Generalized Dimensional Analysis∗†
이러한 규칙은 다음과 같이 간단하게 요약할 수 있습니다.
1. effective Lagrangian 에 포함되는 모든 term 에 대해 f^2/Lambda^2를 포함합니다.
2. Strongly interacting field 는 1/f에 해당 factor 를 가지도록 합니다.
3. dimension을 4로 맞추기 위해 Lambda를 추가합니다.
이 규칙은 NDA의 원래 버전과도 호환되며, 또한 Nc (색상)의 변화에 따라 f/Lambda 의 ratio 가 어떻게 변하는지에 대한 일반화가 가능합니다. 저자는 이 방법이 더 일반적이며 유용한 도구가 될 수 있다고 제안하며, 이를테면 Chiral gauge theories 에서도 이러한 방법을 적용할 수 있다는데 주목한다.
그것은 또한 강자 interaction 이론에서 chiral symmetry breaking scale 이하의 물리 현상에 대한 effective field theory 를 사용하여 coefficient 가 필요할 때는 Dimensional Analysis (NAIVE DIMENSIONAL ANALYSIS) 가 중요하다.
한글 요약 끝
Generalized Dimensional Analysis∗†
arXiv:hep-ph/9207278v1 31 Jul 1992#HUTP-92/A0367/92Generalized Dimensional Analysis∗†Howard GeorgiLyman Laboratory of PhysicsHarvard UniversityCambridge, MA 02138AbstractI describe a version of so-called naive dimensional analysis, a rule for estimating thesizes of terms in an effective theory below the scale of chiral symmetry breakinginduced by a strong gauge interaction. The rule is simpler and more general thanthe original, which it includes as a special case.I also give a simple qualitativeinterpretation of the rule.∗Research supported in part by the National Science Foundation under Grant #PHY-8714654.†Research supported in part by the Texas National Research Laboratory Commission, under Grant #RGFY9206.
In dealing with effective field theories describing physics of mesons below a symmetry breakingscale in a strongly interacting theory, it is important to have a tool for estimating the coefficientsof nonrenormalizable interactions. Naive dimensional analysis (NDA) [1] was proposed as such atool.
It works pretty well in QCD. However, a better instrument is needed for theories in whichthe number of colors and flavors may be very different from what they are in QCD.
In this verybrief note, I describe one. I will give a rule for such dimensional estimates that is both simplerand more general than the original.
This simplicity and generality is obtained by introducing anadditional parameter, the ratio of the Goldstone boson decay constant to the mass of the lightestnon-Goldstone bound-states. I will also give an extremely simple, qualitative argument to interpretthe rule.I will consider only strongly interacting theories that are “QCD-like” — with fermions trans-forming only under the simplest representation of the gauge group, in order to avoid the additionalcomplications of chiral fermions and of dependence on ratios of Casimir operators.
For example,I don’t want to think about “tumbling” [2] because it makes my head hurt. [3] The low energyphysics described by these QCD-like theories is the physics of the light pseudo-Goldstone mesons.The effective field theory is only useful at energies small compared to the scale at which otherbound-states appear.At the end, I speculate on the dependence of the extra parameter on the color and flavorstructure of the theory and discuss possible generalizations of the trivial idea described here.
Thediscussion in this paper is very simple. I do not pretend that it is very deep.
It has probably beenstated in only slightly different form by others. Nevertheless, I think that the very simplicity of thestatement is a virtue.
It strips NDA down to its barest essentials. I think that this is useful in tryingto determine the form the dimensional analysis will take in more interesting effective theories.In order to be able to keep track of things like the number of colors and flavors in the variousstrong groups, I will distinguish the Goldstone boson decay constant, f, from the typical mass of thelow-lying (non-Goldstone) bound states, Λ.
In QCD, f = fπ is the Goldstone boson decay constantand Λ ≈1GeV (or the ρ mass — take your pick) is the typical mass of the light but non-Goldstonebound states.The simple rule to assign a dimensional coefficient of the right size to any term in the effective2
Lagrangian is1. include an overall factor of f 2Λ2;2. include a factor of 1/f for each strongly interacting field;3. add factors of Λ to get the dimension to 4.
(1)It is that simple. The mass Λ now implicitly contains the factor of 4π from the original version ofNDA.
In fact, this simple rule encompasses all the cases discussed in the original version of NDA,including external fields, quark masses, and the like.However, this rule also makes it possible to extend NDA to different numbers of colors, forexample. If Nc is large in QCD, then as Nc changes, f scales with N1/2cwhile Λ does not change.The result of (1) then agrees with more sophisticated analyses so long as you are calculating thecoefficient of a term that is leading in powers of Nc.1 [4]Even for theories in which the large Nc arguments do not apply (QCD may be in this class aswell — we don’t know for sure [5]), this formulation of dimensional analysis makes sense.
The costof this increased generality is that we now have an additional parameter, Λ/f, to fix before we canuse dimensional analysis. Except for QCD, where we can read the answer from the particle databook, we do not really know Λ/f.Why should it work?
The idea is simple. What can the coefficients depend on?
They willclearly depend on the masses of the non-Goldstone bound-states. Nonrenormalizable interactionsamong the Goldstone bosons can be produced by virtual exchange of these bound-states, so theirmomentum dependence, at least, will presumably be set by Λ.
If this were the whole story, therewould be no more story. However, we know that it is not.
We need an independent parameter, 1/f,that measures the amplitude for making a Goldstone boson. The rule, (1), is just the statementthat these two effects are the only things going on.
There are no other large or small parameters inthe strongly interacting theory. There is a dimensional scale set by the strong interactions — thisis Λ.
There is an amplitude for emitting a Goldstone boson — either the dimensional constant,1/f, or if you prefer, a dimensionless number, Λ/f, one for each Goldstone boson, and the rest issimply dimensional analysis with the mass scale Λ.1You can, of course, foul up any scheme for estimating sizes by looking at a term that is suppressed by somesymmetry.3
I should say that there is absolutely nothing special about the Goldstone bosons in this analysisexcept that they are light. If we are willing to extend the effective theory to describe other mesonstates as well, we would expect coefficients consistent with exactly the same rule.
Similarly, thesame rule applied to baryons gives the conventional NDA result that each dimension 3/2 baryonfield gets a factor of1f√Λ. Thus I interpret the inverse Goldstone boson decay constant as a moreor less universal measure of the amplitude for producing a strongly interacting bound state.
Fieldsdescribing weakly interacting particles, on the other hand, behave just like external fields and aresuppressed by the appropriate factors of 1/Λ. Of course, unless there is some reason for the otherstrongly interacting states to be light, it is not obvious that effective theories describing these heavierstates would be of much use.
[6]We do not know very much for certain about the dependence of the ratio, Λ/f on the numberof flavors, Nf, and colors, Nc, in the gauge theory. The bound from the original NDA,Λf <∼4π ,(2)must still be satisfied, because the arguments of [1] and [7] are still valid, but in general there willbe stronger constraints.
For example, we know that for sufficiently large Nf and Nc, it goes like1/N1/2ctimes some function of the ratio, Nf/Nc. [8] The authors of [5], elaborating an argument ofKaplan, suggest that the ratio has the form:Λf ≈min4πaN1/2c, 4πbN1/2f,(3)where a and b are constants of order 1.
This is a useful provisional form. Note that (2) is satisfied.If this generalization of NDA proves to be useful, we will want to know how to generalize itfurther.
Can anything similarly simple be said about chiral gauge theories in which the low energyeffective theory contains light fermions as well as bosons? There are many uncertainties in this kindof generalization.
How do the scales depend on the Casimir operators? Is the analog of f for theproduction of chiral fermions the same as for bosons?
These questions are interesting and difficultfield theory. If nature chooses to make use of chiral gauge theories above the SU(2)×U(1) breakingscale, they may one day become relevant phenomenology.AcknowledgementsI am grateful to Sekhar Chivukula, David Kaplan, and Aneesh Manohar for interesting comments.4
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[5] R. S. Chivukula, M. Dugan and M. Golden, “Electroweak Correction in Technicolor Reconsid-ered,” BUHEP-92-25, HUTP-92/A033, hep- ph/9207249. Phys.
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by Tran Thanh Van, 1977, 113.5
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