Anomalous coupling of scalars to gauge fields
Philippe Brax, Clare Burrage, Anne-Christine Davis, David Seery, Amanda Weltman
aa r X i v : . [ h e p - t h ] J un DESY 10-167
Anomalous coupling of scalars to gauge fields
Philippe Brax
Institut de Physique Th´eorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/Yvette Cedex, France ∗ Clare Burrage
D´epartment de Physique Th´eorique, Universit´e de Gen`eve,24 Quai E. Ansermet, CH-1211, Gen`eve, Switzerland andTheory Group, Deutsches Elektronen-Synchrotron DESY, D-22603, Hamburg, Germany † Anne-Christine Davis
Department of Applied Mathematics and Theoretical Physics,Centre for Mathematical Sciences, Cambridge CB3 0WA, United Kingdom ‡ David Seery
Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, United Kingdom § Amanda Weltman
Astronomy, Cosmology and Gravity Centre, Department of Mathematics and Applied Mathematics,University of Cape Town, Private Bag, Rondebosch, South Africa, 7700 ¶ We study the transformation properties of a scalar–tensor theory, coupled to fermions, under theWeyl rescaling associated with a transition from the Jordan to the Einstein frame. We give a simplederivation of the corresponding modification to the gauge couplings. After changing frames, thisgives rise to a direct coupling between the scalar and the gauge fields.
I. INTRODUCTION
Axions have a coupling to gauge fields resulting fromthe existence of a chiral anomaly. Similarly, string ax-ions must couple to gauge fields in order to cancel gaugeanomalies using a four-dimensional version of the Green–Schwarz mechanism [1]. These gauge interactions of ax-ions and axion-like particles leads to a rich phenomenol-ogy. They give rise to important effects in astrophysics,causing stars to cool at an accelerated rate [2]. At an-other extreme there are interesting optical phenomena,leading to a prospect of detecting axion-like particleswith cavity-based experiments such as ALPS and Gam-meV [3], or experiments such as CAST and the Tokyoaxion helioscope [4].Scalar fields have attracted considerable attention fol-lowing the discovery, less than fifteen years ago, that theexpansion rate of the universe is accelerating [5]. It haslong been known that scalar fields can play a significantrole in a phase of early-universe accelerated expansion, or‘inflation.’ The same is true in the late universe. How-ever, if a scalar ‘dark energy’ field is responsible for theacceleration measured today, there is difficulty. To mod-ify the expansion rate on cosmological scales, the massof the field should be as low as the Hubble rate today, ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] of order 10 − eV. But this would lead to the existenceof a ‘fifth force,’ violating the weak equivalence princi-ple. Fortunately, mechanisms making the small mass ofthe scalar field phenomenologically acceptable on largescales—and preventing the appearance of a fifth force inthe solar system and in the laboratory—have been dis-covered.In certain dark energy models, acceleration is due tothe rolling of a scalar field over a runaway potential [6–9].Such potentials have a long history in attempts to explainthe dark energy scale [10]. Two methods have been in-voked to eliminate the unwanted scalar fifth forces whichaccompany them: (a) either the chameleon mechanism [11] or the Vainshtein effect [12], in which the scalar fieldis screened by massive bodies due to non-linear effects; or(b) the
Damour–Polyakov mechanism [7], where the cou-pling of the scalar field to matter is constructed to vanishdynamically. Similar effects can be achieved in models ofmodified gravity. Indeed, under certain circumstances,modified gravities such as the Dvali–Gabadadze–Porattimodel [13], f ( R ) theories [14], or the Galileon model [15]reduce to the inclusion of a new, scalar degree of free-dom, whose fifth force is either screened by a chameleonmechanism [16] or the Vainshtein effect. In all these the-ories the dynamics of the scalar field are determined bya scalar–tensor theory.Does the scalar field φ couple to gauge fields such asthe photon? If so, the rich optical phenomenology ofaxion-like particles could be extended to models of darkenergy or modified gravity [17–19]. Since φ is a gauge-singlet, this coupling must involve an interaction with thegauge field kinetic term of the form f ( φ ) F ab F ab , where F ab = ∂ [ a A b ] is the field strength associated with thegauge field A a .The possibility of such couplings was investigated byKaplunovsky & Louis [20] in the context of locally su-persymmetric effective quantum field theories. AfterWeyl rescaling, Kaplunovsky & Louis found that a Wess–Zumino term was generated, associated with a super-Weyl anomaly, which necessarily induced an f ( φ ) F ab F ab interaction. The function f ( φ ) was determined entirelyat one-loop level. Kaplunovsky & Louis used these effectsto derive exact threshold corrections [21] to the gaugecouplings from integrating out species above the super-symmetry breaking scale. The study of such thresholdcorrections in string theories has generated a large liter-ature.Kaplunovsky & Louis’ result was framed in the con-text of supergravity. Doran & J¨ackel later observed thatchanging frame would typically lead to a redefinitionof couplings in any effective field theory [22]. In thisbrief note we give an alternative, simpler derivation ofKaplunovsky & Louis’ result which does not make use ofthe methods of supersymmetry or require the applicationof superfield Feynman rules, and which applies in any lowenergy effective theory. Weyl rescaling. —A range of scalar–tensor theories exist.Writing the theory in terms of a metric ˜ g µν defined sothat the new scalar field does not directly interact withmatter—the ‘Jordan frame’—we focus on theories whichcan be written in the form S J = Z d x p − ˜ g " M B ( φ ) ˜ R −
12 ( ∂φ ) − V ( φ ) + S m ( ψ i , ˜ g µν ) , (1)where φ is the new scalar degree of freedom, B ( φ ) is anarbitrary function, M = (8 πG ) − is the reduced Planckmass, and ˜ R is the Ricci scalar constructed using themetric ˜ g µν . The matter action, S m , involves an arbitrarycollection of fields ψ i coupled to g µν , but not φ . Eq. (1)includes a wide class of dark energy models, but does notinclude infrared modifications of Einstein gravity such asthe ‘Galileon’ [15].In this frame, φ is coupled non-minimally to gravityvia B ( φ ) ˜ R . In the ‘Jordan frame,’ this encapsulateshow Einstein gravity is modified. However, it is possibleto describe the same modification of gravity in differentways, by making field redefinitions which do not changethe physics. The Jordan frame action is classically equiv-alent to an ‘Einstein frame’ theory, written in terms of aredefined metric g µν , which satisfies g µν = B − ( φ )˜ g µν . (2)This rescaling is an example of a Weyl transformation. Inthe Einstein frame, the scalar field is minimally coupledto g µν but couples directly to the matter fields ψ i . Thereis no principle which can tell us whether the Einsteinframe or Jordan frame is more fundamental. Which we pick is simply a matter of obtaining the most convenientdescription for the problem at hand. Field-dependent gauge couplings. —It is well-known thatthe Maxwell kinetic term F ab F ab is invariant under Weylrescalings. Classically, therefore, it follows that if this in-teraction is absent in one frame it is absent in all frames,and its inclusion or otherwise is merely a free choice tobe made in model-building.Kaplunovsky & Louis’ result implies that, after quan-tization, the change of variables associated with shiftingfrom one frame to another naturally induces a couplingbetween φ and the kinetic term of any gauge field, ei-ther Abelian or non-Abelian, which is coupled to fermionspecies. Therefore, quantization and change of frame donot commute. From the standpoint of an effective fieldtheory it is unnatural to take the interaction to be absent:this corresponds to picking one choice of frame as ‘fun-damental,’ and all others as ‘derived.’ There is no justifi-cation for such a choice. Instead, the coupling should beincluded and its magnitude constrained by experiment.In § II we make a quantitative estimate of the effect,by explicit calculation of the Jacobian associated withchange of variables in the path integral. This can beaccomplished using a method introduced by Fujikawa tocalculate the chiral anomaly [23]. A related calculation ofthe conformal anomaly in a chameleon model with con-formally coupled scalars was given by Nojiri & Odintsov[24]. Similar anomalous Jacobians were encountered byArkani-Hamed & Murayama [25] while studying exact β -functions in supersymmetric gauge theories. Arkani-Hamed & Murayama referred to the appearance of anon-trivial Jacobian as a “rescaling anomaly.” In § III wesummarize the calculation and discuss our conclusions.We work in a spacetime with signature ( − , + , + , +),and choose units so that ~ = c = 1. Throughout, we sup-press spinor indices. The four-dimensional Dirac matri-ces are γ a , and satisfy the algebra [ γ a , γ b ] = 2 η ab . Withour sign conventions, the conjugate of a Dirac spinor is¯ λ = λ † γ . II. GAUGE COUPLINGS FROM WEYLTRANSFORMATIONS
Our starting point is the scalar–tensor action, Eq. (1).A redefinition of couplings arises on changing frame be-cause it is not possible to define the path integral measurein a way which is invariant under transformations of theform (2). In our formalism, this is the analogue of theWeyl rescaling studied by Kaplunovsky & Louis, given inEq. (2.23) of their paper [20].To define the measure, we formally compactify space-time on a large manifold after performing a Wick ro-tation. This is an intermediate step: our conclusionsare independent of the details of the compactification,and continue to apply if we later revert to a noncompactspace. To simplify the presentation, we have kept theMinkowski-convention signature throughout. We writethe vielbein on this compact manifold as e aµ ; Greek let-ters label spacetime indices, whereas Latin letters labelLorentz indices associated with the tangent bundle. Themassless Dirac equation has a discrete eigenvalue spec-trum λ n , where n = 1 , , . . . ,( /e µ D µ ) ψ n = λ n ψ n , (3)where /q = γ a q a for any tangent-space vector q a , and ψ n is a Dirac spinor, obeying suitable boundary conditionsif necessary. This step is formally valid only after Wickrotation to Euclidean signature.The derivative operator D µ includes appropriate gaugeterms for charged fermion species, D µ = ∂ µ − i eA µ + ω µ , (4)where ω µ is the spin connexion, e is a coupling constant,and ω µ = [ γ a , γ b ] ω abµ . We generally neglect this term inwhat follows, since it is inessential for obtention of theanomalous scalar interactions with gauge fields. In anycase, where the compactification manifold is a flat torus,this gives exact results.The ψ n form a complete, orthogonal set of ba-sis functions. We choose a normalization so that R d x √− g ¯ ψ m ψ n = δ nm . In terms of this basis, a genericmassless spinor λ J , or conjugate spinor ¯ λ J , can be rep-resented by a set of coefficients { a n , ¯ b n } , where we havewritten λ J = X n a n ψ n and ¯ λ J = X n ¯ b n ¯ ψ n . (5)The subscript ‘ J ’ denotes that these are Jordan-framefields. It is a familiar idea that the measure [d λ d¯ λ ] forintegration over λ and ¯ λ can be defined by integrationover the coefficients a n and ¯ b n . We define[d λ d¯ λ ] J = Y n M d a n d¯ b n , (6)where M is mass scale needed to make the measure di-mensionally correct, but which plays little role in theanalysis and will not appear in subsequent expressions.Eq. (6) expresses the functional integral in terms of ameasure on the Jordan-frame fields. We are interested indetermining the transformation law connecting (6) withthe same measure expressed in terms of Einstein-framefields. As we will see, this typically contains local di-vergences which can be absorbed in a redefinition of theEinstein frame Lagrangian.On translation to the Einstein frame, the matter actionacquires interactions with φ . In particular, the fermionkinetic term is not conformally invariant and mixes with φ . To obtain canonically normalized Einstein-frame fieldswe make the change of variable λ E = B / λ J and ¯ λ E = B / ¯ λ J , (7)where we have used that B is real. The S-matrix is invari-ant under field redefinitions, so this transformation does not change the physical content of the theory. Written interms of these fields, the Einstein frame Lagrangian foreach species of fermion satisfies L E ⊇ − ¯ λ E ( /e µ D µ ) λ E + h.c , (8)where ‘h.c.’ denotes the Hermitian conjugate of the pre-ceding term.This field redefinition is associated with a Jacobian, orchange of measure, represented by a fermionic determi-nant. To calculate it, we represent the Einstein-framespinor fields in terms of the basis functions ψ n and ¯ ψ n with coefficients c n and ¯ d n . Using (6), the change ofvariables is [d λ d¯ λ ] J = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( a, ¯ b ) ∂ ( c, ¯ d ) (cid:12)(cid:12)(cid:12)(cid:12) [d λ d¯ λ ] E . (9)where the Jacobian determinant satisfies (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( a, ¯ b ) ∂ ( c, ¯ d ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) det ∂a k ∂c ℓ det ∂ ¯ b m ∂ ¯ d n (cid:19) − = exp tr ln Z d x √− g B / ( ¯ ψ m ψ n + ¯ ψ n ψ m ) , (10)where the trace is over the basis indices m and n .The completeness relation for ψ n and ¯ ψ n readstr ¯ ψ m ( x ) ψ n ( y ) = δ ( x − y ), so Eq. (10) involves the di-vergent quantity δ (0) and it is clear that this trace willrequire regularization.In general, Eq. (10) represents a complicated redefini-tion of each local operator in the Lagrangian. However, ina mean field approximation where φ = φ + δφ , its valuecan be calculated perturbatively in δφ . This approxima-tion is reasonable in the physical situations of interest foroptical phenomena associated with coupling of the scalarto gauge fields, including those in astrophysics and lab-oratory experiments. Working to first order in δφ anddiscarding an infinite δφ -independent prefactor, we find (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( a, ¯ b ) ∂ ( c, ¯ d ) (cid:12)(cid:12)(cid:12)(cid:12) ∝ exp tr 3 α Z d x √− g δφ ( ¯ ψ m ψ n + ¯ ψ n ψ m ) , (11)where we have defined a coefficient α , satisfying α = d ln B d φ (cid:12)(cid:12)(cid:12)(cid:12) φ , (12)which has dimension [mass] − . In what follows it willsometimes be useful to identify this mass scale explicitly,writing M α = α − .Eq. (11) can be evaluated using a standard regulariza-tion method introduced by Fujikawa [23]. We write thetrace over basis indices in (11) as δ ( x − x ), and make thesubstitution δ ( x − x ) → Tr exp (cid:16) /D x µ (cid:17) Z d k (2 π ) e i k · ( x − y ) (cid:12)(cid:12)(cid:12) y → x . (13)The trace is over spinor indices, and the exponential isto be interpreted in a matrix sense. In this expressionwe have specialized to flat spacetime, so that /D = γ a D a where D a is the flat gauge-covariant derivative. This isthe only case which has yet been required in phenomenol-ogy, although the metric coupling could be reintroducedusing /e µ D µ if desired. The subscript ‘ x ’ on /D indicatesthat the differential operator applies only to the explicit x -dependence, and it is understood that the limit y → x is to be taken after allowing /D x to act on terms to itsright. For finite µ , Eq. (13) softly suppresses the contri-bution of ultraviolet modes with k ≫ µ , while maintain-ing gauge-invariance: this is the key virtue of Fujikawa’smethod. At the end of the calculation one removes theregulator by taking µ → ∞ .The operator /D can be written /D = D − i e γ a , γ b ] F ab . (14)We write the Jacobian as a perturbative shift of the ac-tion, δS , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( a, ¯ b ) ∂ ( c, ¯ d ) (cid:12)(cid:12)(cid:12)(cid:12) = e i δS . (15)Making use of the elementary identity e − i k · x f ( ∂ a )e i k · x = f ( ∂ a + i k a ), rescaling the momentum integration so that k a → k a /µ , and expanding the exponential, we findi δS = − αµ Z d x δφ Z d k (2 π ) e − k × (cid:18) − e µ Tr[ γ a , γ b ][ γ c , γ d ] F ab F cd + · · · (cid:19) , (16)where ‘ · · · ’ denotes operators of lower engineering dimen-sion which we have not written explicitly, but which van-ish on taking the trace, together with terms suppressedby more powers of 1 /µ which will not contribute in thelimit µ → ∞ .The first term in round brackets ( · · · ) in Eq. (16) isindependent of F ab . It is an infinite renormalization ofthe Einstein-frame cosmological constant. This is just arestatement of the usual cosmological constant problem.Therefore, discarding this term is harmless. In a moregeneral theory, we would find infinite renormalizationsof each relevant local operator. The remaining term isfinite. It is a new contribution which we must absorbin the Einstein frame action. To evaluate it we Wickrotate to Euclidean signature, making the substitution k → − i k E , where q is the timelike component of thefour-vector k a , k E is its Euclidean counterpart, and thesign is fixed by the requirement that the Euclidean actionbecomes a Boltzmann weight of the correct sign. Carry-ing out the trace over Dirac indices and assembling allnumerical coefficients, we find δS ⊇ e α π Z d x δφ F ab F ab . (17) In the language of Kaplunovsky & Louis, this is the renor-malization of the Wilsonian gauge coupling [20].A contribution of this form arises for each fermionspecies. In a theory with N f species of fermion, the to-tal shift in the Einstein frame action can be written asa new, local dimension-five operator coupling δφ and thegauge-kinetic term, δ L E ⊇ δφM F ab F ab , (18)where the mass-scale M satisfies M = 16 π e N f M α . (19)Although we have given the calculation only for a mass-less fermion and Abelian gauge field, it applies for a non-Abelian field after straightforward modifications. Massterms for the fermions were discussed by Arkani-Hamed& Murayama [25], and can be incorporated perturba-tively in the present framework. Contributions of order δφ and higher could be retained, if desired, by retain-ing higher-order terms in the Taylor expansion of B inEq. (11) and subsequent expressions. III. DISCUSSION
Eqs. (18)–(19) represent a purely quantum effect as-sociated with canonical normalization of the chargedfermion species. Before changing variables, there is nointeraction involving φ and the gauge field alone; theonly interaction arises from the fermion kinetic term L ⊇ B ¯ λ ( /e µ D µ ) λ + h.c. After expanding around themean field φ , this gives rise to φA ¯ λλ vertices involvingan arbitrary number of φ quanta, a single gauge field,and a fermion/antifermion pair.Changing variables to canonically normalized fields in-troduces a new dimension-five operator, Eq. (18), whichyields a contact interaction between φ and two quantaof the gauge field. This eliminates the φA ¯ λλ vertex. Ifthe fermion has a mass term, however, then conformalinvariance is broken and a φ ¯ λλ interaction persists evenafter canonical normalization. By computing triangle di-agrams, one can show that a contact interaction of theform (18) is introduced at energies sufficiently small thatthe fermion λ cannot be resolved. This contribution en-ters with the same sign as (18), and its associated massscale is [26] M ′ = 48 π e N > f M α , (20)where N > f indicates the number of charged fermionspecies whose mass thresholds have been passed. Theseheavy fermions do not appear in an effective theory validat energies below the threshold; Eq. (20) summarizestheir residual influence. We note that, in principle, themass scale M α which appears in (19) and (20) can bespecies-dependent but in a minimal theory we can ex-pect approximately the same scale for each species.Kaplunovsky & Louis gave an interpretation of the φ -dependent gauge coupling as a one-loop effect, in whichEq. (18) arises from diagrams with λ -quanta circulatingin the interior of the loops. However, we emphasize thatEq. (18) must be absorbed into the Einstein-frame actionwhether or not the massive fermion species are integratedout. Therefore, although these two effects are relatedthey are not the same. The first is a consequence ofthe non-invariance of the fermionic path integral measureunder a Weyl rescaling. The second is due to the pathintegral over fermions whose masses explicitly break Weylinvariance.What are the consequences for an effective theory de-scribing the interactions of φ with gauge fields at lowenergies? Such a theory governs the relevance of φ foroptical interactions in astrophysics or the laboratory.Eqs. (18), (19) and (20) show that when studying theEinstein-frame phenomenology of a theory with N f totalfermion species, at energies below the mass threshold of N > f of these, we should augment whatever bare contactterm of the form (18) exists by a shift of 1 /M ′′ , wherethe mass scale M ′′ is determined by M ′′ = 16 π e M α N f + N > f → π e N f M α , (21)and the limit on the right-hand side corresponds to verylow energies, where all fermion species are integratedout. If the bare contribution corresponds to a very largemass scale, this shift will dominate the resulting coupling.Typically one would expect a coupling to be generated atleast by loops of virtual gravitons, although presumably strongly suppressed. For a large number of species, wefind that the coupling scale to matter M m can be muchlarger than M ′′ . IV. CONCLUSION
We have shown that, in a theory where a scalar field φ couples to fermionic matter charged under a gauge sym-metry, a coupling between φ and the gauge fields is au-tomatically generated via the noninvariance of the pathintegral measure under rescaling of the fermion fields.This opens up a rich phenomenology previously associ-ated with axion physics. Since scalar fields couple to thegauge field strength and can be subject to the chameleonmechanism, physical predictions will differ between mod-els, as discussed in Refs. [17–19]. ACKNOWLEDGMENTS
CB is supported by the German Science Foundation(DFG) under the Collaborative Research Centre (SFB)676 and by the SNF. ACD was supported in part bySTFC and the Perimeter Institute of Theoretical Physics.DS was supported by the Science and Technology Fa-cilities Council [grant number ST/F002858/1]. AW ac-knowledges support from NRF, South Africa. AW andDS were supported by the Centre for Theoretical Cosmol-ogy, Cambridge, during the early part of this work. Wewould like to thank Andreas Ringwald for drawing ourattention to Ref. [20] and Mark Goodsell for suggestingthe relevance of Ref. [25]. [1] E. Witten, Phys. Lett.,
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