gauge-protected clockwork and ﬂavourful axions · 2019. 11. 4. · gauge-protected clockwork and...

Gauge-protected clockwork and flavourful axions

Quentin Bonnefoy(DESY Hamburg)

Theory Institute in Particle Physics and Cosmology

Warsaw, 17/10/2019

Based on 1804.01112 and 1909.05336 with Emilian Dudas and Stefan Pokorski

WHY GAUGE-PROTECTED?

We want to build theories with accidental global symmetries.



• in the SM : only accidental/approximate/anomalous global symmetries


• from string theory/quantum gravity : no global symmetry - broken by wormholes

- no black hole global hair

- no global symmetry in string theory or AdS/CFT



Banks, Seiberg, …

Hawking, Giddings, Strominger, …

Banks, Dixon, Harlow, Ooguri…


• from string theory/quantum gravity : no global symmetry - broken by wormholes

- no black hole global hair

- no global symmetry in string theory or AdS/CFT



Banks, Seiberg, …

Hawking, Giddings, Strominger, …

Banks, Dixon, Harlow, Ooguri…

• not relevant for gauge theories : robust starting point to get accidental global symmetries

WHAT GAUGE-PROTECTED?

Relevance for phenomenology : any model with a highquality global symmetry

WHAT GAUGE-PROTECTED?

Relevance for phenomenology : any model with a highquality global symmetry

In this talk : QCD axion, Froggatt-Nielsen mechanism

HOW GAUGE-PROTECTED?Disclaimer : here, assumed

« gravity-induced » breaking = all gauge-invariant Planck-suppressed local terms

in the EFT



in the EFT

For instance, too naive for closed string axionsfor a recent trial at an estimation, see 1909.02002 by Fichet & Saraswat



in the EFT


The quality problem now concrete :

LKSVZ � y� L R +c

MP�5

| {z }U(1)PQ

+h.c.

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in the EFT


The quality problem now concrete :

LKSVZ � y� L R +c

MP�5

| {z }U(1)PQ

+h.c.

<latexit sha1_base64="D7gNMCysQQ5g3kfwvhQAv7Z6zUg=">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</latexit>

solves the strong-CP problem if c 10�18

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(with fa ⇠ 1010GeV)<latexit sha1_base64="lflZbKVHO8QvU5C3vaxbr4W5LNQ=">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</latexit>

Barr, Seckel, Kamionkowski, March-Russel, Holman, Hsu, Kephart, Kolb, Watkins, Widrow…

GAUGE-PROTECTED: CONSEQUENCES

In this talk :

• effective coupling scales in models with Stückelberg mixing (ex: clockwork axions)

• phenomenological predictions in flaxion/axiflavon models (ex: gauged chiral Froggatt-Nielsen models)

EXAMPLE 1 : CLOCKWORK AXIONS

(4D) clockwork models : Choi, Im, Kaplan, Rattazzi, Giudice, McCullough…



Mas

ses

Field space

N copies of a heavy sector



Mas

ses

Field space


Not in this talk



Mas

ses

Field space


Not in this talk

Exponentially localised zero mode


Clockwork axion models : start with global symmetries carried by N+1 axions, and select one by



• a suitable explicit breaking of

• gauging

Choi, Im, Kaplan, Rattazzi, Giudice, McCullough…

Coy, Frigerio, Ibe, Ahmed, Dillon, Choi, Im, Shin… + QB, Dudas & Pokorski 1804.01112

the N other symmetries



• a suitable explicit breaking of

• gauging

Choi, Im, Kaplan, Rattazzi, Giudice, McCullough…

Coy, Frigerio, Ibe, Ahmed, Dillon, Choi, Im, Shin… + QB, Dudas & Pokorski 1804.01112

the N other symmetries

In both cases : exponentially localised physical massless axionBut : different parameter space in the EFTs


Global clockwork Gauged clockwork



Sym. breaking

AxionCouplings to :gauge bosons

fermions



Sym. breaking L � c0�q0�1 + ...+ h.c.

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fermions





Introducing Ai=1...Nµ

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fermions







Axion a / ✓0 + q✓1 + ...+ qN✓N<latexit sha1_base64="IJMOwiV2NmVXaKnrCQvvM9fFPMY=">AAAC8HicjVHLSsQwFD3W9/gademmOAjCQGlV0KXoxpWM4DzA0SGtUYudpqapIOJHuHMnbv0Bt/oV4h/oX3gTM+AD0ZS25557z0lubpglca58/6XP6R8YHBoeGS2NjU9MTpWnZxq5KGTE65FIhGyFLOdJnPK6ilXCW5nkrBsmvBmebup885zLPBbprrrI+H6XHafxURwxRVSnXGXtTIpMCbetTrhiHb96ZlFQ9TyvenawbePtTrnie75Z7k8QWFCBXTVRfkYbhxCIUKALjhSKcAKGnJ49BPCREbePS+IkodjkOa5QIm1BVZwqGLGn9D2maM+yKcXaMzfqiHZJ6JWkdLFAGkF1krDezTX5wjhr9jfvS+Opz3ZB/9B6dYlVOCH2L12v8r863YvCEdZMDzH1lBlGdxdZl8Lcij65+6krRQ4ZcRofUl4Sjoyyd8+u0eSmd323zORfTaVmdRzZ2gJv+pQ04OD7OH+CxpIXLHtLOyuV9Q076hHMYR6LNM9VrGMLNdTJ+xoPeMSTI50b59a5+yh1+qxmFl+Wc/8OmN2gGg==</latexit>

Couplings to :gauge bosons

fermions









fermions

=qia

32⇡2p

1 + ...+ q2NfFµ⌫ F̃

µ⌫

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✓i32⇡2f

Fµ⌫ F̃µ⌫

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fermions

=qia

32⇡2p

1 + ...+ q2NfFµ⌫ F̃

µ⌫


✓i32⇡2f

Fµ⌫ F̃µ⌫


@µ✓if

�µ�5 =qi@µap

1 + ...+ q2Nf �µ�5

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fermions

=qia

32⇡2p

1 + ...+ q2NfFµ⌫ F̃

µ⌫


✓i32⇡2f

Fµ⌫ F̃µ⌫


@µ✓if

�µ�5 =qi@µap

1 + ...+ q2Nf �µ�5


Clockwork properties









fermions

=

p1 + ...+ q2Na

32⇡2fFµ⌫ F̃

µ⌫

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=qia

32⇡2p

1 + ...+ q2NfFµ⌫ F̃

µ⌫


✓i32⇡2f

Fµ⌫ F̃µ⌫


✓0 + q✓1 + ...

32⇡2fFµ⌫ F̃

µ⌫

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@µ✓if

�µ�5 =qi@µap

1 + ...+ q2Nf �µ�5











fermions

=

p1 + ...+ q2Na

32⇡2fFµ⌫ F̃

µ⌫


=qia

32⇡2p

1 + ...+ q2NfFµ⌫ F̃

µ⌫


✓i32⇡2f

Fµ⌫ F̃µ⌫


✓0 + q✓1 + ...

32⇡2fFµ⌫ F̃

µ⌫


@µ✓if

�µ�5 =qi@µap

1 + ...+ q2Nf �µ�5


Clockwork properties Generic for gauged models




=

p1 + ...+ q2Na

32⇡2fFµ⌫ F̃

µ⌫


✓0 + q✓1 + ...

32⇡2fFµ⌫ F̃

µ⌫


Generic for gauged models

Recently revisited in more generality : noenhancement of the decay constant in Stückelberg models

Choi, Shin & Yun 1909.11685


Let us compare : - (q,N) generic - N=1, whatever q - q=1, whatever N

Barr, Seckel, Phys. Rev. D46, 539 (1992)

Hill, Leibovich, Phys. Rev. D66, 016006+075010 (2002)

Model (in)dependence :


(q,N) generic N=1, whatever q q=1, whatever N




First symmetry breaking operator


�0�q1

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�0�1...�N<latexit sha1_base64="hXcjNg6CaWXHVvQqm/7jC9AtoAg=">AAAC2nicjVHLSsQwFD3W1/iuiitFioPgqrS60OWgG1ei4KjgyNDG6ITpizQVZHAjuBC3/oBb/QH/RPwDxZ/wJlZQB9GUJifn3nOSmxtmkciV5z33WL19/QODlaHhkdGx8Ql7cmovTwvJeJ2lUSoPwiDnkUh4XQkV8YNM8iAOI74ftjd0fP+My1ykya46z/hRHJwm4kSwQBHVtGcaWUs0PTP7rusasNW0q57rmeF0A78E1drc487b1fzjdmo/oYFjpGAoEIMjgSIcIUBO3yF8eMiIO0KHOElImDjHBYZJW1AWp4yA2DbNp7Q7LNmE9tozN2pGp0T0S1I6WCRNSnmSsD7NMfHCOGv2N++O8dR3O6c1LL1iYhVaxP6l+8z8r07XonCCNVODoJoyw+jqWOlSmFfRN3e+VKXIISNO42OKS8LMKD/f2TGa3NSu3zYw8ReTqVm9Z2VugVd9S2qw/7Od3WBv2fVX3OUd6vQ6PkYFs1jAEvVzFTVsYht18u7gDvd4sBrWpXVt3XykWj2lZhrfhnX7DkDzmtY=</latexit>

�0�q1...�

qN

N<latexit sha1_base64="25M9KkEbNXHiOXD3AsqFw6rvBL0=">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</latexit>



Dimension of the first symmetry breaking

operator1 + q + ...+ qN

<latexit sha1_base64="x/zZRunjVVRxCd+CQcdWlh9QrAE=">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</latexit>

1 + q<latexit sha1_base64="VkKs0Iid4tXZx+3BwiOU3oa+zZw=">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</latexit>

N + 1<latexit sha1_base64="L+PKom9AeTTTuUmIiyLx3iS5rRI=">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</latexit>





operatordO

<latexit sha1_base64="wEDNBASeHzllWMu5ii9hvMz9xjc=">AAACzXicjVHLTsJAFD3UF+ILdemmkZi4Ii2a6JLoxh2YyCMCIe0wYENpm3ZqQhC3/oBb/S3jH+hfeGccEpUYnabtmXPvOTP3XjfyvURY1mvGWFhcWl7JrubW1jc2t/LbO/UkTGPGayz0w7jpOgn3vYDXhCd83oxi7oxcnzfc4bmMN255nHhhcCXGEe+MnEHg9T3mCKKue91Jmzm+WZl28wWraKllzgNbgwL0qob5F7TRQwiGFCNwBBCEfThI6GnBhoWIuA4mxMWEPBXnmCJH2pSyOGU4xA7pO6BdS7MB7aVnotSMTvHpjUlp4oA0IeXFhOVppoqnylmyv3lPlKe825j+rvYaEStwQ+xfulnmf3WyFoE+TlUNHtUUKUZWx7RLqroib25+qUqQQ0ScxD2Kx4SZUs76bCpNomqXvXVU/E1lSlbumc5N8S5vSQO2f45zHtRLRfuoWLo8LpTP9Kiz2MM+DmmeJyjjAlXUyDvAI57wbFSM1Lgz7j9TjYzW7OLbMh4+AIWAkxk=</latexit>

dO<latexit sha1_base64="wEDNBASeHzllWMu5ii9hvMz9xjc=">AAACzXicjVHLTsJAFD3UF+ILdemmkZi4Ii2a6JLoxh2YyCMCIe0wYENpm3ZqQhC3/oBb/S3jH+hfeGccEpUYnabtmXPvOTP3XjfyvURY1mvGWFhcWl7JrubW1jc2t/LbO/UkTGPGayz0w7jpOgn3vYDXhCd83oxi7oxcnzfc4bmMN255nHhhcCXGEe+MnEHg9T3mCKKue91Jmzm+WZl28wWraKllzgNbgwL0qob5F7TRQwiGFCNwBBCEfThI6GnBhoWIuA4mxMWEPBXnmCJH2pSyOGU4xA7pO6BdS7MB7aVnotSMTvHpjUlp4oA0IeXFhOVppoqnylmyv3lPlKe825j+rvYaEStwQ+xfulnmf3WyFoE+TlUNHtUUKUZWx7RLqroib25+qUqQQ0ScxD2Kx4SZUs76bCpNomqXvXVU/E1lSlbumc5N8S5vSQO2f45zHtRLRfuoWLo8LpTP9Kiz2MM+DmmeJyjjAlXUyDvAI57wbFSM1Lgz7j9TjYzW7OLbMh4+AIWAkxk=</latexit>






operatorNumber of gauge groups required to generate the

protection




1<latexit sha1_base64="oUia0qInSK3saNye7yiRnX2IFlc=">AAACxHicjVHLSsNAFD2Nr1pfVZdugkVwVZIq6LIoiMsW7ANqkSSd1tC8mJkIpegPuNVvE/9A/8I74xTUIjohyZlz7zkz914/i0IhHee1YC0sLi2vFFdLa+sbm1vl7Z22SHMesFaQRinv+p5gUZiwlgxlxLoZZ17sR6zjj89VvHPHuAjT5EpOMtaPvVESDsPAk0Q13Ztyxak6etnzwDWgArMaafkF1xggRYAcMRgSSMIRPAh6enDhICOujylxnFCo4wz3KJE2pyxGGR6xY/qOaNczbEJ75Sm0OqBTIno5KW0ckCalPE5YnWbreK6dFfub91R7qrtN6O8br5hYiVti/9LNMv+rU7VIDHGqawippkwzqrrAuOS6K+rm9peqJDlkxCk8oDgnHGjlrM+21ghdu+qtp+NvOlOxah+Y3Bzv6pY0YPfnOOdBu1Z1j6q15nGlfmZGXcQe9nFI8zxBHZdooKW9H/GEZ+vCiixh5Z+pVsFodvFtWQ8fw7uPOg==</latexit>


logq(dO)<latexit sha1_base64="P5mdhvIdIMWdBd/MoYFaY45rdqo=">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</latexit>






protectionGauge charges required

to generate the protection






d1/NO

<latexit sha1_base64="IH2J9ksAMDSkQeXZfwC+I1027Po=">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</latexit>
















d1/NO













Number of fermions required to generate a coupling to gauge fields






d1/NO











Intermediate conclusion I :

• gauge protection is restrictive : decay constants are reduced in Stückelberg models




• gauge protection enforces model-independent features : once the protection is fixed, the number of heavy fermions in the UV completion is also fixed





Those two remarks follow from anomaly cancellation

EXAMPLE 1I : GAUGED FROGGATT-NIELSEN MECHANISM

W � Yu,ijQiUjHu + Yd,ijQiDjHd<latexit sha1_base64="gR4f8iptNCtqnLN1KeQB+eIWhR8=">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</latexit>

mu

mt⇡ �8 ,

mc

mt⇡ �4 ,

md

mt⇡ �7 ,

ms

mt⇡ �5 ,

mb

mt⇡ �3 , ...

VCKM ⇡

0

@1 � �3

� 1 �2

�3 �2 1

1

A

<latexit sha1_base64="zVAXNUrq04zC+d/WoR8EzlM5KZY=">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</latexit>

� ⇡ 0.2<latexit sha1_base64="0o7284uYMTHiYZeJq9fsqDvKo4I=">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</latexit>



Yd ⇡

0

@�7 �6 �6

�6 �5 �5

�4 �3 �3

1

A

<latexit sha1_base64="BSIFDC02FSirL/KA2KovLqoSQ1M=">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</latexit>

Yu ⇡

0

@�8 �5 �3

�7 �4 �2

�5 �2 1

1

A

<latexit sha1_base64="TW17bkBSh2IMg69ABP4FE8aXed4=">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</latexit>

� ⇡ 0.2<latexit sha1_base64="0o7284uYMTHiYZeJq9fsqDvKo4I=">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</latexit>



Yu,ij = yu,ij

✓h�iM

◆nu,ij

<latexit sha1_base64="Rb5Z4huhnesLcurTq/bZpHUCfLI=">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</latexit>

Yd,ij = yd,ij

✓h�iM

◆nd,ij

<latexit sha1_base64="AT9PV/XfNHag0l4ul0QQQ30dkCc=">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</latexit>

� ⇡ h�iM<latexit sha1_base64="KGs4uG6lQTficX3WJCcTHVaegdY=">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</latexit>

Froggatt, Nielsen, Nucl. Phys. B147 (1979)

W � yu,ij

✓h�iM

◆nu,ij

QiUjHu + yd,ij

✓h�iM

◆nd,ij

QiDjHd<latexit sha1_base64="66ximYSWTT6PT5mGbw5kmCZ2Dlo=">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</latexit>


• the FN symmetry is anomalousIbanez, Ross, Jain, Shrock, Binetruy, Ramond, Dudas, Pokorski, Savoy, …

| det(YuY�2d Y 3

e )| ⇠ �32 (A1+A2�2A3) ⇠ �30

<latexit sha1_base64="KkuKJSLiMbc6ivYJkxjSyfzz5n8=">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</latexit>

W � yu,ij

✓h�iM

◆nu,ij

QiUjHu + yd,ij

✓h�iM

◆nd,ij




May be nice : flaxion/axiflavon scenarioWilczek, Ema, Hamaguchi, Moroi, Nakayama, Calibbi, Goertz, Redigolo, Ziegler, Zupan, …


• to gauge it, one way out : anomaly cancellation via the (4D version of the) Green-Schwarz mechanism

(in a heterotic string EFT, nice interplay with gauge-couplingunification)

Green, Schwarz, Phys. Lett. 149B (1984)


W � yu,ij

✓h�iM

◆nu,ij

QiUjHu + yd,ij

✓h�iM

◆nd,ij



• to gauge it, an other way out : spectator fermions(but two independent sectors, and impossible for SUSY)


W � yu,ij

✓h�iM

◆nu,ij

QiUjHu + yd,ij

✓h�iM

◆nd,ij


Alonso, Carmona, Dillon, Kamenik, Martin Camalich, Zupan, Smolkovic, Tammaro + QB, Dudas, Pokorski 1909.05336


• here : use a chiral FN heavy sector

W � yu,ij

✓h�iM

◆nu,ij

QiUjHu + yd,ij

✓h�iM

◆nd,ij




W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij

QiDjHd<latexit sha1_base64="+L8XQg7w/tUkDQIOYKjjmiqhfGI=">AAADhHicrVFdS9xAFL0x1trt17Y+9iW4CJbKkqwt7VMR64PQFwXjCsaGyWR2d3Q2CfNRWML+Nn+G9B+0/8I701nwg1IfnJDk3HPvOTN3btEIrnQc/wqWwuUnK09Xn3Wev3j56nX3zdtjVRtJWUprUcuTgigmeMVSzbVgJ41kZFoINiwuvtn88CeTitfVkZ417GxKxhUfcUo0Unn3cpgp0yimo1nemi1+Ps8EG+nNbCQJbTNBqrFgWTPheZJJF8xvsYMFm0k+nuj3P9rK+8wPc57m5/u5+YDW5SNZlwvrPWtddvJuL+7HbkX3QeJBD/w6qLtXkEEJNVAwMAUGFWjEAggofE4hgRga5M6gRU4i4i7PYA4d1BqsYlhBkL3A7xijU89WGFtP5dQUdxH4SlRGsIGaGuskYrtb5PLGOVv2X96t87Rnm+G/8F5TZDVMkP2fblH5UJ3tRcMIvrgeOPbUOMZ2R72LcbdiTx7d6EqjQ4OcxSXmJWLqlIt7jpxGud7t3RKX/+0qLWtj6msN/LGnxAEnd8d5HxwP+sl2f3D4sbez60e9Cu9gHTZxnp9hB/bhAFKgwUbwPTgK0nAl3Aq3w09/S5cCr1mDWyv8eg3wgdgd</latexit>


W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij


indicates a chiral heavy sector coupled as

W � Qi 1Hu + 1 ̃1�2 + ̃1 2�1 + ...+ ̃NUj�1<latexit sha1_base64="F9DOfcmL6cxcNIdrBxQGZeL8vCs=">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</latexit>


If is a Dirac mass (see Jure Zupan’s slides), no axion�1<latexit sha1_base64="UJ9Yqdat4bfrjKy9Q3t0f2qvIJM=">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</latexit>

W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij



If is a Dirac mass (see Jure Zupan’s slides), no axionIf is a Dirac mass, no axion and gauge anomalies�1

<latexit sha1_base64="UJ9Yqdat4bfrjKy9Q3t0f2qvIJM=">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</latexit>

�2<latexit sha1_base64="6ZvfEwunfA+pXezd4YGuciLX9Gc=">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</latexit>

W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij



If is a Dirac mass (see Jure Zupan’s slides), no axionIf is a Dirac mass, no axion and gauge anomalies

We use both fields (minimal SUSY two-fields FN model) and get a protected flaxion/axiflavon

�1<latexit sha1_base64="UJ9Yqdat4bfrjKy9Q3t0f2qvIJM=">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</latexit>

�2<latexit sha1_base64="6ZvfEwunfA+pXezd4YGuciLX9Gc=">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</latexit>

W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij



W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij


Global FN + axion Gauged FN + axion


W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij



E

N<latexit sha1_base64="YnvApu5xHD/o2lYO9GfotEFFHkk=">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</latexit>

fijfa

<latexit sha1_base64="PsATSsWxLy7S3oCgTWMtWM1vff0=">AAAC13icjVG7TsMwFD2E97vAyBKBkFioUhhgA8EAY5EoDxVUOa5bDGkSOQ5qVVVsiJUfYIXfYUJMrPAXXLtBAioEjmIfn3vPsa+vHwcy0Z730uf0DwwODY+Mjo1PTE5N52ZmD5MoVVyUeBRE6thniQhkKEpa6kAcx0qwhh+II/9yx8SProRKZBQe6FYszhqsHsqa5EwTVcnNntYU4+1apS0vOh1aWaeSW/Tynh1uLyhkYHFr8/WpubKxW4xyzzhFFRE4UjQgEEITDsCQ0FdGAR5i4s7QJk4RkjYu0MEYaVPKEpTBiL2kuU67csaGtDeeiVVzOiWgX5HSxRJpIspThM1pro2n1tmwv3m3rae5W4tWP/NqEKtxTuxfus/M/+pMLRo1bNgaJNUUW8ZUxzOX1L6Kubn7pSpNDjFxBlcprghzq/x8Z9dqElu7eVtm428207Bmz7PcFO/mltTgws929oLD1XxhLb+6T53eRneMYB4LWKZ+rmMLeyiiRN5N3OMBj86Jc+3cOLfdVKcv08zh23DuPgARwprR</latexit>


W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij



E


fijfa


fixed by the mass matrix (close to 8/3)


W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij



E


fijfa



F ({nij})<latexit sha1_base64="OyKXMrmdru5TMUqifS1HlLXK5uc=">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</latexit>


W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij



E


fijfa



F ({nij})<latexit sha1_base64="OyKXMrmdru5TMUqifS1HlLXK5uc=">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</latexit>

F ({nij})⇥x22 + x2

1�2

x22 + x2x1�2

<latexit sha1_base64="jZHk5kBhrA89wgLl5Z78lDkHEik=">AAADCnicjVFNT9VAFD1UUEDUpy5Z0EhIMCYvbV3owgViJCwx4QEJ5dXpvHmPkX5lOjWQ5v0D/olx446w5Q+4MugfkN/AxjtDEZAYmKbtmXPvOTP33rhIZKk973jEuTM6dvfe+MTk/akHDx+1Hj9ZK/NKcdHheZKrjZiVIpGZ6GipE7FRKMHSOBHr8c47E1//LFQp82xV7xViK2WDTPYlZ5qoqPV+aT6ss6iWn4bh8HmoZSrKsK8Yr3ejoBu82I38bhAmZNhj3WD4lw0ocEFHrVmv7dnlXgd+A2YX3r75+nHmdHklb/1AiB5ycFRIIZBBE07AUNKzCR8eCuK2UBOnCEkbFxhikrQVZQnKYMTu0HdAu82GzWhvPEur5nRKQq8ipYs50uSUpwib01wbr6yzYf/nXVtPc7c9+seNV0qsxjaxN+nOM2+rM7Vo9PHa1iCppsIypjreuFS2K+bm7qWqNDkUxBnco7gizK3yvM+u1ZS2dtNbZuO/baZhzZ43uRVOzC1pwP6/47wO1oK2/7IdfKBJL+JsjWMazzBP83yFBSxjBR3y/oLv+Ilfzr7zzTlwDs9SnZFG8xRXlnP0B+nprho=</latexit>


W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij


Minimality of the model :

• only chiral possibility with two input scales and an axion


W � yu,ij

✓h�1ih�2i

◆nu,ij

QiUjHu + yd,ij

✓h�1ih�2i

◆nd,ij


Minimality of the model :

• only chiral possibility with two input scales and an axion

• With the chiral heavy sector, need for no/less spectator fields : larger range of perturbativity (good for pheno : smaller decay constant)


Intermediate conclusion II : • possible to gauge SUSY FN with the usual field

content. If extended scalar sector : accidental flaxion/axiflavon




• in the minimal model : qualitative predictions unchanged (even though -mostly formal- effects of the chiral heavy fields)





- Possible SM gauge coupling unification : anomalous axion couplings, SM spectrum and gauge coupling running entangled





- Possible SM gauge coupling unification : anomalous axion couplings, SM spectrum and gauge coupling running entangled

- Perturbativity up to the Planck scale (at one loop) compatible with axion dark matter, still one-two orders of magnitude away from the flavour constraints

• possible to gauge SUSY FN with the usual field content. If extended scalar sector : accidental flaxion/axiflavon

• in the minimal model : qualitative predictions unchanged



Those two remarks follow from anomaly cancellation

CONCLUSIONS

gauge-protected clockwork and ﬂavourful axions · 2019. 11. 4. · gauge-protected clockwork and...

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