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Mainzer Markenzeichen Anwendung komplementärer Methoden, Ansätze und Experimente zur Verfolgung gemeinsamer Ziele Concepts and Applications of Effective Field Methodische Vielfalt Theories: Flavor Physics and Beyond Breiter Bereich der physikalischen Skalen; „von den niedrigsten zu den höchsten Energien“ Matthias Neubert Mainz Institute for Theoretical Physics (MITP) Johannes Gutenberg University, Mainz, Germany TRIGA Ionenfallen MAMI LHC IceCube ! BCVSPIN-MSPF-MITCHELL Joint School 2014 0 4 14 -18 -13 10 10 10 10 10 Las Hadas Golf Resort & Marina ultrakalte Colima, HadronenHiggs, SUSY, Antiprotonen Manzanillo, Mexico, 8-13 December 2014 Neutrinos Neutronen 01.02.2010 struktur Extra-Dimensionen Energie [GeV] Effective Field Theory Effective field theories are a very powerful tool in quantum field theories: • systematic formalism for the analysis of multi-scale problems • simplifies practical calculations, often makes them feasible (“Taylor expansion of Feynman graphs”) • particularly important in QCD, where short-distance effects are calculable perturbatively, but long-distance effects are not • provides new perspective on renormalization • basis of factorization (i.e. scale separation) and resummation of large logarithmic terms Effective Field Theory Useful reviews: • E. Witten, Nucl. Phys. B 122 (1977) 109 • S. Weinberg, Phys. Lett. B 91 (1980) 51 • L. Hall, Nucl. Phys. B 178 (1981) 75 • J. Polchinsky, hep-th/9210046 • A. Buras, hep-ph/9806471 • M. Neubert, hep-ph/0512222 Effective Field Theory Lecture I: • General concepts of EFTs • Scale separation, integrating out high-energy modes, low-energy effective Lagrangian • Modern view of QFTs and general principles Lecture II: • Applications • The Standard Model as an effective field theory • Interesting insights Lecture I: Concepts of Effective Field Theory uch calculations feasible. As we will discuss, EFT also provides a new enormalization”. Derivation of the effective Lagrangian idea of EFT is simply stated: Consider a quantum field theory with cale M. This could be the mass of a heavy particle, or some large (E Consider a QFT with a characteristic (fundamental) high-energy ansfer. Suppose we are interested in physics at energies E (or momenta scale M M. How can we expand scattering or decay amplitudes in powers of E/ We areproceeds interested performing question in in several steps:experiments at energies E ⌧ M < Mfields Step Λ 1: < Choose a cutoff and divide quantum a cutoff M and divide⇤ the of thealltheory intofields low-frequ into high- and low-frequency components ( ! > ⇤ and ! < ⇤ ): uency modes, φ = φL + φH , ! M ⇤ Recall: the Fourier modes with frequency ω < Λ, while φH con L contains Z ⇣ We can think of⌘ the cutoff as a “ g modes with frequency dω3 k> Λ. † ik·x ik·x (x) =that we 3may pretend ak e + know ak e nothing about the t ance” in the sense to (2⇡) 2Ek bove Λ (which is indeed often the case). By construction, low-energ E bed in terms of the φL fields. Everything we ever wish to know a Feynman diagrams, scattering amplitudes, cross sections, decay rates, ed from vacuum correlation functions of these fields. These correlato uch calculations feasible. As we will discuss, EFT also provides a new fundamental scale This could be orthe mass of a heavy particle, or The som maller than M. How canM. we expand scattering decay amplitudes in powers of E/M? enormalization”. nswer this question proceeds in several momentum transfer. Suppose we are interested in physics at energies E( often to makes such calculations feasible. Assteps: we will discuss, EFT also provides a new, modern Derivation of the effective Lagrangian idea of EFT is simply stated: Consider a quantum field theory with meaning to “renormalization”. smaller than M. How can we expand scattering or decay amplitudes p 1. Choose a cutoff Λ < M and divide the fields of the theory into low-frequency in and The ideacould of EFTbe is simply stated:of Consider a quantum a large cale M.main This the mass a heavy particle,fieldortheory somewith large (E high-frequency modes, answer to this question proceeds in several steps: fundamental scalea M. This could be the mass of a heavy particle, or some large (Euclidean) Consider QFT with a characteristic (fundamental) high-energy φ =inφLphysics + φH , at energies E (or momenta (2) ansfer. Suppose we are interested momentum transfer. Suppose we are interested in physics at energies E (or momenta p) much scale M 1. Choose a expand cutoff Λexpand < modes M and ofpowers theory where φL M. contains thewe Fourier with frequency ω fields < Λ, in while φpowers contains the M. How can we scattering ordivide decay amplitudes inthe ofinto E/ smaller than How can scattering or decaythe amplitudes E/M? The H of remaining modes with frequency ω > steps: Λ.experiments We can think at of energies the cutoff as a⌧ “threshold answer to this question proceeds in several high-frequency modes, E M We are interested in performing question proceeds in several steps: of ignorance” in the sense that we may pretend to know nothing about the theory for φof=theφLtheory + φHinto , low-frequency and 1. Choose a cutoff Λ < M and divide the fields scales above Λ (whichaiscutoff indeed ⇤ often the and case).divide By construction, low-energy physics < M Step 1: Choose all quantum fields a cutoff Λ <in M and divide the fields of the theory into low-frequ modes, is high-frequency described terms of the φ fields. Everything we ever wish to know about the L where φ contains the Fourier modes with frequency ω < Λ, wh ! > ⇤ ! < ⇤ L into highand low-frequency components ( and ): φ = φ + φ , (2 L H cross sections, decay rates, etc.) can theory (Feynman diagrams, scattering amplitudes, uency modes, remaining modescorrelation with frequency ωof > Λ.fields. We These can think of the cu be where derived from vacuum functions these correlators can be φ contains the Fourier modes with frequency ω < Λ, while φ contains the L H φ = φ + φ , ! L we H of ignorance” in the sense that may pretend to know nothing obtained using remaining modes with frequency ω > Λ. We can think of the cutoff as a “threshold ! " !E "while ofscales ignorance” in the that is weindeed may pretend to the know nothing about the theory fo above Λ sense (which often case). By construction # the Fourier modes with frequency ω < Λ, φ con ⌧ ⇤ Physics (any Green function) at low energies is entirely L contains H δ δ 1often the case). # scales above Λ (which is indeed By construction, low-energy physics −i . . . −i Z[J ] ,wish (3) ⟨0| T {φ (x ) . . . φ (x )} 0⟩ = # L L 1 L n is described in terms of the φ fields. Everything we ever described in terms of the fields ; Green functions of these g modes with frequency ω > Λ. We can think of the cutoff as a “ L Z[0] δJ (x ) δJ (x ) J =0 L L 1 L n is described in terms of the φL fields. Everything we ever wish to know about the fields can be derived from thepretend generating functional: theory (Feynman diagrams, scattering amplitudes, cross sections, dt ance” in the sense that we may to know nothing about the theory (Feynman diagrams, scattering amplitudes, cross sections, decay rates, etc.) can where $ % D fields.of bebe derived from vacuum correlation functions offunctions these These correlators can be derived vacuum correlation these fields. Thes bove Λ (which is from indeed often the case). By low-energ iS(φL ,φH )+i d construction, x JL (x) φL (x) Z[JL ] = DφL DφH e (4) obtained using bed inobtained terms ofusing the φL fields. !Everything we ever wish to know a % " ! " # action, D D is the generating functional of the theory. Here S(φ , φ ) = d x L(x) is the δ δ 1 L H !cross " Z[J ! ]## rates, Feynman diagrams, scattering amplitudes, sections, decay −i . . . −i (3 ⟨0| T {φ (x ) . . . φ (x )} 0⟩ = L light ,fields, L 1 of space-time, L n δ δ 1 is the dimension andZ[0] we have δJ only included sources J for the δJL (xnL) JL =0 L (x1 ) −i . . . −i ⟨0| T {φtoLcompute (x φL (x )} 0⟩ =functions ed from correlation functions of these correlato 1 ) . . . the n as this vacuum suffices correlation in (3).fields. These % to this question proceeds in several steps: iS(φL ,φH )+i dD x JL (x) φL (x) Z[JL ] =$ here DφL DφH e % D al ofDerivation the theory. Here S(φ the , Lagrangian φ fields ) =of the d theory x L(x) the of effective = Choose a cutoff ΛZ[J <L ]the M and divide intoislow-freq is the generating functional of is the action % igh-frequency modes, time, and we have only included sources for the lig is the dimension of space-time, and we have only included sources for the light fie L D JJ L the generating functional of the theory. Here S(φL , φH ) = d x L(x) is the action φ =functions φL + φHin,do Step 2: Since the high-frequency fields as this suffices to compute the correlation (3).not appear in the % dD x JL (x) φL (x) % D H DφL DφH L e the theory. Here S(φL , φH ) = d x L(x) iS(φL ,φH )+i dimension of space-time, and we have in only(3). included sources JL for the light fiel ethethe correlation functions generating functional, we can “integrate them out” in the path sIn this to compute thethe correlation functions in (3). where φnext the Fourier modes with ω < Λ, fields. whileThis φH yic thesuffices step, we perform path integral overfrequency the high-frequency L contains integral: modes with frequency ω > Λ.over We can think of thefields. cutoff asyie a $ integral %the nemaining the the next step, we perform the path high-frequency This rm path integral over the high-frequency fields. T iSΛ (φL )+i dD x JL (x) φL (x) Z[JL ]that ≡$ we Dφmay ! f ignorance” in the sense pretend to know ,nothing about the Le % D $ iSΛ (φ d x JL (x) φL (x) L )+icase). cales above Λ (which isL ]indeed often the By construction, low-ene Z[J ≡ Dφ e , % L where ! $D x J (x) φ (x) iS (φ )+i d s[Jdescribed in terms of φ fields. Everything we ever wish to know Λthe L L iS(φLL L iS (φ ) ,φ ) Λ L H ] ≡ Dφ e , e = Dφ e L L H here where $ amplitudes, cross sections, decay rate heory (Feynman diagrams, scattering iSΛ (φL ) iS(φL ,φH ) e = Dφ e eis derived vacuum correlation fields. this These correla called thefrom “Wilsonian effective action”.functions NoteHthat,ofbythese construction, action depe ! on the choice of the cutoff Λ used to define the split between low- and high-freque $ btained using called the “Wilsonian effective action”. µ Note that, by construction, this action depen modes. S is non-local on scales ∆x ∼ 1/Λ, because high-frequency fluctuations h Λ (φ ) iS iS(φ ,φ ) iscutoff called the Wilsonian action and high-frequen ! " ! " Λ of L Ltheeffective H nbeen theand choice the Λ used to define split between low# e DφThe e removed from= the theory. process of removing these modes is often referred H µ1 δ δ # modes. S is non-local on scales ∆x ∼ 1/Λ, because high-frequency fluctuations h Λ L (x1 ) out” −i in the functional . . . −i Z[JL ]# T {φ . . . φLthe (xnhigh-frequency )} 0⟩ = as⟨0| “integrating fields integral. Z[0] δJ δJ (x 1 ) condition L the n ) referred een removed from the The process of removing these modes often ⇤ enters Dependence ontheory. the cutoff viaL (x the onis s “integrating out”of thethe high-frequency fields inconstruction, the functional integral. ffective action”. Note by this action frequencies fields that, where $ % ff Λ used to define the split between iS(φ ,φ )+i dlowx J (x)and φ (x) high-f Z[JL ] = DφL DφH e L µ H D L L Derivation of the effective Lagrangian Step 3: Effective action is non-local on the scale t ⇠ 1/! , corresponding to the propagation of high-energy modes that have been removed from the Lagrangian the final step, we expand the non-local action functional in terms of local opera the final step, we expand the non-local action functional in terms of local opera ! < ⇤ Since the remaining fields have energies , the non-local mposed of light fields. This process is called the (Wilsonian) operator-product ex mposed of lightaction fields. can Thisbe process is called the (Wilsonian) operator-product ex effective expanded in an infinite series of local n (OPE). This expansion is possible because E ≪ Λ by assumption. The result n (OPE). This expansion is possible because E ≪ Λ by assumption. The result operators: cast in the form ! cast in the form ! D eff SΛ (φL ) = d Dx LΛeff (x) , ! SΛ (φL ) = d x LΛ (x) , ere where: here " " g Q (φ (x)) . Leff (x) = i i L LΛeff (x) = g Q (φ i i L (x)) . Λ i i is object is called the “effective Lagrangian”. It is an infinite sum over local opera his object is called the “effective Lagrangian”. It is an infinite sum over local opera multiplied by coupling constants g i , which are referred to as Wilson coefficients multiplied by coupling constants gi, which are referred to as Wilson coefficients eral, all by symmetries the theory theory are generated in in constants localof built out of neral, all operators operators allowed allowedcoupling by the the symmetries ofoperators the are generated (Wilson coefficients) fields in their derivatives L and struction of the effective Lagrangian and appear this sum. nstruction of the effective Lagrangian and appear in this sum. Dimensional analysis Does a Lagrangian consisting of an infinite number of interactions and hence an infinite number of (renormalized) coupling constants give us any predictive power? • Not if one adopt an old-fashioned view about renormalization and renormalizable QFTs • But not all is lost... Can use naive dimensional analysis to estimate the size of individual terms in the infinite sum to a given matrix element here of the effective Lagrangian and uction "appear in this sum. eff LΛ (x) = gi Qi (φL (x)) . analysis i ere isDimensional always an infinite number of such operators, the question arises: Ho low-energy theorythebe“effective predictive? This is where the simple, powerful his object is called Lagrangian”. It is an infinite sumbut over local opet by coupling constants gAs are referred to asin Wilson coefficien nsional comes to play. is physics, common practice high-energy p As isanalysis” common practice in particle we adopt units i multiplied i , which −1 generated eneral, allwhere operators by of = the[x−1 theory ~ =h̄c = c1allowed where ,=such thatthe in units 1. Then [m]symmetries = [E] = [p] ] = [tare ] are are all all me onstruction the and appear in thisofsum. measured ineffective the (mass units) units. We of denote by same [gi ] Lagrangian = units −γi the mass dimension the effective coupli at e there is always an infinite number of such operators, the question arises: Ho −γ i Denote by [gi ] = mass dimension of the coupling i the g = C M i i This is where the simple, but powerful t ctive low-energy theory be predictive? constants in the effective Lagrangian dimensional analysis”Ccomes to play. As is common practice ina high-energy ph sionless coefficients . Since by assumption there is only single funda i −1 −1 work in units where h̄ = c = 1. Then [m] = [E] = [p] = [x ] = [t ]asarethe allhyp mea the theory, we expect that C = O(1). This assertion is known Since by assumption thei theory has only a single fundamental ame units. We denote by [gi ] = −γi the mass dimension of the effective couplin ess”.scale Unless is a specific that could explain the smallness M, there it follows that: mechanism ws that ss numbers Ci , we should assume those numbers to be of O(1). The prese −γ i g = Ci M 6 −6i rge (e.g. 10 ) or small (e.g. 10 ) numbers in a theory would appear “unna mensionless coefficients Ci . Since by assumption further explanation. where by naturalness we expect that Ci = there O(1) is only a single fundam in the(E theory, thatcontribution Ci = O(1). This is knownQas in thethe hypo nergy ≪ Λwe<expect M), the of aassertion given operator eff i uralness”. Unless there is a specific mechanism that could explain the smallness to an observable (which for simplicity we assume to be dimensionless) is ex onless numbers Ci , we should assume those numbers to be of O(1). The prese −6 ly large (e.g. 106 ) or small (e.g. 10⎧ ) numbers in a theory would appear “unna ⎪ # $ O(1) ; if γ i = 0, γ ⎨ i l for further explanation.E with dimensional dimensionless coefficients Ci .comes Since by is only a single fundament ive analysis” toassumption play. Asthere is common practice in scale M in the theory, we expect that Ci = O(1). This assertion is known as the−1 hypothes Dimensional analysis us work in units where h̄ a=specific c = 1. Then [m] = [E]explain = [p]the=smallness [x ] =of [tt of “naturalness”. Unless there is mechanism that could he same units. WeCidenote byassume [gi ] =those −γinumbers the mass the e dimensionless numbers , we should to bedimension of O(1). Theof presence 6 −6 unusually large (e.g. 10 ) or small (e.g. 10 numbers in a theory ollows thatenergy, it follows that the) contribution At low of a would givenappear term “unnatura and call for further explanation. −γi g = C M gi low Qi to an observable (which for simplicity i At energy (E ≪ Λ < M), the contribution of ia we givenassume operator to Q be in the effecti i dimensionless) scales like: Lagrangian to an observable (which for simplicity we assume to be dimensionless) is expect hto dimensionless coefficients C . Since by assumption there is only a i scale as ⎧ le M in the theory, we expect Ci = ⎪ # $γthat O(1) ; O(1). if γi = 0,This assertion is know ⎨ i E Ci is a specific = ≪ 1 ; mechanism (1 if γi > 0, “naturalness”. Unless there that could explain ! ⎪ M ⎩ ≫ 1 ; if γthose i < 0. numbers to be of O( mensionless numbers Ci , we should assume ! It followslarge that only operators whose couplings 0 are important low energy. Th usually (e.g. 106 ) or small (e.g. have 10−6γ)i ≤numbers in a at theory would fact what only makes the OPE a useful tool. Depending on the precision goal, one m d very callTherefore, forisfurther explanation. 0 E ⌧ M operators with i are important for truncate the series in (8) at a given order in E/M. Once this is done, only a finite (oft At low energy (E ≪QΛ < M), the contribution of a given operato small) number of operators and couplings g need to be retained. i i This is what makes the effective Lagrangian useful! grangian tothrough an observable (which for assume to be Assumin dimen Let us go the above arguments oncesimplicity again, being we slightly more careful. 1 weak coupling, we on can the use the free action to assign scaling behaviorthe withinfinite E to all fields an scaleDepending as precision goal, oneacan truncate ⎧ 1 Interactions canterms change the γ , as we will see this #dimensions $operators O(1) ; later. if γFor 0,reason, sum over bynaive onlyscaling retaining whose value is the γi a i = γi i ⎪ ⎨ referred to as “anomalous dimensions”. E smaller than a certain value Ci = ≪ 1 ; if γi > 0, ⎪ M ⎩ ≫ 1 ; if γi < 0. Dimensional analysis Since the Lagrangian has mass dimension D = dimensionality of spacetime (the action is dimensionless), it follows that ! i = [Qi ] = D + i Hence we1:can summarize: Table Classification of operators and couplings in the effective Lagrangian ! ! ! ! Dimension Importance for E → 0 δi < D, γi < 0 grows δi = D, γi = 0 constant δi > D, γi > 0 falls Terminology relevant operators (super-renormalizable) marginal operators (renormalizable) irrelevant operators (non-renormalizable) Only a finite number of relevant and marginal operators exist! couplings in the low-energy effective theory. Consider scalar φ4 theory as an example. The action is " # ! 2 1 m 2 λ 4 D µ Dimensional analysis Table 1: Classification of operators and couplings in the effective Lagrangian Comments: ! Dimension Importance for E → 0 δi < D, γi < 0 grows ! δi = D, γi = 0 constant δi > D, γi > 0 falls ! ! Terminology relevant operators (super-renormalizable) marginal operators (renormalizable) irrelevant operators (non-renormalizable) • “relevant” operators are usually unimportant, since they are 4 couplings in the low-energy effective theory. Consider scalar φ theory an example. forbidden by some symmetry (otherwise they giveasrise to a The action is # 2 hierarchy problem) ! D " 1 m 2 λ 4 µ S= d x ∂µ φ ∂ φ − φ − φ . 2 2 4! • “marginal” operators are all there is in renormalizable QFTs (11) Using that x ∼ E −1 and ∂ µ ∼ E, and requiring that the action scale like O(1) (in units of D • “irrelevant” operators are the interesting since by δi most the mass dimension ofones, an operator Qi , then h̄), we see that φ ∼ E 2 −1 . If we denote about the fundamental scale M γi =they δi − D.tell Forus thesomething operators in the Lagrangian (11) we find: δi γi Coupling Example: - theory at weak δi > D, γi >40 falls (renormalizable) irrelevant operators coupling (non-renormalizable) Use the free Lagrangian to derive the mass dimension of all 4 fields and couplings, assuming the theory uplings in the low-energy effective theory. Consider scalarisφweakly theory coupled: as an example. The tion is ! S= ! D d x " # 2 1 m 2 λ 4 µ ∂µ φ ∂ φ − φ − φ . 2 2 4! (11) sing that x ∼ E −1 and ∂ µ ∼ E, and requiring that the action scale like O(1) (in units of D In D dimensions, If follows we denotethat: by δi the mass dimension of an operator Qi , then , we see that φ ∼ E 2 −1 . it = δi − D. For the operators D in the Lagrangian (11) we find: ! Hence: • The mass [ ]= 2 1, [m] = 1 , [ ]=4 D δi γi Coupling ∂µ φ ∂ µ φ D 0 1 2D − 4 D − 4 λ ∼ Λ4−D φ4 term is 2a relevant operator 2 φ D−2 −2 m ∼ Λ2 • The interaction term is marginal in D=4 (relevant in D<4) ore generally, an operator with n1 scalar fields and n2 derivatives has " # " # D D δi = n1 − 1 + n2 , γi = (n1 − 2) − 1 + (n2 − 2) . 2 2 (12) δi > >0 0 δi > D,D, γi γ> i4 Example: (renormalizable) (renormalizable) irrelevant irrelevantoperators operators (non-renormalizable) (non-renormalizable) falls falls - theory at weak coupling Use the free Lagrangian to derive the mass dimension of all couplings in the low-energy effective theory. Consider scalar φ4 4theory as an example. The fields and couplings, assuming the theory uplings in the low-energy effective theory. Consider scalarisφweakly theory coupled: as an example. The action is tion is ! " # 2 " 1 m2 2 λ 4 # µ S= D d x 1 ∂µ φ ∂µ φ − m φ 2− λφ 4 . S = d x 2∂µ φ ∂ φ − 2 φ −4! φ . 2 2 4! µ ! D (11) (11) ! Using that x ∼ E −1 and ∂ ∼ E, and requiring that the action scale like O(1) (in units of −1 D µ sing that x ∼ E and requiring that thedimension action scale O(1) (in we and denote by δi the mass of anlike operator Qi , units then of h̄), we see that φ ∼ D E 2∂−1 . ∼If E, In D dimensions, −1 it follows that: 2 . If we by δi the dimension of an operator Qi , then , γwe see that φ ∼ E = δ − D. For the operators in denote the Lagrangian (11)mass we find: i i = δi − D. For the operators D in the Lagrangian (11) we find: ! [ ]= δi [m] =γi1 , Coupling [ ]=4 2 δD γ0i Coupling ∂µ φ ∂ µ φ 1 i ∂µ φ ∂φµ4 φ 2DD− 4 D − 0 4 λ ∼ Λ14−D 2 2 4−D 42 φ D − 2 −2 m ∼ Λ 2D − 4 D − 4 λ ∼ Λ φ Hence: 1, D • An operator containing n1 fields and 2n2 derivatives has 2 2 φ n1 scalar D − 2fields −2 m ∼ Λ has More generally, an operator with and n derivatives 2 dimension: " # " # D with n1 scalar fields and n2 derivatives D ore generally, an operator has δi = n1 − 1 + n2 , γi = (n1 − 2) − 1 + (n2 − 2) . (12) " 2 # "2 # ! D D δi = nD 1 few + noperators γihave = (nγ1 − 2) − 1 + (n2 − 2) . (12) 1 > 2− 2, It follows that for only ≤ 0. i • For D>2, adding fields or derivatives increases the dimension! 2 2 A summary of these considerations is presented in Table 1. The common terminology Comments Examples of effective field theories: ! ! ! ! High-energy theory Fundamental scale Standard Model MW ∼ 80 GeV MGUT ∼ 1016 GeV GUT String theory MS ∼ 1018 GeV 11-dim. M theory ... ... ... Low-energy theory Fermi theory Standard Model QFT String theory ... 5 GeV NRQCD ! arguments justQCD The presented providemab ~new perspectiveHQET, on renormalization. Instead o a paradigm of renormalizable theories M based on the concept of systematic “cancellations o ChPT ChSM ~ 1 GeV ! nfinities”, we should adopt the following, more physical point of view: • SM andphysics GUTs depends are perturbative QFTs structure of the fundamental theor • Low-energy on the short-distance via relevant and marginal couplings, and possibly through some irrelevant coupling • Fermi theory contains only irrelevant operators (4 fermions) provided measurements are sufficiently precise. • String/M theory: fundamental theory is non-local and even • “Non-renormalizable” interactions not forbidden; on the contrary, irrelevant opera spacetime breaks down atare short distances tors always contribute at some level of precision. Their effects are simply numericall Comments Examples of effective field theories: ! ! ! ! High-energy theory Fundamental scale Standard Model MW ∼ 80 GeV MGUT ∼ 1016 GeV GUT String theory MS ∼ 1018 GeV 11-dim. M theory ... ... ... Low-energy theory Fermi theory Standard Model QFT String theory ... 5 GeV NRQCD ! arguments justQCD The presented providemab ~new perspectiveHQET, on renormalization. Instead o a paradigm of renormalizable theories M based on the concept of systematic “cancellations o ChPT ChSM ~ 1 GeV ! nfinities”, we should adopt the following, more physical point of view: • QCD at physics low energy: with strong coupling, the theor • Low-energy dependsexample on the short-distance structure of thewhere fundamental degrees of freedom energy (hadrons) are viarelevant relevant and marginal couplings, at andlow possibly through some irrelevant coupling provided measurements sufficiently precise. different from the are degrees of freedom of QCD • “Non-renormalizable” interactions are not forbidden;yet on ChPT the contrary, irrelevant opera • Low-energy theory is strongly coupled, is useful tors always contribute at some level of precision. Their effects are simply numericall Running couplings / Wilson coefficients Often the fields H correspond to heavy particles, whose effects become unimportant at low energies But the frequency decomposition implies that high-energy excitations of massless particles (such as gauge bosons) are also integrated out from the low-energy effective theory M ⇤ Consider now the situation where we lower the cutoff ⇤ without crossing the threshold for a heavy particle that could be integrated out: • the structure of the operators Qi in the effective Lagrangian remains the same • hence, the effect of lowering the cutoff must be entirely absorbed into the values of the coupling constants gi Follows that gi = gi (⇤) are running, ⇤-dependent parameters! E Modern quantum field theory “Theorem of modesty”: • no QFT ever is complete on all length and energy scales • all QFTs are low-energy effective theories valid in some energy range, up to some cutoff ⇤ Giving up renormalizability as a construction criterion for “decent” QFTs: • at low energy, any effective theory will automatically reduce to a “renormalizable” QFT, meaning that “non-renormalizable” interactions give rise to small contributions ~(E/M)n • this does not make renormalization irrelevant, but it provides a different point of view (Wilsonian picture of the RG) Modern quantum field theory We should forget the folklore about “cancellations of infinities” Adopt the more physical viewpoint that: • low-energy physics depends on the short-distance dynamics of the fundamental theory only through a small number of relevant and marginal couplings, and possibly through some irrelevant couplings if our measurements are sufficiently precise • this finite number of couplings can be renormalized (i.e., infinities can be removed consistently) using a finite number of experimental data • the criterion of “renormalizability” is automatically fulfilled (approximately) by any effective field theory Modern quantum field theory We should forget the folklore about “cancellations of infinities” Adopt the more physical viewpoint that: • contrary to the old paradigm of strictly forbidding irrelevant interactions, we always expect them to be present and give rise to small effects, which may or may not be observable at a given level of accuracy • this provides an “indirect way” to search for hints of physics beyond the (current) Standard Model: ! ! low-energy, high-precision measurements • e.g.: flavor physics, neutrino physics, (g-2)μ, EDMs, darkphoton searches, … Modern quantum field theory Instead, relevant (“super-renormalizable”) interactions cause problems! 2 2 Consider, e.g., the mass term m in scalar field theory 2 2 2 m ⇠ M ⇠ ⇤ Dimensional analysis suggests that UV But then a light scalar particle should not be present in the lowenergy effective theory! ! Hierarchy problem! The same argument applies for all mass terms in any QFT! And likewise for the cosmological constant! Victor Weisskopf Modern quantum field theory New paradigm: EFTs must be natural in the sense that all mass terms should be forbidden by (exact or broken) symmetries! Indeed: • gauge invariance: forbids mass terms for gauge fields (photons and gluons in the Standard Model) • chiral symmetry: forbids mass terms for fermions (all matter fields in the Standard Model) Explains why the SM is a (broken) chiral gauge theory! • But the Higgs boson exists and causes a naturalness problem! • Supersymmetry: would link the masses of scalars and fermions and hence, in combination with chiral symmetry, forbid mass terms for scalar fields (solves hierarchy problem) Is nature supersymmetric? • The Higgs boson exists and causes a naturalness problem! • Supersymmetry: would link the masses of scalars and fermions and hence, in combination with chiral symmetry, forbid mass terms for scalar fields (solves hierarchy problem) gs&und&das&Weltbild&der&Physik& nde&Idee:&Supersymmetrie& et&jedem&SMVTeilchen&einen& r&unentdeckten&Partner&zu& Die&fieberhage&Suche&nach&diesen& vermutlich&sehr&schweren&Teilchen& am&LHC&war&bisher&leider&erfolglos&& Is nature supersymmetric? Lecture II: Some applications of EFTs Standard Model as an effective field theory Some interesting insights can be gained by considering the Standard Model (SM) as a low-energy effective theory of some more fundamental theory (supersymmetry, extra dimensions, new strongly coupled physics, GUT, ...) We will denote the scale of New Physics by M; this could be as large as 1016 GeV for some applications, but as small as 103 GeV (= 1 TeV) for others The SM Lagrangian should then be extended to an effective Lagrangian, which besides the SM terms contains additional, irrelevant operators These operators must respect the symmetries of the SM (gauge invariance, Lorentz symmetry, CPT) but are otherwise unrestricted QϕW B ϕ τ ϕ Wµν B QϕW BQdW ϕ τ(q̄ϕ drB )τ ϕ WµνQdW Qϕd(q̄p σ (ϕd p σWµν light particles exist, like axions or sterile neutrinos. † I $I µν µν † I(q̄ $ I d )ϕ µν B µνϕ QϕW ϕ τ ϕ W B Q σ Q i( ! Q ϕ τ ϕ W B Q (q̄ σ "B dB p r µν ϕud " dB p µν µν ϕW B In most of beyond-SM theories that have been considered to date, reduction low energies proceeds via decoupling of heavyoperators particles with than masses order Λoto Table 2: Dimension-six the of four-fermion Table 2: other Dimension-six operators a decoupling at the perturbative level is described by the Appelquist-Carazzon effective Lagrangian up to operator dimension operators D=6 reads: This inevitably leads to appearance of higher-dimensional in the SM La 3 The complete set of dimension-five and -si 3 The complete set of dimen are suppressed by powers of Λ " # ! ! This Section to presenting our final results in Secs. Section is(6) devoted to presenting final 5, res6 1 1(derived our 1is devoted (5) This (5) (6) (4) (5) (6) (6) Ck Qof + Ck. operators Qk +independence OQ(5) 3and, Qmeans LSM LSM + k Qindependent 2 of = independent operators and Q Their thatind n n n nΛ n . Their Λ k Λ k of them and their Hermitian conjugates is Hermitian EOM-vanishing up to is total der of them and their conjugates EOM(5) (4) Imposing the SM gauge Imposing symmetry constraints on Q out just a SM SM gauge symmetry constraints n leaves where LSM is the usual “renormalizable” partthe of the Lagrangian. It contains (n) up to Hermitian conjugation flavour assignments. It reads up remaining toand Hermitian conjugation and flavour assignmo and -four operators only.1 In the terms, Q denote dimension-n unique operator (neutrinokmasses): (n) j m dimensionless k T n †coupling T j m† constants Ck stand forQthe corresponding coeffi = ε ε ϕ ϕ (l ) Cl ≡ ( ϕ ! l ) !(llrpk), Q = ε ε ϕϕ )T Clrn ≡(Wilson (ϕ !† lp )T C( ϕ !† νν jk mn p ϕC( jk mn p rνν (n) the underlying high-energy theory is specified, all the coefficients Ck can be d 2 2 Weinberg (1979) where C is the charge conjugation matrix. Q violates the lepton nu where C is the charge conjugation matrix. Q νν ν integrating out the heavy fields. electroweak symmetry breaking, it generates neutrino masses and of mixing electroweak symmetry breaking, it generates neu Our goal in this paper is to find a complete set of independent operators dim can doare the job. Thus, consistency ofjob. thesymmetr SM (as dimension-six terms can Thus, co that are the builtdimension-six out of the SMterms fieldsthe and consistent with the do SMthe gauge andoriginal Tab. 1)analysis with observations crucially on this dimension-five and operators Tab. 1) with observations crucially rely on the of such bydepends Buchmüller and Wylerdepends [3] buttero All the independent operators that allowed by the St All the the outset. independent operators the full classification once againdimension-six from One ofdimension-six theare reasons for repeatin arethat listed in Tabs. 3. in the left of each sho areTheir listednames in 2 and 3.column Their names inblock the left is the fact many linear2 and combinations of Tabs. operators listed in Ref. [3] vanish by with generation indices of theare fermion fields whenever necessary, e.g.,whe Q with generation indices the give fermion fields Of Motion (EOMs). Such operators redundant, i.e.ofthey no contributi indices are always contracted within the brackets, andwithin not displayed. The indices are always contracted the brackets matrix elements, both in perturbation theory (to all orders) and beyond [4–9]. 2been presence of2 several EOM-vanishing combinations in Ref.and [3] has already In the Dirac representation2 CIn =the iγ 2Dirac γ 0 , with Bjorken conventp representation CDrell = iγ [21] γ 0 , phase with Bjorken Standard Model as an effective field theory The light particles exist, like axions or sterile neutrinos. Standard as theories an effective field theory In mostModel of beyond-SM that have been considered to date, reduction The low energies proceeds via decoupling of heavy particles with masses of order Λ o a decoupling at the perturbative level is described by the Appelquist-Carazzon effective Lagrangian up to operator dimension operators D=6 reads: This inevitably leads to appearance of higher-dimensional in the SM La are suppressed by powers of Λ " # ! ! 1 1 1 (5) (5) (6) (6) (4) Ck Qk + 2 Ck Qk + O , LSM = LSM + 3 Λ k Λ k Λ (4) where LSM is the usual “renormalizable” part of the SM Lagrangian. It contains X ϕ and ϕ D ψ ϕ (n) 1 and -four operators only. In the remaining terms, Q denote 59 operator (2499 flavordimension-n q. numbers) o k incl. Q f G G G Q (ϕ ϕ) Q (ϕ ϕ)(¯l e ϕ) !(n) Q f CG G stand G Q for (ϕthe ϕ)!(ϕcorresponding ϕ) Q (ϕ ϕ)(q̄ udimensionless ϕ) ! coupling constants (Wilson coeffi k " # " # ϕD ϕ ϕD ϕ Q ε W W W Q Q (ϕ ϕ)(q̄ d ϕ) (n) Buchmüller, Wyler (1986) high-energy theory is specified, all the coefficients Ck can be d $ W underlying Q ε the W W Hagiwara et al. (1987 & 1993) outψ Xϕ the heavy fields. Xintegrating ϕ ψ ϕ D Grzadkowski, Iskrzynski, Misiak, Rosiek (2010) ¯l σ e )τ ϕW ¯l γ l ) Q ϕ ϕ GOur G Q ( Q (ϕ i D ϕ)( goal in this paper is to find a complete set of independent operators of dim ! Q ϕ ϕG G Q (¯l σ e )ϕB Q (ϕ i D ϕ)(¯l τ γ l ) that are built out of! Gthe QSM (ϕfields and are consistent with the SM gauge symmetr Q ϕ ϕW W Q (q̄ σ T u )ϕ i D ϕ)(ē γ e ) $ Won the analysis ofi D such by Buchmüller and Wyler [3] but r Q ϕrely ϕW Q original (q̄ σ u )τ ϕ !W Q (ϕ ϕ)(q̄ γ q operators ) Q ϕ ϕB B Q (q̄ σ u )ϕ !B Q (ϕ i D ϕ)(q̄ τ γ q ) the full classification once again from the outset. One of the reasons for repeatin ! Q ϕ ϕB B Q (q̄ σ T d )ϕ G Q (ϕ i D ϕ)(ū γ u ) is the factQ that many linear combinations of operators listed in Ref. [3] vanish by Q ϕ τ ϕW B (q̄ σ d )τ ϕ W Q (ϕ i D ϕ)(d¯ γ d ) $Motion operators Q ϕOf τ ϕW B Q (EOMs). (q̄ σ d )ϕ B Such Q i(ϕ ! D ϕ)(ū γ d ) are redundant, i.e. they give no contributi matrix elements, both perturbation theory (to all orders) and beyond [4–9]. Table 2: Dimension-six operators other than the in four-fermion ones. Operators other than four-fermion operators presence several EOM-vanishing combinations in Ref. [3] has been already p The complete set of of dimension-five and -six operators 3 G ! G W " W ABC Aν µ Bρ ν Cµ ρ ϕ ABC Aν µ Bρ ν Cµ ρ ϕ! IJK Iν µ Jρ ν Kµ ρ IJK Iν µ Jρ ν Kµ ρ 2 ϕG ! ϕG ϕW " ϕW ϕB ! ϕB ϕW B "B ϕW 3 6 3 † † 2 † ⋆ µ † Aµν † A µν Aµν † I µν Iµν † I µν Iµν µν µν µν µν † I I µν µν † I I µν µν eW eB p uG p uW p uB p dW p dB µν r A r µν p dG µν µν µν µν p µν I r r r (1) ϕl (3) ϕl µν r A µν ϕe I I µν (1) ϕq r A I µν µν (3) ϕq r A µν ϕu I I µν ϕd µν ϕud p r † dϕ 2 µν p p r † uϕ µ 3 † eϕ 2 A µν † 2 † 2 † † ϕD 4 p r 2 ↔ µ ↔ † I µ ↔ † µ ↔ † µ ↔ † I µ ↔ † µ ↔ † µ † † µ p µ r I µ p p p µ µ r r r I µ p p p p µ µ µ r r r r light particles exist, like axions or sterile neutrinos. Standard as theories an effective field theory In mostModel of beyond-SM that have been considered to date, reduction low energies proceeds via decoupling of heavy particles with masses of order Λ o a decoupling at the perturbative level is described by the Appelquist-Carazzon effective Lagrangian up to operator dimension operators D=6 reads: This inevitably leads to appearance of higher-dimensional in the SM La are suppressed by powers of Λ " # ! ! 1 1 1 (5) (5) (6) (6) (4) Ck Qk + 2 Ck Qk + O , LSM = LSM + 3 Λ k Λ k Λ The (4) where LSM is the usual “renormalizable” part of the SM Lagrangian. It contains (L̄L)(L̄L) (R̄R)(R̄R) (n) 1 (L̄L)(R̄R) and -four operators only. In the remaining terms, Q denote Q (¯l γ l )(¯l γ l ) Q (ē γ e )(ē γ e ) Q (¯l γ l )(ē γ e ) 59 operator (2499 flavordimension-n q. numbers) o k incl. Q (q̄ γ q (n) q) Q γ u )(ū γ u ) Q (¯l γ l )(ū γ u ) Ck)(q̄ γ stand for(ū¯ the corresponding dimensionless coupling constants (Wilson coeffi ¯ ¯ ¯ Q (q̄ γ τ q )(q̄ γ τ q ) Q (d γ d )(d γ d ) Q (l γ l )(d γ d ) (n) Buchmüller, Wyler (1986) Q (¯l the γ l )(q̄ γunderlying q) Q (ē γ high-energy e )(ū γ u ) Q (q̄ γ q )(ē γ e ) theory is specified, all the coefficients Ck can be d Hagiwara et al. (1987 & 1993) Q (¯l γ τ l )(q̄ γ τ q ) Q (ē γ e )(d¯ γ d ) Q (q̄ γ q )(ū γ u ) integrating out the heavy fields. Q (ū γ u )(d¯ γ d ) Q (q̄ γ T q )(ū γ T u ) Grzadkowski, Iskrzynski, Misiak, Rosiek (2010) Our goal inγ this Q (ū T u )(d¯ γpaper T d ) Q is to (q̄ γfind q )(d¯ γ a d ) complete set of independent operators of dim γ T q )(d¯ γ T d ) that are built out of the QSM(q̄ fields and are consistent with the SM gauge symmetr (L̄R)(R̄L) and (L̄R)(L̄R) B-violating Flavor by observables are and crucial in order to r ! " ! of such " rely on the original analysis operators Buchmüller Wyler [3] but ¯ ¯ Q (l e )(d q ) Q ε ε (d ) Cu (q ) Cl explore this enormous parameter space! ! "! " Q (q̄the u )ε (q̄full d) Q ε ε (q ) Cq again (u ) Ce from the outset. One of the reasons for repeatin classification once ! "! " Q (q̄ T u )ε (q̄ T d ) Q ε ε ε (q ) Cq (q ) Cl is the factQ thatε many linear combinations of operators listed in Ref. [3] vanish by ! "! " Q (¯l e )ε (q̄ u ) (τ ε) (τ ε) (q ) Cq (q ) Cl ! "! " Motion (EOMs). are redundant, i.e. they give no contributi Q (¯l σ Of e )ε (q̄ σ u) Q ε (dSuch ) Cu (uoperators ) Ce matrix elements, Table 3: Four-fermionboth operators. in perturbation theory (to all orders) and beyond [4–9]. Four-fermion operators presence of several combinations in Ref. [3] has been already p isospin and colour indices in the upper part of Tab. 3. EOM-vanishing In the lower-left block of that table, ll p µ r s (1) qq p µ r s (3) qq p µ (1) lq (3) lq I r s p µ r s p µ I r s µ t µ t µ I µ t t µ I t µ ee p µ r s uu p µ r s dd p µ r s eu p µ r s ed p µ r s (1) ud (8) ud p µ r p µ A le p µ r s t lu p µ r s t ld p µ r s t qe p µ r s t (1) qu p µ r s t (8) qu µ µ µ µ s r t s µ µ A p µ (1) qd t (8) qd j s t j p r ledq (1) quqd jk k s t r jk k s (1) lequ j p r jk k s t (3) qqq (3) lequ j p µν r jk k µν t s duu (8) quqd j p r j p A αβγ duq A αβγ qqu t (1) qqq αβγ αβγ I jk α T p jk αj T p jk mn jk αβγ I βk r αj T p mn α T p s p µ r s αj T p A r γj T s k t γ T s t βk r β r r p µ β r A γm T s βk r γ T s s µ µ µ µ µ µ µ µ t t t t t A t t A t n t γm T s n t t colour indices are still contracted within the brackets, while the isospin ones are made explicit. Standard Model as an effective field theory We will discuss a couple of interesting aspects of SM physics from the perspective of this construction: • neutrino masses and the see-saw mechanism • effective weak interactions in the quark sector • anomalous magnetic moment of the muon • proton decay • conservation of baryon and lepton numbers (accidental symmetries) • Higgs production at the LHC Neutrino masses and the see-saw mechanism The discovery of non-zero neutrino masses is often described as a departure from the SM But this is no longer true if we consider the SM as an effective low-energy theory Without a right-handed neutrino (which indeed is not part of the SM), it is impossible to write a neutrino mass term at the level of relevant or marginal operators However, it is possible to write a gauge-invariant neutrino mass term at the level of irrelevant operators of dimension ≥5: Lneutrino mass g ˜T = lL M ⇤ C ˜ lL Neutrino masses and the see-saw mechanism However, it is possible to write a gauge-invariant neutrino mass term at the level of irrelevant operators of dimension ≥5: ! ! Lneutrino mass g ˜T = lL M ⇤ C ˜ lL After electroweak symmetry breaking, this gives rise to a Majorana mass term of the form: ! ! Lneutrino mass = v2 g T ⌫˜L C ⌫L 2M The SM as an effective field theory predicts that neutrinos 2 should be massive, with m⌫ ⇠ v /M suppressed by the fundamental scale of some BSM physics Neutrino masses and the see-saw mechanism Experiments hints at the fact that the fundamental scale relevant for the generation of neutrino masses is very heavy, 14 ! M ⇠ 10 GeV which is not far from the scale of grand unification Extensions of the SM containing heavy, right-handed neutrinos (with masses that are naturally of order M) provide explicit examples of fundamental theories which yield such a Majorana mass term when the heavy, right-handed neutrinos are integrated out (see-saw mechanism) Weak interactions at low energies (flavor physics) Fermi’s description of the weak interactions at low energy is a prime example of an effective field theory, which has provided first evidence for the scale of electroweak symmetry breaking At the low energies relevant for neutron β-decay, kaon physics, charm physics or B-meson physics (few MeV - few GeV), we can integrate out the heavy W and Z bosons as well as the top-quark and Higgs boson from the SM This gives rise to a low-energy effective theory containing 4-fermion interactions (Fermi theory) and dipole interactions between fermions and the photon and gluon This effective Lagrangian successfully describes the huge phenomenology of flavor-changing processes Weak interactions at low energies (flavor physics) Example: Lagrangian for b→s pseudoscalar FCNC transitions all other B Effective decays into two light, flavour-nonsinglet mesons. Using (see Buras lectures derivation) unitarity relation −λt = λufor + λa write c , we the " into two # mesons. Using the all otherG B ! decays pseudoscalar ⇤ light, flavour-nonsinglet ! F p = pVps Vpb p (CKM matrix elements) √ λ C Ci Qi + C7γ Q7γ + C8g Q8g + h.c. , (1) H = 2Q eff unitarity relationp −λ1tQ=1 + λuC+ λ2c ,+we write 2 p=u,c i=3,...,10 GF √ left-handed where H = the from λp C1 current–current Qp1 + C2 Qp2 + operators Ci Qarising Q7γW+-boson C8g Qexchange, i + C7γ 8g + h.c. , p=u,c i=3,...,10 operators, and Q7γ and Q8g are the Q3,...,6 and Q7,...,102 are QCD and electroweak penguin Qp1,2 eff are ! " # ! electromagnetic and chromomagnetic dipole operators. They are given by p (1) where Q1,2 are the left-handed current–current operators arising from W -boson exchange p p = (p̄b) (s̄p) , Q (p̄i bj )V −A (s̄j pi )V −A ,and Q7γ and Q8g are the Q3,...,6Qand Q are QCD and electroweak penguin operators, V −A V −A 1 2 = 7,...,10 !chromomagnetic dipole operators.! electromagnetic and They by Q = (s̄b) (q̄q) , Q = (s̄ b ) (q̄ qare ) given , 3 V −A q V −A 4 p Q = (p̄b) Q5 =1(s̄b) (q̄q)VV +A −A,, V −A V −A q (s̄p) 3 = (s̄b) q (q̄q) V −A , , Q7 Q =3(s̄b) V +A V −A V −A q 2 eq (q̄q) ! ! Q9 =5(s̄b)V −A V −Aq 32 eq (q̄q) , V −A q V +A Q = (s̄b) Q7γ where (q̄q) q j i V −A p Q (s̄i )jVp+A Q6 = (s̄i2bj= )V (p̄ −Aj q i )V,−A , −Ai bj )V q (q̄ ! ! ! i j V −A , ! ! 3 Qi4bj= (s̄i bj )Vq −A Q8 = (s̄ )V −A e (q̄ qq )(q̄j qi ),V −A , 2 q j i V +A ! ! 3 (s̄i6bj )V −Ai j Vq −A e (q̄ q ) , 2 q j qi V j−Ai V +A ! Q10 = Q = (s̄ b ) (q̄ q ) , ! ! −g −e 3µν 3 s µν e (q̄q) , Q = (s̄ b ) ebq (q̄ , Q mb s̄σ (1 + γ )F b , Q = m s̄σ (1 + γ )G , (2) =7 =2 (s̄b) q V +A 8 i j V −A q j qi )V +A V −A q µν 8g b µν 5 25 2 8π 8π 2 3 Q = (s̄b) , 9 = q̄ γ V(1 −A q 2,ei, q (q̄q) (q̄1 q2 )V ±A ± γ5)q j areV −A colour 1 µ 2 ! 3 Q = (s̄ b ) eq (q̄j qi )ofV −A 10 e are i the j V −A indices, electricq 2charges the, q ! Weak interactions at low energies (flavor physics) Example: Effective Lagrangian for b→s FCNC transitions (see Buras lectures for a derivation) u s u s d s W W g u,c,t W u,c,t g d u d u q q (a) SM diagrams involving virtual heavy-particle exchanges contributing to the low-energy effective weak Lagrangian d s (b) d s W b s u,c,t u,c,t u,c,t W γ,Z q W W t t g,γ γ,Z q q q (c) d (d) b,s d s W W u,c,t t t u,c,t γ,Z b,s W (e) d l l (f) Weak interactions at low energies (flavor physics) Example: Effective Lagrangian for b→s FCNC transitions (see Buras lectures for a derivation) From the fact that the leading operators in the low-energy effective theory have dimension 6, it follows that the corresponding couplings are irrelevant and proportional to MW2, indeed: ! ! GF g22 p = 2 8M 2 W The strong suppression of these contributions at low energies explains why we refer to these interactions as the weak interactions, even though the coupling constants of the SU(2)L⊗U(1)Y electroweak interactions is about as large as the electromagnetic coupling constant discuss the most important experimental observables, ⇤ and3, uncertainties. the numerical analysis. start • We use the information on BIn !section Kuncertainties. Bswe ! perform form factors from most precise LCSR In section 3, wethe perform the We numerica which of theoretical uncertainties, if underestimated, could account calculation [13, 16], sources taking into account the correlations between the uncertainties of which all sources of theoretical uncertainties, if underesti within thedi↵erent SM.even Weq 2then proceed model-independent di↵erent formeven factors and at values. ThisSM. iswith particularly important to estimate within the Wea then proceed with aanalysis modelstudying the allowed regions for the NP Wilson coefficients. In section 4, we the uncertainties in angular observables that ratios of form factors. studying theinvolve allowed regions for the NP Wilson coeffici model-independent findings imply for the findings Minimalimply Supersymmetric Standard model-independent for the Minimal Supe Our paper is organized as follows. In section 2, we define the e↵ective Hamiltonian and for models with a new neutral We summarize and conclu forheavy models with gauge a new boson. heavy neutral gauge boson. We discuss the most important experimental observables, detailing our treatment of theoretical Several appendices contain all our SM predictions for the observables of int Several appendices contain all our SM⇤predictions for uncertainties. In section 3, we of perform the numerical analysis. We start by investigating our treatment form factors, and plotss of constraints on Wilson coefficients. our treatment of form factors, and plots of constraints which sources of theoretical uncertainties, if underestimated, could account for the tension (⇤) + even within the SM. We then proceed with a model-independent analysis beyond the SM, 2. Observables Wilson and uncertainties studying the allowed regions for the NP2. coefficients. Inand section 4, we discuss what the Observables uncertainties model-independent2.1. findings imply for the Minimal Supersymmetric Standard Model as well as E↵ective Hamiltonian 2.1. boson. E↵ective Hamiltonian for models with a new heavy neutral gauge We summarize and conclude in section 5. e↵ective Hamiltonian for b ! s transitions can beofwritten as details on 4 Several appendicesThe contain all our SM The predictions the observables 4 e↵ectivefor Hamiltonian for b ! sinterest, transitions can be 2 our treatment of form factors, and plots of constraints on Wilson coefficients. X 4 GF 0 0 ⇤ e X p H = V V (C O + C O⇤i ) e+2 h.c. 4 G i i F e↵ tb ts 3 i 3 2 p Vtb Vts 16⇡ He↵ = (Ci O 2 i 16⇡ 2 2 i 2. 2Observables and uncertainties 2 and we consider NP e↵ects in the following set of dimension-6 operators, SMwe operators: and consider NP e↵ects in the following set of dime m mb 2.1.1 E↵ective Hamiltonian O = b (s̄ P b)F µ⌫ , 0 O = (s̄ µ⌫ PL b)F µ⌫ 1 m 7 µ⌫ R 7 µ⌫ b e O7 = (s̄ µ⌫ PR b)F , e O The e↵ective Hamiltonian for b ! s transitions can be written as e µ 0 µ O9 = (s̄ µ PL b)(`¯ `) , O = (s̄ µ PR b)(`¯ `) , µ9 0 ¯ 0 O 2 X µ O9 = (s̄ µ PL b)(` 0`) , µ 4G e ¯ ¯ F 0 0 ⇤ O = (s̄ P b)( ` `) , O = (s̄ P b)( ` ` p 10Vtb Vts µ L2 He↵ = (Ci5O Oi +=C(s̄ h.c.`¯ µ10 `) , µ R (1)5O i Oiµ)P+ b)( 10 5 L 16⇡ 2 -1 -1 i the complete -4 -3 -2 Of -1 0 1 set of dimension-6 operators invariant under the strong and -4 -3 -2 -1 0 1 Of the complete set of dimension-6 operators invariant NP Opposite chirality operators: gauge groups, this set does not include and we consider Re(C NP e↵ects in the following set of dimension-6 operators, ) Re(C9NP ) 9 gauge groups, this set does not include m mb b µ⌫ 0 •NPFour-quark operators (includingOcurrent-current, and (2) elec NP )-Re(C 0 ) plane NP NP ) plane NP 0 NP O = (s̄ P b)F , = (s̄ µ⌫ PLQCD b)F µ⌫penguin, , Figure 5:Figure Allowed regions in the Re(C (left) and the Re(C )-Re(C 5: Allowed regions in the 7 9 Re(C9 )-Re(C 9 10 µ⌫ 10R) plane 9 ) plane (left) and the79Re(C9 )-Re(C • Four-quark (including current-current, e operators). e These operators only operators contribute to the observables considerQ (right). (right). The blue contours correspond to the 1toand best regions from the The blue contours correspond the 21 and 2 fitbest fit regions fromglobal the global operators). operators contribute to(3)th O =if (s̄ Ponly b)( `¯ µ `)mixing , O90These = (s̄listed b)(`¯ µonly `) , through µbranching µ PRabove Lthrough sis into the operators and higher o fit. The fit. green redand contours correspond to the to 1 and regions onlyif Theand green red contours correspond the 12 and 2 9regions branching ⇤ + ⇤ + 0 mixing into the sis through listed above ratio data or data only or data ondata B !onKBµ!µK angular observables intoPaccount. ratio only µ µ angular observables into account. ¯ µ 5 `) , Ois10taken =is(s̄taken O10 = (s̄ µ PR b)(`¯ µoperators (4) µ L b)(` 5 `) , Weak interactions at low energies (flavor physics) Re(C9 ) NP Re(C10 ) A global analysis of experimental data on B ! X , B ! K , and Bd! K µ µ decay distributions provides information about various operator coefficients (defined to vanish in SM): Altmannshofer, Straub:1411.3161 Of the complete set of dimension-6 operators invariant under the strong and electromagnetic tension SM prediction tension with thewith SM the prediction gauge groups, this set does not include 3 3 RK = • Four-quark operators (including current-current, QCD penguin, and electroweak penguin operators). These operators only contribute to the observables considered in this analyThe theoretical of the SM prediction is completely negligible compared the current The theoretical error oferror the SM prediction is completely negligible compared to thetocurrent sisprediction throughand mixing into the operators listed above and through higher order corrections. experimental uncertainties. The tension between the SM the experimental data + BR(B BR(B ! Kµ+! µ Kµ )[1,6]µ )[1,6] +0.090 +0.090 SM RK = = 0.745 0.036 ± ,0.036 , RK + e=) 0.745 0.074 ±0.074 + BR(B ! Ke [1,6] BR(B ! Ke e )[1,6] SM 'RK 1.00'. 1.00 . A first hint of New Physics? (23) (23) experimental uncertainties. The tension between the SM prediction and the experimental data is driven by the reduced B + ! Kµ+ µ branching ratio, while the measured B !+ Ke+ e Anomalous magnetic moment of the muon In a celebrated calculation that was the birth of modern QFT, Schwinger computed the anomalous magnetic moment of the electron in 1948 and found: ! ! ge ge 2 ↵ µe = = + ... , with ae = 2me 2 2⇡ How will this result be affected if the SM is considered as an effective field theory? Anomalous magnetic moment of the muon In a celebrated calculation that was the birth of modern QFT, Schwinger computed the anomalous magnetic moment of the electron in 1948 and found: ! ! ge ge 2 ↵ µe = = + ... , with ae = 2me 2 2⇡ How will this result be affected if the SM is considered as an effective field theory? Add unique dimension-5 operator ( = 5 , gv ¯ M2 µ⌫ F = µ⌫ factor v required by EWSB 1): Anomalous magnetic moment of the muon In a celebrated calculation that was the birth of modern QFT, Schwinger computed the anomalous magnetic moment of the electron in 1948 and found: ! ! ge ge 2 ↵ µe = = + ... , with ae = 2me 2 2⇡ This adds g/M to µe and hence: ! ! ↵ gme v ae = + + ... 2 2⇡ M me the additional term will be very small, and As long as M by comparing a measurement of µe with theory we can constrain M Anomalous magnetic moment of the muon Analogous discussion (with me replaced by mµ ) holds for the muon In this case, there is presently a 3.6 σ discrepancy between theory and experiment: ! aSM µ ! aexp ⇡ µ 2.8 · 10 9 Interpreting this effect in terms of our irrelevant operator implies that: contains loop factor (small) M⇠ p g ⇥ 100 TeV One of the best hints for BSM physics! Proton decay Suppose you know the gauge symmetry SU(3)c⊗SU(2)L⊗U(1)Y of the SM but nothing else (no GUTs). What could you say about proton decay? The effective Lagrangian must contain at least three quark fields (change baryon number by 1 unit) and one lepton field (change lepton number by 1 unit) Hence: ! Lproton decay g ⇠ 2 qqq` M Since the lowest-dimension operators have dimension 6 (corresponding to i = 2 ), the proton can be made sufficiently long-lived by raising the fundamental scale M into the 1016 GeV range Proton decay Now imagine that you do not know about the existence of quarks (no one has seen any) but you do know about protons and pions Then an effective Lagrangian giving proton decay could be: ! Lproton decay ⇠ g ⇡ ¯e p This is a marginal operator, and hence proton decay would not be suppressed by any large mass scale! In some sense, we see that the longevity of the proton provides a hint for a substructure of the proton: replacing a fundamental field by a composite of several fields raises the dimension of the operators and hence gives rise to additional suppression Proton decay The same trick can be applied to other fine-tuning problems For example, the hierarchy problem can be solved by supposing that the Higgs boson is not an elementary scalar particle but instead a composite of a pair of elementary fermions If this is the case, then the Higgs mass term corresponds to a 4-fermion operator, which is irrelevant This is the main idea of composite Higgs and technicolor theories Baryon and lepton number conservation In the construction of the SM, the conservation of baryon and lepton number is not imposed as a condition There are no corresponding U(1) symmetries of the Lagrangian How can we understand that in nature we have not seen any hints of baryon- or lepton-number violating processes? Baryon and lepton number conservation In the construction of the SM, the conservation of baryon and lepton number is not imposed as a condition There are no corresponding U(1) symmetries of the Lagrangian How can we understand that in nature we have not seen any hints of baryon- or lepton-number violating processes? The answer is that it is impossible to construct any relevant or marginal operator that would respect the gauge symmetries of the SM and violate baryon or lepton number! Hence, at the level of renormalizable interactions, baryon- and lepton-number conservation are accidental symmetries of the SM Higgs production at the LHC The protons collided at the LHC contain only light quarks (u,d, and a little bit of s), which in the SM have negligible couplings to the Higgs boson, and gluons, which do not couple to the Higgs boson at all How, then, is the Higgs boson produced in pp collisions at the LHC? Higgs production at the LHC The protons collided at the LHC contain only light quarks (u,d, and a little bit of s), which in the SM have negligible couplings to the Higgs boson, and gluons, which do not couple to the Higgs boson at all How, then, is the Higgs boson produced in pp collisions at the LHC? We can gain insight by assuming (as seems to be the case) that the Higgs boson is lighter than the top quark We can then construct an effective low-energy theory for Higgs physics, in which the top quark is integrated out Higgs production at the LHC In this effective low-energy theory, direct couplings of the Higgs boson to pairs of gluons and photons arise at the level of irrelevant dimension-5 operators, with coefficients that scale like 1/mt , e.g.: ! ! Lhgg yt ↵s a µ⌫,a p = h Gµ⌫ G 2mt 12⇡ These operators appear first at one-loop order, via the exchange of a virtual top-quark g t q (n) h qt (n) q (n) t g The effective hgg interaction provides the dominant production mechanism for the Higgs Figure 1: Effective couplings of the Higgs boson to two gluons boson in gluon-gluon fusionofat the LHC KK quarks. that it has unit area. The η-dependence of the Yukawa coupling Summary Effective field theories are a very powerful tool in quantum field theory The are of great practical use, but also provide the conceptual tools to understand scale separation (factorization) and renormalization in a physical and systematic way Effective field theories are abundant, since any QFT can be considered as an effective low-energy theory of some more fundamental theory, which is often not yet known Because of this fact, effective field theories provide the tools to perform indirect searches for new physics beyond the Standard Model Contact: [email protected] activities 2015 Mainz Institute for Theoretical Physics ScientiFic ProgrAmS JgU campus mainz effective theories and Dark matter crossroads of neutrino Physics Vincenzo Cirigliano LAnL, Richard J. Hill Univ. chicago, Achim Schwenk, tU Darmstadt, Tim M.P. Tait Uc irvine Steen Hannestad Aarhus Univ., Patrick Huber Virginia tech, Alexei Smirnov mPi-K Heidelberg, Joachim Kopp JgU mainz march 16-27, 2015 July 20-August 14, 2015 Amplitudes, motives and Beyond Fundamental Parameters from Lattice QcD Francis Brown iHeS Paris, Marcus Spradlin Brown Univ., Don Zagier mPi-m Bonn, Stefan Weinzierl JgU mainz, Stefan Müller-Stach JgU mainz Gilberto Colangelo Bern Univ., Heiko Lacker HU Berlin, Georg von Hippel JgU mainz, Hartmut Wittig JgU mainz August 31-September 11, 2015 may 26-June 12, 2015 Stringy geometry Higher orders and Jets at the LHc Matteo Cacciari LPtHe Paris, Paolo Nason inFn milan, Giulia Zanderighi cern & Univ. oxford Eric Bergshoeff Univ. groningen, Gianfranco Pradisi University of rome „tor Vergata“, Fabio Riccioni inFn rome „La Sapienza“, Gabriele Honecker JgU mainz June 29-July 17, 2015 September 14-25, 2015 toPicAL WorKSHoPS JgU campus mainz challenges in Semileptonic B Decays the Ultra-Light Frontier Paolo Gambino Univ. turin, Marcello Rotondo inFn Padua, Christoph Schwanda ÖAW Vienna, Andreas Kronfeld Fermilab, Sascha Turczyk JgU mainz Surjeet Rajendran Univ. Stanford, Dmitry Budker JgU mainz April 20-24, 2015 Quantum Vacuum and gravitation Higgs Pair Production at colliders Manuel Asorey Univ. Zaragoza, Emil Mottola LAnL, Ilya Shapiro Univ. Juiz de Fora, Andreas Wipf Univ. Jena June 15-19, 2015 Daniel de Florian Univ. Buenos Aires, Christophe Grojean cern, Fabio Maltoni Univ. Leuven, Aleandro Nisati inFn rome, José Zurita JgU mainz June 22-26, 2015 April 27-30 2015 For more details: http://www.mitp.uni-mainz.de Applications via http://indico.mitp.uni-mainz.de mainz institute for theoretical Physics PRISMA Cluster of Excellence Johannes Gutenberg-Universität Mainz, Germany www.mitp.uni-mainz.de