m=1 ; 3=-½(md-mu)/(mu+md) ; 0 =-½

  38. Note that since in the KSVZ modelm=1 the axion mass is given by the formula: ma=mast[vF / fa] 6.3 [106 GeV / fa] eV The calculation of K a in this model is equally straightforward. To the contribution of the superheavy quark in the em anomaly [3eQ2], one must add that coming from the mixing of the axion with the ° and  [ 3 + 5/3 0 ] This gives, finally, K a= 3eQ2 – (4md +mu)/3(mu+md)

  39. I will not go through the analogous calculation for the DFSZ model, but just quote the results It proves convenient to define X1=2v22/vF2, X2=2v12/vF2 , where vF= √(v12 + v22) and v1 and v2 are Higgs VEVs, and to rescale fa fa/2Ngto make m≡1, so that also in the DFSZ model ma=mast[vF / fa] 6.3 [106 GeV / fa] eV One then finds 3= ; 0= (1-Ng)/ 2Ng K a=

  40. Although KSVZ and DFSZ axions are very light, very weakly coupled and very long-lived, they are not totally invisible Astrophysics gives bounds on ma since axion emission, through e ae and Primakoff processes causes energy loss ~1/ fa affecting stellar evolution. Other upper bounds on ma come from SN1987a, since axion emission through NNNNa in core collapse affects neutrino spectrum. Typically bounds allow axions lighter than ma ≤ 1-10-3 eV

  41. Remarkably, cosmology gives a lower bound on axion mass (upper bound on fa ) [Preskill Wise Wilczek; Abbott Sikivie; Dine Fischler] Physics is simple to understand. When Universe goes through PQ phase transition at T~ fa >>ΛQCD anomaly ineffective and <aphys> is arbitrary. Eventually, when Universe cools to T~ΛQCD the axion gets a mass and <aphys> 0. Coherent pa=0 axion oscillations towards minimum contribute to Universe’s energy density and act as cold dark matter. WMAP data provides bound on CDM and axions: Ωah2≤ 0.12

  42. Quote result of a recent calculation of axion contribution to Universe’s energy density by Fox Pierce Thomas: Ωah2=0.5[(fa/)/1012 GeV]7/6[i2 +2] Here  is coefficient of PQ anomaly, Iis initial misallignment angle and  its mean square fluctuation and  is a possible dilution factor For =1, and using for i an average angle i2= <2>= 2/3 and neglecting fluctuations, WMAP data gives the following cosmological bound for the PQ scale: fa/ < 3 x 1011 GeV or 2.1 x 10-5 eV < ma

  43. Concluding Remarks • After more than 25 years, preferred solution to strong CP problem remains having a U(1)PQ symmetry in the theory and its concomitant axions • Although Fermi scale axions have been ruled out, invisible axions models are still viable and axion oscillations could account for the dark matter in the Universe • No totally compelling invisible axion models exist, but it is encouraging that experimentalists are actively searching for axions

  44. Appendix: -vacua and chirality • The gauge matrices n can be obtained by compounding: n =[1]n. It follows thus that on an n-vacuum state 1|n>=|n+1> • Hence n-vacua are not gauge invariant, but the -vacuum is; 1|>=  e-in 1 |n>=  e-in |n +1>= ei | > • In a theory with N massless quarks there is a conserved but gauge variant chiral current Jc5 = J5 -g2N/322 K

  45. The associated time independent chiral charge Qc5 = d3x Jc5o, as a result, shifts under gauge transformations which change the n-vacua 1 Qc51-1=Qc5 + N Consider 1e i/NQc5|> = 1e i/NQc5 1-11 |> = e i( + ) e i/NQc5|> which shows that a chiral rotation changes the -vacuum [Jackiw Rebbi] e i/NQc5|> = | +  >

More Related