Lorentz Invariance Violation Matrix from a General Principle
aa r X i v : . [ h e p - ph ] S e p November 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
Lorentz Invariance Violation Matrix from a General Principle ∗ ZHOU LINGLI
School of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, [email protected]
BO-QIANG MA
School of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, [email protected]
Received (Day Month Year)Revised (Day Month Year)We show that a general principle of physical independence or physical invariance of math-ematical background manifold leads to a replacement of the common derivative operatorsby the covariant co-derivative ones. This replacement naturally induces a backgroundmatrix, by means of which we obtain an effective Lagrangian for the minimal standardmodel with supplement terms characterizing Lorentz invariance violation or anisotropyof space-time. We construct a simple model of the background matrix and find that thestrength of Lorentz violation of proton in the photopion production is of the order 10 − . Keywords : Lorentz invariance, Lorentz invariance violation matrixPACS Nos.: 11.30.Cp, 03.70.+k, 12.60.-i, 01.70.+w
1. Introduction
Lorentz symmetry is one of the most significant and fundamental principles inphysics, and it contains two aspects: Lorentz covariance and Lorentz invariance.Nowadays, there have been increasing interests in Lorentz invariance Violation (LV)both theoretically and experimentally (see, e.g., Ref. 1). In this paper we find out ageneral principle, which provides a consistent framework to describe the LV effects.It requires the following replacements of the ordinary partial derivative ∂ α andcovariant derivative D α by the co-derivative ones ∂ α → M αβ ∂ β , D α → M αβ D β , (1)where M αβ is a local matrix. In the following, we introduce this general principleat first, and then explore its physical implications and consequences. ∗ Published in Mod. Phys. Lett. A 25 (2010) 2489 - 24991 ovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Zhou L. L., B.-Q. Ma
2. Principle of Physical Invariance
Principle: Under any one-to-one transformation X → X ′ = f ( X ) on mathematicalbackground manifold, the transformation ϕ ( · ) → ϕ ′ ( · ) of an arbitrary physical field ϕ ( X ) should satisfy ϕ ′ ( X ′ ) = ϕ ( X ) . (2)Most generally, this principle can be handled in geometric algebra G (or Clif-ford algebra) and geometric calculus (see, e.g., Refs. 2, 3). The general element ingeometric algebra is called a multivector, and addition and various products of twomultivectors are still a multivector, i.e. geometric algebra is closed. Different vari-ables in physics, such as scalar, vector, tensor, spinor, twistor, matrix, etc., can bedescribed by the corresponding types of multivectors in a unified form in geometricalgebra (see, e.g., Refs. 2, 3, 4, 5 for details). ϕ ( X ) ∈ G is a multivector-valuedfunction of a multivector variable X ∈ G , and it can be decomposed in an arbitrarylocal coordinate system with the basis vectors { e , e , . . . , e n } as ϕ ( X ) = ϕ + n X α =1 ϕ α e α + · · · + X α < ··· <α n ϕ α ··· α n e α ∧ · · · ∧ e α n ≡ X J ϕ J e J , where “ ∧ ” is called wedge product, e J ≡ e α ∧ · · · ∧ e α r , with J = { α , · · · , α r } for r = 1 , · · · , n , and e ≡ J = 0. Here ϕ J is the components of ϕ ( X ) and J denotes tensor index. We should not be frustrated by the operation algorithms ingeometric algebra, but could consider a general multivector ϕ just as an ordinaryvector and that two multivectors ϕ and ψ are not commutable, namely ϕψ = ψϕ ,somehow like matrices. Remembering these two points will suffice the following dis-cussions. What should be kept in mind is that in this new framework, the commonlyused tensor notation ϕ α ··· α n only represents the component of ϕ with respect to aspecific coordinate basis. Conventionally one simply means a multivector ϕ by itscomponent ϕ α ··· α n and regards the transformation rules for these components fromone basis to another as the transformation of a multivector. However, a multivectorshould be coordinate-independent. For instance, ϕ = ϕ α e α is a multivector anddoes not change under any coordinate transformation but its component ϕ α does.This general principle actually makes the field ϕ ( X ) represent a physical dis-tribution, rather than a common mathematical function. Although the uniquenessof reality can be mathematically described in many ways like ϕ ( X ), ϕ ′ ( X ′ ), and ϕ ′′ ( X ′′ ), · · · , the physics behind remains unchanged, saying independence or invari-ance. So (2) claims Physical Independence or Physical Invariance (PI) on mathe-matical background manifold.For the field ϕ ( X ) satisfying (2), its derivative field might be naively defined as π ( X ) = ∂ X ϕ ( X ) . But does this definition still fulfil (2)? If π ( X ) is a physical field, there must be theovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Lorentz Invariance Violation Matrix from a General Principle condition π ′ ( X ′ ) = π ( X ) according to PI. However, π ( X ) = ∂ X ϕ ( X ) = ∂ X ϕ ′ ( X ′ )= ∂ X f ( X ) ∗ ∂ X ′ ϕ ′ ( X ′ )= F ( ∂ X ′ ) ϕ ′ ( X ′ ) = π ′ ( X ′ ) , where we define F ( · ) ≡ ∂ X f ( X ) ∗· , and F ( · ) is linear to “ · ”. So the above-mentioneddefinition of π ( X ) fails to satisfy PI. The problem arises from the derivative withrespect to X . Therefore we redefine the derivative field as π ( X ) ≡ M ( ∂ X ) ϕ ( X ) , where M ( · ) is linear to “ · ” and has the covariant transformation property M ( · ) → M ′ ( · ) = M ( F ( · )) . Thus we have π ′ ( X ′ ) = M ′ ( ∂ X ′ ) ϕ ′ ( X ′ )= M ( F ( ∂ X ′ )) ϕ ′ ( X ′ )= M ( ∂ X f ( X ) ∗ ∂ X ′ ) ϕ ′ ( X ′ )= M ( ∂ X ) ϕ ( X )= π ( X ) . According to (2), π ( X ) is now indeed a physical field. We realize the principle of PIvia the introduction of M ( ∂ X ) and the replacement of ∂ X → M ( ∂ X ) .
3. Background Matrix
The derivations in Section 2 are coordinate-free. If concrete calculations are con-cerned, we may specify a coordinate system with basis { e I } , and the completenessrelation leads to M ( e J ) = e I e I ∗ M ( e J ) = e I M IJ , where M IJ ≡ e I ∗ M ( e J ). We call M IJ the Background Matrix (BM) field.If we choose another coordinate frame { e I ′ } , the coordinate transformation is e I ′ = T ( e I ) = e J e J ∗ T ( e I ) = e J T JI , with T JI ≡ e J ∗ T ( e I ) being the coordinatetransformation matrix. Therefore, M IJ is accordingly transformed to M IJ → M I ′ J ′ = e I ′ ∗ M ( e J ′ ) = T ( e I ) ∗ M ( T ( e J ))= ( e K T IK ) ∗ M ( e L T JL )= T IK ( e K ∗ M ( e L )) T JL = T IK M KL T JL . ovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Zhou L. L., B.-Q. Ma
So the component M IJ is coordinate-dependent and transforms in the same way asa common tensor.Furthermore for ∂ X , we have M ( ∂ X ) = M ( e J ∂ J ) = M ( e J ) ∂ J = e K M KJ ∂ J , where the second and third items result from the facts that ∂ J is a scalar operatorand M ( · ) is linear to “ · ”. So the replacement for the component ∂ J of a multivector ∂ X is ∂ K → M KJ ∂ J . We now turn our attention to the physical implications of the BM and show thatit contains LV and information of anisotropy of spacetime. But ahead of that, forthe integrity and consistence of a complete framework, we further provide anotherprinciple aside from the one of PI for the discussion of covariant derivatives andvarious gauge fields related with local symmetries.If a symmetry group is local to manifold, we must define a covariant derivativeoperator to maintain the covariance of the Lagrangian under gauge transformations.If there exists a scalar operator D J , which applies to two arbitrary fields ϕ ( X ) and ϕ ( X ), D J ( ϕ ( X ) ϕ ( X )) = D J ϕ ( X ) ϕ ( X ) + ϕ ( X ) D J ϕ ( X ) ,D J ϕ ( X ) = ∂ J ϕ ( X ) , if ϕ ( X ) is a scalar , (3)we demand the principle of covariance: Under the transformation ϕ ( X ) → R ( X ) ϕ ( X ) , or ϕ ( X ) → R ( X ) ϕ ( X ) R − ( X ) , there is a corresponding transformation, D J → D J ′ , such that D J ′ ( R ( X ) ϕ ( X )) = R ( X ) D J ϕ ( X ) , or D J ′ ( R ( X ) ϕ ( X ) R − ( X )) = R ( X ) D J ϕ ( X ) R − ( X ) , where R ( X ) is an invertible multivector in G , standing for various local symmetries,with specific matrix representations like SU(N) or SO(N). The operator D J is namedas covariant derivative, and the principle of covariance can determine the forms of D J and further introduce gauge fields with respect to the local symmetry R ( X ) inthe framework of geometric algebra. D J in geometric algebra may have different forms in the Standard Model (SM),differential geometry, and general relativity, just as a symmetry group may havedifferent representations. What we want to emphasize is that all definitions areequivalent and can be unified in geometric algebra. For example, from the viewpointof general relativity, for a general multivector ϕ ( X ), we have D J ϕ ( X ) = D J ( ϕ K e K ) = D J ϕ K e K + ϕ K D J e K = ∂ J ϕ K e K + ϕ K Γ IJK e I = ( ∂ J ϕ I + Γ IJK ϕ K ) e I , ovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Lorentz Invariance Violation Matrix from a General Principle where D J e K = Γ IJK e I , with the coefficient Γ IJK named as connection. Usually wedefine D J ϕ I ≡ ∂ J ϕ I + Γ IJK ϕ K as the covariant derivative in general relativity.However in geometric algebra, we abandon this specific definition, and the property(3) and the principle of covariance give an alternative choice.Till now, we have all the physical and mathematical preparations ready. Letus pause here and briefly sum up the basic ideas in our paper. (i) When requiringthe property of PI for an arbitrary field, we must introduce a local matrix M IJ to modify a derivative ∂ I to a co-derivative M IJ ∂ J . (ii) When promoting a globalsymmetry to a local one, we have to change a derivative ∂ I to a covariant derivative D I and acquire gauge fields. Altogether these two considerations straightforwardlylead us to a new covariant co-derivative operator ∂ I → M IJ ∂ J → M IJ D J or ∂ I → D I → M IJ D J . This generation is the essence for the origin of the LV termsin the SM.When considering both the principles of PI and of covariance, we get M ( D X ) = M ( e J D J ) = M ( e J ) D J = e K M KJ D J , with the coordinate-free covariant multivector derivative D X ≡ e J D J , and D J beingthe component of D X . So we arrive at our replacement for the covariant multivectorderivative D X , D X → M ( D X ) , (4)and the replacement for its component D J , D K → M KJ D J . (5)The replacements (4) and (5) are the consequences of the principles of bothPI and covariance. The first principle indicates the existence of the BM, and thesecond is important to introduce covariant derivatives, local symmetries and gaugefields. For the goal of this paper, to explore the BM and its physical implications,the principle of PI is enough. But for completeness and clearness, we simply discussthe principle of covariance. Now, we move on to spacetime, which can be part ofgeneral geometric algebra space. So X is replaced by spacetime coordinate x , andthe indices are explicitly denoted by α, β instead of I, J .
4. Standard Model Supplement
Section 3 provides the essentials to construct a mathematical-background-manifold-free and coordinate-free framework for physics. One of the significant results is thatin order to satisfy the principle of PI, the common derivative ∂ α and covariantderivative D α must be generalized to M αβ ∂ β and M αβ D β , with M αβ being theBM. Except that, other basic fields remain untouched, because they do not involvewith these two derivatives. In this section, we follow this scheme and focus on thephysical implications and consequences from these new introduced co-derivatives M αβ ∂ β and M αβ D β .ovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Zhou L. L., B.-Q. Ma
The effective Lagrangian of the minimal SM L SM consists of the following fourparts L SM = L G + L F + L H + L HF , L G = − F aαβ F aαβ , (6) L F = i ¯ ψγ α D α ψ, (7) L H = ( D α φ ) † D α φ + V ( φ ) . (8)Here ψ is fermion field, φ is Higgs field, and V ( φ ) is its self-interaction. F aαβ ≡ ∂ α A aβ − ∂ β A aα − gf abc A bα A cβ , D α ≡ ∂ α + igA α , and A α ≡ A aα t a , with g being thecoupling constant, f abc the structure constant, and t a the generator of gange groupsrespectively. L HF is the Yukawa coupling between fermion and Higgs fields, whichis not related to ∂ α and D α , so it keeps unchanged under the replacement (1). Thechiral difference and the summation of chirality and gauge index are omitted herefor simplicity.We divide M αβ into two parts M αβ = g αβ + ∆ αβ , with g αβ as the metric ofspacetime. (This decomposition will be fully discussed in the next section.) Under(1), the Lagrangians in (6), (7), and (8) become L G = −
14 ( M αµ ∂ µ A aβ − M βµ ∂ µ A aα − gf abc A bα A cβ ) × ( M αµ ∂ µ A aβ − M βµ ∂ µ A aα − gf abc A bα A cβ )= − F aαβ F aαβ + L GV , (9) L F = i ¯ ψγ α M αβ D β ψ = i ¯ ψγ α D α ψ + L FV , (10) L H = ( M αµ D µ φ ) † M αν D ν φ + V ( φ )= ( D α φ ) † D α φ + V ( φ ) + L HV , (11)with the condition M αβ being real matrix to maintain the Lagrangian hermitian.The last three terms L GV , L FV , and L HV in (9), (10), and (11) are the supplementterms for the minimal SM, reading L GV = −
12 ∆ αβ ∆ µν ( g αµ ∂ β A aρ ∂ ν A aρ − ∂ β A aµ ∂ ν A aα ) − F aµν ∆ µα ∂ α A aν , (12) L FV = i ∆ αβ ¯ ψγ α ∂ β ψ − g ∆ αβ ¯ ψγ α A β ψ, (13) L HV = ( g αµ ∆ αβ ∆ µν + ∆ βν + ∆ νβ )( D β φ ) † D ν φ. (14)Thus L SM is modified to an effective Lagrangian of the SM with supplementterms (SMS) L SMS , L SMS = L SM + L LV , with L LV ≡ L GV + L FV + L HV . ovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Lorentz Invariance Violation Matrix from a General Principle L SMS satisfies the invariance of gauge group SU(3) N SU(2) N U(1) and the invari-ance of PI. All the terms above in the Lagrangians are Lorentz scalars at morefundamental level than the minimal SM. But for the SM, ∆ αβ is treated as cou-pling constants or background influences from this more fundamental theory, andall the other fields for the SM are what we are studying, so L LV is not Lorentz in-variant under the observer’s Lorentz transformation on these fields. From this pointof view, we call the supplement term L LV the Lorentz invariance violation term,and it contains the information of LV or anisotropy of spacetime in the SM.To achieve a deeper insight and clearer understanding for the SMS here, let usmake a comparison with the commonly used Standard Model Extension (SME) 6and try to figure out the relations of the various coupling constants. Keeping theconventions in Ref. 6 and omitting detailed derivations, we summarize our resultsin Table 1. Table 1. Comparison of the Standard Model Supplement (SMS) and the Standard Model Ex-tension (SME) in Ref. 6. The notation h·i means the vacuum expectation value. The subscripts A and B denote the flavors of particles, and G , W , and B mean SU(3), SU(2), and U(1) gaugefields respectively.SMS SME L FV L CPT − evenlepton + L CPT − oddlepton , L CPT − evenquark + L CPT − oddquark L GV L CPT − evengauge + L CPT − oddgauge L HV L CPT − evenHiggs + L CPT − oddHiggs h ∆ µν i δ AB ( c L ) µνAB , ( c R ) µνAB , ( c Q ) µνAB , ( c U ) µνAB , ( c D ) µνAB g h ∆ µν A ν i δ AB ( a L ) µAB , ( a R ) µAB , ( a Q ) µAB , ( a U ) µAB , ( a D ) µAB h ( g γρ ∆ γβ ∆ ρν g αµ − ∆ αβ ∆ µν ) i ( k G ) βµνα , ( k W ) βµνα , ( k B ) βµνα g λν h ∂ α ∆ µα i k ) κ ǫ κλµν , 2( k ) κ ǫ κλµν , 4( k ) κ ǫ κλµν , 2( k AF ) κ ǫ κλµν We find: (i) ∆ αβ provides the most equivalent coupling constants in the SMEof the LV items in the sectors of fermion, gauge, and Higgs fields; (ii) The variouscombinations of ∆ αβ as coupling constants own a different CPT property. For exam-ple, ∆ µν , ∆ µν A ν , g γρ ∆ γβ ∆ ρν g αµ − ∆ αβ ∆ µν , and ∂ α ∆ µα are CPT-even, CPT-odd,CPT-even, and CPT-odd respectively. The SME in Ref. 6 includes all the possibleLV terms of spontaneous symmetry breaking for the SM and it is mentioned thatall these LV terms may origin from a fundamental theory. Thus what we performin this paper shows a fundamental way for the LV terms in the SM from basicprinciples.
5. Lorentz Invariance Violation Matrix
Now let us turn to the local BM M αβ , of which the vacuum expectation value isused for the coupling constants in (12), (13) and (14). We decompose M αβ into twoparts M αβ = g αβ + ∆ αβ , ovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Zhou L. L., B.-Q. Ma where g αβ is the metric of spacetime. Since all the elements of M αβ or ∆ αβ aredimensionless, they naturally encode the strength of LV or the degree of anisotropyof spacetime 7, i.e. ∆ αβ = 0 , no LV , → , small LV , = otherwise , large LV . Hence we call ∆ αβ the Lorentz invariance Violation Matrix (LVM), and its entrieswill be constrained with the help of laboratory experiments 8 and astronomicalobservations 9 , , , , , ,
15. Generally speaking, ∆ αβ depend on the types ofparticles. While ϕ ( x ) can be re-scaled to absorb one of the 16 degrees of freedom in∆ αβ , so that only 15 are left physical. Thus in this paper, we assume M = g ,or ∆ = 0.As a result of the LVM, we may attain various modified dynamical equations offields, as well as dispersion relations from the effective Lagrangian L SMS . Here asa preliminary test of our construction, we take the Dirac equation for free fermionfield ψ ( x ) as an example. First, we replace ∂ α to M αβ ∂ β ,( iγ α M αβ ∂ β − m ) ψ ( x ) = 0 . Second, we multiply ( iγ α M αβ ∂ β + m ) on both sides,( g αµ M αβ M µν ∂ β ∂ ν + m ) ψ ( x ) = 0 . With the Fourier transformation ψ ( x ) = R ψ ( p ) e − ip · x dp , the extended dispersionrelation becomes p + g αµ ∆ αβ ∆ µν p β p ν + 2∆ αβ p α p β = m . (15)We see that the left-hand side of (15) is not invariant under the observer’s Lorentztransformation on p , reflecting the influences from the fundamental theory. So weclaim that the last two items of the left-hand side of (15), which are the extensionsof the ordinary mass-energy relation p = m , contain the information of LV.Systematic LV effects from the general form of ∆ αβ still need further studies,but here we merely employ a special SO(3) invariant model of LVM to demonstrateour mechanism. So we assume ∆ αβ = ξ ξ
00 0 0 ξ . (16)Substituting (16) into (15) gives the extended dispersion relation for free fermionfield in this simple case, E = (1 − δ ) ~p + m , δ ≡ ξ − ξ . (17)ovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Lorentz Invariance Violation Matrix from a General Principle
6. Comparison with Experimental Data
We could utilize proton to determine the upper bound of ξ . The photopion produc-tion of nucleon in the Greisen-Zatsepin-Kuz’min (GZK) cutoff 16 ,
17 observationsgives an available energy threshold E ≈ eV (see, e.g., Ref. 1). The dominantchannel for this production begins with p + γ → ∆ + (1232 MeV). We concentrateon the head-on collision of the proton in cosmic rays and the photon from the Cos-mic Microwave Background (CMB). The dispersion relations for p , γ , and ∆ + aresimilar as that in (17), with the corresponding δ ’s denoted by δ p , δ γ , and δ ∆ + . Asconsidering the LV of the high energy protons from cosmic rays, we are allowed toassume δ γ = δ ∆ + = 0. So in this channel, we have p + ( p p + p γ ) , with p p = ( E p , ~p p ), p γ = ( ω, ~p γ ), and p ∆ + = ( E ∆ + , ~p ∆ + ) being the 4-momenta of p , γ , and ∆ + respectively. Using (17), we obtain m + ( E p + ω ) − ( ~p p + ~p γ ) = E p − ~p p + ω − ~p γ + 2 ωE p − ~p p · ~p γ = m p − δ p ~p p + 2 ωE p − ~p p · ~p γ = m p − ξ p E p + 4 ωE p . For high energy protons, E p ≃ ~p p , and ~p p · ~p γ = − ωE p due to the head-on collision.We keep only 2 ξ p in δ p since the quadratic term − ξ p is negligible, so the finalexpression for the energy E p of high energy protons satisfies the inequality2 ξ p E p − ωE p + m + − m p . In case of no LV for high energy protons, namely ξ p = 0, we have E p > ( m + − m p ) / (4 ω ) = 5 . × eV, which is the common threshold energy for the GZKcutoff. (Here m ∆ + = 1232 MeV, m p = 938 MeV, and the mean energy of thephotons in the CMB is taken as ¯ ω ≃ × − eV.) For high energy photons fromthe CMB, we take ω = 5¯ ω = 3 meV for calculations. A small positive ξ p will increasethe threshold energy, which can be higher than 5 . × eV, and the constrainton ξ p is ξ p ω m + − m p = 2 . × − . This constraint is consistent with our previous estimate in Ref. 18.
7. Conclusion
With a general requirement of the physical independence or physical invariance ofmathematical background manifold, we introduce the background matrix M αβ , andthe replacement of the common derivative operators by the covariant co-derivativeones. This replacement gives rise to supplement terms in the minimal standardmodel. We introduce a Lorentz invariance violation matrix ∆ αβ , which is able toovember 6, 2018 8:10 WSPC/INSTRUCTION FILE liv-mpla-r3 Zhou L. L., B.-Q. Ma characterize the Lorentz invariance violation or spacetime anisotropy. Thus we havea feeling that the principles of physical invariance and covariance are more funda-mental than Lorentz invariance or spacetime isotropy.
Acknowledgements
This work is partially supported by National Natural Science Foundation of China(No. 10721063 and No. 10975003), by the Key Grant Project of Chinese Ministryof Education (No. 305001), and by the Research Fund for the Doctoral Program ofHigher Education (China).
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