# Time variation of proton-electron mass ratio and fine structure constant with runaway dilaton

###### Abstract

Recent astrophysical observations indicate that the proton-electron mass ratio and the fine structure constant have gone through nontrivial time evolution. We discuss their time variation in the context of a dilaton runaway scenario with gauge coupling unification at the string scale . We show that the choice of adjustable parameters allows them to fit the same order magnitude of both variations and their (opposite) signs in such a scenario.

###### pacs:

98.80.Cq^{†}

^{†}preprint: KUNS-2041 RESCEU-25/06

## I Introduction

In unified theories of fundamental interactions, a variety of fundamental constants are not necessarily “constant” but can vary as a function of spacetime. Therefore, many experiments and observations have been done to test the constancy of various fundamental constants uzan .

Among them, several groups report nonvanishing time variation of some of the fundamental constants. For example, Murphy et al. report time variability of the fine structure constant by use of absorption systems in the spectra of distant quasars webb . They found that the fine structure constant was smaller in the past,

(1) |

for the redshift range ,^{1}^{1}1In webb2 , the
data is updated but the time variation has almost the same value . though similar
observations of other groups do not necessarily reproduce this result
SCP ; LCM . The linear interpolation of such a change yields the
rate of the change,^{2}^{2}2In this paper, we use the units of
.

(2) |

where the dot represents the time derivative and is the reduced Planck scale.

The observations of spectral lines in the Q 0347-383 and Q 0405-443 quasars also suggest a fractional change in the proton-electron mass ratio ,

(3) |

for a weighted fit mpme , which implies that it has decreased over the last 12 Gyr. The linear interpolation of such a change yields the rate of the change,

(4) |

Thus, though there are still large uncertainties, the hints of the time variation of fundamental constants are found.

On the other hand, from the theoretical point of view, it is natural to allow time and space dependence of fundamental constants. In fact, superstring theory, which is expected to unify all fundamental interactions, predicts the existence of a scalar partner (called dilaton) of the tensor graviton, whose expectation value determines the string coupling constant witten . The couplings of the dilaton to matter induces the violation of the equivalence principle and hence generates deviations from general relativity. Therefore, though the dilaton is predicted to be massless at tree level, it is usually assumed that it acquires a sufficiently large mass, associated with supersymmetry breaking, to satisfy the present experimental constraints on the equivalence principle.

However, Damour and Polyakov proposed another possibility which can naturally reconcile a massless dilaton with experimental constraints DP . They pointed out that full string-loop effects modify the four dimensional effective low-energy action as

(5) |

Here () are dependent coupling functions. String scale is given by . In the weak coupling limit (), they are expanded into

(6) |

which comes from genus expansion of string theory with . Assuming a universality of the dilaton coupling functions, that is, the existence of a value of which extremizes all the coupling functions , it has been shown that, during a primordial inflationary stage, the dilaton evolves towards the special value , at which it decouples from matter (so-called “Least Coupling Principle”) DP . Subsequent (slight) change of the dilaton induces the time variation of fundamental constants. Note that the dilaton becomes almost homogeneous in space during inflation so that spatial variations of fundamental constants are expected to be much smaller than their time variations.

On the other hand, in the infinite bare coupling limit (), it is suggested that all the coupling functions have smooth finite limits IBC ,

(7) |

with . Since () is expected to be positive in the “large N”-type toy model of IBC , we assume that all ’s are positive in this paper. All the coupling functions are extremized (minimized) at . In this case, also, the dilaton evolves towards its fixed point at infinity during inflation so that it decouples from matter GPV ; DPV . In Ref. DPV , assuming the dilaton coupling to dark matter and/or dark energy, the magnitude of the time variation of the fine structure constant is estimated.

In this paper, we discuss time variations of the proton-electron mass ratio and the fine structure constant in the context of a dilaton runaway scenario with gauge coupling unification at the string scale . We show that our model can account for the putative time variation of these constants of the same magnitude with the opposite signs.

The rest of the paper is organized as follows. In the next section, we
calculate time variation of the proton-electron mass ratio and that of
the fine structure constant by taking into account effects associated
with thresholds in renormalization group running and variations in the
vacuum expectation value (VEV) of the Higgs field.^{3}^{3}3For related
works in the context of GUT, see varying . We then apply it to
the specific case of a runaway dilaton scenario in §III. In §IV we
show that our model can not only fit the observations well but also is
testable by the experiments to verify the equivalence principle.
Finally §V is devoted to the conclusion.

## Ii Renormalization group analysis of the time variation of the fundamental constants

In this section, we calculate time variation rates of fundamental constants such as the proton-electron mass ratio and the fine structure constant from a fundamental point of view using renormalization group analysis. As for the particle contents, we concentrate on the standard model and its minimal supersymmetric extension, although our discussions could proceed in the same way for more extended models as well.

### ii.1

First of all, we focus on the time variation of the proton-electron mass ratio . The time variation of is given by

(8) |

Though the proton mass depends not only on the QCD scale
but also on the masses of the up quark and the down
quark, we set to be proportional to
because these quark masses are much smaller than .
Then assuming , the time variation of is given
by^{4}^{4}4Note that the light quark masses may contribute to the
proton mass, which may lead to the slight change of the coefficient in
Eq.(12) Dent .

(9) |

The QCD scale can be extracted from the Landau pole of the renormalization group equations as

(10) | |||||

Here the parameters are given by , , and . is the gauge coupling unified at the string scale , is the supersymmetry (SUSY) breaking scale, and are the masses of top, bottom, and charm quarks, respectively. Reduction to the case of nonsupersymmetric theory would be obvious, that is, we take . Then, the time variation of the QCD scale is given by

(11) | |||||

Here and hereafter, for simplicity, we assume the universality of the
time dependence of fermion masses, that is, the time variation of all
the fundamental fermion masses except three light quarks, u, d, and
s,^{5}^{5}5The current masses of these light quarks are irrelevant to
our analysis. is identical, which is denoted by . Later we present a set of sufficient conditions to realize such a
universality in the context of a dilaton runaway scenario. Under such a
universality, the last two terms of the right hand side of the above
equation are dropped. Then, the time variation of the proton mass is
given by

(12) |

After all, the time variation of the proton-electron mass ratio becomes

(13) |

where we have used the universality of the time variation of fermion masses .

### ii.2

Next, we discuss the time variation of the fine structure constant . The renormalization group equations yield the scale dependence of the gauge couplings ( = 1,2)

(14) |

where is the electroweak symmetry breaking scale. Assuming GUT, for example, the fine structure constant can be related to the gauge couplings ( = 1,2) at any scale as

(15) | |||||

for . After the electroweak symmetry breaking, charged fields acquire their masses. Therefore, taking their mass thresholds into account, the fine structure constant is given by

(16) |

Here, the third term in the right hand side corresponds to fermion mass thresholds, where , and denotes beta-function coefficients for fermion mass thresholds between and .

Then, the time variation of the fine structure constant is given by

(17) |

where we have used the universality of the time variation of fermion masses except three light quarks u, d, s, and because light quarks are confined and effectively have a mass of the order of .

We have assumed that the gauge couplings are unified to . The gauge coupling unification is consistent with experimental values on the gauge couplings within the framework of the minimal supersymmetric standard model, but not in the (non-SUSY) standard model. For the latter case, we need some corrections to gauge couplings at . Such corrections can appear from gauge kinetic functions, which depend on moduli fields other than the dilaton. We assume that moduli-dependent corrections to the gauge couplings do not vary while only the dilaton varies in the time range relevant to our analysis.

### ii.3 Universality of Time Variation of Fermion Masses

The four dimensional effective low-energy action related to the generation of fermion masses is given by

(18) | |||||

where , represents the covariant derivative, and is the generation index. Canonically normalizing all fields yield effective Yukawa couplings and fermion masses,

(19) |

Uplike and downlike fermions acquire masses and through the Higgs mechanism, respectively. Here is a Yukawa coupling constant and and are the VEV of the up-type and down-type Higgs fields, respectively. Note that is replaced by the VEVs of the standard Higgs field in the case of the nonsupersymmetric standard model with a single Higgs doublet.

Then, the time variation of fermion masses is given by

(20) |

Thus, the universality of the time variation of fermion masses is realized, for example, if the following conditions are satisfied,

(21) |

where are generation indices and the last equality comes from . Note that the second condition is unnecessary in the case of the nonsupersymmetric minimal standard model. Here, we have implicitly assumed that the scale dependence of the Yukawa couplings is negligible, that is, the time dependence of the Yukawa couplings is dominated by their dilaton dependence.

Here, we comment on radiative corrections on Yukawa couplings. Renormalization group effects on Yukawa couplings are the same among quarks except the top quark. Thus, radiative corrections do not violate the universal time variation of Yukawa couplings among quarks except the top quark. Furthermore, the mass ratios of other quarks to the top quark do not change drastically between and , i.e. . Hence, even including radiative corrections, the universal time variation of quark Yukawa couplings would be a reasonable assumption, and such corrections on the time variation on would be sufficiently small. The same discussion holds true for the universal time variation of Yukawa couplings only among leptons. However, radiative corrections on quark Yukawa couplings are different from those on lepton Yukawa couplings, because of corrections from . Such difference would be estimated as , where , and it has some effect on Eq.(17), but it can be neglected compared with the first term in Eq. (17).

### ii.4 Electroweak scale, Higgs VEV, and SUSY scale

Before investigating the time dependence of the VEV of the Higgs fields , we discuss the time dependence of the electroweak symmetry breaking scale characterized by the gauge boson mass ,

(22) |

where , and and are and gauge coupling constants respectively. Then, the time variation of the electroweak symmetry breaking scale is given by

(23) |

Inserting Eq. (14) into this equation yields

(24) |

where we have used and .

We now investigate the time variation of the VEV of the Higgs fields, . First, we consider the nonsupersymmetric minimal standard model. The Lagrangian density related to the standard Higgs field is expected to read

(25) | |||||

with and . Assuming that is intrinsic and has no time dependence, the time variation of the VEV of the standard Higgs field is given by

(26) |

Next, we consider the SUSY model. The neutral components of up and down sector Higgs fields, and , have the following potential,

(27) |

with

(28) |

where are SUSY breaking scalar masses squared of Higgs fields, , and is the supersymmetric mass parameter. In addition, the parameter is also SUSY breaking parameter with mass dimension two, that is, the so-called -term.

By using the stationary conditions, , we obtain

(29) | |||

(30) |

where . Furthermore, for a moderate and/or large value of , i.e. , these equations reduce to

(31) | |||

(32) |

That implies that if the time variation of mass parameters is the same, i.e.

(33) |

does not vary, that is, . The above assumption (33) might be plausible for mass parameters at tree level, but has a significant radiative correction due to stop mass,

(34) |

where is SUSY breaking stop mass. Thus, in general, the value of varies in time, and the time variations of and are different. To take into account this aspect, we have to consider the situation that the time dependence of up-type quark masses are different from those of down-type quark masses and lepton masses, and we have to introduce another parameter to represent such difference. Such extension of our analysis is straightforward and would enlarge a favorable parameter space. (Note that because of the SUSY model has more degrees of freedom than the non-SUSY standard model.) Similarly, the time variation of also depends on those of several values, , , , as well as the gauge couplings. To simplify our analysis, we use the same parameterization as the non-SUSY model Eq.(26).

Now let us discuss the time dependence of the SUSY breaking scale . Although it strongly depends on the SUSY breaking model, we give one example based on the gaugino condensation and gravity mediation model. We consider a hidden sector, in which a hidden gauge group with a coupling blows up and hence gauginos condensate at some scale , which breaks the SUSY. Then, repeating the same argument as the case of the QCD scale, the condensation scale is given through the RG flow by

(35) |

where is the beta-function coefficient which, for example, is given by for the gauge group . If this breaking is transmitted to the visible sector through the gravitational interaction, the SUSY breaking scale is given by

(36) |

In this case, the time variation of the SUSY breaking scale is given by

(37) |

## Iii Time variation in a runaway dilaton scenario

### iii.1 Runaway Dilaton

Now, we estimate the time variation of the proton-electron mass ratio and the fine structure constant in the context of a runaway dilaton scenario. From the four dimensional effective low-energy action (5), we have the following relations,

(38) |

Since the coupling functions are given in Eq. (7), the dilaton dependence of the gravitational coupling and the unified gauge coupling is given by

(39) |

which leads to

(40) |

where we set without generality and we have assumed . We regard the string scale as fundamental and hence it has no time dependence .

In the context of the dilaton runaway scenario, the sufficient conditions for the universality of the time variation of Yukawa couplings, (21), are satisfied, for example, in the case that the dilaton dependent functions have the following properties,

(41) |

Hereafter we assume that the dilaton dependent functions satisfy the above conditions (41). In this case, the universal time variation of fermion masses is given by

(42) |

where the universal time variation of Yukawa couplings is estimated as

(43) |

with

(44) |

and (26) reads

(45) |

Note that can take either a positive value or a negative one, which is an important point to account for the decline of .

After all, the universal time variation of fermion masses is given by

(46) |

### iii.2 and in the Runaway Dilaton Scenario

Finally, we show that the choice of adjustable parameters allows them to fit the observed time variation of the proton-electron mass ratio and the fine structure constant in the dilaton runaway scenario, and give constraints on the parameters.

In the case of the non-SUSY model (), the time variation of the proton-electron mass ratio reads

(47) | |||||

where we have used Eqs. (40) and (46) in the second equality. As is seen in Eq. (44), can be negative. On the other hand, the time variation of the fine structure constant is given by

(48) | |||||

As mentioned in the introduction, observations indicate and , which impose constraints on the parameters,

(49) |

These constraints can be easily satisfied if and . Though the time variations of the proton-electron mass ratio and the fine structure constant depend on the evolution of the dilaton field , from Eqs. (1), (3), (47), and (48), we can expect that observed variation can be fitted by taking our model parameters appropriately around order of unity.

In the case of the SUSY model, the time variation of the proton-electron mass ratio becomes

(50) | |||||

where we have used Eqs. (37), (40) and (46). On the other hand, time variation of the fine structure constant is given by

(51) | |||||

In this case, too, from Eqs. (1), (3), (50), and (51), we can expect that our parametrization can fit the observed variation naturally. In fact, the SUSY model has more degrees of freedom than the non-SUSY standard model because of . Furthermore, if we introduce the time variation of the ratio of to as well, we have a wider parameter space to fit the observed variation.

## Iv Comparison with observations

Having formulated the time variation of the fundamental constants in the particle-physics context and given their explicit form in the dilaton runaway scenario, we now solve cosmological evolution of the dilaton field to show our model can account for the observed variation. Before giving an explicit result, however, we must consider other experimental consequences of the dilaton coupling which impose a stringent constraints on the parameter space.