Is the Local Lorentz Invariance of General Relativity Implemented by Gauge Bosons That Have Their Own Yang-Mills-Like Action?

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Is the local Lorentz invariance of general relativity implemented by gauge bosons that have their own Yang-Mills-like action?

Kevin Cahill∗ Department of Physics and Astronomy University of New Mexico Albuquerque, New Mexico 87131 (Dated: December 10, 2020) General relativity with fermions has two independent symmetries: general coordinate invariance and local Lorentz invariance. General coordinate invariance is implemented by the Levi-Civita connection and by Cartan’s tetrads both of which have as their action the Einstein-Hilbert action. It is suggested here that local Lorentz invariance is implemented not by a combination of the Levi-Civita connection and Cartan’s tetrads known as the spin connection, but by independent Lorentz bosons ab L i that gauge the Lorentz group, that couple to fermions like Yang-Mills ﬁelds, and that have their own Yang-Mills-like action. Because the Lorentz bosons couple to fermion number and not to mass, they generate a static potential that violates the weak equivalence principle. If a Higgs mechanism makes them massive, then the static potential also violates the inverse-square law. Experiments put upper bounds on the strength of such a potential for masses mL < 20 eV. These upper limits imply that Lorentz bosons, if they exist, are nearly stable and contribute to dark matter.

I. INTRODUCTION Just as the Yang-Mills connection Ai is a linear combi- α α α nation Ai = − it Ai of the matrices t that generate General relativity with fermions has two indepen- the internal symmetry group, so too the spin connection 1 ab dent symmetries: general coordinate invariance and local ωi is a linear combination ωi = − 8 ω i [γa, γb] of the 1 Lorentz invariance. General coordinate invariance is the matrices − i 4 [γa, γb] that generate the Lorentz group. well-known, defining symmetry of general relativity. It So I ask: Does the independent symmetry of local acts on coordinates and on the world indexes of tensors Lorentz invariance have its own, independent gauge field but leaves Dirac and Lorentz indexes unchanged. It is im- 1 ab plemented by the Levi-Civita connection and by Cartan’s Li = − 8 L i [γa, γb] (4) tetrads. Local Lorentz invariance is a quite different symmetry. with its own field strength Fik = [∂i + Li, ∂k + Lk] and It acts on Dirac and Lorentz indexes but leaves coordi- Yang-Mills-like action nates and world indexes unchanged. In standard formu- Z lations [1–5], the derivative of a Dirac field is made co- 1 † ik √ 4 SL = − 2 Tr Fik F g d x? (5) variant (∂i + ωi) ψ by a combination of the Levi-Civita 4f connection Γj and Cartan’s tetrads ca known as the ki j and should the Dirac covariant derivative be spin connection 1 ab 1 ab Diψ = ∂i − 8 L i [γa, γb] ψ (6) ωi = − 8 ω i [γa, γb] (1) in which instead of the standard form (1–3)

ab a bk j a bk j ω = c c Γ + c ∂ic , (2) 1 a bk a bk i j ki k Diψ = ∂i − 8 c j c Γ ki + c k ∂ic [γa, γb] ψ? (7) a and b are Lorentz indexes, and i, j, k are world indexes. ab Because it acts on Lorentz and Dirac indexes but If so, then the Lorentz bosons L i couple to fermion leaves world indexes and coordinates unchanged, local number and not to mass and lead to a Yukawa potential Lorentz invariance is more like an internal symmetry than that violates the inverse-square law and the weak equiv- like general coordinate invariance. In theories with local alence principle. Lorentz invariance and internal symmetry, the covariant Experiments [6–29] have set upper limits on the strength of such Yukawa potentials for Lorentz bosons derivative Di of a vector of Dirac fields ψ has the spin of mass less than 20 eV. These upper limits imply that connection ωi and a matrix Ai of Yang-Mills fields side by side Lorentz bosons, if they exist, are nearly stable and contribute to dark matter. Whether fermions couple to Diψ = (∂i + ωi + Ai) ψ. (3) Lorentz bosons Li with their own action SL or to the spin connection ωi is an open experimental question. This paper outlines a version of general relativity with fermions in which the six vector bosons of the spin con- ∗ [email protected] ab ab nection ω i are replaced by six vector bosons L i that 2 gauge the Lorentz group and have their own Yang-Mills- III. LOCAL LORENTZ INVARIANCE like action. The theory is invariant under general coordinate transformations and independently under local The Einstein action (8) has a trivial symmetry under Lorentz transformations. local Lorentz transformations that act on Lorentz indexes Section II sketches the traditional way of including but leave world indexes and coordinates unchanged. This fermions in a theory of general relativity. Section III a symmetry becomes apparent when Cartan’s tetrads c i describes the local Lorentz invariance of a theory with b and c are used to write the metric gik in a form Lorentz bosons. Section IV says why general-coordinate k invariance and local-Lorentz invariance are independent a b gik(x) = c (x) ηab c (x) (13) symmetries. Section VI describes the Yang-Mills-like ac- i k ab tion of the gauge fields L i of the Lorentz group. Sec- that is unchanged by local Lorentz transformations tions VII and VIII suggest ways to make gauge theory and general relativity more similar to each other. 0a a b c i(x) = Λ b(x) c i(x). (14) Section IX discusses Higgs mechanisms that may give ab masses to the gauge bosons L i of the Lorentz group. The Levi-Civita connection (10) and the action (8) are Section X describes some of the constraints that experi- defined in terms of the metric and so are also invariant mental tests [6–29] of the inverse-square law and of the under local Lorentz transformations. weak equivalence principle place upon the proposed the- More importantly, the two Dirac actions (11) and (12) ory. Section XI discusses the stability and masses of L have a nontrivial symmetry under local Lorentz trans- bosons and suggests that they may be part or all of dark formations. Under such a local Lorentz transformation, a 1 1 matter. Section XII summarizes the paper. Dirac field transforms under the ( 2 , 0) ⊕ (0, 2 ) represen- tation D(Λ) of the Lorentz group with no change in its coordinates x II. GENERAL RELATIVITY WITH FERMIONS 0 −1 ψα(x) = Dαβ (Λ(x)) ψβ(x). (15) A century ago, Einstein described gravity by the action The Lorentz-boson matrix L = − 1 Lab [γ , γ ] makes 1 Z √ 1 Z √ i 8 i a b S = R g d4x = gik R g d4x (8) ∂ + L a covariant derivative E 16πG 16πG ik i i 2 ∂ + L0 = D−1(Λ)(∂ + L )D(Λ). (16) in which G = 1/mP is Newton’s constant, the metric i i i i is (−, +, +, +), letters from the middle of the alphabet In more detail with Λ = Λ(x), the matrix L transforms are world indexes, g = | det gik| is the absolute value of i the determinant of the space-time metric, and the Ricci as ` tensor Rik = R i`k is the trace of the Riemann tensor 0 −1 −1 Li = D (Λ) ∂iD(Λ) + D (Λ) Li D(Λ) j j j j m j m R i`k = ∂`Γ ki − ∂kΓ ì + Γ `m Γ ki − Γ km Γ ì (9) −1 1 −1 ab = D (Λ) ∂iD(Λ) − 8 D (Λ) L i [γa, γb] D(Λ) in which −1 1 ab c d = D (Λ) ∂iD(Λ) − 8 L i Λa Λb [γc, γd] (17) k 1 kj k Γ = g (∂`gji + ∂igj` − ∂jgi`) = Γ (10) i` 2 ì c −1c in which Λa = Λ a. Since is the Levi-Civita connection which makes the covariant derivative of the metric vanish [30]. a b a b a b Tr [γ , γ ][γc, γd] = 16 δ δ − δ δ , (18) The standard action of general relativity with fermions d c c d is the sum of the Einstein-Hilbert action (8) and the ac- its components transform as tion of matter fields including the Dirac action Z L0ab = − 1 Tr(L0 [γa, γb]) (19) ¯ a i √ 4 i 2 i −ψ γ ca (∂i + ωi + Ai) ψ g d x. (11) a b cd 1 −1 a b = Λc Λd L i − 2 Tr D (Λ)∂iD(Λ)[γ , γ ] . In what follows, it is proposed to replace the spin con- The trace is cyclic, so nection ωi in the standard Dirac action (11) with an in- 1 ab dependent gauge field L = − L [γ , γ ] that has its 0ab a b cd 1 a b −1 i 8 i a b L = Λ Λ L − Tr ∂ D(Λ)[γ , γ ]D (Λ) . own action (5) and to use i c d i 2 i Z Under an infinitesimal transformation ¯ a i √ 4 SD = −ψ γ ca (∂i + Li + Ai) ψ g d x (12) 1 a b Λ = I + λ and D(λ) = I − 8 λab[γ , γ ], (20) as the action of a Dirac field. This change reflects the independence of general coordinate invariance and lo- the Lorentz bosons transform as cal Lorentz invariance and makes general relativity and 0ab ab cb a ad b ab quantum field theory somewhat more similar. L i = L i + L iλc + L iλd + ∂iλ . (21) 3

The components of the spin connection obey similar which has only 6 [31]. equations, and the conventional Dirac action (11) also Special relativity offers another temptation. In special is invariant under local Lorentz transformations. relativity, global Lorentz transformations Λ act on the Local Lorentz transformations operate on the Lorentz spacetime coordinates and on the indexes of a Dirac field indexes a, b, c, . . . of the tetrads, of the spin connection 0a a b ωab , and of the gamma matrices γa[γ , γ ], and also on x = Λ bx i b c (26) the Dirac indexes α, β, γ of the gamma matrices and of 0 0 −1 ψα(x ) = Dαβ (Λ) ψβ(Λx). the Dirac fields ψ,¯ ψ, but not upon the world index i or the spacetime coordinates x. In this sense, the invariance This global Lorentz transformation leaves the specially of Dirac’s action SD under local Lorentz transformations relativistic Dirac action density unchanged is like an internal symmetry. † 0 a 0 † −1† 0 a −1 0 − iψ γ γ ∂aψ = − iψ D γ γ D ∂aψ † 0 a −1 c = − iψ γ Dγ D Λa ∂cψ IV. LOCAL LORENTZ INVARIANCE AND † 0 b a c INVARIANCE UNDER GENERAL = − iψ γ γ Λb Λa ∂cψ (27) COORDINATE TRANSFORMATIONS ARE = − iψ†γ0γbδc∂ ψ = −iψ†γ0γb∂ ψ. INDEPENDENT SYMMETRIES b c b But in general relativity with fermions, Cartan’s i The Dirac action SD is invariant both under a local tetrads ca allow the action to be invariant under a local Lorentz transformation Λ(x) and under a general coordi- Lorentz transformation without a corresponding general 0 −1 nate transformation x → x . Under a local Lorentz trans- coordinate transformation. The matrix Dαβ (Λ(x)) repre- formation Λ(x), the coordinates are unchanged, x0 = x, sents a local Lorentz transformation and acts (15) on the and the fields transform as spinor indexes of the Dirac field but not on its spacetime

0 −1 coordinates. Since ψα(x) = Dαβ (Λ(x)) ψβ(x) a −1 a a0 −1 a0 0a a b Dγ D = Λa0 γ and DγaD = Λ aγa0 , (28) c i(x) = Λ b(x) c i(x) 0ab a b cd a local Lorentz transformation (17) does not change L i(x) = Λc (x)Λd (x)L i(x) 1 −1 a b a −1 0i b a c i b c i b i − 2 Tr D (Λ(x))∂iD(Λ(x))[γ , γ ] Dγ D ca = γ Λb Λa cc = γ δb cc = γ cb. (29) 0 Ai(x) = Ai(x) (22) But the effect of a local Lorentz transformation (16) on a −1a the Lorentz matrix Li is in which Λc (x) = Λ c. Under a general coordinate transformation, the fields transform as 0 −1 ∂i + Li = D (Λ)(∂i + Li)D(Λ). (30) 0 0 ψα(x ) = ψα(x) so that ∂xk c0a (x0) = ca (x) D(∂ + L0 )D−1 = ∂ + L . (31) i ∂x0i k i i i i ∂xk A local Lorentz transformation therefore leaves the Dirac L0ab (x0) = Lab (x) i ∂x0i k action density invariant k ∂x 0 A0 (x0) = A (x). (23) − iψ†γ0γaci (∂ + L ) ψ i ∂x0i k a i i = − iψ†D−1†γ0γac0i (∂ + L0 ) D−1ψ −1 a i i The two transformations, ψα(x) → D (Λ(x)) ψβ(x) and αβ = − iψ†D−1†γ0γac0iD−1D (∂ + L0 ) D−1ψ x → x0, are different and independent; the coordinates x0 a i i † 0 a 0i −1 and Λ(x)x are unrelated. = − iψ γ Dγ ca D (∂i + Li) ψ (32) Every conventional, local Lorentz transformation is = − iψ†γ0γaci (∂ + L ) ψ. a general coordinate transformation, so one might be a i i tempted to imagine that every general coordinate trans- The symmetry under local Lorentz transformations is formation is a conventional, local Lorentz transformation. independent of the symmetry under general coordinate But one can see that this is not the case by comparing the transformations. They are independent symmetries. infinitesimal form of a general coordinate transformation ∂x0i dx0i = dxk (24) V. HOW DOES THE SPIN CONNECTION GO? ∂xk which has 16 generators with that of a conventional, local Under a local Lorentz transformation, the spin connec- Lorentz transformation tion

0a a b a a a b ab a bk j a bk dx = Λ b dx = dx + (r · R b + b · B b) dx (25) ω i = c j c Γ ki + c k ∂ic (33) 4

VI. ACTION OF THE GAUGE FIELDS Li with their own action (34) as

Since local Lorentz symmetry is like an internal sym- Diψ = (∂i + Li + Ai) ψ (43) metry, its gauge ﬁelds L = − 1 Lab [γ , γ ] should have i 8 i a b rather than in terms of the spin connection (1) an action like that of a Yang-Mills ﬁeld 1 ab Z ωi = − ω i [γa, γb] 1 † ik √ 4 8 SL = − Tr F F g d x (34) (44) 4f 2 ik 1 a bk j a bk = − 8 c j c Γ ki + c k ∂ic [γa, γb] in which as (3)

Fik = [∂i + Li, ∂k + Lk]. (35) (∂i + ωi + Ai) ψ. (45)

In terms of the gamma matrices One reason is that the symmetry of local Lorentz transformations is independent of the symmetry of general 0 1 0 σi γ0 = − i , γi = −i , coordinate transformations. So local Lorentz invariance 1 0 − σi 0 (36) should have its own gauge ﬁeld Li and action SL inde- 1 0 pendent of the tetrads and the Levi-Civita connection of and γ5 = , 0 −1 general coordinate transformations. A second reason to prefer the Lorentz connection Li 1 ab to the spin connection ω is that the L-boson covariant the commutators in Li = − 8 L i [γa, γb] are for spatial i a, b, c = 1, 2, 3, derivative

c a 5 1 ab [γa, γb] = 2iabcσ I and [γ0, γa] = −2σ γ . (37) ∂i − 8 L i [γa, γb] ψ (46)

So setting is simpler than the spin-connection covariant derivative

a 1 bc a a0 1 a bk j a bk r i = 2 abcL i and b i = L i, (38) ∂i − 8 (c j c Γ ki + c k ∂ic ) [γa, γb] ψ. (47) the matrix of gauge ﬁelds Li is A third reason is that using the Lorentz connection (42), the Dirac covariant derivative (43), and the action L = − 1 Lab [γ , γ ] = −i 1 r · σ I − 1 b · σ γ5. (39) i 8 i a b 2 i 2 i (34) for the Lorentz connection, makes general relativity with fermions more similar to the gauge theories of the Its ﬁeld strength (35) is standard model.

Fik = [∂i + Li, ∂k + Lk] (40) 1 = − i 2 (∂irk − ∂kri + ri × rk − bi × bk) · σ I VIII. MAKING GAUGE THEORY MORE 1 5 SIMILAR TO GENERAL RELATIVITY − 2 (∂ibk − ∂kbi + ri × bk + bi × rk) · σ γ , and its Yang-Mills-like action density (34) is Under a local Lorentz transformation, the spin connection ωi changes more naturally, more automatically than 1 † ik does the Lorentz connection L . The automatic feature of SL = − Tr F F (41) i 4f 2 ik the spin connection is that its definition (2) implies that 1 h under infinitesimal (20) and finite local Lorentz transfor- = − (∂ r − ∂ r + r × r − b × b ) 4f 2 i k k i i k i k mations it transforms as i k k i i k i k · ∂ r − ∂ r + r × r − b × b 0ab ab cb a ad b ab ω i = ω i + ω iλc + ω iλd + ∂iλ (48) + (∂ibk − ∂kbi + ri × bk + bi × rk) i and as · ∂ibk − ∂kbi + ri × bk + bi × rk . 0ab a b cd a cb ω i = Λc Λd ω i + Λc ∂iΛ . (49)

ab a cb VII. MAKING GENERAL RELATIVITY MORE The terms ∂iλ and Λc ∂iΛ occur automatically SIMILAR TO GAUGE THEORY without the need to put in by hand a term like −1 D (Λ)∂iD(Λ). −1 There are three reasons to deﬁne the covariant deriva- Terms like D (Λ) ∂iD(Λ) are a common feature of tive of a Dirac ﬁeld in terms of Lorentz bosons gauge theories whether abelian or nonabelian. We can make them occur automatically in local Lorentz trans- 1 ab ab Li = − 8 L i [γa, γb] (42) formations if we add to the Lorentz connection L i 5

a bα the term u α ∂i u in which the four Lorentz 4-vectors in which the n n-vectors uαγ are orthonormal uaα(x) obey the condition ∗ uβαu = δβγ . (59) aα bβ ab γα u ηαβu = η , (50) An SU(n) transformation and α = 0, 1, 2, 3 is a label, not an index. It follows then from this condition (50) on the quartet of vectors uaα † Ai → gAig , u → gu, and ψ → gψ (60) ab that the augmented Lorentz connection Lnew i ab ab a bα would then change the covariant derivative (∂i +Anew i)ψ Lnew i = L i + u α ∂i u (51) to automatically changes under a local Lorentz transforma- 0 † † † a −1a [(∂i+Anew i)ψ] = (∂i + gAig + g u ∂i(u g ))gψ tion Λc = Λ c to † † † † = g(∂i + g ∂ig + Ai + (∂ig )uu g + u ∂i u )ψ 0ab a b cd a c b dα Lnew i = Λc Λd L i + Λc u α ∂i (Λd u ) † † † = g(∂i + g ∂ig + Ai + (∂ig )g + u ∂i u )ψ a b cd c dα c dα a b = Λc Λd (L i + u α ∂i u ) + u αu Λc ∂i Λd † = g(∂i + Ai + u ∂i u )ψ = g(∂i + Anew i)ψ. (61) a b cd cd a b = Λc Λd Lnew i + η Λc ∂i Λd = Λ aΛ b Lcd + Λ a ∂ Λcb c d new i c i IX. POSSIBLE HIGGS MECHANISMS a b cd −1a cb = Λc Λd Lnew i + Λ c ∂i Λ (52) without the need to explicitly add the last term The actions SL and SD (5 & 12) leave the gauge bosons −1a cb L massless, but a Higgs mechanism is possible. An inter- Λ c ∂i Λ by hand. a In matrix form, the condition (50) is the requirement action with a ﬁeld v that is a scalar under general coordinate transformations but a vector under local Lorentz aα bβ ab u ηαβu = η (53) transformations has as its covariant derivative

aα a a a b that the matrix formed by the quartet of vectors u be Div = ∂iv + L biv . (62) a Lorentz transformation 0 aα bβ ab If the time component v has a nonzero mean value in x u η u y = x η y . (54) 0 a i a αβ b a b the vacuum h0|v |0i= 6 0, then the scalar −Div D va ab contains a mass term The augmentation of the Lorentz connection L i → ab a bα Lnew i by the addition of the term u α ∂i u , which is a 0 0i a0 0 a0i 0 a bk − L 0iv La v0 = − L iv L v (63) similar to the tetrad term c k ∂ic of the spin connection (2), makes its change (52) under local Lorentz transfor- s s0 that makes the boost vector bosons b i = L i massive mations as automatic as that (49) of the spin connection. s 1 tu but leaves the rotational vector bosons r i = 2 stuL i The use of a more automatic connection makes gauge massless. On the other hand, if the spatial components theory more similar to general relativity with fermions. have a nonzero mean value, h0|v|0i 6= 0, then the mass We can extend the use of such terms to internal sym- term is metries and so make the inhomogeneous terms appear automatically rather than by hand or by ﬁat. For in- a s si s0s s s0si s 0s s 0si s −L siv La vs = −L iv L v +L iv L v . (64) stance, we can augment the abelian connection Ai to −iφ(x) iφ(x) Adding the two mass terms (63 and 64), we ﬁnd Anew i(x) = Ai(x) + e ∂ie (55) a b 0i 0s a 0si s0s s s0si s in which φ(x) is an arbitrary phase. A U(1) transforma- − L 0iv La vb = L iv L va − L iv L v (65) tion which makes all six gauge bosons massive as long as the e−iφ(x) → e−i(θ(x)+φ(x)) and ψ(x) → e−iθ(x)ψ(x) (56) mean value is timelike

a would then change the covariant derivative (∂i +Anew i)ψ h0|v va|0i < 0 (66) to s 0 −i(θ+φ) i(θ+φ) −iθ and at least two spatial components h0|v |0i 6= 0 have [(∂i+Anew i)ψ] = (∂i + Ai + e ∂ie )e ψ nonzero mean values in the vacuum. This condition holds −iθ −iφ iφ a1 a2 a3 = e (∂i − i∂iθ + Ai + i∂iθ + e ∂ie )ψ (57) in all Lorentz frames if three vectors v , v , and v have diﬀerent timelike mean values in the vacuum. = e−iθ(∂ + A + e−iφ∂ eiφ)ψ = e−iθ(∂ + A )ψ. i i i i new i In the vacuum of ﬂat space, tetrads have mean values a a Similarly, we can augment the nonabelian connection that are Lorentz transforms of ci = δi and that produce α α the Minkowski metric (13) Ai = − it Ai for SU(n) to ∗ a b Anew i(x) = Ai(x) + uαβ(x)∂iuαγ (x) (58) gik = δi ηab δk = ηik. (67) 6

So it is tempting to look for a Higgs mechanism that a mass of mSi = 28.085 u, so FSi mP/mSi = 4.573 and a 38 2 uses the covariant derivatives D` c k of the tetrads. For αSi = − 9.988 × 10 f . Platinum has FPt = 663 and j a a m = 195.084 u, so α = − 9.474 × 1038 f 2. Tungsten Γ k` = 0 and c k = δk , the term Pt Pt has FW = 626 and mW = 183.84 u, so αW = − 9.510 × 1 2 i k a 1 2 bi k a c 38 2 2 −39 − m (D c ) Di c = − m L c L c 10 f . For such test masses, f ≈ |α| × 10 . 2 L a k 2 L a b ci k (68) 1 2 i ab The Riverside group [6] placed on the strength |αAu| = − 2 mL Lab L i an upper limit (95% confidence) that drops from |αAu| . 19 16 −8 makes the rotational bosons rs = 1 Ltu massive but 10 to |αAu| . 10 as the length λ rises from 10 m i 2 stu i −8 s s0 to 4 × 10 m. The IUPUI group [7] put an upper limit makes the boost bosons b i = L i tachyons. If weakly (95% confidence) on the strength |αAu-Si| that drops from coupled tachyons are unacceptable, then the Higgs mech- 16 5 |αAu-Si| . 10 to |αAu-Si| . 10 as the length λ rises from anism (62–65) that uses three world-scalar Lorentz vec- −8 −6 tors with different time-like mean values in the vacuum 4 × 10 m to 8 × 10 m. These results of the Riverside a a a and IUIPUI groups are plotted in Fig. 1 from Chen et v1 , v2 , v3 is a more plausible way to make the gauge bosons Lab massive. al. [7]. i Other short-distance experiments [8, 9, 11–20, 26–28] have tested the inverse-square law at the slightly longer −6 −3 X. TESTS OF THE INVERSE-SQUARE LAW distances of 2×10 < λ < 3×10 m. The Washington group [8] used test masses of platinum. The [9] used test In the static limit, the exchange of six Lorentz bosons masses of tungsten. The upper limits (95% confidence) ab on the strength |α| are shown for platinum in Fig. 2 from L i of mass mL would imply that two macroscopic bod- 0 Lee et al. [8] and for tungsten in Fig. 3 from Tan et al. [9]. ies of F and F fermions separated by a distance r would 6 The upper limit on the strength |α| falls from |α| . 10 contribute to the energy a static Yukawa potential −6 4 −6 at λ ∼ 2 × 10 m to |α| . 10 at λ ∼ 8 × 10 m 0 2 3 −5 3 FF f −m r and then from |α| . 10 at λ ∼ 10 m to |α| . 1 at V (r) = e L . (69) −5 −3 −3 L 2πr λ ∼ 4 × 10 m and to |α| . 10 at λ ∼ 2 × 10 m. This potential is positive and repulsive (between fermions and between antifermions) because the L’s are vector bosons. It violates the weak equivalence principle because 1019 it depends upon the number F of fermions (minus the Riverside number of antifermions) as F = 3B + L and not upon Region 15 excluded by experiments their masses. The potential VL(r) changes Newton’s po- 10 G tential to auged IUPUI (2005-2007) b aryons 0 11 mm −r/λ 10 IUPUI VNL(r) = − G 1 + α e (70) | (2014) Yale  Str

r | a

n ge mod ulus in which the coupling strength α is 107 Gluo n mo Stanford 0 2 0 2 2 dulus 3FF f 3FF mPf α = − 0 = − 0 , (71) 3 2πGmm 2πmm 10 Washington Dilaton and the length λ is λ = ~/cmL. Couplings α ∼ 1 are Radion of gravitational strength. Experiments [6–29] that test 10-1 the inverse-square law and the weak equivalence principle 10-7 10-6 10-5 10-4 have put upper limits on the strength |α| of the coupling  (meters) for a wide range of lengths 10−8 < λ < 1013 m and −20 masses 2 × 10 < mL < 20 eV. FIG. 1: Upper limits (95% conﬁdence) on the strength Experiments that tested the inverse-square law at very |αAu| of Yukawa potentials that violate the inverse-square short distances, between 10 nm and 3 mm, were done law at distances 10−8 < λ < 2 × 10−4 m. (Fig. 4 of Chen with masses of gold [6], of gold and silicon [7], of plat- et al. [7]) inum [8], and of tungsten [9]. For a mass m of N atoms of gold which has FAu = 670 fermions (quarks and elec- trons) in each atom of mass mAu = 196.966 u, the ratio Other groups [13, 20–25, 28, 29] have tested the F mP/m that appears in the coupling α (71) is inverse-square law over a huge range of longer distances, 10−3 < λ < 3×1015 m. In 2012 the HUST group [28] put NFAu mP FAu mP 670 mP = = = 4.458 × 1019. (72) an upper limit of |α| 10−3 for 7 × 10−4 < λ < 5 × 10−3 Nm m 196.966 u . Au Au m, while in 1985 the Irvine group [20] put an upper limit 38 2 −3 −3 −1 So the coupling strength is αAu = − 9.490 × 10 f of |α| . 10 for lengths 7 × 10 < λ < 10 m. for gold. An atom of silicon has FSi = 98 fermions and Fischbach and Talmadge [29] and Adelberger et al. [13] 7

FIG. 4: Upper limits (95% conﬁdence) on the strength FIG. 2: Upper limits (95% conﬁdence) on the strength |α| of Yukawa violations of the inverse-square law at large |αPt| of Yukawa potentials that violate the inverse-square distances 10−2 < λ < 1015 m [13, 20, 23, 24, 29] (Fig. 10 law at sub-mm distances [8–20]. (Fig. 5b of Lee et al. [8]) of Adelberger et al. [13]).

periments to look for Yukawa potentials that violate the weak equivalence principle in the range of distances 0.3 < λ < 109 m [13]. They have put upper limits (95% confidence) on the strength |α| of the coupling to B, Z, and N ≡ B − L but not explicitly on the coupling to fermion number F = 3B + L. For B, their upper limit runs from −5 −1 −8 5 |α| . 10 at 10 m to |α| . 6 × 10 at 7 × 10 m and −10 7 13 then falls to |α| . 10 for 10 < λ < 10 m as shown by the dashed lines in Fig. 5 from Bergéet al. [10]. For Z −6 −1 and N, their upper limit runs from |α| . 10 at 10 m −9 6 −11 to |α| . 6×10 at 10 m and then falls to |α| . 2×10 for 107 < λ < 109 m [13]. More recent satellite measurements by the MICRO- SCOPE mission have lowered the upper limit on the FIG. 3: Upper limits (95% confidence) on the strength strength |α| of Yukawa potentials that violate the weak |α | of Yukawa potentials that violate the inverse-square equivalence principle by about an order of magnitude for W 7 9 law at mm and sub-mm distances [7, 9, 11, 14, 15, 17– 10 < λ < 10 m [10]. The upper limit for coupling to B −11 7 9 20, 26–28]. Light lines are theory [13, 22] (Fig. 6 of Tan is |α| . 10 for 10 < λ < 10 m as shown in Fig. 5 et al. [9]). from Bergéet al. [10]. Their limit for coupling to N is −13 7 9 even lower: |α| . 4 × 10 for 10 < λ < 10 m [10]. Some of these important results [6–29] are summarized in broad-brush fashion in Fig. 6. The upper bound (95% have reported tests of the inverse-square law for distances confidence) on |α| is the solid dark-blue curve which falls in the range 10−2 < λ < 1015 m [13, 20, 23, 24, 29]. As from 1019 for λ = 7 × 10−8 m to 10−11 at λ = 109 m. shown in Fig. 4 from Adelberger et al. [13], the upper The (λ, |α|) region above this curve is excluded. Points in limit lies between |α| < 3 × 10−4 and |α| < 2 × 10−3 the allowed region that are below the blue-green dotted −2 4 −4 for 10 < λ < 10 m but drops from |α| . 10 to line correspond to L bosons with lifetimes longer than −10 4 8 |α| . 10 as the length increases from 10 to 10 m. the age of the universe. Those that also are between the −9 The upper limit is about |α| . 5 × 10 on planetary vertical dashed lines denote effectively stable L bosons scales 1010 < λ < 5 × 1011 m. whose masses could account for between 1 and 100% of The Washington group have used torsion-balance ex- dark matter. 8

XI. LORENTZ BOSONS AS DARK MATTER

Analysis of the double galaxy cluster 1E0657-558 (the “bullet cluster” at z = 0.296) suggests [33, 34] that dark matter interacts weakly, perhaps with gravitational strength |α| ∼ 1. As of now, there has been no accepted detection of dark matter in a laboratory. The experiments [6–29] sketched in Sec. X put no upper limits on the mass mL = h/cλ of L bosons and no lower limits on their coupling |α|. The proposed L bosons are electrically neutral. If their mass is heavy enough and if their coupling is sufficiently weak, then they would be an effectively stable part of dark matter. Because they couple to fermion number and not to mass, their coupling f 2 is much weaker than |α| by a factor related to Avogadro’s number. For the metals (Au, Si, Pt, and W) used in many of the experiments [6–28], FIG. 5: Upper limits (95% confidence) on the strength |α| the relation is of Yukawa potentials that violate the weak equivalence 2 −39 principle at long distances [10, 13, 20, 21, 23, 24] (Fig. 1 f ∼ |α| × 10 . (73) of Bergéet al. [10]). Even for the highest upper limit |α| < 1019 shown in 2 −20 Fig. 6, the coupling of the L bosons is only f . 10 . Because they interact so weakly, L bosons decay slowly. The decay width of the Z boson is ΓZ = 2 2 3.7 e mZ c /4π = 2.5 GeV, and its lifetime is τZ = −25 ~/ΓZ = 2.6 × 10 s. The analog of the electromagnetic coupling e2/4π ∼ 1/137 for L bosons is f 2/4π. In terms 15 10 of f 2 and |α| (73), the decay width of an L boson of mass mL is roughly 1010 2 2 2 b f mLc b |α| mLc −39 ΓL ∼ = × 10 (74) 105 4π 4π

100 in which b is a fudge factor, 0.1 . b . 10, that depends on the decay channels. The L boson lifetime then is

10-5 8.3 × 1024 1.9 107 τ = ~ = s = t (75) Γ b |α| m c2[eV] b|α| m c2[eV] 0 10-10 L L L 17 in which t0 = 4.356 × 10 s is 13.8 billion years, the age 10-15 10-9 10-7 10-5 10-3 10-1 101 103 105 107 109 of the universe. An L boson of mass mL < (19/b|α|) MeV is effectively stable in that its lifetime exceeds the age of the universe. If the fudge factor b is taken to be unity, FIG. 6: The upper bound (95% confidence) on the then L bosons of wavelength λ are effectively stable for strength |α| of Yukawa potentials that violate the inverse- couplings square law or the weak equivalence principle at various 13 distances λ is the solid dark-blue curve [6–29]. The re- |α| . 1.5 × 10 λ [m] (76) gion under the dotted green line denotes L bosons with lifetimes greater than the age of the universe for the case which is the dotted green line in Fig. 6. Points below it in which the fudge factor b = 1. The region below that denote effectively stable L bosons. line and between the vertical dashed blue lines denotes If the lightest fermion has mass mlightest, then L bosons L bosons that are between 1 and 100% of dark matter. of mass less than 2 mlightest would be absolutely stable. The thin vertical gray solid line marks the wavelength of The dash-dotted gray vertical line in Fig. 6 is the wave- eff −7 an L boson whose mass is 2 mν = 2.2 eV [32]. length λ = 5.6 × 10 m of twice the upper limit on the e (eff) effective mass of the electron neutrino, 2 mνe [32]. The mass density of cold dark matter is ρcdm = (2.2414 ± 0.017) × 10−27 kg m−3 [35, col. 7, p. 15]. So 9 if all of cold dark matter were made of L bosons of mass present number density of each kind of L boson would mL, then their number density would be be 7 −3 ρ 1.26 × 109 n(t0) = 7.5 × 10 m . (85) cdm −3 (77) nL = = 2 m . mL mLc [eV] Let us further assume that all six L bosons get the same mass m . In this case, if their mass density is not To estimate the present number density of each kind L to exceed the density of dark matter, then the inequality of L boson, I’ll assume that the L bosons are effectively −27 −3 stable and have not interacted since they dropped out of 6 mL n(t0) < ρcdm = 2.24 × 10 kg m (86) equilibrium in the very early universe. 2 implies that the mass mL must be less than At temperaures kT mLc so high that the weakly interacting L bosons were in thermal equilibrium, the m < 4.9 × 10−36 kg = 2.8 eV/c2. (87) number density of each of the six L bosons is given by L the Planck distribution as The lifetime of an L boson (75) would then be 3 7 3 3ζ(3)(kT ) 9.609 × 10 T 3.4 6 n(T ) = = . (78) τL > × 10 t0 (88) π2(~c)3 (m K)3 b|α| 6 In the limit of vanishing coupling |α| → 0, the number which for b|α| < 1.7 × 10 exceeds the age t0 of the n(t)a3(t) of L bosons within a fixed comoving box does universe. The range λL = h/mLc of the corresponding 3 not change with time. So the number now n(t0)a (t0) = Yukawa potential is 3 n(t0) is the number at any earlier time multiplied by a (t) −7 λL > 4.5 × 10 m. (89) 3 n(t0) = n(t)a (t). (79) Points (λ, |α|) below the dotted green line in Fig. 6 and between its vertical dashed blue-green lines denote L At very early times, we may approximate the integral for bosons constituting between 1 and 100% of the dark mat- the time as a function of the scale factor a as [36] ter. The upper limit on the effective mass of the electron (eff) a 1 Z dx neutrino is mνe < 1.1 eV [32]. The thin gray vertical t(a) = √ line labels L bosons of mass m = 2 meff. 2 −1 −2 L νe H0 0 ΩΛ x + Ωk + Ωm x + Ωr x 1 Z ax dx a2 ≈ √ = √ . (80) H0 0 Ωr 2H0 Ωr XII. CONCLUSIONS

−5 The Hubble constant and the fraction Ωr = 9.0824×10 General relativity with fermions has two indepen- then give us the scale-factor as dent symmetries: general coordinate invariance and local q Lorentz invariance. General general coordinate invariance p −10p a(t) = 2H0 Ωr t = 2.04 × 10 t[s]. (81) acts on coordinates and on the world indexes of tensors but leaves Dirac and Lorentz indexes unchanged. Local If N types of particles made up the radiation at very Lorentz invariance acts on Dirac and Lorentz indexes but early times, then the time and the temperature were re- leaves world indexes and coordinates unchanged. It acts lated by [37] like an internal symmetry. s General coordinate invariance is implemented by the √ 3c2 Levi-Civita connection Γj and by Cartan’s tetrads ca . N t T 2 = = 3.25924 × 1020 s K2 (82) ki i 16πGar In the standard formulation of general relativity with fermions, local Lorentz invariance is implemented by the in which ar is the radiation constant same fields in a combination called the spin connection ab a bk j a bk ω = c c Γ + c ∂ic . These fields all have the 2 4 i j ki k π k −16 −3 −4 same action, the Einstein-Hilbert action R. ar = = 7.56577(5) × 10 J m K . (83) 15~3c3 Because local Lorentz invariance is different from and independent of general coordinate invariance, it is sug- So in terms of the number density (78), the scale-factor gested in this paper that local Lorentz invariance is im- (81), and the time-temperature relation (82), the number ab plemented by different and independent fields L i that density is roughly gauge the Lorentz group and that have their own Yang- 4.8 × 109 Mills-like action. n(t ) = n(t)a3(t) = m−3. (84) The replacement of the spin connection with Lorentz 0 N 3/4 bosons moves general relativity closer to gauge theory In the standard model, N = 126, but the actual number and simplifies the standard covariant derivative relevant at high temperatures may be much higher. If j 4 −3/4 1 a bk a bk we assume that N = 4 , then N = 1/64, and the ∂i − 8 c j c Γ ki + c k ∂ic [γa, γb] ψ (90) 10 to obvious requirement that they could make up all of dark matter but not more, we can infer a crude theoretical 2 1 ab upper limit on their mass of mL 2.8 eV/c if all 6 are ∂i − L [γa, γb] ψ. (91) . 8 i stable and have the same mass. The discovery of a violation of the inverse-square law Whether the Dirac action has the spin-connection form by future experiments would not be enough to establish (90) or the Lorentz-boson form (91) is an experimental the existence of L bosons because the violation could be question. due to the physics of a quite different theory. Because the proposed action (12) couples the gauge If L bosons are discovered, physicists will decide how ab fields L i to fermion number and not to mass, it violates to think about the force they mediate. The force might the weak equivalence principle. It also leads to a Yukawa be considered to be gravitational because it arises in a potential (69) that violates Newton’s inverse-square law. theory that is a modest and natural extension of general Experiments [6–29] have put upper limits on the relativity. But the force is not carried by gravitons. It is strength |α| of the Yukawa potentials (70) that violate the carried by L bosons, and they implement a symmetry, inverse-square law and the weak equivalence principle for local Lorentz invariance, that is independent of general distances 10−8 < λ < 109 m. The upper limit ranges from coordinate invariance. So the force is new and might be |α| < 1019 at λ = 10−8 m to |α| < 103 at λ = 10−5 m and called a Lorentz force. to |α| < 10−11 at λ = 109 m. There are no experimental lower limits on the coupling at any distance, so L bosons could have lifetimes that exceed the age of the universe. ACKNOWLEDGMENTS There are no experimental upper limits on the masses of L bosons. Long lived, massive, weakly interacting, neu- I am grateful to E. Adelberger, R. Allahverdi, D. tral L bosons would contribute to dark matter. From the Krause, E. Fischbach, and A. Zee for helpful email.

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