FFTPI-MINN-09/32, UMN-TH-2812/09
Non-Abelian Strings and Axions M. Shifman
William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455
Abstract.
Axion-like fields can have a strong impact on non-Abelian strings. I discuss axionconnection to such strings and its implications in two cases: (i) axion localized on the strings, and(ii) axions propagating in the four-dimensional bulk.
Keywords:
Axion, non-Abelian string, topological solitons.
PACS:
INTRODUCTION
Axions were introduced by Weinberg [1] and Wilczek [2] in a bid to save naturalness of P and T parity conservation in QCD. Shortly after, the axion construction evolved into“an invisible axion” of the first [3, 4] or the second [5, 6] kind. Moreover, already in theearly days of string theory people realized that axion-like particles are an unavoidablefeature of string theory and are abundant. Since then, axions acquired a life of theirown (Fig. 1), a big part of which is due to Pierre’s unconditional commitment to thisfundamental topic which goes as far as participation in experimental axion searchesstarting from the early stages of experiment design! I think it is fair to say that he shapedresearch in this area. Pierre is the true and ultimate Axionman (Fig.2). Happy birthday,Pierre, and many new findings in your exciting career!In the last 10 years I was only marginally connected with the development of ideasin the area of the axion phenomenology. The last active effort in this direction was areview paper [7], written with Gregory Gabadadze, in which we explored consequencesof a newly acquired knowledge of nonperturbative aspects of the QCD vacuum in axionphysics. Needless to say, such great occasion as Pierre’s birthday calls for presentation offresh results. My current work is focused on non-Abelian strings, a construction whichemerged recently [8, 9, 10, 11] (for a detailed review see [12]). These strings couldplay the role of the cosmic strings [13], which would be a very appropriate topic today,but, unfortunately, this idea is not yet fully implemented in viable phenomenologicalconstructions. Therefore, I will talk today about a kind of axions which serve as atheoretical laboratory in the explorations of flux tubes (strings) and other topologicalsolitons, rather than the “real” axions which are likely to be a part of our world (thefavorite object of Pierre). Most of the results to be reported today were obtained withGorsky and Yung [14]. Talk at the Workshop
Axions 2010 , in honor of the 60 th birthday of Pierre Sikivie, University of Florida,January 15–17, 2010. a r X i v : . [ h e p - t h ] M a r IGURE 1.
Axion research on industrial basis.
BRIEFLY ON NON-ABELIAN STRINGS
The Abrikosov flux tube (string) in the Abelian U(1) gauge theory is known from the1950’s [15]. What is the difference between our good old acquaintance, the Abrikosovstring, and the new arrival, non-Abelian string? Of course, the non-Abelian strings areusually found in non-Abelian gauge theories, but this is not their main defining feature.Of most importance is the fact that additional moduli – (classically massless) fieldsdescribing internal degrees of freedom – exist on the string world sheet. The mostpopular example refers to orientational moduli localized on the string [8, 9, 10, 11].If the bulk theory has the U ( N ) gauge symmetry and SU ( N ) flavor symmetry, with theappropriate choice of the Higgs sector [17], the bulk theory is fully Higgsed, while stillpreserving a color-flavor locked global SU ( N ) symmetry. The latter is broken down toSU ( N − ) × U ( ) on any given string solution. As a result, moduli living in a coset spaceSU ( N ) / ( SU ( N − ) × U ( )) emerge. Their interaction is described by two-dimensionalCP( N − ) model (Fig. 3). Relativistic generalization was given in [16].
IGURE 2.
Pierre Sikivie, the Axionman.
The simplest example is provided by the following (nonsupersymmetric) model: S = (cid:90) d x (cid:26) g (cid:16) F a µν (cid:17) + g (cid:0) F µν (cid:1) + Tr ( ∇ µ Φ ) † ( ∇ µ Φ ) + g (cid:104) Tr (cid:16) Φ † T a Φ (cid:17)(cid:105) + g (cid:104) Tr (cid:16) Φ † Φ (cid:17) − ξ (cid:105) + i θ π F a µν ˜ F a µν (cid:27) , (1)where the gauge group is assumed to be U(2), F a µν and F µν are the SU(2) and U(1)gauge field tensors, with the coupling constants g and g , respectively, ξ is a constantof dimension m triggering the Higgsing of the theory, θ is the vacuum angle, and,finally, the field Φ is a 2 × Φ = { ϕ kA } where k is the SU( N ) gauge index while A is the flavor index, k , A = ,
2. On the worldsheet we get the CP(1) model L + = ( + ¯ φ φ ) (cid:18) g ∂ α ¯ φ ∂ α φ + θ π i ε αβ ∂ α ¯ φ ∂ β φ (cid:19) (2)plus a free field theory for translational moduli. Non-Abelian strings and bulk four-dimensional theories which support them are discussed in detail in the book [12] towhich I refer the interested reader. In this talk I will focus on applications involvingaxion-like particles. IGURE 3.
Orientational moduli on the string world sheet are depicted by arrows.
AXION ON THE STRING
In the first part I will consider models with “axion” localized on the string, and theimpact of such axion.
The simplest model
In the simplest scenario axions (a massless or nearly massless field defined on thecircle) is the only modulus on the string (except the translational moduli, of course).The simplest model of this type is obtained from Witten’s superconducting string model[18] by its reduction. Namely, we will downgrade one of two U(1)’s of the Witten modelto a global symmetry, rather than local, L = − g F µν F µν + | D µ φ | − λ φ (cid:16) φ − v φ (cid:17) + | ∂ µ χ | − λ χ (cid:16) χ − v χ (cid:17) − β φ χ . (3)This model is a crossbreed between those used in [19, 20]. If the constants λ φ , χ and β are appropriately chosen, the field φ condenses in the vacuum, Higgsing the gauge U(1)symmetry and, simultaneously, stabilizing the field χ . Then in the vacuum (cid:104) χ (cid:105) vac = χ phase rotations remainsunbroken. The theory (3) obviously supports a string which is almost the Abrikosovstring. There is an important distinction, however. In the string core φ =
0, and the β φ χ term stabilizing χ is switched off. Having χ = χ (cid:54) = ∈ U ( ) – an axion – is localized on the string. The world-sheet theory becomes S = (cid:90) dt dz (cid:8) T (cid:2) ( ∂ µ x ) + ( ∂ µ y ) (cid:3) + f ( ∂ µ α ) (cid:9) (4) FIGURE 4.
A torus made of the bent Abrikosov flux tube. The dashed line along the large period of thetorus is the line of constant α (say, α = where T is the string tension, f is a (dimensionless) axion constant which can beexpressed in terms of the bulk parameters, t is time, z is the coordinate along the stringwhile x and y are perpendicular coordinates. They can be combined as x ⊥ = { x , x } ,where x ⊥ depends on t and z , x ⊥ = x ⊥ ( t , z ) . Moreover, α ( t , z ) is the phase field on the world sheet, α ↔ α ± π ↔ α ± π ... . Inother words, the target space of α is the unit circle.Now, let us take a long Abrikosov string and and bend it into a circle of circumference L , see Fig 4 (I assume L (cid:29) (cid:96) where (cid:96) is the string thickness). If α is constant along z (say, α = L becomes of the order of (cid:96) , and then the string will annihilate. However,one can stabilize it by forcing α to wind along z in such a way as to make the full 2 π winding when z changes from 0 to L , α ( t , z ) = π z / L . (5)Note that α linearly depending on z goes through the equation of motion on the worldsheet, ∂ α =
0. It is not difficult to estimate the value of L . Indeed, the string energy is E = T L + ( π f ) L (6)Minimizing (6) with respect to L we get L = π f / √ T . (7)Making f large enough we can always force L to be much larger than the flux tubethickness (cid:96) which is roughly speaking of the order of 1 / √ T . Note that for k windings L = π f k / √ T .The soliton of the type discussed above was first constructed in [19] where it goesunder a special name “vorton” (in the context of cosmic strings; for a recent review anda rather extended list of references see [21]). Its classical stability is due to a nontrivialHopf topological number [22]. In the limit L (cid:29) (cid:96) the Hopf soliton is also stable withregards to the quantum tunneling annihilation.A similar Hopf soliton, albeit with a richer internal structure, was obtained in [23] inthe framework of N = agnetic Field Electric Field Q torus
FIGURE 5.
Twisted torus constructed in [23]. This Hopf soliton is also topologically stable.
Axion-induced deconfinement of kinks in two dimensions
Now I will pass to the axion impact [14] on “genuinely” non-Abelian strings, withthe orientational moduli on the world sheet described by the CP ( N − ) model. In thegauged formulation the CP ( N − ) model can be written as L = g (cid:104) ( ∂ α + iA α ) n ∗ (cid:96) ( ∂ α − iA α ) n (cid:96) − λ (cid:16) n ∗ (cid:96) n (cid:96) − (cid:17)(cid:105) , (8)where n (cid:96) is an N -component complex filed, (cid:96) = , , ..., N , subject to the constraint n ∗ (cid:96) n (cid:96) = . (9)This constraint is implemented by the Lagrange multiplier λ in Eq. (8). The field A α inthis Lagrangian is also auxiliary, it enters with no derivatives and can be eliminated byvirtue of the equations of motion, A α = − i n ∗ (cid:96) ↔ ∂ α n (cid:96) . (10)Substituting Eq. (10) in the Lagrangian, we rewrite it in the form L = g (cid:104) ∂ α n ∗ (cid:96) ∂ α n (cid:96) + ( n ∗ (cid:96) ∂ α n (cid:96) ) − λ (cid:16) n ∗ (cid:96) n (cid:96) − (cid:17)(cid:105) . (11)The coupling constant g is asymptotically free, and defines a dynamical scale Λ of thetheory by virtue of the dimensional transmutation, Λ = M exp (cid:18) − π Ng (cid:19) , (12)where M uv is the ultraviolet cut-off and g is the bare coupling.At first, let us forget for a while about the axion and outline the solution of the“axionless” CP ( N − ) model (8) at large N [24]. To the leading order it is determined byne loop and can be summarized as follows: the constraint (9) is dynamically eliminatedso that all N fields n (cid:96) become independent degrees of freedom with the mass term Λ .The photon field A µ acquires a kinetic term L γ kin = − e F µν , e = π Λ N , (13)and also becomes “dynamical.” The quotation marks here are used because in twodimensions the kinetic term (13) does not propagate any physical degrees of freedom; itseffect reduces to an instantaneous Coulomb interaction. This is best seen in the A = ( ∂ z A ) while the interactionis A α J α = A J , J α = n ∗ (cid:96) ↔ ∂ α n (cid:96) . (14)Since A enters in the Lagrangian without time derivative, it can be eliminated by virtueof the equation of motion leading to the instantaneous Coulomb interaction J ∂ − z J . (15)In two dimensions the Coulomb interaction is proportional to | z | , implying linear con-finement acting between the n , ¯ n “quarks” [24]. Only n ¯ n pairs are free to move alongthe string.The axion part of the Lagrangian can be written as follows: L a = f a ( ∂ µ a ) + a π ε αγ ∂ α A γ , (16)where A γ is defined in Eq. (10), and f a is a (dimensionless) axion constant. I willcontinue to assume that f a (cid:29) − F µν + e π f a a ε αγ ∂ α A γ + ( ∂ µ a ) + e A α J α . (17)The expression for e is given in (13). The axion field represents a single degree offreedom. The role of the “photon" is that upon diagonalization we get a massive spin-zero particle, with mass of the order of f − a Λ N − / . Indeed, taking account of thephoton-axion mixing amounts to summing the infinite series of tree graphs, e J J (cid:40) p + p (cid:18) e π f a (cid:19) p µ + ... (cid:41) = − e J α J α p µ − (cid:16) e π f a (cid:17) , (18)where p is the spatial component of the momentum transfer p µ , and I used Eqs. (15) and(17), and the current conservation. The “ex-photon” mass is determined by the positionof the pole in (18).As a result, the long distance force responsible for confinement disappears , givingplace to deconfinement at distances (cid:29) m − a .he axion-induced liberation of the n fields at distances (cid:29) m − a demonstrated aboveis a two-dimensional counterpart of domain-wall deconfinement in four-dimensions [25,7]. The parallel becomes even more pronounced in the (string-inspired) formalism whichascends to [26] (in connection with walls it was developed in [25] and discussed in [27]in another context). In this formalism one introduces an (auxiliary) antisymmetric three-form gauge field C αβ γ , while the four-dimensional axion is replaced by an antisymmetrictwo-form field B µν (the Kalb–Ramond field). In four dimensions the gauge three-form field has no propagating degrees of freedom while the Kalb–Ramond field B µν presents a single degree of freedom. The domain walls are the sources for C αβ γ , muchin the same way as the kinks are the sources for A in two dimensions. The fieldstrength four-form built from C αβ γ is constant (cf. F in two dimensions). The C αβ γ B µν mixing produces one massive physical degree of freedom, a four-dimensional massiveaxion. Simultaneously, the domain-wall confinement is eliminated at distances (cid:29) m − a .Everything is parallel to the two-dimensional CP ( N − ) world-sheet theory. FOUR-DIMENSIONAL AXION AND NON-ABELIAN STRINGS
Now, I address a different problem: a non-Abelian string soliton coupled with a four-dimensional axion existing in the bulk. We introduce a four-dimensional axion in thebulk theory which supports non-Abelian strings; confined monopoles are seen as kinksin the world-sheet theory (CP ( N − ) ). What’s the impact of this four-dimensional axionon dynamics of strings/confined monopoles? The bulk model with non-Abelian strings
The appropriate bulk theory (nonsupersymmetric) is given in Eq. (1) where T a standsfor the generator of the gauge SU(2) group, ∇ µ Φ ≡ (cid:18) ∂ µ − i √ N A µ − iA a µ T a (cid:19) Φ , (19)and θ is the vacuum angle, to be promoted to the axion field, θ → θ + a → a ( x ) . (20)The last term forces Φ to develop a vacuum expectation value (VEV) while the last butone term forces the VEV to be diagonal, Φ vac = (cid:112) ξ diag { , } . (21)This VEV results in the spontaneous breaking of both gauge and flavor SU(2)’s. Adiagonal global SU(2) survives, however, namelyU ( ) gauge × SU ( ) flavor → SU ( ) diag . (22)The vacuum is color-flavor locked.ne can combine the Z center of SU(2) to get a topologically stable string solution[8, 9] possessing both windings, in SU(2) and U(1) since π ( SU ( N ) × U ( ) / Z N ) (cid:54) = . Their tension is 1 / ( N − ) . The four-dimensional axioninteraction is added in (8) through the term − θ + a π ε nk ∂ n n ∗ ∂ k n , (23)where θ coincides with the four-dimensional θ while a ( t ,(cid:126) x ) is the four-dimensionalpseudoscalar field propagating in the bulk. Monopole-antimonopole “mesons" vs. axion clouds
What happens with the monopole-antimonopole meson on the non-Abelian stringin the presence of the four-dimensional axion? Given the discussion above one mightsuspect that the four-dimensional axion induces deconfinement of monopoles localizedon the non-Abelian string, much in the same way as the two-dimensional axion. Now Iwill argue that this does not happen.The classical action of the four dimensional bulk axion field is L a = (cid:90) d x (cid:20) f a ( ∂ a ) + ia π F a µν ˜ F a µν (cid:21) , (24)where in the case at hand f a has dimension of mass. The axion has a small massgenerated by four-dimensional bulk instantons m a ∼ Λ f a (cid:32) Λ (cid:112) ξ (cid:33) b − , f a (cid:29) Λ , (25)where b is the first coefficient of the β function in the theory (1), but this mass plays norole in what follows.The impact of the bulk axion on the non-Abelian string is two-fold. First, the axiongets coupled to the translational moduli of the string. Assuming that the string collectivecoordinates adiabatically depend on the world-sheet coordinates we get for this coupling L ( ) a ∼ ξ (cid:90) d x a ( x ) ε i j ε αβ ∂ i x α ⊥ ∂ j x β ⊥ δ ( ) ( x − x string ( t , z )) , (26)where the indices i , j = , α , β = , B µν ( x ) =0 k=0 (cid:47) a=0k= (cid:239) FIGURE 6.
The monopole-antimonopole meson together with the axion cloud. The region of the stringbetween the monopole and antimonopole is not exited because the value of the axion field is nonvanishinginside the axion cloud, a = π . but this is not necessary.) The coupling (26) is not specific for non-Abelian strings, it isgenerated in the case of the Abrikosov strings as well.Now, let us discus orientational moduli. It is easy to see that no mixed n - x ⊥ termsappear in the axion Lagrangian (at least, in the the quadratic order in derivatives). Thebulk axion generates a quadratic in n coupling, as is clearly seen from Eq. (23). Theimpact of this term in the axion Lagrangian can be summarized as follows: L ( ) a ∼ (cid:90) d x a ( x ) ε nk ∂ n n ∗ ∂ k n δ ( ) ( x − x string ( t , z )) , (27)For a short while forget about axions and consider the monopole-antimonopole pairattached to the string. The energy of this monopole-antimonopole meson is of order of ( Λ / N ) L , where L is the distance between the monopole and antimonopole along thestring. What happens upon switching on the four-dimensional axion field?Logically speaking, the axion field could develop a nonvanishing expectation value a = π on the string between the monopole and antimonopole positions, equalizingthe string energies inside and outside the pair, and screening the confinement force.This is exactly what happened with the two-dimensional axion. To see whether ornot a similar effect occurs with four-dimensional axions we have to examine a fieldconfiguration in which (cid:104) a (cid:105) = (cid:104) a (cid:105) = π , which would let (anti)monopolesmove freely along the string, with no confinement along the string.It is not difficult to estimate the energy of the axion cloud. The transverse size ofthe cloud (in two directions perpendicular to the string) must be of order of m − a . Thelongitudinal dimension is L , see Fig. 6. Assuming that L (cid:29) m − a we get E cloud ∼ f a L , (28)to be compared to the energy ( Λ / N ) L of the monopole-antimonopole meson. Since f a is supposed to be very large compared to Λ we see that the energy of the axioncloud (28) is much larger than the monopole-antimonopole meson energy. Developing acompensating axion cloud is energetically disfavored. Therefore we conclude that thereis no monopole deconfinement driven by four-dimensional axions. osmic non-Abelian string and axion emission Hashimoto and Tong suggested [13] to consider non-Abelian strings as cosmic stringcandidates. It is worth discussing possible signatures of such non-Abelian strings inthe context of axion physics. Obviously, both, translational and orientational modescan be excited in collisions. In the latter case one can think of production of energeticmonopole-antimonopole pairs attached to the string and bound in mesons by the confin-ing potential along the string, as I described above. On the part of the string between themonopole and antimonopole (the kink and antikink) the state of the string is describedby a quasivacuum with k = −
1. In this state (cid:104) ε nk ∂ n n ∗ ∂ k n (cid:105) ∼ Λ / N . (29)The topological charge density is localized in the domain of the excited part of thestring, and is approximately constant in this domain. Therefore, as is clear from Eq. (27),this interval, whose length L oscillates in accordance with the monopole-antimonopolemotion, will serve as a source term in the equation for the axion field. Assume thatthe energy of the kink-antikink pair E (cid:29) Λ so that they can be treated quasiclassically.The distance L between the kink and antikink will oscillate between − L and L where L ∼ E / Λ with the frequency ω ∼ Λ / E , L ( t ) = L e i ω t . (30)Therefore, for a distant observer the monopole-antimonopole meson is seen as a point-like source with the interaction term Λ (cid:90) d x a ( x ) L ( t ) δ ( r − r ) , (31)where r is a position of the meson on the string. The intensity of the axion radiationfrom this point-like source can be estimated as I a ∼ ω Λ L f a r ∼ ω E f a r , (32)where r is the distance to the observer.Of course the string produces axion radiation also due to coupling with translationalmodes, Eq. (26). This radiation is seen as coming from a linear source, and can beestimated (per unit length) as I a ∼ ξ / f a E (cid:96) ρ . (33)Here ρ is the distance from the string to the observer in the plane orthogonal to the string, E is the total excitation energy and (cid:96) is the length of the excited part of the string. Thisradiation is not specific for non-Abelian strings. Abelian strings produce this radiationas well.We see that the non-Abelian string is seen by a distant observer as a linear source ofthe axion radiation (33), with additional point-like sources of the axion radiation (32)ocated on the linear source at the positions of the monopole-antimonopole mesons. Therate of the axion radiation depends of f a . The oscillating kink-antikink pair will shakeoff energy until exhaustion. The time duration of the monopole-antimonopole mesonde-excitation can be estimated as T ∼ E f a . CONCLUSIONS
The existence of axion-like particles is almost unavoidable in the framework of stringtheory. The impact of axions on various field-theoretic strings (flux tubes) is multi-faceted. Two-dimensional axions can stabilize toric Hopf solitons, liberate kinks, con-fined in the absence of axion, and do other equally remarkable jobs. Introducing abulk axion in the “benchmark" nonsupersymmetric model [17] supporting non-Abelianstrings we observe that the four-dimensional axion does not lead to monopole decon-finement. In the context of cosmic strings, the axion emission due to excitations of non-Abelian strings occurs in a different way compared to that from Abelian string. Energeticpairs of confined and oscillating (anti)monopoles act as an additional pointlike sourcespecific to non-Abelian strings.In coclusion, I would like to mention that a new study of toric Hopf solitons stabilizedby axion-like fields is under way now [28].
ACKNOWLEDGMENTS
I am grateful to S. Bolognesi, A. Gorsky and A. Yung for useful discussions andcollaboration. I would like to thank A. Feldshteyn for providing Fig. 1. This work wassupported in part by DOE grant DE-FG02-94ER408.
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