Baryogenesis in Lorentz-violating gravity theories

Physics Letters B 773 (2017) 186–190

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Physics Letters B www.elsevier.com/locate/physletb

Baryogenesis in Lorentz-violating gravity theories Jeremy Sakstein ∗ , Adam R. Solomon Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd St., Philadelphia, PA 19104, USA

a r t i c l e

i n f o

Article history: Received 6 June 2017 Accepted 18 August 2017 Available online 24 August 2017 Editor: S. Dodelson

a b s t r a c t Lorentz-violating theories of gravity typically contain constrained vector fields. We show that the lowestorder coupling of such vectors to U(1)-symmetric scalars can naturally give rise to baryogenesis in a manner akin to the Affleck–Dine mechanism. We calculate the cosmology of this new mechanism, demonstrating that a net B − L can be generated in the early Universe, and that the resulting baryon-tophoton ratio matches that which is presently observed. We discuss constraints on the model using solar system and astrophysical tests of Lorentz violation in the gravity sector. Generic Lorentz-violating theories can give rise to the observed matter–antimatter asymmetry without violating any current bounds. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

1. Introduction Why is there so much more matter than antimatter? One most likely cannot appeal to initial conditions, as these would be washed away by inflation. The standard model can provide such an asymmetry during the electroweak phase transition, but cannot produce enough to accommodate observations [1]. It seems probable, then, that a dynamical generation mechanism, or baryogenesis, arises from new physics beyond the standard model.1 In this paper we point out that Lorentz violation might play a key role in this new physics. While Lorentz invariance is extraordinarily well tested in the matter sector, the possibility of gravitational Lorentz violation remains relatively unconstrained. If boosts are broken but rotational invariance is maintained—i.e., if gravity picks out a preferred rest frame—then the low-energy physics is described by Einstein-æther theory, a vector–tensor theory in which the vector field is constrained to have a fixed, timelike norm. This is the general effective field theory when boosts are broken [8]; for example, a special case of Einstein-æther arises in the low-energy limit of Hoˇrava–Lifschitz gravity [9,10], a putative UV completion of general relativity which relies on the existence of a preferred foliation. We demonstrate that if a U(1) B − L scalar couples to the vector of Einstein-æther theory, then the lowest-order interactions between the two can lead to baryogenesis. This operates in a manner

*

Corresponding author. E-mail addresses: [email protected] (J. Sakstein), [email protected] (A.R. Solomon). 1 For reviews of baryogenesis, see, e.g., Refs. [1–7].

qualitatively similar to Affleck–Dine baryogenesis [11] or the recent model of Ref. [12], in which the U(1) B − L symmetry is broken at early times due to a tachyonic mass proportional the Hubble parameter H appearing in the effective scalar potential.2 In purely metric theories this is difficult to achieve, as H is not a spacetime scalar. Breaking boosts cures this difficulty, and indeed in Einsteinæther H is simply proportional to the divergence of the timelike vector field. As Einstein-æther is the most general low-energy effective theory for broken boosts, our conclusion can be stated as follows: if the Universe contains a U(1) B − L scalar with softly broken symmetry and spontaneous Lorentz violation, a working baryogenesis mechanism comes for free.3 This paper is organized as follows. In section 2 we introduce the model of scalar-æther baryogenesis and discuss known constraints on the theory. In section 3 we derive the cosmology and verify that this model can yield the observed baryon-to-photon ratio with sensible parameters, and we conclude in section 4. 2. Model The model we will consider is the constrained vector (or “æther”) u μ of Einstein-æther theory coupled to a new U(1) B − L scalar φ . At leading order, the most general action we can write down is

2 In the former model, this arises due to a coupling of the scalar to the inflaton, while in the latter, it arises from a Weyl coupling to dark matter. 3 Baryogenesis in Lorentz-violating theories has also been studied in various other contexts in Refs. [13–15]. Our method is distinct from these. In particular, we do not μ consider a coupling of the vector to the baryon current of the form u μ j B , which would give rise to spontaneous baryogenesis.

http://dx.doi.org/10.1016/j.physletb.2017.08.039 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

J. Sakstein, A.R. Solomon / Physics Letters B 773 (2017) 186–190





√

d4 x − g

S=

2 M Pl

2



μν

R − Kα β ∇μ u α ∇ν u β + κ u μ u μ + m2

λ ε ε† 4 − ∂μ φ∂ μ φ † − m2φ |φ|2 − |φ|4 − φ 4 − φ † 2 4 4  α 2 + |φ| ∇μ u μ ,



like mH rather than H 2 , which can lead to novel and interesting new features. In what follows, we will calculate the cosmology of this model, paying special attention to the generation of a net B − L.

(1)

3

where μν

μ

μ

Kα β = c 1 g μν g α β + c 2 δα δβν + c 3 δβ δαν + c 4 u μ u ν g α β

(2)

is the most general kinetic term for the Lorentz-violating vector u μ , and κ is a Lagrange multiplier that ensures that the vector is timelike and of fixed norm, u μ u μ = −m2 . We have included some U(1) B − L -violating terms proportional to ε in order to generate a net B − L. The last line is the leading-order interaction one can write down between φ and u μ given the symmetries.4 This is the general low-energy effective theory with φ , broken boosts, and softly-broken U(1) B − L . The coupling between u μ and φ can give rise to baryogenesis using a mechanism akin to (but distinct from) that of Affleck and Dine [11] or similar generalizations [12]. In particular, the effective potential for φ is





1 V eff (φ, φ † ) = m2φ − α ∇μ u μ |φ|2 3

λ ε ε† 4 + |φ|4 + φ 4 + φ † 2

4

(3)

4

so that ∇μ u μ acts as a tachyonic mass term for φ . We can see this explicitly by considering a homogeneous and isotropic cosmological setting, so that the Universe is described by a Freidmann– Lemaître–Robertson–Walker metric (in cosmic time)

ds2 = −dt 2 + a2 (t )dx2 .

(4)

The on-shell condition u μ u μ = −m2 and symmetry imply that the vector must be of the form [22–24]

u μ = (m, 0, 0, 0) .

(5)

In this case, the divergence of u μ is ∇μ u μ = m∂t ln so that the effective potential becomes





V eff (φ, φ † ) = m2φ − αmH |φ|2 +

λ 2

√

ε

ε†

4

4

|φ|4 + φ 4 +

− g = 3mH , 4

φ† .

187

(6)

This potential leads to baryogenesis similarly to the well-known Affleck–Dine mechanism. In the early Universe, the U(1) B − L symmetry is broken as the tachyonic mass term is more important than the bare mass, while at late times the symmetry is restored. During the broken-symmetry phase, the motion of the angular component of the scalar generates a net B − L due to the 4

symmetry-breaking terms ε φ 4 + ε † φ † . When the symmetry is restored, the B − L is stored in the field, and can be transferred to the standard model through sphaleron processes [25], although one must first transfer the asymmetry to left-handed standard model particles. The details of the transfer were discussed for models such as ours in Ref. [12], where the neutrino portal [26,27] was identified as one promising mechanism. We note that our model differs quantitatively from Affleck–Dine: the tachyonic mass scales

4 Couplings between a scalar and ∇μ u μ were first introduced in Ref. [16], and have also been considered in, e.g., Refs. [17–21]. One could also consider a term |φ|2 u μ u μ , but this can always be absorbed into the mass and Lagrange multiplier when the vector is on-shell.

2.1. Constraints In this subsection, we briefly summarize observational and theoretical constraints on the parameters in our model. Most of these will apply to Einstein-æther theory or to its coupling to a real scalar. We will use the notation c 12 = c 1 + c 2 , c 123 = c 1 + c 2 + c 3 , etc. Experimental constraints on Einstein-æther theory tend to place upper bounds on the æther vacuum expectation value (VEV) m, 2 with the result c i m2  M Pl for generic values of the c i parameters. We note that any of these constraints can be weakened or removed entirely by tuning the c i parameters, although these tunings cannot all be done simultaneously. We refer the reader to Sec. V.D of Ref. [19] for a more comprehensive summary of constraints on the æther. ˘ The strongest constraints come from gravitational Cerenkov radiation: high-energy cosmic rays could lose energy to subluminal æther-graviton modes, leading to a degradation in cosmic ray propagation which has not been observed, constraining m/ M Pl < 3 × 10−8 [28]. These constraints can be avoided by tuning the c i or allowing for superluminal propagation in æther-graviton modes; since this is an explicitly Lorentz-breaking theory, superluminality may not be as deadly as one normally expects. The preferred-frame parameters α1,2 in the parametrized post-Newtonian formalism are modified by the æther, constraining m/ M Pl < 6 × 10−4 in the absence of tuning c i [29,30]. Note that the tuning which eliminates ˘ gravitational Cerenkov radiation (c 3 = −c 1 , c 2 = c 1 /(1 − 2c 1 )) also sets α1 = 0, so that the dominant constraint comes from α2 , in which case the strongest constraint on m is rather mild, m/ M Pl  10−2 . 2 Under the assumption that c i m2  M Pl , we are justified in ignoring the mixing with gravity [23], in which case there are a few constraints on the c i from flat space perturbation theory. In the vector sector, the absence of ghosts requires c 1 > 0 [23]; coupling a scalar to ∇μ u μ , as in this paper, does not modify the vector perturbations around flat space [19]. The no-ghost condition for the spin-0 piece of u μ is the same, while gradient stability requires c 123 > 0. Some authors require the spin-0 æther mode to propagate subluminally, which would imply c 123 < c 1 [23], although the scalar coupling relaxes this bound to c 123 < c 1 + x for some x > 0 [19]. If we require the sound speed of tensors to be subluminal then we would require c 13 > 0 [23]. The æther-scalar coupling can lead to a gradient instability, placing an upper bound on α [19]. Writing φ(x) = √1 ρ (x)e i θ(x) , 2

the real scalar ρ interacts with the æther through a potential V (θ, ρ ) = 12 m2φ ρ 2 + 14 λρ 4 − 16 α ρ 2 , where ≡ ∇μ u μ . Gradient stability around flat space requires5





2 2 V ρ ≤ 2c 123 V ρρ + k ,

(7)

where V ρ = ∂ ∂ρ V and V ρρ = ∂ρ2 V are evaluated at the background values of and ρ . Applying this to our potential, we find that the constraint is trivially satisfied in the unbroken symmetry

5

While the analysis of Ref. [19], in contrast to our model, assumed a single real

scalar, the field θ decouples from ρ and u μ around the background ∂μ θ¯ = 0. In principle a non-zero ∂μ θ¯ = c μ could modify the constraint by shifting the effective mass for ρ fluctuations, m2φ → m2φ + c 2 .

188

J. Sakstein, A.R. Solomon / Physics Letters B 773 (2017) 186–190

Fig. 1. The motion of the complex scalar in field space. The left panel corresponds to a model with mφ /αm = 10−4 , mφ /αm = 7 × 10−3 , αm/ M Pl = 2 × 10−6 , and T R = 1013 GeV.

Fig. 2. n B − L /s as a function of z = ln(αmt ). The left panel corresponds to a model with mφ /αm = 10−4 , 7 × 10−3 , αm/ M Pl = 2 × 10−6 , and T R = 1013 GeV.

phase (ρ = 0), while in the broken symmetry phase (ρ = ρ¯ ) we have, in the k → 0 limit,6

α 2 ≤ 54c123 λ,

(8)

where we have assumed mφ  ρ¯ , as we will throughout this paper. This √ constraint places a mild upper bound on the coupling, α  λ. We expect λ  O(1), otherwise the theory is strongly coupled. Only the combination αm is relevant for baryogenesis, and this constraint then implies that we cannot simultaneously take m < M Pl to satisfy the constraints above whilst having αm parametrically larger.

αm/ M Pl = 10−2 , and T R = 1012 GeV. The right panel has

αm/ M Pl = 10−2 , and T R = 1012 GeV. The right panel has mφ /αm =

We can see from the potential (6) that the U(1) symmetry is broken at early times and restored at late times, when

¯ ≡ H
m2φ

αm

(9)

.

¯ there is a time-dependent symmetry-breaking miniWhen H > H mum at

|φmin |2 =

αmH − m2φ λ

≈

αmH λ

,

(10)

where we have assumed that the small symmetry-violating terms 4

3. Cosmology In this section we will assume a homogeneous and isotopic cosmological background and derive a simple estimate (19) for the baryon-to-photon ratio generated by our model. In order to verify that the approximations we make are valid, we also solve the equations of motion numerically, with the results plotted in Figs. 1 and 2. Strictly speaking the k → 0 limit is not physical, as we will be dealing with cosmological spacetimes, which only resemble flat space for k H . A more exact ¯ , where H¯ is the value of the Hubble paramcondition is obtained by setting k = H eter below which the symmetry is restored, as discussed in the next section. This only significantly relaxes √ the constraint (8) if mφ αm, in which case the constraint becomes α 2 ≤ 18c 123 λmφ /m. 6

(ε φ 4 + ε † φ † ) are negligible, or, equivalently, have chosen the coefficients ε so that this is the case at early times. ¯, The field tracks this minimum nearly adiabatically until H ≈ H at which point the symmetry is restored and the field begins to oscillate around the symmetry-restored minimum at φ = 0. When this occurs, the U(1) B − L -violating terms play an important role. We would like these to become important around the time that the symmetry is restored. Expanding ε = ε¯ e i ψ , the correction to the potential is

1

 V eff = ε¯ |φ|4 cos (4θ + ψ) , 2

(11)

where φ = |φ|e i θ . The B − L charge density is ↔

n B − L = J 0 = i ( B − L )(φ † ∂ 0 φ) = 2( B − L )|φ|2 θ˙ ,

(12)

J. Sakstein, A.R. Solomon / Physics Letters B 773 (2017) 186–190

so we see that the motion of the angular field is responsible for generating a net B − L. This means that θ should not sit at its minimum in the early Universe, and, indeed, one expects it to be frozen at some initial value due to Hubble damping. We would like it to begin rolling around the time of symmetry restoration in order to generate a net B − L before the field settles into the new minimum at φ = 0, which will be the case if the canonically-normalized ¯ 2 . If mθ < H¯ the anfield’s mass7 m2θ =  V eff θθ /|φ|2 is of order H gular field will not roll after symmetry restoration and no B − L ¯ the field starts rolling long will be generated. Similarly, if mθ > H before symmetry restoration, and the value of B − L is set by tun¯ implies that ing the initial conditions. Setting mθ ∼ H

ε¯ ∼ λ

 m 2 φ

αm

(13)

,

where we have used |φ| = |φmin | at the time of symmetry restoration. Using the angular field’s equation of motion,



˙ θ˙ = ε¯ |φ|4 sin(4θ + ψ), |φ|2 θ¨ + 3H θ˙ + 2|φ||φ|

(14)

one has

n˙ B − L + 3Hn B − L = 2( B − L )¯ε |φ|4 sin(4θ + ψ).

(15)

Making the approximation n˙ B − L ≈ Hn B − L [31,32], which we will verify numerically later, one finds

n B − L ∼ ε¯

|φmin |4 , ¯ H

(16)

where we have omitted factors of order unity. Using equations (9), (10), and (13) we can estimate the B − L conserved charge density as

m4φ

n B −L ∼

λα m

(17)

.

We do not directly observe n B − L , but rather the baryon-tophoton ratio nb = n B − L /s, where s is the entropy density. This introduces some model dependence; for concreteness, and to minimize the number of free parameters, we will focus on a minimal model in which the Universe reheats instantaneously after inflation, and the U(1) B − L symmetry is restored shortly thereafter. The assumption of instantaneous reheating yields

s=

4ρ I 3T R

2 ∼ H¯ 2 M Pl TR,

(18)

where T R is the reheat temperature. Combining this with equation (17) we find

nb =

n B −L s

∼

10−10

λ



TR 108 GeV



αm M Pl

 .

(19)

We see that this new mechanism can produce the observed baryon-to-photon ratio, nb ∼ 10−10 , with sensible choices for the reheat temperature and model parameters. Note that the parameters α and m only appear in the combination αm, while it is m2 (in combination with the c i ) which is constrained by experimental tests of Lorentz violation, as discussed in section 2.1. We expect λ  O(1), otherwise the theory is strongly coupled, and as discussed above, we should have α 2  O (10)λ to ensure gradient stability around flat space. √

7 Recall that Lθ / − g ⊃ |φ|2 (∂θ)2 , necessitating the factor of |φ|−2 in the canonical normalization.

189

In order to verify the approximations we have made above, we have numerically integrated the scalar field equations assuming a radiation-dominated Universe. In Fig. 1 we plot the motion of the complex scalar for two different models. One can see the behavior we predicted qualitatively above: the field tracks its timedependent minimum at early times before the angular field begins to roll when the symmetry is restored, giving rise to a spiral trajectory. In Fig. 2 we plot the baryon-to-photon ratio n B − L /s for the same models. One can see that our numerical results agree well with our prediction (19). 4. Conclusions In this paper we have studied baryogenesis in Lorentz-violating theories of gravity, which, at low energies, are naturally described by a constrained vector so that there is a preferred frame. Baryogenesis requires a field charged under U(1) B − L , the simplest choice being a complex scalar. We have demonstrated here that the lowest-order interaction between the scalar and vector can give rise to a tachyonic mass term for the scalar proportional to the Hubble parameter so that the U(1) B − L symmetry is broken at early times. Inverse phase transitions such as these can generate a net B − L through the coherent motion of the scalar when the symmetry is restored at late times, and we have shown here that Lorentz-violating theories can successfully generate the observed baryon-to-photon ratio using this phenomenon. Furthermore, this can be achieved for parameter choices that are not ruled out by current constraints. Our theory differs from the quintessential paradigm—the Affleck–Dine mechanism—in that the tachyonic mass is proportional to H rather than H 2 , which gives rise to new features and a qualitatively different cosmology, which we have calculated in detail. Lorentz-violating gravity theories continue to be important in the study of dark energy and quantum gravity. Here, we have shown that they may also shed light on the origin of the matter–antimatter asymmetry. Acknowledgements We are grateful to Mark Trodden for enlightening discussions. JS and ARS are supported by funds provided to the Center for Particle Cosmology by the University of Pennsylvania. This paper is dedicated to the memory of Little Pete’s diner, 1978–2017. Benedictio caro cum caseo sit. References [1] D.E. Morrissey, M.J. Ramsey-Musolf, New J. Phys. 14 (2012) 125003, arXiv:1206.2942 [hep-ph]. [2] M. Trodden, Rev. Mod. Phys. 71 (1999) 1463, arXiv:hep-ph/9803479. [3] M. Trodden, in: Proceedings, 32nd SLAC Summer Institute on Particle Physics: Cosmic Connections, SSI 2004, Menlo Park, California, August 2–13, 2004, eConf C040802 (2004) L018, arXiv:hep-ph/0411301. [4] A. Riotto, M. Trodden, Annu. Rev. Nucl. Part. Sci. 49 (1999) 35, arXiv:hepph/9901362. [5] M. Dine, A. Kusenko, Rev. Mod. Phys. 76 (2003) 1, arXiv:hep-ph/0303065. [6] J.M. Cline, in: Les Houches Summer School – Session 86: Particle Physics and Cosmology: The Fabric of Spacetime, Les Houches, France, July 31–August 25, 2006, 2006, arXiv:hep-ph/0609145. [7] R. Allahverdi, A. Mazumdar, New J. Phys. 14 (2012) 125013. [8] C. Armendariz-Picon, A. Diez-Tejedor, R. Penco, J. High Energy Phys. 10 (2010) 079, arXiv:1004.5596 [hep-ph]. [9] P. Horava, Phys. Rev. D 79 (2009) 084008, arXiv:0901.3775 [hep-th]. [10] D. Blas, O. Pujolas, S. Sibiryakov, J. High Energy Phys. 04 (2011) 018, arXiv:1007.3503 [hep-th]. [11] I. Affleck, M. Dine, Nucl. Phys. B 249 (1985) 361. [12] J. Sakstein, M. Trodden, arXiv:1703.10103 [hep-ph], 2017. [13] O. Bertolami, D. Colladay, V.A. Kostelecky, R. Potting, Phys. Lett. B 395 (1997) 178, arXiv:hep-ph/9612437. [14] S.M. Carroll, J. Shu, Phys. Rev. D 73 (2006) 103515, arXiv:hep-ph/0510081.

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