New Chapter 3 The Universal Constants

New Chapter 3 The Universal Constants 3. Our Set of Universal Constants The ten dimensionless universal onstants to be used here have already been listed at the beginning of.. In this hapter we desribe what these onstants are. The set of universal onstants adopted depends upon the most fundamental physial theory onsidered. Our ambitions are modest. The history of the universe will be onsidered only after the first milliseond. Attention an then be restrited to what an be derived using relatively simple, low energy, theory. Quantum field theory (QFT) will be avoided. Unfortunately this means that the true basis of the standard model of partile physis, whih is a gauge field theory, will not be addressed. Compensation for this takes two forms. Firstly, the partile zoo upon whih the standard model is based is desribed without mathematis in Appendix B. Seondly, the 6 universal onstants whih our in the standard model will be desribed in this hapter. Most important are the three so-alled oupling onstants whih define the strengths of the eletroweak and strong fores. These also our in our redued set of 0 onstants. Providing a quantitative definition of the oupling onstants really does require a mathematial setting. In lieu of the full QFT formalism, an explanation of the oupling onstants is offered in terms of the energies of interating partiles (see Appendies C, D, E and F). Whilst impreise, it is hoped that this provides an intuitive feel for what these oupling onstants mean. Finally, in the interests of honesty, it is not possible to ignore one partiular impliation of the standard QFT model entirely. This is that the oupling onstants are not truly onstant but vary with energy. They are referred to as running oupling onstants. This is also desribed in simple physial terms, in Appendix. However, in the low energy regime whih is required to desribe the universe after the first milliseond, the running of the oupling onstants will be unimportant in pratie. The disadvantage of starting the lok at about one milliseond is that ertain properties of the universe whih are imprinted upon it before this time will appear instead to be initial onditions. This is not a disadvantage in pratie beause, as theory stands at present, onsideration of the physis at earlier times does not lead to any redution in the number of free parameters (though it may do eventually). An exeption is the resolution of the flatness and horizon problems, but these will be addressed in hapter? The other disadvantage of missing out the first milliseond is that we loose the opportunity to disuss the origin of the universe. What happened at time zero? Compensation for this is provided by a highly simplified summary of urrent osmologial ideas in Appendix A. The main advantage of starting the lok at around a milliseond is that the physis is really quite simple at that time, and for some time to ome thereafter. In ontrast, That is, the theory is simple, not neessarily the appliation. Stars and galaxies are as ompliated as the depth with whih you wish to understand them. Even simple theories produe very ompliated appliations. Reall that the Newtonian 3-body problem is not analytially soluble, and that it is a hallenge to understand a boiling sauepan. Page of 3

during the first milliseond the physis is omplex. Either large numbers of strongly interating partiles are present or irreduibly quantum field theoreti phase hanges indue qualitative hanges in the nature of the ontent of the universe. In any ase, it appears that we need fewer universal onstants to desribe the universe after the first milliseond than before, at least as theory urrently stands. So it is arguably the optimal time to start. In this book we shall adopt a set of just 3 fundamental onstants, whih omprises the 0 dimensionless onstants listed in. plus three dimensionful onstants. The latter are really just equivalent to defining a set of units, so it is the set of 0 dimensionless onstants that matters. This is explained further below. There are nine onstants from physis, as follows:- [] The speed of light in a vauum,. [] Plank s onstant,. (Stritly, this is Plank s onstant divided by, but it will be referred to simply as Plank s onstant to avoid unneessary repetition). [3] Newton s universal gravitational onstant,. [4] The eletri harge of the proton (equal to minus that of the eletron), e. [5] Fermi s weak nulear oupling onstant, F. [6] The low energy effetive oupling onstant for the strong nulear fore, g s. [7] The mass of the neutron (M n ). [8] The mass of the proton (M p ). [9] The mass of the eletron (m e ). The four oupling onstants,, e, F and g s, are desribed in greater detail in Appendies C, D, E and F. In addition, there are the four osmologial parameters whih define the large sale struture of the universe and whih have already been introdued in hapter :- [0] The ratio,, of the number of nuleons to the number of photons in the universe. (It may not be obvious, but this ratio is essentially onstant). [] The average density of dark matter ( d as a fration of the ritial density); [] The average density of dark energy ( as a fration of the ritial density); [3] A parameter Q representing the deviation of the primordial universe from perfet homogeneity (and isotropy). It is possible that developments in theoretial understanding will eventually lead to these last four osmologial onstants being derivable from mirosopi physis. Page of 3

The numerial values of all 3 onstants are given in Table 3., in MKSA units and in Plank units and in partile physiist s MeV units where relevant. The 98% onfidene limit errors ( standard deviations) in the last digit(s) are shown in brakets. Seven of the nine physial onstants are known to high preision. The strong oupling is only roughly defined. This is not beause of impreision in experimental data involving the strong fore, but beause the desription of the low energy strong fore in terms of a single parameter is theoretially impreise. Even at higher energies a unique oupling onstant is not defined, sine the oupling is running, i.e. energy dependent. Exepting g s, the least well known of the physial onstants is. The four osmologial parameters are far less preisely known than the physial onstants, although they are a very great deal better defined now than a deade ago, thanks to COBE and WMAP [see for example Spergel et al (006)]. The ritial density in the urrent epoh is ~0-6 kgm -3 (equivalent to ~6 H atoms per m 3 ). As a fration of the ritial density, the amount of ordinary (baryoni) matter is 0.04 ± 0.005. The average density of old dark matter plus neutrinos as a b fration of the ritial density is dm 0.8 ± 0.04, and the dark energy density parameter is 0.74 ± 0.04. The total density parameter ( b dm ) is onsistent with unity to within ~%, and hene onsistent with a spatially flat universe. The photon:baryon ratio ( ) is ~.6 x 0 9. The so-alled salar flutuation amplitude, Q, i.e. the frational deviation of the osmi mirowave bakground (CMB) from homogeneity, ~ x 0-5. The fat that all four quantities,,, dm and Q are known to within about 0%, or better, is a triumph of observational osmology over the last ten years. 3. The 0 Dimensionless Constants Of the 3 universal onstants, three may be hosen to normalise the rest and make them dimensionless. Three are required, of ourse, to replae the three dimensions of mass, length and time. Physiists generally hoose and as two of these normalising onstants. In partile physis this results in everything being measured in a suitable power of MeV. Normalising to dimensionless quantities requires a third parameter to set a sale for mass or energy. One hoie is Plank units, in whih the third onstant hosen for normalisation purposes is the gravitational onstant,. The resulting Plank units are, Plank Units Quantity Definition Value (MKS) Plank length, L P 35.66x0 m 3 Plank time, t P 44 5.39x0 s 5 Plank mass, M Plank Plank temperature, T P k B 5.765x0 8 kg.468 x 0 3 K Page 3 of 3

Plank energy, E P Plank density, P 5.09 x 09 ev.956 x 0 9 J 5 5.5749 x 0 96 kg/m 3 In terms of Plank units, our first three universal onstants (, and ) would be defined as unity, leaving 0 onstants in our list now in dimensionless form. The hoie of Plank units is generally regarded as most fundamental. For example, the Plank length and time may effetively define the quanta of spaetime, or at least the size sale at whih spaetime eases to behave as a lassial ontinuum. However, even if the Plank sale is indeed truly fundamental, an alternative mass sale may be more onvenient. There is, in any ase, a pratial reason why metrologists would not wish to adopt Plank units. Plank units fail to do justie to the preision with whih many of the universal onstants are known. The reason is that the gravitational onstant,, is amongst the least well known of the onstants. By adopting Plank units, the unertainty in is ommuniated to all the other onstants. The normalising mass sale to be used here is the proton mass, M p. Hene,, and M p drop out of our set of onstants, but we retain, now in its dimensionless form M P as. Consequently our list of 3 universal onstants is really only 0 dimensionless onstants. The numerial values of the 0 dimensionless onstants are given in the final olumn of Table 3.. 3.3 Parameter Spae and What Varying a Constant Means If the dimensionful onstants,, or M p, were varied whilst holding all the dimensionless onstants fixed, would there be any hange in the physial world? The answer is, no, but this is a matter whih an ause onfusion. The ruial requirement is that all the dimensionless onstants remain unhanged. In that ase, a hange in the numerial magnitude of, or M p amounts merely to a hange in the units in whih, or M p are being measured. There is no hange in the physial world. For example, the numerial magnitude of may be doubled simply by taking a haksaw to the standard metre in Paris and utting it in half. All lengths would be numerially doubled in terms of this (halved) standard length. So would inrease by times 4, for example. In general, quantities with dimensions M a L b T would hange by a fator b. So, the quantum of harge, e, would inrease by. Hene, the dimensionless fine struture onstant,, would be unhanged. Inevitably, all 4 dimensionless onstants will be unhanged beause their overall b is zero. Barrow (003) has expressed this as follows:- The last important lesson we learn from the way that pure numbers like define the world is what it really means for worlds to be different. The pure number that we all the fine struture onstant and denote by is a ombination of the eletron harge, e, the speed of light,, and Plank s onstant,. At first sight we might be tempted to think that a world in whih the speed of light was slower would be a different world. e Page 4 of 3

But this would be a mistake. If e, and were all hanged so that the values they have in metri (or any other) units were different when we look them up in our tables of physial onstants, but the value of remained the same, this new world would be observationally indistinguishable from our world. The only things that ount in the definition of the world are the values of the dimensionless onstants of Nature. If all the masses are doubled in value you annot tell beause all the pure numbers defined by the ratios of any pair of masses are unhanged. This is learly true in any example where we hange one or more of the definitions of metre, seond or kilogram. All dimensionful quantities then hange in suh a manner that, neessarily, the dimensionless quantities are unhanged and the world itself is unhanged. However, oneptual diffiulty an arise if a hange is speified as, say, double the speed of light without also speifying how all the other onstants are to hange. Magueijo (003), Chapter 0, desribes very niely the onfusion that an ensue. The whole point of the work that Magueijo (003) was undertaking was based upon the postulate that the speed of light might have been different in the past. But did we not just laim that a hange in the dimensionful parameters makes no differene to the world? No! A hange in the dimensionful parameters makes no differene to the world as long as none of the dimensionless onstants hange. In Magueijo s ase, was hanging, and this had to be attributed to a hange in at least one of e, and - and was the hosen ulprit in this theory. Let us suppose that we onsider a hange in whilst holding the other onstants as listed above fixed (and noting that all nine of the physial onstants are dimensionful). What differene does this make to the world? The key to this question is to onsider whih of the dimensionless onstants is affeted by the hange in. There are only p F p, w 3 M P M and three. They are. Thus, if is doubled, holding the other onstants fixed, the strength of gravity and the strength of the eletromagneti fore are both halved, whereas the strength of the weak nulear fore is doubled. On the other hand, the strong nulear fore, whih depends upon g s e 4 s, is unhanged as are all the other dimensionless onstants. 4 Could the hange in be neutralised as regards its impat on the world by onsidering appropriate hanges to other quantities? Clearly, the answer is yes, as we have already seen. For example, attributing the hange in to an underlying hange in the units of length and/or time will aomplish this automatially. Another way of ahieving this is to require that halves when doubles, and that the Fermi onstant, F, redues by a fator of /6. This also results in all the dimensionless onstants remaining invariant, as does the world. Is there something fundamental about the fat that gravity, eletromagnetism and the weak nulear fore hange if hanges, but the strong nulear fore does not? No. This is merely a onsequene of the hoie of onstants. For example, we ould employ a new definition of the gravitational onstant, suh as /. Adopting instead of now leads to gravity being invariant when hanges. But this is only for the trivial reason that, being one of our set of onstants, is being held fixed by fiat. Page 5 of 3

The moral is that we should never talk of varying onstant X but rather of varying onstant X whilst holding the set of onstants {Y i } fixed. It is just as important to define the invariant set {Y i } as it is to define X. Otherwise the atual nature of the hange envisaged has not been defined at all. However, this is not the most general type of hange that may be onsidered. The orret way to think about hanges in the onstants is to onsider the whole N- dimensional parameter spae (where N = 0 for our redued set of dimensionless onstants). A variation onsists of defining a vetor in this spae: C C C, where C represents the N-dimensional set of onstants in this universe, and C is the new set of onstants being onsidered. The variation therefore onsists of a magnitude, C, and a diretion C. The latter omprises N- degrees of freedom. A speial ase is when the diretion of hange is hosen parallel to one of the oordinate system axes. In other words, when one onstant is varied and the other N- onstants are held fixed. But this is a speial ase. In general, variations in any diretion should be onsidered. After all, an alternative set of universal onstants ould have been hosen, and these would result in a different oordinate system spanning the same parameter spae. We have already presented an example. Thus, holding fixed and varying leads to a hange of gravity, but varying whilst holding fixed does not. They are simply hanges in different diretions in parameter spae. It will be seen that onsidering hanges in more than one of the onstants simultaneously is ruial to a proper understanding of the nature of fine tuning. 3.4 The 6 Constants of the Standard Model of Partile Physis The fundamental partiles whih appear in the standard model of partile physis are desribed in Appendix B. The omplete set of 6 onstants of the standard model is listed in Table 3.. This more omplete set of onstants will not be used in this book, and are inluded only for ompleteness. There are different ways of presenting the 6 onstants. The sheme that has been adopted in Table 3. is to maximise the number of onstants whih are partile masses. This has the advantage that these onstants then need no explanation. As a result, the first 6 onstants in Table 3. are dimensionful (5 being masses). An alternative, and more ommon, onvention is to use a set of onstants all but one of whih is dimensionless. The dimensionful onstant is generally hosen to be v, the vauum expetation value of the Higgs field. The first mass parameters would then be replaed by the so-alled Yukawa oupling onstants (denoted u, d, et in Table 3.). But v uniquely determines the v. low-energy strength of the weak nulear fore, i.e., the Fermi onstant F = Our set of onstants uses F rather than v sine F is more transparent, relating diretly to measurable quantities. Similarly, the dimensionless weak oupling onstant, g, and the Weinberg angle, W, would be onventionally hosen, whereas Table 3. favours the masses of the W and Z bosons as being more readily understood. The last nine onstants in Table 3. are more diffiult to interpret, being rather deeply buried in the quantum field theoretial formalism. They omprise two sets of four, plus one more. The first set of four onstants desribes how the quarks are mixed. The seond set desribes how the neutrinos are mixed. The final onstant is the soalled CP-violating QCD vauum phase. I need to swot up on these and give a brief Page 6 of 3

desription..put some words of explanation in here. Wasn t there a good Physis World artile on this? For pratial purposes it is more onvenient to employ the neutron and proton masses in the set of onstants (as in our Table 3.). In priniple, these are derivable from the standard model parameters, but the required lattie QCD alulations are extremely omputationally hallenging. Impressive progress has been made in reent years in deriving hadron masses from the standard model [see Christine Davies (006) and add other Refs]. This is very important in terms of onfirming the fundamental soundness of the standard model, but it offers no advantage for our purposes. Table 5. lists a rather small upper bound for the neutrino masses (<0.4eV). This is derived from WMAP data, augmented by other astronomial data, see oobar et al (006) and Spergel et al (006). The Partile Data roup still list far bigger upper bounds for the neutrino masses (Referene). However, measurements of neutrino osillations give even smaller values for the mass differenes (stritly m ). m Whilst this is still ompatible in priniple with large, but nearly equal, neutrino masses, this does not seem likely. It seems most likely that the neutrinos have masses less than 0.07 ev [see Mohapatra et al (005)]. Note that the eletromagneti fine struture onstant ( ), or equivalently the quantum of harge, e, do not feature in the list of 6 onstants. This is beause we have hosen to inlude F and the W and Z masses as fundamental onstants, and, in eletroweak theory, the harge e an be expressed in terms of the Fermi onstant and the W and Z masses. Thus, e 5 4 M W F Table 3.3 shows how dependent parameters are derived from the set of 6 onstnats in Table 3.. Most obviously, and lamentably, the standard model requires more onstants rather than less. The 6 onstants of Table 3. are represented in our set of onstant, Table 3., by just 5 physial onstants. Having said this, we may heat a little and oasionally sneak the up and down quark masses, and the pion mass, into our disussions. 3.5 Other Cosmologial Constants Table 3. ontains only 4 parameters to define the large sale struture of the universe. Whilst these are undoubtedly the four most signifiant parameters, several more may be required to define the finer detail of osmologial observations. For example, Tegmak, Aguirre, Rees and Wizek (006) list parameters. One is the dark energy equation of state, whih measures the relative ontribution of the dark energy density (w) and the dark energy pressure. Based on WMAP data [Spergel (006)] w appears to be lose to -, the value expeted if dark energy an be represented by a osmologial onstant (see hapter?). Another parameter measures the urvature of spae. As we have already noted, this is urrently onsistent with spae being flat (though this might mean that the radius of urvature is very large). M M W Z Page 7 of 3

Another possibly parameter might be the ontribution of neutrinos to the mean density of the universe. Table 3. ontains the density parameter for ombined old dark matter and neutrinos, d, but not a density for neutrinos alone. The number of thermal neutrinos originating from the Big Bang is known, but only so long as the number of neutrino flavours is known (urrently believed to be 3). The other unertainty regarding the neutrino ontribution is whether the universe might have a non-zero lepton number. If so, there might be more neutrinos (or antineutrinos) than is implied by a blak body distribution. So the neutrino density is another possible osmologial parameter. Finally, there are several more parameters than an be added to refine the desription of the all-important anisotropies in the osmi mirowave bakground radiation. Having made these observations, there will be no further onsideration of these refinements in this book. Page 8 of 3

Table 3.: Our 3 Low Energy Universal Constants Those hosen to define the system of units shown blue. Numbers in brakets give 98% onfidene error bars on last digit(s) Constant Value in MKS or MKSA Value in Plank units Value in MeV x Alternative Dimensionless Form.9979458 x 0 8 ms - - -.054577(±4) x 0-34 Js - - 6.674(±) x 0 - m 3 kg - s - - M P = 5.906 x 0-39 F.43584(±3) x 0-6 Jm 3 FM PL.6637(±) x 0-5 ev - 33 FM p.7386 x 0 p 3 w -5.068 x 0 e 5.384384 x 0 Jm e - e 0.308 e.60765(±3) x 0-9 Coulomb 4 37.035999 g s 3.5 () (dimensionless) - - g s s 4 () 4.4 M n.674965(±3) x 0-7 kg 7.6955 x 0-0 939.56536(±6) MeV M n / M p n ; n 0.00378 M p.6767(±6) x 0-7 kg 7.6849 x 0-0 938.703(±6) MeV - m e 9.09380(±3) x 0-3 kg 4.853 x 0-3 0.509989(±) MeV m e / M p = /836.5 b (4) (5) 0.5 m -3 ~4 x 0 8 m -3.06 x 0-05.0 x 0-96 - - =.6(±0.) x 0 9 b = 0.04(±0.00) d+ (3) 0.0 x 0-6 kgm -3 0.39 x 0-3 - d+ = 0.8(±0.04) (6) 0.76 x 0-6 kgm -3.47 x 0-3 - = 0.74(±0.04) Q x 0-5 (dimensionless) - - (±0.) x 0-5 () This is the MKSA harge divided by 0, where 0 8.8548787 x 0 - Fm -. The values for e here relate to low energies. Its value inreases at larger energies (see Setion 5.6). () This is the low energy value relevant to the inter-nuleon fore. At energies equal to the Z boson mass (~9eV) g s redues to. ( s = 0.86). g s falls below at energies above 300 ev. 3 Page 9 of 3

(3) Density of dark matter and neurinos is not onstant. Dedued using d+ and a ritial density at the urrent epoh of.00 x 0-6 kgm -3 (equivalent to ~6 H atoms per m 3 ). (4) Average number density of baryons (nuleons) in the present epoh. This orresponds to a mass density normalised by the ritial density of b = 0.04 (±0.005). The sum of the baryon, dark matter and dark energy densities equals the ritial density to within ~%. (5) Average number density of photons in the present epoh. (6) Dark energy appears to have an equation of state of lose to -, implying that it is assoiated with a pressure whih is equal and opposite to this density (times ). The dark energy density is equivalent to ~4.5 H-atoms per m 3. Page 0 of 3

Table 3.: The 6 Universal Constants of the Standard Model of Partile Physis (=) Constant Desription Algebrai Equivalent Value in MKS Value in MeV x M u Up quark mass v u /.7 to 7. x 0-30 kg.5 to 4 MeV M d Down quark mass v d / 7. to 4. x 0-30 kg 4 to 8 MeV M s Strange quark mass v s /.4 to.3 x 0-8 kg 80 to 30 MeV M Charm quark mass v /.0 to.4 x 0-7 kg.5 to.35 ev M b Bottom quark mass v b / 7.3 to 7.8 x 0-7 kg 4. to 4.4 ev (MS sheme) M t Top quark mass v t / 3.(±) x 0-5 kg 74 ± 5 ev m e Eletron mass v e / 9.09380(±3) x 0-3 kg 0.509989(±) MeV M Muon mass v /.8835306(±) x 0-8 kg 05. 65837(±) MeV M Tau mass v / 3.678(±5) x 0-7 kg.7770(±3) ev M e Eletron-neutrino mass v e / <7 x 0-37 kg (see text) <0.4 ev (see text) M Muon-neutrino mass v / <7 x 0-37 kg (see text) <0.4 ev (see text) M Tau-neutrino mass v / <7 x 0-37 kg (see text) <0.4 ev (see text) m H Higgs boson mass /.8 to 4.5 x 0-5 kg Unknown: Perhaps 00-50 ev M W W boson mass vg/.4337(±3) x 0-5 kg 80.45(±76) ev M Z Z boson mass vg/.os w.655(±) x 0-5 kg 9.88(±4) ev F Fermi onstant v.43584(±3) x 0-6 Jm 3.6637(±) x 0-5 ev - For energy < tens of MeV: g s = 3.5 ( s = 4.4) g s Strong fore oupling For energy ~640 MeV: g s = 3.545 ( s =.0) - (dimensionless) For energy = M Z (~9 ev): g s =. ( s = 0.86) For energy ~300 ev: g s =.0 ( s = 0.07958) Page of 3

Table 3. (ontinued): The 6 Universal Constants of the Standard Model of Partile Physis Constant Desription Value (dimensionless) sin Quark CKM matrix angle 0.43(6) sin Quark CKM matrix angle 0.043(5) 3 sin Quark CKM matrix angle 0.0037(5) 3 Quark CKM matrix phase.05(4) 3 sin Neutrino MNS matrix angle 0.55(6) sin Neutrino MNS matrix angle >0.94 3 3 sin Neutrino MNS matrix angle <0. 3 Neutrino MNS matrix phase unknown qd CP-violating QCD vauum phase < 0-9 Page of 3

Table 3.3: Derived Constants from the Standard Model of Partile Physis Constant Desription Algebrai Equivalent Value - Quadrati Higgs potential oeffiient m H Unknown: perhaps x 0 4 to. x 0 5 ev v Higgs vauum expetation value 46.7 ev F Quarti Higgs potential oeffiient Unknown: perhaps ~ (dimensionless) e g w w e 4 e Weak oupling onstant Weak interation strength Weinberg angle Eletromagneti interation strength (fine struture onstant) Eletromagneti oupling onstant (dimensionless quantum of harge) v M At M w Z : 0.658(±4) v At 0 energy: 0.645(±0) g M At M Z : 0.0338(±5) w p w At 0 energy: 0.0385(±) 4 M os M p M w Z At M Z : 0.506(±4) At 0 energy: 0.4908(±0) w sin At M W Z : /7.98(±36) At 0 energy: /37.035999(±3) 4 At M Z : 0.3349(±44) At 0 energy: 0.308(±) Page 3 of 3

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