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Axions - Theory

Axions - Theory. Roberto Peccei UCLA Helen Quinn Symposium SLAC April 16, 2010. Axions - Theory. Recollections The U(1) A Problem of QCD The QCD Vacuum The Strong CP Problem Solutions to the Strong CP Problem U(1) PQ and Axions Axion Dynamics Invisible Axion Models

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Axions - Theory

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  1. Axions - Theory Roberto Peccei UCLA Helen Quinn Symposium SLAC April 16, 2010

  2. Axions - Theory • Recollections • The U(1)A Problem of QCD • The QCD Vacuum • The Strong CP Problem • Solutions to the Strong CP Problem • U(1)PQ and Axions • Axion Dynamics • Invisible Axion Models • Galactic Hints for Axions • Concluding Remarks

  3. Recollections • The 1976-77 academic year was a special year at Stanford, enlivened by the presence of Steve Weinberg in the Department and Gerard ‘t Hooft at SLAC • For me, the highlight was the presence of Helen Quinn, as a visitor from Harvard to the Institute of Theoretical Physics in the Department • We worked happily together on the then hot topic of instantons, whose consequences we labored hard to understand

  4. In this period we wrote 3 papers together at the end of January, March, and May 1977 (Two of which are well known, and a third which should be!) Our first joint paper [Some Aspects of Instantons, Nuovo Cimento 31A, 307 (1977) ] is how we really learned about instantons -how these Euclidean solutions impact gauge theories in Minkowski space - the bearing that they have on the validity of perturbation theory - their relation to the violation of quantum numbers like B+L, broken by anomalies

  5. Our two better known papers [CP Conservation in the Presence of Pseudoparticles, Phys. Rev. Lett. 38, 1440 (1977); Constraints Imposed by CP Conservation in the Presence of Pseudoparticles, Phys. Rev. D16, 1791 (1977)] introduce a global U(1) symmetry to explain why CP is conserved in the strong interactions Steve Weinberg’s persistence in asking questions was a great motivator for the U(1)PQ solution Remarkably, neither Helen nor I, or Steve (or the referees of our papers) realized that the U(1)PQ solution implied the existence of a nearly massless particle, the axion!

  6. Still remember clearly the sinking feeling I got in late summer 1977 when Helen told me that Weinberg (by then in Texas) called her to tell us that there was a Goldstone Boson associated to our “solution”. Shortly thereafter joint papers by Weinberg and Wilczek appeared in which they discussed how U(1)PQleads to axions and how the axions get a small mass, giving also some suggestions on how to search for such a light, very long-lived particle In this talk in honor of Helen I will discuss the theoretical basis for axions and argue that they indeed must exist!

  7. 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

  8. 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 form and breakU(2)Adown spontaneously and no mixed parity multiplets

  9. However, because U(2)A is a 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

  10. In the language of Chiral Perturbation Theory, the QCD dynamics for the (, )- sector needs to be augmented by an additional term which breaks explicitly U(1)A, beyond the breaking term induced by the quark mass terms. Defining  = exp i/F [aa +] and including a symmetry breaking pion mass m2 ~ (mu+ md) one has: Leff = -¼F2 Tr  † + ¼F2 m2 Tr ( + † ) - ½M2o 2 Provided M2o >>m2this allows m2>> m2 , but what is the origin of this last term?

  11. The QCD Vacuum • 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 is CP not very badly broken in QCD ?

  12. 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

  13. 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!

  14. 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!

  15. 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

  16. 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

  17. 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 • The 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= - = g2/322d3r K0 |t=+ t= -

  18. 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>-

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

  20. Because one cannot neglect the UA(1) anomaly in  ≠ 0 sectors, the topological charge density Q = ;  =  d4x Q in effect, acts as a dynamical parameter. Thus [Di Vecchia Veneziano] Q should also be added to the Chiral Lagrangian describing the low energy behavior of QCD: Leff = ¼F2 Tr  † + ¼F2 m2 Tr ( + † ) ½ i QTr [ln -ln † ] + [1/ F2M2o] Q2 +…

  21. The 3rd term in Leff is included to take into account the anomaly in the UA(1) current, while the 4th term is the lowest order term in a polynomial in Q2 Q is essentially a background field and can be eliminated through its equation of motion: Q = -i/4 [F2M2o] Tr [ln -ln † ] = ½ [FM2o]  +... Hence the last two terms in Leff reduce, effectively, to: ½ i QTr [ln -ln † ] + [1/ F2M2o] Q2 →-½M2o2 +which serve to provide an additional gluonic mass term for the  meson, solving the UA(1) problem

  22. This emerges more directly from lattice QCD calculations, which show that, indeed, m → constant as m → 0

  23. The Strong CP Problem • The resolution of the U(1)A problem, by recognizing the complicated nature of QCD’s vacuum, however, engenders another problem: the strong CP problem • As we saw, effectively, the QCD vacuum structure addsand extra term to LQCD • This term violatesP and T, but conserves C, and thus can produce a neutron electric dipole moment of order dn e mq/Mn2 

  24. The strong bound on the neutron electric dipole momentdn<1.1 x 10-26 ecm requires the angle to be very small  < 10-9 -10-10 [Baluni; Crewther Di Vecchia Veneziano Witten] • Why  should be this small is the strong CP problem • Problem is actually worse if one considers the effect of chiral transformations on the -vacuum • One can show that chiral transformations, because of the anomaly, change the -vacuum [Jackiw Rebbi ]: eiQ5 |  > = |  +  >

  25. The proof is a bit involved but worth going thru • The gauge matrices n associated with the n vacua can be obtained by compounding: n =[1]n, where 1|1>=|2>.It follows thus that on an n-vacuum state 1|n>=|n+1> • Hence n-vacua, as expected, 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

  26. As a result, the associated time independent chiral charge Qc5 = d3x Jc5oshifts 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 indeed changes the -vacuum [Jackiw Rebbi] e i/NQc5|> = | +  >

  27. If besides QCD one includes the weak interactions, in general the quark mass matrix is non-diagonal and complex LMass = -qiR Mij qjL + h. c. To diagonalize M one must, among other things, perform a chiral transformation by an angle of Arg det M which, as a result of the Jackiw Rebbi result, changes  into total =  + Arg det M • Thus, in full generality, the strong CP problem can be stated as: Why is the angle total, coming from the strong and weak interactions, so small? [Weinberg’s persistent question to Helen and me!]

  28. Solutions to the Strong CP Problem • Thirty years later, there are only three possible “solutions”to the strong CP problem: • Anthropically totalis small • CP is broken spontaneously and the induced totalis small • A chiral symmetry drives total→ 0 • In my opinion, only iii. is a viable solution and it necessitates introducing in the Standard Model a new global, spontaneously broken, chiral symmetry U(1)PQ

  29. i. Anthropic solution • It is, of course, possible that, as a result of some anthropic reasons total =  + Arg det M just turns out to be of O(10-10). There are, after all, such small ratios in the SM [e.g. me/mt~10-6] • What make me doubt this “explanation” is that the physics of the QCD vacuum (and hence ) and that of the quark mass matrix (Arg det M) seem totally unrelated. So, why should totalbe a small CP violating phase?

  30. iiSpontaneously broken CP solution • This second possibility is more interesting. In fact, if CP were a symmetry of nature which is then spontaneously broken, one can set =0 at the Lagrangian level. • However, since CP must be spontaneously broken, gets induced back at the loop-level. [Beg Tsao; Georgi; Mohapatra Senjanovic ] • But to get  < 10-9 one needs, in general, also to insure that 1-loop=0. This is not easy and, furthermore, there are other problems associated with spontaneously broken CP violation

  31. Theories with spontaneously broken CP need complex Higgs VEVsand, as Zeldovich, Kobzarev and Okun pointed out, these give rise to different CP domains in the Universe • These different CP domains in the Universe are separated by walls which have substantial energy density and, being 2-dimensional, dissipate slowly as the Universe cools wall ~ T • Indeed,if these domains existed, the energy density in the walls would badlyoverclose the Universe now, unless the scale of spontaneous CP violation was greater than Tinflation~ 1010 GeV

  32. Theories where CP is violated at high scales and which have 1-loop=0 exist [Nelson, Barr], but they are recondite and difficult to reconcile with experiment • Typically, the CP violating phases generated at high scales induce small phases at low energy: eff ~ [Mw/MHS]n  and this is not what one sees experimentally. • All experimental data is in excellent agreement with the CKM Model– a model where CP is explicitly, not spontaneously, broken and where the CP violating phasesare of O(1)

  33. iii. A chiral symmetry drives total→ 0 • This is a very natural solution to the strong CP problem since it, effectively, rotates  -vacua away e-iQ5 |  > = | 0 > • Two suggestions has been put forth for this chiral symmetry: • The u-quark has no mass mu = 0 [Kaplan Manohar] • The SM has an additional global U(1) chiral symmetry [Peccei Quinn ] • Want to argue that only PQ solution is tenable

  34. The “ solution” mu = 0 is disfavored by current algebra analysis [ Leutwyler]. Furthermore, it is difficult to understand why Arg det M = 0 What is the origin of this chiral symmetry? • The ratio of quark masses is computable in Chiral Perturbation Theory. Correcting for electromagnetic effects, one arrives at leading order to the famous Weinberg formula

  35. MILC Collaboration rules out mu=0 at 10 This result is confirmed by calculations on the Lattice Theoretical predictions summarized in the Figure Leutwyler mu0

  36. 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

  37. 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

  38. 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 Much simpler proof than in original PQ paper!

  39. 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 the NGB a mass] • Calculation of axion mass first done explicitly by current algebra techniques [Bardeen Tye], but an effective Lagrangian derivation [Bardeen Peccei Yanagida] is easier and also readily gives axion couplings to matter

  40. 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)

  41. 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

  42. 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: =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 introduce, as before, 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∂ † ∂ 

  43. 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 m )2Tr[ΣAM+(ΣAM)†] where and under PQ-transformations

  44. 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

  45. 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

  46. 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]

  47. 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 and one finds K a=Ng(x +1/x)[mu/(mu+md)]

  48. 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

  49. 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

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