Lorentz Violation from the Higgs Portal
CCERN-PH-TH/2010-049IFUP-TH/2010-1
Lorentz Violation from the Higgs Portal
Gian F. Giudice a , Martti Raidal b,c , Alessandro Strumia a,d (a) CERN, Theory Division, CERN, CH-1211 Geneva 23, Switzerland(b) National Institute of Chemical Physics and Biophysics, Ravala 10, Tallin, Estonia(c) Department of Physics, P.O.Box 64, FIN-00014 University of Helsinki(d) Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italia Abstract
We study bounds and signatures of models where the Higgs doublet has an inhomo-geneous mass or vacuum expectation value, being coupled to a hidden sector that breaksLorentz invariance. This physics is best described by a low-energy effective Lagrangianin which the Higgs speed-of-light is smaller than c ; such effect is naturally small becauseit is suppressed by four powers of the inhomogeneity scale. The Lorentz violation in theHiggs sector is communicated at tree level to fermions (via Yukawa interactions) andto massive gauge bosons, although the most important effect comes from one-loop dia-grams for photons and from two-loop diagrams for fermions. We calculate these effectsby deriving the renormalization-group equations for the speed-of-light of the StandardModel particles. An interesting feature is that the strong coupling dynamically makesthe speed-of-light equal for all colored particles. a r X i v : . [ h e p - ph ] J un Introduction
The sector responsible for the electroweak symmetry breaking still leaves open theoreticalquestions and is experimentally unknown. Its most plausible explanation relies on the ideaof spontaneously broken gauge symmetry, although the Higgs mechanism introduces its ownproblems. The main puzzle is associated with the presence of a mass parameter for theHiggs field, which sets the scale for the electroweak phenomena. At the quantum level thismass term is quadratically sensitive to short-distance physics. Actually, being the only super-renormalizable interaction in the Standard Model, this mass term can be viewed as a windowopen towards the influence of new and unknown high-energy or hidden sectors of the theory.New scalars M ( x ), neutral under the SM gauge group, can have renormalizable couplings tothe Higgs H : M ( x ) | H | . (1)This aspect was discussed by several authors and was dubbed “Higgs portal” in [1]. In thispaper we consider the possibility that this Higgs portal connects the Standard Model withsome hypothetical sector that breaks Lorentz invariance, such that M ( x ) has a space-timedependent vacuum expectation value (vev) varying on a characteristic small length-scale (cid:96) .Violation of Lorentz invariance is not uncommon in certain theories of quantum gravity, asin the presence of a space-time foam, and even in string theory. Alternatively the Higgs fielditself might be ‘foamy’, existing only in tiny islands of space-time. Or maybe its vev might be‘foamy’, being non-zero only in some regions, giving rise to a small average vev from a largerfundamental vev. In both cases an apparently constant Higgs vev is obtained at low energy,i.e. after averaging over length-scales much bigger than (cid:96) . Here we study the low energy signalsof these kinds of scenarios, performing concrete computations from the interaction in eq. (1).In section 2 we compute the Lorentz non-invarant dispersion relation satisfied by a scalaror a fermion with a non-constant mass M ( x ). In section 3 we develop a general technique toobtain the full effective Lagrangian. In section 4 we write RGE equations for the speed-of-lightof the various SM particles, finding how Lorentz-breaking in the Higgs sector propagates atloop level to all other particles, and how a strong coupling can dynamically restore the Lorentzsymmetry. In section 5 we consider the signals and constraints, and in section 6 we conclude. One of the consequences of the scenario we consider is a non-constant Higgs vev, and con-sequently a non-constant mass for SM particles. In order to obtain some physical intuitionabout our setting, we start by considering the propagation of a complex scalar particle or ofa Dirac fermion with masses that vary periodically in space.1 .1 Scalar
We study the case of a complex scalar H ( x ) with a squared mass M ( x ) that depends onlyon one spatial coordinate x . We assume that M ( x ) has period (cid:96) , and constant values M and M within intervals of length r (cid:96) and r (cid:96) (with 0 < r , < r + r = 1): M ( x ) = (cid:40) M for 0 < x (mod (cid:96) ) < r (cid:96)M for r (cid:96) < x (mod (cid:96) ) < (cid:96) . (2)According to the Floquet-Bloch theorem [2], as a result of the periodicity of M ( x ), thesolution of the Klein-Gordon equation is of the form H ( x, t ) = e − i ( Et − kx ) u ( x ) , (3)where u ( x ) also has periodicity (cid:96) . The Klein-Gordon equation for u ( x ) is given by (cid:34) d dx + 2 ik ddx + E − k − M ( x ) (cid:35) u ( x ) = 0 , (4)and has the solution u ( x ) = (cid:40) A e i ( k − k ) x + B e − i ( k + k ) x for 0 < x < r (cid:96)A e i ( k − k ) x + B e − i ( k + k ) x for r (cid:96) < x < (cid:96) , (5) k , ≡ (cid:113) E − M , . (6)Continuity of the function u ( x ) and of its first derivative at the matching points x = 0 and x = r (cid:96) imposes four constraints. Three of them determine the integration constants A , and B , up to an overall normalization, while the fourth equation defines the dispersion relation:cos( k(cid:96) ) = cos( k r (cid:96) ) cos( k r (cid:96) ) − k + k k k sin( k r (cid:96) ) sin( k r (cid:96) ) . (7)This equation describes the relation between energy E and momentum k . We see that k isfixed up to a 2 π/(cid:96) ambiguity.Since we are assuming that Lorentz violation is related to phenomena at very short dis-tance, we are interested in particle propagation for momenta much smaller than 1 /(cid:96) . In thislimit, the dispersion relation in eq. (7) can be written in the familiar form E = k c + m c , (8)where m ≡ M r + M r + O ( (cid:96) ) , c ≡ − r r (1 + 2 r r )360 ( M − M ) (cid:96) + O ( (cid:96) ) . (9)Therefore, when the particle is observed at momenta much smaller than 1 /(cid:96) , the effect ofa space-varying mass can be absorbed in a redefinition of its mass and of its “light speed”(or, more appropriately, of the maximal attainable velocity in the massless limit). While the2ass redefinition is unobservable (unless we have a theory in which particle masses can bepredicted), the redefinition of the “light speed” can be experimentally measured when theparticle propagation is compared with another particle with different value of c . Thus, ascalar particle with a space-varying mass, when viewed at low energies, appears as a particlewith a constant mass, given by the square root of the average of M ( x ), but with a modifiedrelation between energy and momentum.The Lorentz-violating effect trivially disappears when M → M , since the source ofLorentz violation vanishes in this limit. More interestingly, Lorentz violation also disappearswhen (cid:96) →
0. In this limit, the characteristic length of the mass variation becomes infinitelysmaller than the de Broglie wavelength of the particle. Since the source of Lorentz violation M ( x ) has dimension of mass squared, the adimensional correction to c must be suppressed bythe high scale Λ = 2 π/(cid:96) . Thereby, in our scenario, high-scale physics generates small Lorentz-breaking effects. This is unlike a generic Lorentz-breaking scenario (such as ‘space-time foam’),where one typically expects order unity deviations from c = 1 even from Lorentz-violation atthe Planck scale. The correction to c in eq. (9) is always negative and thus the “light speed”of a scalar is smaller than the canonical value. We can now repeat the discussion in the case of a fermion. Suppose that its Dirac massdepends on x with period (cid:96) , being constant within intervals of lengths r (cid:96) and r (cid:96)M ( x ) = (cid:40) M for 0 < x (mod (cid:96) ) < r (cid:96)M for r (cid:96) < x (mod (cid:96) ) < (cid:96) . (10)Again, we are considering only one space dimension. Exploiting the Floquet-Bloch theorem,we can decompose the fermion field as ψ ( x, t ) = e − i ( Et − kx ) u (+)1 ( x ) u ( − )2 ( x ) u ( − )1 ( x ) u (+)2 ( x ) , (11)where the 2-component spinors u ( ± ) are periodic functions. In the Weyl basis, the Diracequation becomes (cid:32) i ddx + E − k (cid:33) u (+) ( x ) = − M ( x ) u ( − ) ( x ) (cid:32) i ddx − E − k (cid:33) u ( − ) ( x ) = M ( x ) u (+) ( x ) . (12) Phenomenological analyses assume that c = 1 and focus on effects from higher-dimensional operators thatgrow with some unknown power of energy, although such effects at loop level also give rise to a power-divergentcorrection to c , and more generically to some of the Lorentz-violating operators of [3]. In our case we insteadneglect effects that grow with energy, because they are suppressed by more powers of (cid:96) . u (+) ( x ) = (cid:40) A e i ( k − k ) x − B M k + E e − i ( k + k ) x for 0 < x < r (cid:96)A e i ( k − k ) x − B M k + E e − i ( k + k ) x for r (cid:96) < x < (cid:96) (13a) u ( − ) ( x ) = (cid:40) − A M k + E e i ( k − k ) x + B e − i ( k + k ) x for 0 < x < r (cid:96) − A M k + E e i ( k − k ) x + B e − i ( k + k ) x for r (cid:96) < x < (cid:96) , (13b)where k , are again given by eq. (6). The conditions of continuity of the functions u ( ± ) atthe matching points x = 0 and x = r (cid:96) determine, up to their normalization, the integrationconstants A , and B , and the dispersion relationcos( k(cid:96) ) = cos( k r (cid:96) ) cos( k r (cid:96) ) + M M − E k k sin( k r (cid:96) ) sin( k r (cid:96) ) . (14)In the limit of small momenta, the energy-momentum relation is given by the familiar expres-sion E = k c + m c , where m ≡ M r + M r + O ( (cid:96) ) , c ≡ − r r M − M ) (cid:96) + O ( (cid:96) ) . (15)Again, at small momenta, the total effect of the space-varying mass can be parametrized by adistortion of the “light speed”. Notice that the “light speed” of the scalar deviates from thecanonical value at order (cid:96) while, for space-varying fermion masses, the effect comes alreadyat order (cid:96) . This is because the Lorentz-violating effect is always suppressed by two powers ofthe mass inhomogeneity, which amounts to ( M − M ) for the scalar and ( M − M ) for thefermion. Dimensional arguments then determine the different powers of (cid:96) . As in the case ofthe scalar, the correction to c in eq. (15) is negative and thus the maximal attainable velocityis smaller than the ordinary light speed. After having clarified the physical meaning of a particle with space-varying mass, we canproceed in the analysis of the Standard Model with a Lorentz-violating Higgs mass parameter.Our goal is to construct an effective theory valid at energies below a cutoff scale Λ, obtainedby integrating out the high-frequency modes. Here 1 / Λ represents the typical length of thevariations of the Higgs mass. Since Λ is the energy scale at which the Lorentz violation,originating in a hidden sector, is communicated to the Higgs field, we assume that Λ is muchlarger than the TeV scale. The space dependent mass M ( x ) mixes the low-frequency modes(with Fourier momentum k (cid:28) Λ) with the high-frequency ones. By integrating out the high-frequency modes, their effects is described at low energy by Lorentz-violating operators. Letus explain the procedure to derive the effective theory.
We denote with H the Higgs doublet and introduce in the Lagrangian a space-time dependentcomponent for its mass, L = − H † ( x ) (cid:104) ∂ + M + µ F (ˆ x ) (cid:105) H ( x ) . (16)4ere M is a mass parameter of the order of the electroweak scale, µ is a mass parameter(much smaller than the electroweak scale) parametrizing the amplitude of the space-timevarying component, and F is a generic order unity dimensionless function that modulatesthe space-time dependence. For simplicity we take F to depend on a single combination ofspace-time coordinates, ˆ x ≡ x · a − a , (17)where a is a fixed 4-vector. It is convenient to normalize ˆ x with − a because we have inmind a space-like fluctuation of the Higgs mass ( a < a > F is a real function with the followingthree properties. (i) It is periodic: F (ˆ x + 2 πn ) = F (ˆ x ) for any integer n . (ii) It is bounded: | F (ˆ x ) | ≤ (iii) It averages to zero within one period: (cid:82) π d ˆ xF (ˆ x ) = 0.Being periodic, the function F can be expanded in an infinite sum of Fourier modes, F (ˆ x ) = + ∞ (cid:88) n = −∞ f n e in ˆ x with f n ≡ π (cid:90) π d ˆ x F (ˆ x ) e − in ˆ x . (18)The Fourier coefficients f n are such that f ∗ n = f − n , because F is real, and such that f = 0,because of the property (iii) above. Moreover, the f n are real (imaginary) when F is an even(odd) function of ˆ x .It is convenient to work in Fourier space and express the Higgs field H ( x ) as H ( x ) = 1(2 π ) (cid:90) d k e ikx H ( k ) . (19)We can decompose the quadri-momentum k in terms of the quantities k (cid:107) ≡ k · a | a | , k ⊥ ≡ k − k · aa a, | a | ≡ √− a , (20)which have been defined such that k ⊥ · a = 0 and k = k ⊥ − k (cid:107) . Note that k (cid:107) is positive(negative) for space-like (time-like) fluctuations of the Higgs mass. The integration over k (cid:107) ofa generic function g ( k ) can be decomposed into an infinite sum of integrations within shellsof momenta ( n − / / | a | < k (cid:107) < ( n + 1 / / | a | for any arbitrary integer n : (cid:90) + ∞−∞ dk (cid:107) g ( k ⊥ , k (cid:107) ) = + ∞ (cid:88) n = −∞ (cid:90) (2 | a | ) − − (2 | a | ) − dk (cid:107) g (cid:32) k ⊥ , k (cid:107) + n | a | (cid:33) . (21)Using this expansion, the Higgs action in eq. (16) becomes S = (cid:90) d k ⊥ (cid:90) (2 | a | ) − − (2 | a | ) − dk (cid:107) + ∞ (cid:88) n = −∞ H † ( k (cid:107) + n | a | ) k ⊥ − (cid:32) k (cid:107) + n | a | (cid:33) − M H ( k (cid:107) + n | a | ) − µ ∞ (cid:88) n,m = −∞ f n − m H † ( k (cid:107) + n | a | ) H ( k (cid:107) + m | a | ) , (22)where the dependence of H on k ⊥ is understood.5he first line of eq. (22) is just the usual SM Lagrangian, that gives the ordinary Lorentz-invariant dispersion relation for each mode with momentum k . The term proportional to µ inthe second line of eq. (22) introduces a mixing between the different modes of the Higgs field.In particular, the zero mode of the Higgs field ( n = 0) mixes with every high-frequency mode n with a coefficient µ f n . Notice that the term proportional to µ generates only off-diagonalmixings, since f = 0. A non-periodic M ( x ) would give a continuous (rather than discrete)mixing, but the final result would be the same as long as the Fourier transform of M ( x )vanishes fast enough at k →
0, so that an unambiguous splitting between low-momentum andhigh-momentum modes still exists.Coming back to the periodic M ( x ), the low-energy effective theory is obtained by inte-grating out all modes of the Higgs field H with n (cid:54) = 0. This procedure leads to a non-trivialresult because of the mixing of the high-frequency modes with the zero mode. To obtain theeffective theory, it is convenient to express the high-frequency modes through their equationsof motion at first order in µ , H ( k (cid:107) + n | a | ) = µ f n k ⊥ − (cid:16) k (cid:107) + n | a | (cid:17) − M H ( k (cid:107) ) + high-frequency terms . (23)Replacing eq. (23) into eq. (22), retaining only terms involving zero modes ( n = 0) andexpanding the result for small | a | , we obtain the low-energy effective theory for the Higgsfield: S eff = (cid:90) d k H † ( k ) (cid:104) Z ( k − M ) − ∆ M − δc k (cid:107) (cid:105) H ( k ) , (24) Z = 1 − δc , ∆ M = 2 µ a ∞ (cid:88) n =1 | f n | n , δc = − µ a ∞ (cid:88) n =1 | f n | n . (25)In eq. (24), the integration is only over momenta k smaller than the cutoff Λ = 1 / | a | .While Z and ∆ M can be absorbed in the wave-function and mass definitions, δc leadsto a physical effect. Note that, for space-like variations of the Higgs mass ( a < M isnegative and one could imagine scenarios in which the electroweak breaking is triggered solelyby Lorentz-violating effects.In coordinate space, the new effect is described by a Lorentz-violating term: S eff = (cid:90) d x L eff , L eff = | ∂ µ H | − M | H | − δc H † ( a · ∂ ) a H. (26)This is a renormalizable interaction. Its coefficient δc ∼ µ / Λ is however suppressed by fourpowers of the cutoff scale, at which the Lorentz violation is communicated to the Higgs sector.This operator modifies the Higgs kinetic term in such a way that the “light speed” for theHiggs field along the direction identified by the quadri-vector a becomes c = 1 + δc. (27)By making specific assumptions on the function F (ˆ x ) that modulates the space-time de-pendence of the Higgs mass, we can explicitly calculate the expression of ∆ M and δc from6q. (25). For instance, if F (ˆ x ) = cos(ˆ x ), the Fourier coefficients are f ± = 1 / f n = 0 for n (cid:54) = ±
1. Hence, we obtain∆ M = − µ , δc = − µ Λ for F (ˆ x ) = cos(ˆ x ) . (28)Another example is the square-wave function F (ˆ x ) = (cid:40) +1 for 2 nπ < ˆ x < (2 n + 1) π − n + 1) π < ˆ x < (2 n + 2) π . (29)In this case f n = i [( − n − / ( πn ) and thus we obtain∆ M = − π µ , δc = − π µ for square wave F (ˆ x ) . (30)Notice that this expression of δc coincides with the result in eq. (9) obtained by solving theKlein-Gordon equation with variable mass, after the replacement r , = 1 / (cid:96) = 2 π/ Λ and | M − M | = 2 µ .So far we have studied the case in which the Lorentz violation identifies one special directionin space-time, but our results can be easily generalized. Actually the derivation of the effectivetheory used a generic 4-vector a and can be adapted to different cases. For instance, takinga 4-vector a with vanishing space components corresponds to the rotationally invariant case,which leads to a Lorentz-violating Lagrangian term2 δc H † (cid:126) ∇ H, δc ∼ − µ / Λ . (31)When M ( x ) has the symmetry of a cube (corresponding to the octahedral group), thedimension-4 effective operator is still of the form of eq. (31), exhibiting rotational symme-try. This is because the usual δ ij is the only two-index invariant tensor of both the octahedralgroup and the full SO(3) rotation group. The breaking of the rotational symmetry will ap-pear only in higher dimensional operators. Another case leading to eq. (31) is the one inwhich M ( x ) is a randomly-varying function. This gives the rotationally invariant effectiveoperator, just like the random motion of molecules gives, on average, a rotationally invariantrefraction index of air. In the following, just for simplicity, we focus on the rotationally-symmetric case, which contains all the important features of Higgs-induced Lorentz violation.Finally note that, after taking into account gauge corrections, the operator in eq. (31) getsgauge-covariantized as usual, (cid:126) ∇ → (cid:126)D = (cid:126) ∇ + ig (cid:126)A . The construction of the low-energy effective Lagrangian for the Higgs field can now be extendedto the full Standard Model. Each Standard Model field is expanded in Fourier space and thehigh-frequency modes are integrated out. This procedure can be carried out with the helpof the equations of motion, as discussed above, or, more simply, with the Feynman diagramtechnique. The propagator of the n -mode Higgs field, expanded for small | a | , is given by ia n (cid:34) − k (cid:107) | a | n − (4 k (cid:107) + k − M ) a n + O ( a ) (cid:35) . (32)7 H (cid:72) n (cid:76) H Ψ HH (cid:72) n (cid:76) Ψ (cid:72) n (cid:76) H (cid:72) n (cid:76) H Ψ Figure 1:
The diagrams generating the effective Lorentz-violating interactions, obtained af-ter integrating out the high-frequency modes, for the terms with two Higgs bosons (a) andtwo fermions (b). Single lines denote propagators of low-frequency modes for the Higgs bo-son (dashed line) and the fermion (solid line). Double lines denote propagators of the high-frequency modes, as given by eq. (32) for the Higgs boson and by eq. (34) for the fermion. Thedot denotes the mixing between high- and low-frequency modes, as given by eq. (33).
The mixing between the zero-mode and any n -mode of the Higgs field corresponds to a massinsertion iµ f n . (33)A summation (cid:80) + ∞ n = −∞ is required in diagrams with high-frequency modes in the internal lines.Using these rules, we can easily recover the Lagrangian in eq. (24) from the Feynmandiagram in fig. 1a. We can then extend the calculation to the other Standard Model fields.The Lorentz violation, originally residing in the Higgs mass term, is communicated to quarksand leptons through the tree-level diagram of fig. 1b. The propagator of the n -mode fermionfield, expanded for small | a | , is given by i (cid:54) an + ia n ( (cid:54) k + m ) − i | a | (cid:54) ak (cid:107) n + O ( a ) , (34)where m is the mass of the fermion field. Hence, we obtain that the diagram in fig. 1b generatesthe following effective operator connecting two Higgs ( H ) and two fermion fields ( ψ ), i(cid:15) ψ ( H † ¯ ψ ) (cid:54) aa · ∂a ( ψH ) , (cid:15) ψ = 12 λ † λµ a ∞ (cid:88) n =1 | f n | n . (35)Here λ is the corresponding Yukawa coupling. If ψ is a weak singlet, contractions of SU(2) L indices give | H | ; if ψ is a weak doublet, ψ (cid:104) H (cid:105) is the component of ψ that gets mass from theYukawa λ .In the special cases in which the modulating function F (ˆ x ) is a cosine or a square wave, (cid:15) ψ becomes (cid:15) ψ = − λ † λ µ Λ × (cid:40) F (ˆ x ) = cos(ˆ x )51 π / F (ˆ x ) . (36)The operator in eq. (35), after electroweak symmetry breaking, gives a modification of the“light speed” of the fermion along the direction identified by a : c = 1 + (cid:15) ψ v . (37)8ere v = (cid:104) H (cid:105) is the Higgs vacuum expectation value. In sect. 2 we found that a space-dependent fermion mass gives a distortion to the “light speed” already at order a . Instead,here we are finding that, if the Lorentz violation originates in the Higgs mass, the effect forthe fermions starts only at order a . The reason is that, in such a case the inhomogeneity in v and consequently in the fermion mass m = λv , is itself suppressed by two powers of a , due tothe effect of the Higgs kinetic term. From an effective theory point of view, the Higgs vacuumexpectation value is constant and does not break Lorentz invariance. Therefore fermions canfeel the effect of Lorentz violation only through higher dimensional operators, like the one ineq. (35), induced by the mixing between the zero mode and the high-frequency modes of theHiggs field.At tree level, the Lorentz violation is communicated to gauge fields through diagramsinvolving Higgs and gauge particles. These diagrams have the effect of making the derivativescontained in eq. (26) covariant under the gauge group and also generate some new higher-dimensional operators.In summary, we have considered the effects of a space-time varying Higgs mass in theStandard Model. The source of Lorentz violation is expressed in terms of two parameters: µ , which characterizes the amplitude of the mass square variations, and | a | (or 1 / Λ), whichdefines the wavelength (or frequency) of these variations. The most appropriate language toaddress the problem is that of an effective low-energy field theory, valid below the scale Λ.In the effective theory, all the effects are induced by the mixing of the Higgs modes, which isproportional to µ / Λ . The Higgs vacuum expectation value and all masses are constant in theeffective theory, but the Higgs kinetic term is modified by a Lorentz-violating operator. Thisoperator is renormalizable, but its coefficient is proportional to µ / Λ , and thus suppressed byfour powers of the cutoff scale. All other effects can be written in terms of higher-dimensionaloperators, suppressed by additional powers of Λ. For instance, the Lorentz-violating effectsin the fermion kinetic term are proportional to µ v / Λ . These conclusions are based ontree-level considerations. Now we turn to discuss the effects of quantum corrections. Let us consider the effective Lagrangian containing the dominant rotationally invariant dimension-4 Lorentz-violating operators, in the rest frame of the Lorentz-breaking sector L = L SM − δc H | ( (cid:126) ∇ + ig (cid:126)A ) H | − (cid:88) A δc A F i − (cid:88) ψ δc ψ i ¯ ψ (cid:126)γ · ( (cid:126) ∇ + ig (cid:126)A ) ψ. (38)Here A = { Y, W a , G a } describes the SM gauge bosons, ψ are the 15 SM Weyl (chiral) fermionmultiplets ( L, E, Q, U, D , appearing in 3 generations) and H is the scalar Higgs. The coef-ficients δc are the corrections to their ‘speed-of-light’. Each of the various δc can be set to The most generic Lorentz-breaking Lagrangian in the notations of [3] can be reduced to this form setting( k φφ ) = − δc H , ( k F ) i i = ( k F ) i i = − ( k F ) ii = − ( k F ) i i = δc A , ( c ψ ) = δc ψ . All other tensors vanishand all other components of these tensors vanish. In general, such tensors describe non-isotropic Lorentzviolation [3]. Performing Lorentz transformations on eq. (38) the other components of the c ψ , k F , k φφ tensorsare generated as dictated by their Lorentz structure. AH Ψ ΨΨ A Figure 2:
Loop diagrams feeding the original Lorentz violation into the gauge and fermionsectors.The dots denote all possible insertions of Lorentz-violating operators, to be performedone-by-one. zero by rescaling time: t → (1 + δc/c ) t ; the difference between the δc of different particles hasphysical meaning. At tree-level only δc H is non-zero, and it only affects the Higgs velocityand the W, Z masses, which are negligibly probed by experiments.At one-loop level δc H induces a correction to the speed-of-light of SU(2) L and hyperchargeelectroweak vectors A The propagator of these vectors, taking into account the one loopcorrection of fig. 2a, is − i Π µν = [ p µ p ν − p η µν ] c A − b H g A (4 π ) (cid:18) (cid:15) + ln µ Λ (cid:19) [ p µ p ν − p η µν ] c H , (39)that differs from the standard expression only because we specified the speed of light to beused in the various terms, e.g. p = E /c − (cid:126)p . The group theory coefficient is b H = 1 / L and U(1) Y (normalized such that the H hypercharge is 1 / δc (cid:39) g π δc H ln Λ m h , δc (cid:39) g Y π δc H ln Λ m h (40)where m h is the physical Higgs mass and c γ = c cos θ W + c sin θ W for the photon. After afurther loop correction, one also gets a δc for the SM fermions (fig. 2b): δc ψ ∼ δc , g (4 π ) ln Λ m h . (41) The loop effects are best described by a system of RGE for the speed-of-light of the variousSM particles, that allows us as usual to re-sum the log-enhanced corrections.We consider a generic theory with particles p (gauge vectors A , Weyl fermions ψ and scalars H ) interacting among them with gauge couplings g A , Yukawa couplings λ . The quartic scalarcouplings do not enter in the RGE under consideration. We find that the RGE equations fortheir maximal speeds are: (4 π ) d c A d ln µ = 2 g A (cid:88) p b p ( c A − c p ) (42a) RGE equations for Lorentz-violating tensors have been already computed for QED in ref. [4]. Renormal-izability of theories with higher-dimensional Lorentz-violating operators has been studied in [5]. π ) d c ψ d ln µ = 163 (cid:88) A g A C A ( c ψ − c A ) + (cid:88) ψ (cid:48) ,H λ ψψ (cid:48) H ( c ψ − c ψ (cid:48) − c H π ) d c H d ln µ = 4 (cid:88) A g A C A ( c H − c A ) + (cid:88) ψ,ψ (cid:48) λ ψψ (cid:48) H (2 c H − c ψ − c ψ (cid:48) ) (42c)where b p are the well known coefficients that enter in the RGE for the gauge couplings:(4 π ) dg A d ln µ = b A g A b A = (cid:88) p b p = − T + 23 T / + 13 T . (43)The group factors are defined as Tr T a T b = T δ ab ( T = 1 / n ),and T = n for the adjoint) and as ( T aA T aA ) ij = C A δ ij ( C = q for a U(1) charge; C = 3 / C = 4 / c p = c is RGE invariant.In the Standard Model, summing over the three generations of E, L, U, D, Q and using theGUT normalization g = (cid:113) / g Y , the RGE are:(4 π ) d c H d ln µ = 35 g ( c H − c ) + 3 g ( c H − c ) (44a)(4 π ) d c d ln µ = g [8 c − c Q − c U − c D ] (44b)(4 π ) d c d ln µ = g [25 c − c L − c Q − c H ] / π ) d c d ln µ = g [41 c − c H − c D − c E − c L − c Q − c U ] / π ) d c E d ln µ = 165 g ( c E − c ) (44e)(4 π ) d c L d ln µ = 45 g ( c L − c ) + 4 g ( c L − c ) (44 f )(4 π ) d c Q d ln µ = 649 g ( c Q − c ) + 445 g ( c Q − c ) + 4 g ( c Q − c ) (44g)(4 π ) d c U d ln µ = 649 g ( c U − c ) + 6445 g ( c U − c ) (44h)(4 π ) d c D d ln µ = 649 g ( c D − c ) + 1645 g ( c D − c ) . (44 i )We neglect here the effect of Yukawa couplings. After the breaking of the electroweak sym-metry, the SM fermions acquire Dirac masses m . In the non-relativistic limit, the speed-of-light c L and c R of the left and right handed components become the fermion speed-of-light c = ( c L + c R ) / ∼ ( c L − c R ) (cid:126)p · (cid:126)σ . At first orderin perturbation theory around the Lorentz-symmetric state, (cid:126)p · (cid:126)σ changes the energy of onegiven state by an amount proportional to its matrix element, which vanishes being odd in (cid:126)p .11 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Μ in GeV ∆ c (cid:72) Μ (cid:76) (cid:144) ∆ c H (cid:72) (cid:76) (cid:76) H U (cid:72) (cid:76) Y SU (cid:72) (cid:76) L SU (cid:72) (cid:76) c LEQUD Γ e Ν
3u d
Figure 3:
Renormalization group evolution of the the speed-of-light of the various SM particlesfrom
Λ = 10 GeV to the weak scale and then down to the QCD scale.
In sect. 3.1 we have found that Lorentz violation from the Higgs portal predicts, at tree leveland at the RGE scale Λ ∼ /a , a negative correction to the Higgs speed-of-light, δc H ≈− µ / Λ . Fig. 3 shows how the original Lorentz violation feeds into the various SM particlesthrough the RG evolution from Λ = 10 GeV to the weak scale and then down to the QCDscale Λ
QCD .We note an interesting generic feature of the RGE: in the limit in which a gauge couplingbecomes strong, all values of c of particles charged under the gauge group become equal. Inparticular, when the QCD coupling becomes strong, g → ∞ at µ ∼ Λ QCD , all colored parti-cles reach a common c : speed differences get exponentially suppressed by exp ( − k (cid:82) g d ln µ )factors, where k is a numerical constant. To compute the common c , one notices that thestrong coupling does not renormalize the combination 16 c +9 (cid:80) q c q (summed over light quarks q ). This means that Lorentz invariance can be dynamically emergent if all SM particles felt atsome energy a strong coupling. This could be possible in a SU(5) model such that the unifiedcoupling runs to a large enough value.Fig. 4 shows the predictions for the modifications of the speed-of-light of the stable SMparticles in units of the correction relative to the Higgs, as functions of Λ. The pattern isqualitatively similar for all values of Λ, and the main effect is a slower speed-of-light for thephoton than for other SM particles: c γ < c ν < c e < c n,p . This pattern is mainly probed by the following observations, and fig. 5 summarizes the result-ing bounds on the Higgs portal parameters Λ = 1 /(cid:96) and µ :12 (cid:37) (cid:37) (cid:37) (cid:37) (cid:37) inhomogeneity scale (cid:76) in GeV ∆ c (cid:144) ∆ c H Γ e Ν p Figure 4:
Higgs-portal predictions for the speed-of light δc p = c p − c of the stable SM particles p = { γ, e, ν, p } in units of the correction to the Higgs speed-of-light δc H ∼ − µ / Λ . • Proton vacuum ˇCerenkov radiation, namely p → pγ decays, would become kinematicallyallowed at E p > m p / √ c p − c γ [6]. Protons have been observed in cosmic rays up to E p ∼ TeV, and thus c p − c γ < . × − [6, 7]. This bound is plotted as a dashedline: the change in its slope arises because we only consider cosmic ray energies belowthe cut-off Λ of our theory. • Similarly, electron vacuum ˇCerenkov radiation, namely e → eγ decays, would becomekinematically allowed at E e > m e / √ c e − c γ , but cosmic ray electrons have been observedup to 2 TeV, so that c e − c γ < − [8]. • A stronger constraint arises from the γγ (cid:48) → e − e + process, in which energetic γ areabsorbed when traveling in a background of low energy γ (cid:48) . The process becomes kine-matically allowed if E (cid:48) γ > m e /E γ + E γ ( c e − c γ ) /
2. Observations of cosmic rays photonsup to E γ ∼
20 TeV imply c e − c γ < m e /E γ ∼ . × − [9, 8]. • Furthermore, the agreement of electron synchrotron radiation at the LEP acceleratorwith its standard expression implies | c e − c γ | < × − [10]. A numerically similarbound can be deduced from astrophysical observations of Inverse Compton and syn-chrotron radiation [11]. • Stability of various types of spectral lines despite the motion of the earth implies strongbounds on Lorentz-violating operators [12], but not on the δc operators present in ourscenario (also considered in [6]), at leading order in δc . Subdominant bounds are listedin ref. [13]. Theoretical plausibility suggests a generic much stronger bound on Lorentz-violating scenarios, including µ of theHiggs mass: a) µ at the weak scale; b) µ such that its contribution to the Higgs mass in thelow-energy effective theory ∆ M ∼ µ (cid:96) (negative for space-like inhomogeneities) of eq. (25)is at the weak scale. In such a case electroweak symmetry breaking could be a byproduct ofinhomogeneities; the result ∆ M (cid:28) µ holds because inhomogeneities in the Higgs vev aresuppressed by the Higgs kinetic term | ∂ µ H | ∼ v /(cid:96) .We also considered sub-leading effects, suppressed by powers of E/ Λ, which lead to vari-ations in the speed-of-light that depend on the energy E . The main experimental constraintson such effect are: | c γ ( E ) − c γ ( E (cid:48) ) | < ∼ (cid:40) − at E, E (cid:48) ∼ . − at E, E (cid:48) ∼ TeV [16] (45)Such bounds are not competitive. Furthermore, in our scenario (and actually more in gen-eral) also the dominant effects depends on energy, due to the logarithmic RGE running of c . Since | δc H | (cid:29) | δc γ | in our model, c γ has a sizable RGE running above the weak scale, d ln c γ /d ln µ ∼ − , and a much slower running at lower energies below Higgs decoupling, d ln c γ /d ln µ ∼ − . The limits in eq. (45) thereby imply a bound | δc γ | < − , which isagain not competitive with the constraints previously described.The main qualitative point is that Lorentz violation in the Higgs sector must be suppressedby a scale well above the electroweak scale. This means that various possible solutions to theHiggs mass hierarchy problem that one can invent using Lorentz violation (e.g. assuming thatthe Higgs is a 2d field localized on strings that fill the space; or adding spatial gradients | (cid:126) ∇ H | to the Lagrangian) are experimentally too strongly constrained to make the weak scalenaturally small. Lorentz violation is often phenomenologically studied by considering only non-renormalizableoperators leading to corrections to the speed-of-light of the form δc ∼ ( E/ Λ) p , where p = 1or 2 and Λ is some high-energy scale, maybe the Planck scale. On the theoretical side, onceLorentz symmetry is broken, one expects that the renormalizable terms are also stronglyaffected (at least after that quantum corrections are taken into account), such that there areorder-unity differences in the speed-of-light of different particles, δc ∼
1, in dramatic contrastwith the experimental bound | δc | < − .In this paper we have considered a specific and well-defined source of Lorentz violation. Itoriginates in a hidden sector and it is communicated to the Standard Model through the Higgs the one we considered. As emphasized in [14], whatever breaks Lorentz invariance has an energy density whichcouples to gravity, but cosmological observations suggest the presence of a Lorentz-invariant vacuum energydensity, ρ ∼ meV . Such a small cosmological constant poses a puzzle even to Lorentz invariant scenarios,and the only known way out is a cancellation requiring a very fine tuning. In presence of Lorentz breaking,such cancellations seem to need a fluid with negative energy density. Including quantum corrections to thevacuum energy up to experimentally probed energies around the weak scale v , one needs to impose δc v < ρ such that δc < − . inhomegeneity scale (cid:76) (cid:61) (cid:144) (cid:123) in GeV L o r e n t z (cid:45) b r ea k i ng H i gg s m a ss Μ i n G e V excluded Μ (cid:187) v (cid:68) M (cid:187) v Figure 5:
Bounds from searches for Lorentz-violation on the inhomogeneous Higgs mass µ andon its inhomogeneity scale /(cid:96) . The green dotted bands indicate weak-scale-related values ofthe Lorentz-breaking Higgs mass µ or of the induced effective Higgs mass. portal: only the Higgs mass term µ | H | violates Lorentz invariance, being inhomogeneouson small scales 1 / Λ. This naturally leads to a small correction to the Higgs speed-of-light, δc H ∼ − ( µ/ Λ) . We computed this effect at tree level in two ways: i) in section 2 wehave solved the propagation equations in a simple inhomogenous background; ii) in section3 we have derived an effective Lagrangian: inhomogeneities lead to mixing between low andhigh-frequency modes, so that the integration out of the high-frequency modes gives a Lorentz-violating effective operator, | ( (cid:126) ∇ + ig (cid:126)A ) H | . Fermions are affected only by higher dimensionoperators, which we have computed.At loop level, δc H propagates to all other SM particles via a system of RGE equations fortheir speed-of-light; fig. 3 shows a typical solution. An interesting feature is that the strongcoupling dynamically drives the speed-of-light of all colored particles to a common value. Thesignals and bounds of our scheme of Lorentz violation were explored in section 4. Acknowledgements
We thank Leonardo Giusti, Riccardo Rattazzi, and Michele Redi fordiscussions. This work was supported by the ESF 8090 and SF0690030s09.
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