JHEP10(2007)066 hep102007066.pdf October 9, 2007 October 17, 2007 enesis, in some September 10, 2007 Accepted: Published: Received: rly included the amount of in the lepton number current. http://jhep.sissa.it/archive/papers/jhep102007066 /j [email protected] , Published by Institute of Physics Publishing for SISSA Cosmology of Theories beyond the SM, Baryogenesis. We examine recent claims for a considerable amount of leptog [email protected] Theory Group, Department of Physics,1 University University of Station, Texas, C1608,E-mail: Austin, TX 78712, U.S.A. SISSA 2007 c ° inflationary scenarios, through theWe find gravitational that anomaly when thelepton number short generated distances is contributions actually are much prope smaller. Keywords: Willy Fischler and Sonia Paban Leptogenesis from pseudo-scalar driven inflation Abstract: JHEP10(2007)066 7 1 4 3 8 L ) = µ (1.2) (1.1) L J ( µ ∂ d 7 -J) starts with the represents the SU(2) L e ory beyond the Standard metry can be converted to of leptogenesis mechanisms ] stated the three conditions and Sheikh-Jabbari [4, 5] for . eview see [3]. The aim of this ¢ is anomaly equation as i L ˜ e R µ 0 R γ L 2 i L π 3 ¯ e gJ 16 − − i √ = L x l ¢ 3 µ lepton doublets and µ d γ L – 1 – i L L Z l ¯ g J ¡ ≡ − L i √ X ¡ Q µ = and ∂ µ γδ L J νλ R The associated lepton charge 1 αβνλ represents the SU(2) R L l αβγδ ǫ 1 2 = ˜ R . R ˜ R The mechanism of Alexander, Peskin and Sheikh-Jabbari( APS R Notice that our expression disagrees with APS-J who write th 2 1 π 3 16 charged lepton singlets. In this expression where Model. Many candidates have beennecessary examined since to Sakharov [1 generate thishave asymmetry. been In explored addition, since a itbaryon number symmetry was through shown sphaleron [2] processes, thatshort for the paper a lepton is recent asym to r examinea a new recent leptogenesis claim of mechanism. Alexander, Peskin gravitational anomaly of the lepton number current [6]: 1. Introduction Explaining the observed baryon asymmetry is a must of any the Contents 1. Introduction 2. Finite temperature quantum field3. theory One-loop effective action 4. Lepton number density generated by5. the inflaton Conclusions backgroun A. Propagator in de Sitter space JHEP10(2007)066 r (1.5) (1.7) (1.6) (1.4) (1.3) nded ed. Notice e loops. The string t-off, APS-J chose it to ¶ this mass the effect will ert-Einstein action of the 3 lained above, the leptonic k s, it is possible to define a ¶ 3 3 er, through spahleron transi- ) er remains anomalous. Other ¶ k s. This new current, however, k nflationary era, and hence the 3 π ˜ R 3 d ) , since pseudoscalars are copious k (2 R he neutrino mass terms. APS-J’s 3 ) is a function of the inflaton field π P . The net lepton number density t d ) ˜ ( R . P t is the Hubble constant at that time (2 Z M ( F 6 φ M )). Other effects associated with this F H Z P t 1 x R P ( 6 3 + . This computation assumed M φ d M P ˜ 1 R> ( 2 µ N πG 1 M π F Z 8 g R P √ µ 16 2 − 2 π P – 2 – 3 √ = ) = t 16 ˙ dt < R ( )= φHM P e πG 3 1 φ , which in turn will generate a non-vanishing lepton F t 1 = i ( M a ǫHM t 16 2 F Z L 2 N ˜ µ √ 2 R> π dt 4 2 π dQ 3 x N π 4 = 4 16 3 < R d is the slow roll inflation parameter, experimentally of orde = 2 = Z ˜ ) R> P n 2 ˙ φ < R HM is the low energy lepton current and does not include right ha ( 2 1 µ are respectively the times at which inflation started and end L = e J t ǫ , 0 L and J i t is the number of stringy degrees of freedom propagating in th ) is the FRW scale factor during inflation, = is the reduced Planck mass, t ( ), n P N a 2 − M The current This result is very sensitive to the momentum ultraviolet cu ), assumed to be a pseudo-scalar, t (10 ( density through the anomaly equation. In their work and where where neutrinos. At energiesnew above current where that does these not particlesis suffer not get from interesting a gravitational for anomalie mas thetions, since purpose the of associated generating chargecomputation baryon is gives numb explicitly broken by t φ O particular deformation have been considered in [7 – 9]. The key observation ofform: APS-J is that a deformation of the Hilb where scale has been assumed to be of the same order of will generate a non-vanishing generated through this effect obtained by APS-J is varies with time as that the integral over timeresult is is picked at not the sensitive beginning to of the the time i at which inflation ends. be the massdisappear of since the the right-handed currentcurrent neutrino, ceases that to arguing is be converted that through anomalous.computations sphalerons above [10] to As have baryon chosen we numb the exp scale to be closer to JHEP10(2007)066 . (2.1) πnT = 2 , at finite n λ ω ( matter piece), ( vacuum piece) and . 2 mat / vac 1 ergence found is the 2 UV is the product of this ) Λ 2 n d the UV cut-off scale. 2 le was resolved in that ensional operators will m ll explain how a similar λT + e computation of the free ver all possible frequencies dure with some additional effect it will also invalidate 2 s divergent term in the free endent Π p ure one loop renormalization 2 t section. effects, is reminiscent of finite tion procedure. This puzzling interaction at a large but finite tribution, Π 2 n m (1.7) for hort wavelengths only probe the ω = (¯ 4 1 + = 0 counter-terms. In particular, 1 + ω theory, to first order in 2 − 1 λφ 1 p 2 T 4 that represents long distance effects. ω βω + ¯ e 3 λφ 2 4 H ) p 1 ω p 3 π is being computed and should not sensitive d 3 4 ) ) (2 p p ), with a 4 3 π π mat d d Z ˜ (2 (2 M R> – 3 – + Π n Z Z X < R λ λ vac λT . (mass = = 12 = 12 φ ˜ Π = Π R> vac mat Π = 12 Π Π < R , generates at two loops a term of the form T > M A computation of the two point function in a This term involves a mixing between the temperature scale an How can we then make physical sense out of this result? The div [11] , gives: UV divergent contribution and the horizon scale We believe that this resultmixing, reflects between long an distance improper effects renormaliza temperature and the Quantum short Field distance Theory.issue first In appeared the in nextcontext. finite section, Readers temperature familiar we with QFT wi this and issue how can the skip puzz to the2. nex Finite temperature quantum field theory For pedagogical reasons weenergy for will a briefly massive scalar review field this issue. Th at this order inof a the perturbative scalar expansion, field’s theenergy. mass zero squared temperat In , whatdetails. insures follows the we absence briefly of recall thi this standard proce which displays the announced mixing. In this formula This expression is yetnaturally to be separates properly into one renormalized. term The that sum is o temperaturethat is indep finite and temperature dependent. temperature the use ofcontribute perturbation significantly theory to since more and more higher dim in string theory. Although, a higher cut-off will enhance the and divergent and one term containing the Bose-Einstein dis to the global features of the space-time. Yet the expression The UV dependence of this term is taken care of by T result of the ultraviolet behavior oflocal the geometry field around modes. the These point s where JHEP10(2007)066 le ) (3.1) F ν ) as an ∂ t ( F F µ ) , ∂ ( µν · · · g ! 2 + P 2 µν ) ˆ R M F r important, because interest. , as well as derivatives 5 ree energy leads to an ¤ , is hort distance behavior a ( µν λ short distance effects are g 2 2 + We will treat ˆ P R 1 ropagators in two different µν 2 ounds we do expect different M on purposes, in appendix A ˆ g ot be true of the finite terms and ˆ a background ˆ t but void of UV divergences, and finite temperature effects. cern ourselves about this issue. P . 1 7 2 a 1 F M P ˆ R 2 + − M 1 a 2 4 ) βω . a e F )+ Ã µν ν 1 ω F h ∂ + 3 ν ) p ∂ 0, at first order in µν + 3 π µν ˆ g d µν ˆ µν R (2 ˆ g F g – 4 – T > µ µν F Z ∂ ˆ = ˆ µ R ( λ ∂ = 0 mass counter-term is added to the Lagrangian. 2 2 ( µν P P 1 g will be dimensionless. When divergent, the value for 1 1 T , we will compute the one-loop effective action for the a i is the covariant derivative. It is possible that some of M M = 12 a + 6 3 µ a ˜ a R> ˆ ren R D ) is a pseudo-scalar, the one-loop effective action compatib + + t Π ( ( ˆ g < R F − , and p F x ν 4 ∂ d µ D ) + constant Z t ( 2 µν g P M = ˆ → F ) F t ] = ( are background independent [12]. This latter point is rathe ¤ F diffeomorphism invariance i [ To regulate this divergence the Inserting this information in a two loop computation of the f a • • F W acting on them. The coefficients the the renormalization procedure isbackgrounds based to on it. have a Oninsensitive common physical to gr ultra global violet features behaviorwhich of since include the the effects space time.we from have long This included will wavelengths. thebackgrounds: n For explicit flat illustrati expressions (A.4) for and the de graviton Sitter p (A.6). Their dominant s lagrangian given in (1.4) using the background field method. external source and expand the gravitational action around these terms are related by field redefinitions but we won’t con The omitted terms will include higher powers of the fields where Given the symmetries of the inflaton-gravity coupling: expression for the freeit energy doesn’t that contain is the temperature mixingWe dependen between do short expect distance an effects analogous mechanism to be at3. work in One-loop the effective case action of In order to get a value for together with the fact that with these symmetries will be of the form: The complete renormalized self-energy at JHEP10(2007)066 2 (3.3) (3.4) , are: 2 H hallenge is . 2 ontributions that in- nner identical to the ) . mentum cut-off does . ) P , in the de Sitter case, isentangle this peculiar µν 2 ose the reader can think proportional to F ) art to deviate from each ν ext of inflation and pions R tion that we wish to carry , however it obscures the ) higher loop corrections to ∂ F F original terms in the action er case will be proportional . ure (in the case for pions). H/M ∂ e deSitter space. 2 F avitational background, and will be a contribution to the ( izing what the origin for this ut-off dependence multiplied µν ¯ x ν ˆ g for gger. This effect is reflected in field theory computation. The n the flat background, strictly d ∂ . This procedure is general. F Ht F µ f the system at hand will dictate . 2 µ e ∂ ∂ ( ) (3.2) ( + 6 6 P F 2 6 P ν H 4 UV M dt ∂ M Λ − µν = ˆ g 2 F µ ds ∂ – 5 – ( ) and 2 R F ) H 6 P ν F ∂ M ν 4 UV ∂ µν Λ ˆ g µν ˆ F g µ other than H, with M larger than H. The answer to that F ∂ µ ( P ∂ 4 ( 4 H 6 P /M 6 P 2 M UV 2 UV M X Λ Λ = -model for pions at finite temperature. The only additional c σ M , the action will have only one term proportional to ( µν η The action (1.4) is non-renormalizable and the loop computa There are in general several terms that will contribute. The Dimensional regularization is well suited for that purpose The one-loop corrections to the action will, both in the cont The challenge as we explained earlier in section 2, is how to d There will then be left over contributions that in the de Sitt These terms are taken care by counter-terms and for this purp The quadratically divergent term will be taken care of in a ma The astute reader might wonder whether there could be finite c = We are assuming a de Sitter metric of the form 2 µν ˆ out here is only consistentprocedure if it is is similar understoodthe to as an non-linear the effective one followed when one calculates where H is the Hubbleterm constant. is, The when resolution considering liesthen the in evaluating recogn effective it action in the in specific a geometry general at gr hand,that in leads, our when cas evaluated in de Sitter, to the kinetic terms regulating the theory while respecting general covariance mechanism by which theviolate general power covariance, law nevertheless divergences thethe cancel symmetries structure o of out. the counter-terms. A mo mixing of scales. Foraction example, among in others the in case the for form: de Sitter there at finite temperature, includeby terms the that Hubble involve constant power (in law the c case for de Sitter) or temperat to of the effective action for gravitons in flat space. preceding discussion, which then leaves over the finite term volves a scale this term will be multiplied by an infinite series in powers of other at distances of thethe order of expression the for de the Sitter horizon effective and action. bi For example, while i agrees by design. We would expect these two propagators to st g JHEP10(2007)066 . 3 ǫ 2 will − (3.5) (3.8) (3.6) (3.7) ). In F ), in a . t = 4 ( ting pro- F F ) i tensor. d ( F is the only n 2 · · · F − 8 + ∂ 2 nd leaves over an n ¶ . ) R t the field s discussed above is ( µν 2 m dt F ¶ ential during inflation. R he biggest contribution grangian of the form: d ) ) t n that there exists a new F ( µ 3 uadratic in the field 3 ¤ F led by adding the following f pions at finite temperature 2 dt involving a mass scale other ntribution of the interaction ν 3 um of the field R ment )) d ∂ 4 raction will not produce terms s to the action in general and F F µ α wever the discussion about the ( ¤ 2 R + µ ¤ 2 2 ∂ ( . α ( ¶ ] 6 ) ) P ¶ + t t F 1 2 [ ( ( 2 2 c M F F ¶ dt − δ 2 ) δW d t R ( 4 2 µ = c F dt 2 – 6 – 4 in particular. By fiat, such contributions to the + d R ˜ R> R 3 ǫ 1 µ F ) α 1 is related to the dimension of space time as 1 F c ǫ + α < R ¤ − ν ( µ ∂ 4 = µν P 1 ˆ g L stands not only for the scalar curvature but also for the Ricc M F ) ∆ R t ¤ ( µ 3 ∂ ( x a 6 P 4 1 d M Z at one loop ranging from 1 to 3. Following a similar power coun . π n f ] = F . The same procedure takes care of the power law divergences a [ the first finite contribution will be schematically of the for π and 4 W , with T 2 Likewise, the vertex (1.4) will contribute to terms in the la The ultimate goal is to determine which of these terms makes t The part of the renormalized effective action that depends on In dimensional regularization, the first non-vaninshing co The absence of the power law cut-off dependence in the action a In addition, the left over skeptic should consider the case o is the renormalized coupling and There are four derivative on each vertex The notation is schematic, F 4 3 R n 2 agreement with the discussion above,of for the example, form this (3.3) inte but rather cedure R c scale in the problem.counter term The to infinite the contribution lagrangian can be cancel mass scale X betweenIntegrating the out Planck this scale scalemodify and substantially will the the then height potential generate of for contribution the pot concern is no. Ex absurdo, if this was the case, this would mea to: contain eight derivatives, since in the UV region the moment Friedman-Roberston-Walker background, can be written as: effective action for the pionsthan that does not contain any terms automatically realized in dimensionalabsence of regularization. a new Ho scale is still necessary toterm complete (1.4) the to argu the term in the one loop effective action that is q T,T ǫ n ( 4 2 3 1) α a √ ¶ µ ) − · 4 ratio will then be: 2 ( α · π d N – 7 – dt dt < R 100 4 n 5184 e 4 / n n/s t (16 µ 1 i − P − t 1 4 4 ) ¶ Z ! d 2 M ∗ dt 1 ǫ 2 g π − 2 1 N π = 3 3 11 =0 √ 10 X 3 n (16 16 µ Ã 4 ˜ R> P 1 ´ ) = ) = 4 M e 1 n < R α t ( ) 3). The observed baryon density ratio is = ³ t . The predicted n ( = 2254 2 √ 3 / 25 P 3 a s n ˜ ) − R> M P ∆( ( 10 / < R × Λ HM ( √ 56 4 . / 1 ∗ represents the number of e-foldings. This result should be c = g = 1 e 3 . H , (1.7) is: N s n 0 L J To compare this result with the one obtained by APS-J we need t = 2 s = With these asumptions: WMAP [14] has put an upper bound on the scale of inflation to be Λ of the terms in the expression above will give the biggest con dominant contribution will be where be of the form (1.6), and assuming a slow roll approximation, 5. Conclusions A deformation of the Einstein-Hilbert actionscalar of driven inflation, the the form total (1. leptonthe number but observed in net an baryon amo number,favorable even limits when compatible the with free experimental parameter bounds. entropy density at the endthe of reheating reheating. temperature Assuming is that given rehea by the equality where In this and thetime remaining metric computations of we constant should curvature: restrict Minkowski th or de Sitter. F 4. Lepton number density generated by the inflaton backgroun The different dependence inthe the different scale definitions factor ofn between the this current. exp The contribution to is JHEP10(2007)066 ¢ (A.2) (A.1) αβ η µν η ) (A.4) indicates the − αβ να η αβ ¯ δ . δ µν ) (A.3) µβ η ¾ ¯ δ ound the Minkowski λ ) (A.5) αβ − η + Sitter space: ful discussions and the µκ αβ ∂x να µν η ν νβ η η ¯ ∂h δ letion of this work. This e Foundation under Grant µν ∂x µβ η − µα nsion η ng [13] is given by, ¯ δ + ved with the constraint that it − ¡ να j + ] η µ να iǫ dx νβ µβ η i λκ ∂x η η + ν . µβ dx ∂h 2 µα + η ) ij ∂x Ht ′ η µν δ ( + − 2 νβ h − e ) η iǫ t x,x νβ + κ 1 ( ( H η µα a − λ 3). The expressions that will follow are more η µν 2 µν ∂x 2 ( and a bar over the tensor = µα + √ g ) λ η ′ 1 – 8 – H 2 ∂h 2 P ( iǫ ) ) = ˆ 1 ∂x ′ ln[ Ha M ¯ 1 iǫ − . x dt x,x ( ) ) ( ( µν ν − 2 ′ / ′ − ) g = σ k − ′ η − η κ Λ 2 ( +), is ′ ( x 2 η 2 ) , x a a ′ √ i = π λν P ) ∂x ) ηη − + 4 1 2 η µ η , = M ( ( 2 x + (¯ ∂h x,x 2 ( a a ds + P ∂x ( 2 1 µ , H 2 ) λ ) ′ ½ ′ M 1 − π 2 η x )= 4 2 1 i π k 2 4 − − ( = ( )= = P 2 ′ 1 x η P ( ( µν 1 M η is the reduced Planck mass. µν,αβ µν M x,x − λµνκ ( η P = . As always R ≡ − πG ≡ 1 8 µν ) = Ht 2 2 ′ g √ µν,αβ ) e ) ′ ′ D = is the background metric. In this weak field approximation x,x )= ( x,x x,x P t ( ( ( µν λ σ M g a dS µν,αβ In the Harmonic gauge, the form of the graviton propagator ar We will assume the Friedman-Robertson-Walker metric for de In coordinate space it can be written as: D background , ˆ Also in the Harmonic gauge, the graviton propagator followi suppression of the zero components. This propagator is deri easily derived in conformal time. where where where Aspen Center formaterial Physics is for based its uponNo. hospitality work PHY-0455649. supported during by the the comp National Scienc A. Propagator in de Sitter space In our notation perturbation theory is derived from the expa Acknowledgments Paban thanks Arkady Vainshtein and Richard Woodard for help where where ˆ JHEP10(2007)066 gh 0. than (A.6) → Annal. D 68 H , B 174 (1984) 269. y of the . Sov. Phys. Usp. ]. ]. B 234 Phys. Rev. (2007) 377 Phys. Lett. , , . ) must be proportional 170 . The effects of ]. αβ ) . (1967) 24] [ ′ η (A.3) in the limit 5 µν 0. More precisely, the part of hep-ph/0701139 η x,x wski, Nucl. Phys. ( , Gravi-leptogenesis: leptogenesis Leptogenesis from gravity waves in , λ − , → , ′ hep-th/0501153 ) hep-th/0403069 να 1 ηη η 4 /H µβ arXiv:0708.0001 JETP Lett. (1 η , hep-ph/0703087 + µ/ , astro-ph/9812088 Astrophys. J. Suppl. =1+ (1992) 269. , Cosmological signature of new parity-violating νβ (2005) 016 [ 2 , Cambridge Univ. Press, Cambridge U.K. (1989). Leptogenesis and tensor polarisation from a Wilkinson Microwave Anisotropy Probe (WMAP) η ¶ – 9 – 03 µα (2006) 081301 [ (1967) 32 [ 2 η µH B 292 5 (1991) 61]. 96 µ Gravitational anomalies JHEP One loop divergencies in the theory of gravitation (1999) 1506 [ Mode analysis and ward identities for perturbative quantum , 161 Baryogenesis without grand unification ]. 83 (1974) 69. ) = cos ′ Phys. Lett. , is related to the geodesic distance between two points throu A20 Chern-Simons modification of general relativity x,x lectures on leptogenesis ( z z ]. Phys. Rev. Lett. 0. The other terms must either be finite or diverge more slowly , 2006 → Violation of CP invariance, c asymmetry and baryon asymmetr Usp. Fiz. Nauk gr-qc/0308071 Finite temperature field theory Tasi Phys. Rev. Lett. , µH Pisma Zh. Eksp. Teor. Fiz. collaboration, D.N. Spergel et al., as , 1 . The variable − 1 ) (1991) 392] [ z − ) − astro-ph/0603449 z [ three year results: implications for cosmology Poincar´ePhys. Theor. gravity in de Sitter space (2003) 104012 [ interactions WMAP from gravity waves in pseudo-scalar driven inflation models gravitational Chern-Simons term (1986) 45. Chern-Simons gravity on bodies orbiting the earth 34 models of inflation universe − [8] R. Jackiw and S.Y. Pi, [6] L. Alvarez-Gaume and E. Witten, [9] T.L. Smith, A.L. Erickcek, R.R. Caldwell and M. Kamionko [7] A. Lue, L.-M. Wang and M. Kamionkowski, [3] M.-C. Chen, [5] S.H.-S. Alexander, M.E. Peskin and M.M. Sheikh-Jabbari [4] S.H.-S. Alexander, M.E. Peskin and M.M. Sheikh-Jabbari [2] M. Fukugita and T. Yanagida, [1] A.D. Sakharov, [11] J.I. Kapusta, [13] N.C. Tsamis and R.P. Woodard, [12] G. ’t Hooft and M.J.G. Veltman, [14] [10] D.H. Lyth, C. Quimbay and Y. Rodriguez, the propagator that is proportional to ( (1 to (1 References match the flat space propagator in the limit of The de Sitter propagator (A.6) reduces to the flat propagator the relation: