The Rise J and Fall Lof the Cheng-Sher Ansatz

Marc Sher, William & Mary

1. The 2HDM and FCNC

2. The Ansatz

3. Rise

4. Fall

5. Conclusions

2HDM and FCNCThe Standard Model has a single Higgs doublet, Φ, which acquires a vacuumexpectation value

h�i =⇣

0v/p

2

⌘

where v = 246 GeV. The most general Yukawa coupling is given by

LY = yij f̄LifRj� + h. c.When expanding about the vacuum, one gets the mass matrix:

Mij = yijv/p

2

So, when the mass matrix is diagonalized, the Yukawa coupling matrix isautomatically diagonalized, so the Higgs only couples in a flavor-diagonalway. But with 2 doublets:

The most general Yukawa couplings are:

where i and j are generation indices. This gives

Since y1 and y2 are, in general, not simultaneouslydiagonalizable, this will lead to tree level FCNC

These are very problematic-- the dsH couplingwill lead to very large K - K mixing, unless thecoupling is very small or the H is very heavy.

−__

Only way to eliminate tree level FCNC is a discrete symmetry. Paschos-Glashow-Weinberg theorem, applied to a model with doublets and singlets, states that this can only be done if all fermions of a given charge couple to only one Higgs doublet.

Type I: All fermions couple to one doublet,

Type II: The Q=2/3 quarks couple to , the Q=-1/3 quarks and leptons couple to

Paschos, Phys. Rev. D15, 1966 (1977)Glashow, Weinberg, Phys. Rev. D 15, 1958 (1977)

Early 80s

The introduction of an ad hoc Z2 symmetry seemed epicyclic. How necessary was it?

Experimenter in 1980 measuring KL à µ e looked at the bound assuming Higgs exchange and claimed “if the flavor-changing coupling is O(1), we find a lower bound on the Higgs mass of 60 TeV -- this is higher than the energy of the SSC!!” Of course, this ignored mixing, the difference between the two Higgs, etc…..

A more realistic assumption made by Shankar (1980) and by McWilliams and Li (1981). Assume that the flavor-changing coupling was the heaviest fermion of that particular charge times a mixing angle. Since the angle is unknown, assume it is O(1). That still gave a bound of a few TeVfrom ΚL à µ e and an even higher bound of 100 TeV fromΔmK (although there are greater uncertainties in that).

Partly for these reasons (and the rise of SUSY which gave the type II structure), FCNC at tree level was generally ignored for most of the decade.

In the early 80’s, CKM matrix elements weren’t well-known,and there was great interest in Fritzsch type matrices.

✓0 AA B

◆

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If A << B, then the eigenvalues are A2/B and B, so the off-diagonalterm is the geometrical mean of the eigenvalues. If this is thedown quark mass matrix, this leads to the numerically correct result that

Leads to the suggestion that the FCNC couplings should be the geometric mean of the individual Yukawa couplings. Howgeneral is this?

sin ✓c =p

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In ‘86, I moved to Washington Univ. and Ta-Pei Cheng from Missouri, St. Louis was a few miles away. Cheng and Li hadjust been published and I had questions about matrix elements.

We looked at 3x3 Fritzsch matrices and found precisely the same pattern – the FCNC couplings were the geometric meanof the individual couplings. Then Ta-Pei realized it was evenmore general – if you just require that there be no precisecancellations in getting the eigenvalues, it followed.

The ansatz was then written as

yij = �ij

pmimj

v/p2

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where the are O(1). This is order of magnitude – one expects mixing angles, etc.

�ij<latexit sha1_base64="OPip2OqX914mtC9CJ1od1v6bgjg=">AAAB83icbVC7TsMwFL3hWcqrwMhiUZCYqoQFxgoWxiLRh9RUleM4ranjRPYNUhX1N1gYQIiVlZlvYOND2HEfA7QcydLROefqXp8glcKg6345S8srq2vrhY3i5tb2zm5pb79hkkwzXmeJTHQroIZLoXgdBUreSjWncSB5Mxhcjf3mPddGJOoWhynvxLSnRCQYRSv5vrTRkHZzcTfqlspuxZ2ALBJvRsrV4+/3DwCodUuffpiwLOYKmaTGtD03xU5ONQom+ajoZ4anlA1oj7ctVTTmppNPbh6RE6uEJEq0fQrJRP09kdPYmGEc2GRMsW/mvbH4n9fOMLro5EKlGXLFpouiTBJMyLgAEgrNGcqhJZRpYW8lrE81ZWhrKtoSvPkvL5LGWcVzK96NV65ewhQFOIQjOAUPzqEK11CDOjBI4QGe4NnJnEfnxXmdRpec2cwB/IHz9gMYx5Sl</latexit><latexit sha1_base64="6QJzH/FxO24v1n0XyMcu6YPlces=">AAAB83icbVC7TsMwFL0pr9LyKDCyWBQkpiphgTGChbFI9CE1UeU4TmvqOJHtVKqi/gYLAwixsvID/AEbHwIz7mOAliNZOjrnXN3rE6ScKW3bn1ZhZXVtfaO4WSpvbe/sVvb2myrJJKENkvBEtgOsKGeCNjTTnLZTSXEccNoKBlcTvzWkUrFE3OpRSv0Y9wSLGMHaSJ7HTTTE3ZzdjbuVql2zp0DLxJmTqnv89fY+LH/Xu5UPL0xIFlOhCcdKdRw71X6OpWaE03HJyxRNMRngHu0YKnBMlZ9Pbx6jE6OEKEqkeUKjqfp7IsexUqM4MMkY675a9Cbif14n09GFnzORZpoKMlsUZRzpBE0KQCGTlGg+MgQTycytiPSxxESbmkqmBGfxy8ukeVZz7Jpz41TdS5ihCIdwBKfgwDm4cA11aACBFO7hEZ6szHqwnq2XWbRgzWcO4A+s1x8RZ5Yf</latexit><latexit sha1_base64="6QJzH/FxO24v1n0XyMcu6YPlces=">AAAB83icbVC7TsMwFL0pr9LyKDCyWBQkpiphgTGChbFI9CE1UeU4TmvqOJHtVKqi/gYLAwixsvID/AEbHwIz7mOAliNZOjrnXN3rE6ScKW3bn1ZhZXVtfaO4WSpvbe/sVvb2myrJJKENkvBEtgOsKGeCNjTTnLZTSXEccNoKBlcTvzWkUrFE3OpRSv0Y9wSLGMHaSJ7HTTTE3ZzdjbuVql2zp0DLxJmTqnv89fY+LH/Xu5UPL0xIFlOhCcdKdRw71X6OpWaE03HJyxRNMRngHu0YKnBMlZ9Pbx6jE6OEKEqkeUKjqfp7IsexUqM4MMkY675a9Cbif14n09GFnzORZpoKMlsUZRzpBE0KQCGTlGg+MgQTycytiPSxxESbmkqmBGfxy8ukeVZz7Jpz41TdS5ihCIdwBKfgwDm4cA11aACBFO7hEZ6szHqwnq2XWbRgzWcO4A+s1x8RZ5Yf</latexit><latexit sha1_base64="spAX1SuYfjVpKXlfXt+eWLrQ/OA=">AAAB83icbVDLSsNAFL3xWeur6tLNYBFclcSNLotuXFawD2hCmUxu2rGTSZiZCCX0N9y4UMStP+POv3HaZqGtBwYO55zLvXPCTHBtXPfbWVvf2NzaruxUd/f2Dw5rR8cdneaKYZulIlW9kGoUXGLbcCOwlymkSSiwG45vZ373CZXmqXwwkwyDhA4ljzmjxkq+L2w0ooOCP04HtbrbcOcgq8QrSR1KtAa1Lz9KWZ6gNExQrfuem5mgoMpwJnBa9XONGWVjOsS+pZImqINifvOUnFslInGq7JOGzNXfEwVNtJ4koU0m1Iz0sjcT//P6uYmvg4LLLDco2WJRnAtiUjIrgERcITNiYgllittbCRtRRZmxNVVtCd7yl1dJ57LhuQ3v3qs3b8o6KnAKZ3ABHlxBE+6gBW1gkMEzvMKbkzsvzrvzsYiuOeXMCfyB8/kDTleR1g==</latexit>

At the time, the strongest bound on the λij came from ΔmK, andgave (for λij = 1) a lower bound on the exchanged scalar (pseudoscalar)mass of 300 GeV (1 TeV). It ignores contributions from chargedHiggs, and any mixing angles.

The CS ansatz received very little attention for a few years. Then thetop turned out to be heavy, and the B-factories (BELLE/BABAR) began. The ansatz gave experimenters a target (give bounds interms of λij instead of a generic coupling whose value was arbitrary).It also meant that B decays and mixings would have a huge increasein precision, and thus λij =1 was in reach. It received roughly 25citations per year for the next 25 years. Alas, Nature is having thelast word.

RISE

FALL

Over the years, bounds have become much more precise.The best and most recent analysis is Babu and Jana, arxiv:1812.11943.

Strongest bounds are still from meson-meson mixing, butnow we also have D, B and Bs mixing

Bounds on λij obtained from meson mixing, assuming apseudoscalar mass of 500 GeV (bound scales linearly). The bound from scalar exchange is a factor of 3 or so weaker. This assumes real couplings. If there is a CP-violating phase bigger than .005, then the bounds become even worse

Table from Babu and Jana arxiv:1812.11943.

Radiative muon decay

The two-loop diagrams give a bigger contribution(Barr-Zee). The result does depend on the mixingangle, sin(α−β). Assuming the λij are all equal to one:

Taking cos(α−β) =0.4 and amass of H to be 500 GeV,one finds λµe must be less than0.12 to satisfy current bounds.

cos(α−β)

What about Higgs decays? Note: t à hc does currently givesa bound of λtc cos(α−β) < 2, which is quite weak. Better boundswill be available in a couple of decades, but λtc = 1 is beyond reachfor a long time.

From 1903.02718

h à µτ

The branching ratio is 0.0076 λµτ2 cos2(α−β). The current

CMS experimental bound is 0.0025, or λµτ < .6/cos(α−β). Thisgives a weak bound, not yet lethal.

SIDE NOTE: The branching ratio for H à µτ is proportionalto sin2(α−β), which is much larger. Same is true for otherFNCN decays of H.

Sher, Thrasher (2016)Hou, et al (2019)

yij = �ij

pmimj

v/p2

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Any way to avoid these bounds without fine-tuning? Some have suggested replacing v with the smaller vev

effectively rescaling λ by a factor of cos β

But while α−β is basis-independent, β is not. Davidsonand Greiner (2010) chose a basis where one Higgs only couples to the tau, and then the λ is rescaled by cos βτ, butthis angle has nothing to do with the ratio of vevs and thusis completely arbitrary. Davidson and Haber (2005) showedthat IF the scalar self-couplings satisfied a relation, then abasis can be chosen in which the Z2 symmetry appears, andthen tan β has its usual meaning, but that requires tuning.

Conclusions

The Cheng-Sher ansatz parametrizes tree-level flavor-changingneutral currents in terms of coefficients that, in the absence offine-tuning, should be O(1).

Now, 30 years later, data has challenged this ansatz. Five ofthe nine off-diagonal coefficients must be substantial smallerthen 1. It is possible that there might be some wiggle-room,but it appears that the ansatz is no longer viable. It may still be useful in parametrizes and comparing FCNC studies.