# Holographic spin alignment for vector mesons

Xin-Li Sheng<EMAIL_ADDRESS>INFN Sezione di Firenze, Via G. Sansone 1,
I-50019, Sesto F.no (Firenze), Italy  Institute of Particle Physics and Key
Laboratory of Quark and Lepton Physics (MOS),
Central China Normal University, Wuhan 430079, China  Yan-Qing Zhao
<EMAIL_ADDRESS>Institute of Particle Physics and Key Laboratory
of Quark and Lepton Physics (MOS),
Central China Normal University, Wuhan 430079, China  Si-Wen Li
<EMAIL_ADDRESS>Department of Physics, College of Science, Dalian Maritime
University, Dalian 116026, China  Francesco Becattini<EMAIL_ADDRESS>Department of Physics, University of Florence and INFN Via G. Sansone 1,
I-50019, Sesto F.no (Firenze), Italy  Defu Hou<EMAIL_ADDRESS>Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics
(MOS),
Central China Normal University, Wuhan 430079, China

###### Abstract

We develop a general framework for studying the spin alignment $\rho_{00}$ for
flavorless vector mesons by using the gauge/gravity duality. Focusing on the
dilepton production through vector meson decay, we derive the relation between
production rates at each spin channel and meson’s spectral function, which can
be evaluated by holographic models for a strongly coupled system. As examples,
we study $\rho_{00}$ for $J/\psi$ and $\phi$ mesons, induced by the relative
motion to a thermal background, within the soft-wall model. We show that
$\rho_{00}$ in the helicity frame for $J/\psi$ and $\phi$ mesons have positive
and negative deviations from 1/3 at $T=150$ MeV, respectively, which
consequently leads to different properties for their global spin alignments.

##  I Introduction

Relativistic heavy-ion collisions provide a unique opportunity to test our
understanding about the strongly interacting matter. In non-central
collisions, the incoming nuclei carry huge orbital angular momentum (OAM),
which can partly transfer to the Quark Gluon Plasma in the fireball and
consequently induce the polarization of quarks or hadrons. A typical example
is the global spin polarization for $\Lambda$ hyperons, observed by the STAR
collaboration in Au+Au collisions Liang:2004ph ; Voloshin:2004ha ;
Becattini:2007sr ; STAR:2017ckg ; STAR:2018gyt  (see e.g. Becattini:2020ngo ;
Gao:2020lxh ; Huang:2020dtn ; Becattini:2022zvf , for recent reviews). As
spin-one particles, vector mesons should also be globally polarized by the
OAM.

The most successful model quantitatively reproducing $\Lambda$ polarization is
the hydrodynamic-statistical model Becattini:2007sr ; Becattini:2013fla  which
relates spin polarization to the gradient of the hydrodynamic fields
(specifically, thermal vorticity and thermal shear, see ref.Becattini:2022zvf
). However, at least up to non-dissipative terms, this model predicts a spin
alignment of vector mesons which is consistently lower than that measured by
the experiments ALICE:2019aid ; STAR:2022fan . More models have been proposed
to account for the observed spin alignment. A model based on recombination of
polarized quarks Liang:2004xn ; Yang:2017sdk  also leads to somewhat too low
values. Another model was proposed in Refs. Sheng:2019kmk ; Sheng:2020ghv ;
Sheng:2022wsy ; Sheng:2022ffb ; Sheng:2023urn  where it is argued that strong
force field fluctuations are responsible for the vector meson’s spin
alignment; while these fluctuations does not contribute to the polarization of
hyperons due to its vanishing mean value, they generate a spin alignment of
flavor singlet vector mesons through the correlation of spin polarization of
the constituent quark and antiquark Sheng:2022wsy ; Sheng:2022ffb . Other
possible mechanisms for the generation of spin alignment have been proposed in
Xia:2020tyd ; Gao:2021rom ; Muller:2021hpe ; Kumar:2023ghs ; Li:2022vmb ;
Sheng:2022ssp ; Wei:2023pdf ; Dong:2023cng ; Fang:2023bbw , which are still
waiting for a quantitative verification.

The spin state of a vector meson is described by a $3\times 3$ spin density
matrix $\rho_{\lambda_{1}\lambda_{2}}$, with subscripts
$\lambda_{1},\lambda_{2}=0,\pm 1$ denote the spin quantum number along the
quantization direction. Due to the parity conservation in strong and
electromagnetic interactions, the spin polarization of vector meson,
proportional to $\rho_{11}-\rho_{-1,-1}$, cannot be measured by the angular
distribution of their decay products. Instead, it is possible to measure the
spin alignment $\rho_{00}$, denoting the probability for the state with
$\lambda=0$. In experiments, this is achieved through the polar angle
distribution of decay products in either p-wave strong decays such as
$\phi\rightarrow K^{+}+K^{-}$ Liang:2004xn ; Yang:2017sdk ; ALICE:2019aid ;
STAR:2022fan , or dilepton decays such as $J/\psi\rightarrow\mu^{+}+\mu^{-}$
Schilling:1969um ; Faccioli:2010kd ; ALICE:2020iev ; ALICE:2022dyy . For the
later process, the polar angle distribution $W(\theta)$ is usually
parameterized by $\lambda_{\theta}$ as Faccioli:2010kd

$W(\theta)\propto\frac{1}{3+\lambda_{\theta}}(1+\lambda_{\theta}\text{cos}^{2}\theta),$
(1)

where $\theta$ is the angle between the momentum of one daughter particle and
the chosen quantization direction, determined in the rest frame of meson. For
dilepton decays, the $\lambda_{\theta}$ parameter is related to $\rho_{00}$
through the following relation,

$\lambda_{\theta}=\frac{1-3\rho_{00}}{1+\rho_{00}}.$ (2)

Particularly, the two cases with $\lambda_{\theta}=-1$ (corresponding to
$\rho_{00}=1$) and $\lambda_{\theta}=1$ ($\rho_{00}=0$), are called
longitudinally and transversely polarized, respectively. Recent measurements
by the STAR collaboration at RHIC beam-energy-scan energies STAR:2022fan  and
by the ALICE collaboration at the LHC energy ALICE:2022dyy  show that the
$\phi$ and $J/\psi$ mesons are preferably to be longitudinally and
transversely polarized respectively. However, there are no currently
theoretical works to understand the difference between $\rho_{00}$ for $\phi$
and $J/\psi$ systematically and consistently, thus this motivates us to
provide a framework in theory to fill this blank, in particular, by using
AdS/CFT correspondence and holography since it may relate probably to the non-
perturbative property of hadron.

Indeed, the most essential reason to use AdS/CFT and holography to investigate
spin alignment is due to its powerful method to analyze the strongly coupled
system of QFT Maldacena:1997re ; Aharony:1999ti ; Witten:1998qj ;
Witten:1998zw . Specifically, the information of a strongly coupled QFT system
can be obtained by perturbing its dual classical gravity theory in holography.
Since QCD is a typical strongly-coupled QFT, it has attracted great interest
to study hadron physics and QCD in strong-coupled region through AdS/CFT and
holography for a very long time, e.g. top-down approaches based on D-brane,
gauge-gravity duality, and string theory in Babington:2003vm ;
Kruczenski:2003uq ; Sakai:2004cn ; Sakai:2005yt ; Li:2015uea ; Li:2016gtz ;
Li:2022hka ; Li:2023iuf ; Li:2022wwv  or bottom-up approaches in Gubser:2008yx
; Gursoy:2008bu . Remarkably, the prescription to compute the two-point
retarded Green functions in AdS/CFT, which has been justified in many
different ways e.g.Gubser:1998bc ; Son:2002sd ; Policastro:2002se ;
Freedman:1998tz ; vanRees:2009rw ; Iqbal:2008by ; Marolf:2004fy ; Iqbal:2009fd
; Lee:2008xf ; Casalderrey-Solana:2011dxg ; Liu:2009dm ; Li:2023wyb , is the
key part to evaluate the spin alignment. As we will see, once the holographic
action for meson is obtained on the gravity side, it is possible to employ the
standard method in AdS/CFT to compute the two-point Green function of the dual
mesonic current to evaluate the spin alignment. So in this work, we discuss
first the generic form of the Green function associated with the holographic
action for meson, then without loss of generality test our framework with the
QCD soft-wall model and finally attempt to fit the data to understand various
experimental effects related to spin alignment.

The outline of this letter is as follows. In Sec. II, we focus on the dilepton
decay for a flavorless vector meson and build the relation between observed
spin alignment and the meson’s spectral function. Then we provide a general
framework in Sec. III for calculating the spectral function with holographic
models. As enlightening examples, in Sec. IV we apply the soft-wall model to
$J/\psi$ and $\phi$ mesons and discuss the behaviours of their spin alignment.
We finally summarize this paper in Sec. V. Throughout this paper, we use
$\eta_{\mu\nu}=\eta^{\mu\nu}=\text{diag}(-1,1,1,1)$ for the metric in
Minkowski space, and $g_{\mu\nu}$ for a generic metric in curved spacetime.

##  II Dilepton production through vector meson decay

We first consider the dilepton production through the decay of a flavor
singlet vector meson, such as $J/\psi\rightarrow l+\overline{l}$ and
$\phi\rightarrow l+\overline{l}$. The S-matrix element for the process of an
initial state $i$ to a final state $f$ plus a lepton pair $l\overline{l}$ is
Gale:1990pn ,

$S_{fi}=\int d^{4}xd^{4}y\left\langle
f,l\overline{l}\left|J_{\mu}(y)G_{R}^{\mu\nu}(x-y)J_{\nu}^{l}(x)\right|i\right\rangle$
(3)

Phenomenologically, Eq. 3 describes a two-step process: 1) the vector meson is
produced at the spacetime point $y$, and 2) the vector meson decays into
dileptons at point $x$. Here $G_{R}^{\mu\nu}$ is the retarded propagator for
the vector meson which vanishes for $x^{0}-y^{0}<0$ and therefore ensures the
causality. In Eq. 3, $J_{\nu}$ is the current that couples to the vector
meson. It denotes the quark current (hardonic current) when we consider the
quark-gluon plasma (hardon gas) as the initial state. On the other hand,
$J_{\mu}^{l}$ denotes the leptonic current

$J_{\nu}^{l}(x)=g_{Ml\overline{l}}\overline{\psi}_{l}(x)\Gamma_{\nu}\psi_{l}(x)$
(4)

where $\Gamma_{\mu}$ is the effective vertex for the vector meson-dilepton
interaction, $g_{Ml\overline{l}}$ is the coupling strength, and $\psi_{l}$ is
the lepton field. In general, $\Gamma_{\mu}$ could be linear combinations of
all possible $4\times 4$ matrices that behaves like Lorentz vectors, such as
$\overrightarrow{\partial}^{\mu}\pm\overleftarrow{\partial}^{\mu}$,
$\gamma^{\mu}$,
$\sigma^{\mu\nu}(\overrightarrow{\partial}_{\nu}\pm\overleftarrow{\partial}_{\nu})$,
etc. For simplicity, we only consider $\Gamma_{\nu}\approx\gamma_{\nu}$ in
this paper. The transition probability for the process in Eq. (3) is given by
$R_{fi}\equiv|S_{fi}|^{2}/(TV)$, where $TV$ is the spacetime volume of the
considered system, which is assumed to be sufficiently large. We then sum over
the final states and average for the initial state over a thermal bath.
Afterwards, we arrive at the differential production rate $n(x,p)\equiv
dN/(d^{4}xd^{4}p)$ for dileptons as a function of total momentum
$p^{\mu}=(\omega,{\bf p})$,

$\displaystyle n(x,p)$ $\displaystyle=$
$\displaystyle-\frac{2g_{Ml\overline{l}}^{2}}{3(2\pi)^{5}}\left(1-\frac{2m_{l}^{2}}{p^{2}}\right)\sqrt{1+\frac{4m_{l}^{2}}{p^{2}}}p^{2}n_{B}(\omega)$
(5)
$\displaystyle\times\left(\eta_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)G_{A}^{\mu\alpha}(p)\varrho_{\alpha\beta}(p)G_{R}^{\beta\nu}(p)$

where $G_{A}^{\mu\nu}$ is the advanced propagator,
$n_{B}(\omega)=1/(e^{\omega/T}-1)$ is the Bose-Einstein distribution, and

$\varrho_{\alpha\beta}(p)\equiv-\text{Im}D_{\alpha\beta}(p)$ (6)

is defined with the imaginary part of the retarded current-current correlation

$D^{\mu\nu}(p)\equiv\int d^{4}y\
\theta(y^{0})\left\langle\left[J^{\mu}(y),J^{\nu}(0)\right]\right\rangle
e^{-ip\cdot y}$ (7)

Here $\left\langle O\right\rangle$ denotes the thermodynamical average at a
given temperature.

We emphasize that $\varrho_{\alpha\beta}$ in Eq. (6) is interpreted as the
meson’s spectral function in the thermal medium, while $G_{R/A}^{\mu\nu}$ in
Eq. (5) are propagators in vacuum, describing the propagating of vector meson
after the freeze-out. Eq. (5) reduce to the dilepton production rates through
photon decay Burnier:2008ia ; McLerran:1984ay ; Weldon:1990iw ; Gale:1990pn
if we replace $G_{R/A}^{\mu\nu}$ by the vacuum propagator for a virtual
photon, $\eta^{\mu\nu}/p^{2}$. For the dilepton production through vector
meson decay, the propagators take the following form

$G_{R/A}^{\mu\nu}=-\frac{\eta^{\mu\nu}+p^{\mu}p^{\nu}/p^{2}}{p^{2}+m_{V}^{2}\pm
im_{V}\Gamma}$ (8)

where $m_{V}$ is the vacuum mass and $\Gamma$ is the width. On the other hand,
the spectral function can be decomposed using a complete basis of polarization
vectors

$\varrho^{\mu\nu}(p)=\sum_{\lambda,\lambda^{\prime}=0,\pm
1}v^{\mu}(\lambda,p)v^{\ast\nu}(\lambda^{\prime},p)\widetilde{\varrho}_{\lambda\lambda^{\prime}}(p)\,,$
(9)

where the polarization vectors

$v^{\mu}(\lambda,p)=\left(\frac{{\bf
p}\cdot\boldsymbol{\epsilon}_{\lambda}}{M},\boldsymbol{\epsilon}_{\lambda}+\frac{{\bf
p}\cdot\boldsymbol{\epsilon}_{\lambda}}{M(\omega+M)}{\bf p}\right)$ (10)

satisfy the orthonormality conditions
$\eta_{\mu\nu}v^{\mu}(\lambda,p)v^{\ast\nu}(\lambda^{\prime},p)=\delta_{\lambda\lambda^{\prime}}$
and the completeness condition
$\sum_{\lambda}v^{\mu}(\lambda,p)v^{\ast\nu}(\lambda,p)=(\eta^{\mu\nu}+p^{\mu}p^{\nu}/p^{2})$.
Here $M\equiv\sqrt{\omega^{2}-{\bf p}^{2}}$ is the invariant mass for the
vector meson. The three-component vectors $\boldsymbol{\epsilon}_{\lambda}$
denotes the spin direction in the meson’s rest frame, with
$\boldsymbol{\epsilon}_{0}$ being the spin quantization direction and
$\boldsymbol{\epsilon}_{\pm 1}$ being orthogonal to
$\boldsymbol{\epsilon}_{0}$. If the quantization direction is taken as the
direction of ${\bf p}$ , one can find that
$\sum_{\lambda=\pm}v^{\mu}(\lambda,p)v^{\ast\nu}(\lambda,p)$ and
$v^{\mu}(0,p)v^{\ast\nu}(0,p)$ reduce to the transverse and longitudinal
projection operators in textbooks kapusta1989finite , respectively. Then we
are able to separate contributions from different spin states $\lambda=0,\pm$1
to the dilepton production rate in Eq. (5),

$\displaystyle n_{\lambda}(x,p)$ $\displaystyle=$
$\displaystyle-\frac{2g_{Ml\overline{l}}^{2}}{3(2\pi)^{5}}\left(1-\frac{2m_{l}^{2}}{p^{2}}\right)$
(11)
$\displaystyle\times\sqrt{1+\frac{4m_{l}^{2}}{p^{2}}}\frac{p^{2}n_{B}(\omega)\widetilde{\varrho}_{\lambda\lambda}(p)}{(p^{2}+m_{V}^{2})^{2}+m_{V}^{2}\Gamma^{2}}$

The total dilepton number is the summation of all spin states,
$n=\sum_{\lambda=0,\pm 1}n_{\lambda}$. The spin alignment for the vector meson
resonance is defined as the probability of the spin-zero state,

$\rho_{00}(x,{\bf p})\equiv\frac{\int d\omega\ n_{0}(x,p)}{\sum_{\lambda=0,\pm
1}\int d\omega\ n_{\lambda}(x,p)}$ (12)

Since $n_{\lambda}$ is positive definite, the spin alignment is restricted in
the region $0\leq\rho_{00}\leq 1$. We note that in general the width $\Gamma$
is much smaller than meson’s vacuum mass $m_{V}$, thus Eq. (11) behaves as a
narrow peak around $-p^{2}=m_{V}^{2}$ and therefore the spin alignment is
approximate to the ratio
$\widetilde{\varrho}_{00}/\sum_{\lambda}\widetilde{\varrho}_{\lambda\lambda}$
at energy $\omega=\sqrt{m_{V}^{2}+{\bf p}^{2}}$.

##  III General setup for mesonic correlation function

Since the key part of evaluating the spin alignment is to compute the
correlation function as it is discussed in the last section, let us outline
the general setup for the correlation function of vector meson by using
AdS/CFT in this section. The AdS/CFT dictionary,

$Z_{\text{QFT}}\left[A_{\mu}^{\left(0\right)}\right]=Z_{\text{gravity}}\left[A_{\mu}\right],$
(13)

where

$\displaystyle Z_{\text{QFT}}\left[A_{\mu}^{\left(0\right)}\right]$
$\displaystyle=$
$\displaystyle\left\langle\exp\left\\{\int_{\partial\mathcal{M}}J^{\mu}A_{\mu}^{\left(0\right)}d^{4}x\right\\}\right\rangle,$
$\displaystyle Z_{\text{gravity}}\left[A_{\mu}\right]$ $\displaystyle=$
$\displaystyle\exp\left\\{-S_{\text{bulk}}\left[A_{\mu}\right]\right\\},$ (14)

illustrates that the generating functional $Z_{\mathrm{QFT}}$ of the dual QFT
living at boundary $\partial\mathcal{M}$ is equivalent to the partition
function $Z_{\mathrm{Gravity}}$ of the bulk gravity on $\mathcal{M}$. Here
$A_{\mu}$ is a bulk vector on $\mathcal{M}$ and $A_{\mu}^{\left(0\right)}$ is
its boundary value which is recognized as the source of operator $J^{\mu}$ on
$\partial\mathcal{M}$, i.e.
$A_{\mu}^{\left(0\right)}=A_{\mu}|_{\partial\mathcal{M}}$. We note that the
relation presented in Eq. 14 is workable only if the dual QFT is a strongly
coupled theory. In general, the bulk geometry should be AdS-like which takes
the following form of the metric as,

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}+g_{\zeta\zeta}d\zeta^{2}\,,$ (15)

where $g_{\mu\nu},g_{\zeta\zeta}$ only depend on the holographic radial
coordinate $\zeta$ and the indices $\mu,\nu=0,..3$ label the four-dimensional
spacetime. The coordinate $\zeta$ is defined in the region
$0\leq\zeta\leq\zeta_{h}$, where $\zeta=0$ is the location of the holographic
boundary and $\zeta=\zeta_{h}$ refers to horizon determined by
$g_{tt}(\zeta_{h})=0$. The Hawking temperature is given by

$T=\frac{\left|\kappa\right|}{2\pi},$ (16)

where the surface gravity $\kappa$ is given as
$\kappa=\sqrt{-g_{tt}/g_{\zeta\zeta}}(\text{log}(\sqrt{-g_{tt}}))^{\prime}|_{\zeta=\zeta_{h}}$
Hashimoto:2016dfz . Keeping this in hand, it is possible to evaluate the two-
point correlators of $J^{\mu}$ by using the partition function of the bulk
gravity due to

$\left\langle\left[J^{\mu},J^{\nu}\right]\right\rangle=D^{\mu\nu}\propto\frac{\delta^{2}Z_{QFT}}{\delta
A_{\mu}^{\left(0\right)}\delta
A_{\nu}^{\left(0\right)}}=\frac{\delta^{2}Z_{\text{gravity}}}{\delta
A_{\mu}^{\left(0\right)}\delta A_{\nu}^{\left(0\right)}}.$ (17)

As it is known, if QFT is non-perturbative, it would be very challenging to
evaluate analytically the correlation function $D^{\mu\nu}$ by using the QFT
generating function $Z_{\text{QFT}}$. However, according to Eq. 17, we can
achieve this goal by computing the corresponding partition function of the
bulk gravity. In this sense, once the bulk action for $A_{\mu}$ as vector
meson is given, we can compute the correlation function $D^{\mu\nu}$ of the
dual current $J^{\mu}$ by using Eq. 17.

Revisiting various models based on AdS/CFT e.g. Sakai:2004cn ; Sakai:2005yt ;
Karch:2006pv , we find while the exact form of the bulk mesonic action depends
on the holographic setup we chose, it is possible to write down the quadratic
action for vector meson as a generic Maxwell form,

$S_{\text{bulk}}=-\int d^{4}x\,d\zeta\,Q(\zeta)\,F_{MN}F^{MN},$ (18)

where $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$ is the gauge field strength
and $M,N$ runs over the five-dimensional bulk. Note that the exact form of
$Q\left(\zeta\right)$ depends on the model we chose. Since our concern is the
classical partition function $Z_{\text{gravity}}$ and the correlators
$\left\langle\left[J^{\mu},J^{\nu}\right]\right\rangle$ in Eq. 17, we need to
solve the equation of motion for the bulk vector meson $A_{\mu}$, which can be
obtained by varying the action in Eq. 18 as,

$\partial_{M}\left[Q(\zeta)F^{MN}\right]=0,$ (19)

then to find its classical solution and $S_{\text{bulk}}$. By imposing the
radial gauge condition $A_{\zeta}=0$ and the Fourier transformation for the
remaining components,

$A_{\mu}(x,\zeta)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip\cdot
x}A_{\mu}(p,\zeta),$ (20)

Eq. 19 becomes,

$\displaystyle\partial_{\zeta}\left[Q(\zeta)g^{\zeta\zeta}g^{\mu\nu}\partial_{\zeta}A_{\nu}(p,\zeta)\right]-$
$\displaystyle\hskip
28.45274ptp_{\alpha}\left[Q(\zeta)g^{\alpha\beta}g^{\mu\nu}\left(p_{\beta}A_{\nu}(p,\zeta)-p_{\nu}A_{\beta}(p,\zeta)\right)\right]=0,$
$\displaystyle g^{\mu\nu}p_{\mu}\partial_{\zeta}A_{\nu}(p,\zeta)=0,$ (21)

where the four-momentum $p_{\mu}$ is given as $p_{\mu}=(-\omega,\bf{p})$.
Further taking into account the electric fields associated with the gauge
potential

$E_{i}(p,\zeta)\equiv-p_{0}A_{i}(p,\zeta)+p_{i}A_{0}(p,\zeta),$ (22)

it leads to the following relations,

$\displaystyle\partial_{\zeta}A_{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{p_{0}}\left(\delta^{j}_{i}-\frac{p_{i}p^{j}}{p^{2}}\right)\partial_{\zeta}E_{j},$
$\displaystyle\partial_{\zeta}A_{t}$ $\displaystyle=$
$\displaystyle\frac{p^{i}}{p_{0}p^{0}}\left(\delta^{j}_{i}-\frac{p_{i}p^{j}}{p^{2}}\right)\partial_{\zeta}E_{j},$
(23)

Hence the first equation in (21) reduces to a second order differential
equation for $E_{i}$,

$\displaystyle\partial_{\zeta}^{2}E_{i}(p,\zeta)+\frac{\left[\partial_{\zeta}Q(\zeta)g^{\zeta\zeta}\right]}{Q(\zeta)g^{\zeta\zeta}}\left[\partial_{\zeta}E_{i}(p,\zeta)\right]-\frac{p^{2}}{g^{\zeta\zeta}}E_{i}(p,\zeta)$
$\displaystyle+(-p_{0}g_{i\mu}+p_{i}g_{0\mu})\left(\partial_{\zeta}g^{\mu\nu}\right)\left[\partial_{\zeta}A_{\nu}(p,\zeta)\right]=0,$
(24)

where $\partial_{\zeta}A_{\nu}$ in the last line can be expressed in terms of
$\partial_{\zeta}E_{i}$ by using Eq. (III). Afterwards, we need to find a
solution for 21 and III with incoming wave boundary condition. For the
quadratic action 18 we are considering, the correlation function in 17 can be
written as Son:2002sd ; Casalderrey-Solana:2011dxg ,

$D^{\mu\nu}(p)=\lim_{\zeta\rightarrow
0}g^{\zeta\zeta}g^{\mu\alpha}Q(\zeta)\left.\frac{\delta\left[\partial_{\zeta}A_{\alpha}(p,\zeta)\right]}{\delta
A_{\nu}(p,\zeta)}\right|_{A_{\mu}(p,0)=0}\,,$ (25)

where $A_{\mu}(p,\zeta)$ must be determined by solving the equations of motion
in (21). Denoting $\widetilde{E}_{i}(e_{j},\zeta)$ as the solution for
$E_{i}(p,\zeta)$ satisfying the boundary condition

$\lim_{\zeta\rightarrow 0}\widetilde{E}_{i}(e_{j},\zeta)=\delta_{ij},$ (26)

the correlation function (25) can be expressed as

$\displaystyle D^{\mu 0}(p)$ $\displaystyle=$
$\displaystyle-\lim_{\zeta\rightarrow 0}\frac{p_{j}}{p_{0}}\left(g^{\mu
k}-\frac{p^{\mu}p^{k}}{p^{2}}\right)\partial_{\zeta}\widetilde{E}_{k}(e_{j},\zeta),$
$\displaystyle D^{\mu i}(p)$ $\displaystyle=$
$\displaystyle\lim_{\zeta\rightarrow 0}\left(g^{\mu
k}-\frac{p^{\mu}p^{k}}{p^{2}}\right)\partial_{\zeta}\widetilde{E}_{k}(e_{i},\zeta).$
(27)

Obviously, the Ward identity remains to be
$p_{\mu}D^{\mu\nu}=p_{\mu}D^{\nu\mu}=0$ with the above discussion. We note
that the spectral function for the current-current correlation would come from
the imaginary part of $D^{\mu\nu}$,

$\widetilde{\varrho}_{\lambda\lambda}(p)=v_{\mu}^{\ast}(\lambda,p)v_{\nu}(\lambda,p)\text{Im}\,D^{\mu\nu}(p)\,,$
(28)

which is projected onto polarization vectors.

When the correlation function is ready, we will focus on the spin alignment of
vector meson. So, for the reader’s convenience, we briefly summarize the
approach to evaluate the spin alignment of vector meson in the framework of
holography as follows,

  1. 1.

Solving the equations of motion in (III) with incoming wave condition at the
horizon and condition (26) at the boundary.

  2. 2.

Constructing the current-current correlation function using Eq. (III).

  3. 3.

Projecting the correlation function onto polarization vectors as shown in (28)
and substituting into Eq. (11) to obtain the dilepton production rates from
different spin states.

  4. 4.

Averaging over meson’s energy and, if necessary, averaging over some specific
momentum region to obtain the spin alignment.

In the next section, we will test our method by using QCD soft-wall model and
evaluate holographically the spin alignment of vector meson at finite
temperature in the strong coupling region.

##  IV Holographic spin alignment from QCD soft-wall model

In this section, we will use the QCD soft-wall model to investigate the spin
alignment of vector meson in holography as we discussed in the last section.
The holographic background metric in this model is given as,

$ds^{2}=\frac{L^{2}}{\zeta^{2}}\left(-f(\zeta)dt^{2}+dx^{2}+dy^{2}+dz^{2}+\frac{d\zeta^{2}}{f(\zeta)}\right),$
(29)

where $L$ denotes the AdS radius and $f(\zeta)=1-\zeta^{4}/\zeta_{h}^{4}$. The
Hawking temperature is given by $T=1/(\pi\zeta_{h})$. The action for bulk
mesonic field $A_{M}$ takes the form as 18, while the function $Q(\zeta)$ is
given by

$Q(\zeta)=e^{-\Phi(\zeta)}\sqrt{-g}/(4g_{5}^{2}),$ (30)

where $\Phi(\zeta)$ is the background dilaton field playing the role of
infrared cut-off and breaking the conformal symmetry in the soft-wall model.
The coupling constant $g_{5}$ can be determined by matching the data of
current correlators in the dual QFT (see Refs.Karch:2006pv ; Policastro:2002se
; Freedman:1998tz ; Grigoryan:2007my ). The simplest choice for $\Phi$ could
be $\Phi(\zeta)=c\zeta^{2}$ according to Karch:2006pv ; Mamani:2013ssa ;
Braga:2015jca ; Braga:2016wkm , since it can reproduce the linear Regge
behavior $m_{n}^{2}=4c(n+1)$ for the vector meson mass at zero temperature.
While there are some other choices for $\Phi$ Mamani:2022qnf ; Braga:2017bml ;
Braga:2018zlu ; MartinContreras:2021bis , we will not attempt to use them in
this work since they will increase the complexity of our calculation.

###  IV.1 $J/\psi$ meson

We first consider the dimuon decay of the $J/\psi$ meson, which has the vacuum
mass $m_{J/\psi}=3.096$GeV ParticleDataGroup:2018ovx . We take the width of
$J/\psi$ as $\Gamma\approx 100$ MeV instead of its vacuum width, reflecting
the phenomenological dimuon production rates observed at LHC ALICE:2022dyy .
We have checked that varying $\Gamma$ from 100 MeV to its vacuum value does
not have a sizable effect on the meson’s spin alignment. For the dilaton field
in Eq. 30, we take $\Phi(\zeta)=c_{J/\psi}\zeta^{2}$ with the parameter
$c_{J/\psi}=m_{J/\psi}^{2}/4\approx 2.40\text{GeV}^{2}$. When performing the
energy integral in Eq. 12, we restrict the invariant mass in the range
$2.1\text{GeV}\leq M\leq 4.9\text{GeV}$, which is the same as in Ref.
ALICE:2022dyy .

Considering the $J/\psi$ meson passing through a thermal background, a
straightforward choice for the spin quantization direction is
$\boldsymbol{\epsilon}_{0}=\boldsymbol{\epsilon}_{0}^{h}={\bf p}/|{\bf p}|$,
where ${\bf p}$ is the meson’s three-momentum in the rest frame of the thermal
background. Such a choice corresponds to the so-called helicity frame, because
the spatial part of meson’s polarization vector for the longitudinally
polarized state (state with $\lambda=0$) is parallel to the meson’s three-
momentum. Due to the motion relative to the background, the symmetry between
longitudinally ($\lambda=0$) and transversely ($\lambda=\pm 1$) polarized
states is broken, leading to different dilepton decay rates $n_{\lambda}$ and
therefore a nontrivial spin alignment.

In Fig. 1 we plot spectral functions as functions of invariant mass
$M\equiv\sqrt{\omega^{2}-{\bf p}^{2}}$, computed at temperature $T=150$ MeV.
At zero momentum, spectral functions for all spin states are degenerate
because of the rotation symmetry, as shown by the red solid line. For mesons
with finite momentum, we observe a significant separation between spectral
functions for longitudinally polarized state (solid lines) and transversely
polarized states (dashed lines). At a small invariant mass $M$, the vector
meson is preferred to be transversely polarized. The spectral function for
longitudinal polarized state grows and exceeds that for transversely polarized
states at $M\sim 2.8$ GeV. We also find that peak values for spectral
functions decrease with increasing $|{\bf p}|$, reflecting the fact that high-
momentum resonances are harder to be produced than low-momentum resonances.

Figure 1: Spectral functions $\varrho_{\lambda\lambda}(q)$ as functions of
invariant mass $M\equiv\sqrt{\omega^{2}-{\bf p}^{2}}$ and momentum $|{\bf
p}|=0$ (red lines), 5 GeV (blue lines), and 10 GeV (green lines) at
temperature $T=150\,\text{MeV}$, made to be dimensionless by dividing $M^{2}$.

With the help of Eqs. (11) and (12), we are able to calculate the spin
alignment (in helicity frame) for $J/\psi$ using spectral functions. The
results are shown in Fig. 2, plotted as functions of $|{\bf p}|$ at $T=150$
MeV and $200$ MeV, respectively. We observe a positive deviation from 1/3,
corresponding to a negative $\lambda_{\theta}$ parameter. Such a result is
qualitatively different from the experimental results for $J/\psi$ at forward
rapidity in Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV ALICE:2022dyy
. The difference may be because of the $J/\psi$ at forward rapidity region has
a much larger momentum than the considered $|{\bf p}|$ in Fig. 2. The
deviation from $1/3$ is more significant at lower temperatures.

Figure 2:  The spin alignment of $J/\psi$ meson in the helicity frame as
functions of momentum $|{\bf p}|$ at temperature $T=0.15$ GeV (red solid line)
and $T=0.2$ GeV (blue dashed line).

Now we apply our model to heavy-ion collisions by setting the spin
quantization direction as $y$-direction,
$\boldsymbol{\epsilon}_{0}=\boldsymbol{\epsilon}_{0}^{y}=(0,1,0)$, i.e., the
direction of global OAM in heavy-ion collisions, corresponding to the so-
called global spin alignment. Using Eq. (9) and noting the fact that
$\widetilde{\varrho}_{\lambda\lambda^{\prime}}$ is diagonal in the helicity
frame, we find a relation between diagonal elements of spectral functions for
cases with $\boldsymbol{\epsilon}_{0}=\boldsymbol{\epsilon}_{0}^{h}$ and
$\boldsymbol{\epsilon}_{0}=\boldsymbol{\epsilon}_{0}^{y}$,

$\displaystyle\widetilde{\varrho}_{00}^{y}(x,p)=\frac{1}{2}\left\\{1-\widetilde{\varrho}_{00}^{h}(x,p)+(\boldsymbol{\epsilon}_{0}^{y}\cdot\boldsymbol{\epsilon}_{0}^{h})^{2}[3\widetilde{\varrho}_{00}^{h}(x,p)-1]\right\\},$
$\displaystyle\sum_{\lambda}\widetilde{\varrho}_{\lambda\lambda}^{y}(x,p)=\sum_{\lambda}\widetilde{\varrho}_{\lambda\lambda}^{h}(x,p)$
(31)

where the superscripts $h$ and $y$ labels different choices for
$\boldsymbol{\epsilon}_{0}$. The global spin alignment is then calculated
using Eqs. (11) and (12). It is straightforward to show that

$\rho_{00}^{y}(x,{\bf p})=\frac{1}{3}+\frac{3\rho_{00}^{h}(x,{\bf
p})-1}{3|{\bf p}|^{2}}\left(p_{y}^{2}-\frac{p_{x}^{2}+p_{z}^{2}}{2}\right)$
(32)

which qualitatively agrees with the prediction of quark coalescence model
Sheng:2023urn . We plot in Fig. 3 $\rho_{00}^{y}$ as functions of meson’s
azimuthal angle, where the thermal background is assumed to be static while
meson’s transverse momentum is fixed to $p_{T}=2$ GeV and the longitudinal
momentum is determined by the rapidity $Y$ as
$p_{z}=\sqrt{M^{2}+p_{T}^{2}}\,\sinh(Y)$. We also fix the temperature as
$T=150$ MeV. For mesons at center-rapidity $Y=0$, the spin alignment shows a
negative deviation from 1/3 when the azimuthal angle $\varphi=0$, then it
increases with $\varphi$ and reaches the maximum value at $\varphi=\pi/2$,
which is larger than 1/3. At a more forward rapidity $Y=1$, the spin alignment
is always smaller than 1/3. The azimuthal angle dependence can be fitted by
$\rho_{00}^{y}-1/3=c_{1}+c_{2}\cos(2\varphi)$, with the parameters
$c_{1}=0.006125$ ($c_{1}=-0.02205$) and $c_{2}=-0.01838$ ($c_{2}=-0.07892$)
for $Y=0$ ($Y=1$). By averaging over the azimuthal angle, we obtain in Fig. 4
the global spin alignment as functions of the meson’s rapidity, for transverse
momentum $p_{T}=1,\,5,\,10$ GeV, respectively. In order to include the effect
of anisotropic flow $v_{2}$ induced by the initial geometry in collisions, we
have performed the average with a weight function $1+2v_{2}\cos(2\varphi)$ and
set $v_{2}=0.15$. We observed that $\rho_{00}^{y}$ increases with increasing
$p_{T}$. For all considered cases, $\rho_{00}^{y}>1/3$ at center rapidity and
decreases to a negative value at a larger rapidity, which is mainly because
the difference between the spectral function for the longitudinal mode and
that for the transverse modes becomes more significant at larger $|{\bf p}|$.
We can naturally expect that $\rho_{00}^{y}<1/3$ in a more forward rapidity
region $2.5<Y<4$, which qualitatively agrees with the ALICE experiment
ALICE:2020iev .

Figure 3:  The global spin alignment for $J/\psi$ mesons with transverse
momentum $p_{T}=2$ GeV and rapidity $Y=0$ (red solid line) or $Y=1$ (blue
dashed line), as functions of the meson’s azimuthal angle $\varphi$. Figure 4:
The global spin alignment for $J/\psi$ mesons with transverse momentum
$p_{T}=1$ GeV (red line), 5 GeV (blue line), and 10 GeV (green line), as
functions of the meson’s rapidity $Y$.

###  IV.2 $\phi$ meson

Similar to the discussions for the $J/\psi$ meson, we can study the $\phi$
meson within the same framework. Again we apply the dilaton field
$\Phi(\zeta)=c_{\phi}\zeta^{2}$ with $c_{\phi}=m_{\phi}^{2}/4\approx
0.26\text{GeV}^{2}$ determined by the vacuum mass of $\phi$ meson,
$m_{\phi}=1.02\text{GeV}$. We also use the vacuum width $\Gamma=4$ MeV and
restrict the $\phi$ meson’s invariant mass within the region $1\text{ GeV}\leq
M\leq 1.04\text{ GeV}$. We still focus on the dimuon decay, though it is not
the dominant decay channel for the $\phi$ meson, and assume that the spin
alignment does not depend on (or weakly depend on) the decay channel. This
assumption is reliable because $\rho_{00}$ can be approximate by
$\widetilde{\varrho}_{00}/\sum_{\lambda}\widetilde{\varrho}_{\lambda\lambda}$
at $p^{2}=m_{\phi}^{2}$, as demonstrated below Eq. 12, while the spectral
function $\widetilde{\varrho}_{\lambda\lambda}$ is independent to the decay
process.

We plot in Fig. 5 the spin alignment in the helicity frame as functions of
$\phi$ mesons’ momentum $|{\bf p}|$. For the case of $T=0.15$ GeV, shown by
the blue dashed line, $\rho_{00}^{h}$ shows a negative deviation from $1/3$,
whose absolute value becomes larger at larger $|{\bf p}|$. However,
$\rho_{00}^{h}$ at $T=0.1$ GeV (the red solid line) has a non-monotonic
dependence to $|{\bf p}|$: it increases with $|{\bf p}|$ in the region $|{\bf
p}|<1.5$ GeV, then decreases with $|{\bf p}|$ when $|{\bf p}|>1.5$ GeV. For
all considered $|{\bf p}|$, the spin alignment in the helicity frame is larger
at $T=0.1$ GeV than that at $T=0.15$ GeV.

Figure 5:  The spin alignment for $\phi$ mesons in the helicity frame as
functions of momentum $|{\bf p}|$ at temperature $T=0.1$ GeV (red solid line)
and $T=0.15$ GeV (blue dashed line).

Using relation (32) between the global spin alignment (measured along the
$y$-direction) and the spin alignment in the helicity frame, we are now able
to study the global spin alignment for $\phi$ mesons. Fig. 6 shows the
azimuthal angle dependence of $\rho_{00}^{y}$ for $\phi$ mesons with a fixed
transverse momentum $p_{T}=2$ GeV at $T=0.15$ GeV. For mesons at $Y=0$,
$\rho_{00}^{y}$ is larger than 1/3 at $\varphi=0$ and decreases to a minimum
value at $\varphi=\pi/2$, which is smaller than 1/3. The results at $Y=1$ has
a positive shift compared to results at $Y=0$. Then we average over the
azimuthal angle with a weight function $1+2v_{2}\cos(2\varphi)$ with
$v_{2}=0.15$ and plot in Fig. 7 $\rho_{00}^{y}$ as functions of rapidity $Y$
at $p_{T}=1$ GeV, 5 GeV, and 10 GeV, respectively. For all three cases, we
find $\rho_{00}^{y}<1/3$ when $Y<0.5$, and it increases to a positive value
when $Y>0.5$. Absolute values for deviations from 1/3 become larger at larger
$p_{T}$, whose magnitude can be as large as $\mathcal{O}(10^{-2})$, which is
the same magnitude as those observed in STAR experiments STAR:2022fan .

Figure 6:  The global spin alignment for $\phi$ mesons with transverse
momentum $p_{T}=2$ GeV and rapidity $Y=0$ (red solid line) or $Y=1$ (blue
dashed line), as functions of the meson’s azimuthal angle $\varphi$.
Figure 7:  The global spin alignment for $\phi$ mesons with transverse
momentum $p_{T}=1$ GeV (red line), 5 GeV (blue line), and 10 GeV (green line),
as functions of the meson’s rapidity $Y$.

The global spin alignment for $\phi$ mesons obtained in this paper
quantitatively agrees with the prediction by quark coalescence model at 200
GeV Sheng:2023urn . However, Ref. Sheng:2023urn  has used parameters extracted
from experiment measurements for global spin alignment, while calculations in
this paper do not rely on any subjective parameter except the meson’s physical
vacuum mass and vacuum width. Therefore the holographic approach may provide a
possible way to study spin alignment from first-principle calculations.

##  V Summary

We have studied the spin alignment of flavorless vector mesons with a
holographic description. For dilepton decays of vector mesons, we have derived
a relation between the production rate and the imaginary part of the current-
current correlation function, which is the spectral function of vector meson.
According to AdS/CFT dictionary, the correlation function in QFT can be
perturbatively calculated in a dual gravity theory on the AdS space. We have
accordingly derived the equations of motion of the bulk vector meson field,
which are expressed in terms of the corresponding electric fields. The
correlation function have been then calculated by boundary values of fields
and their derivatives with respect to the holographic radial coordinate,
enforcing the incoming wave boundary conditions at the horizon of the bulk.

As an example of the holographic description, we have considered the soft-wall
model in Sec. IV, for both $J/\psi$ and $\phi$ mesons. Without incorporating
external fields like magnetic or vorticity fields, the spin alignment is
purely induced by the motion of vector meson relative to the background. As a
consequence, the spin alignment in the helicity frame $\rho_{00}^{h}$ deviates
from 1/3 when the meson has a non-vanishing momentum. Surprisingly, the model
prediction shows that $J/\psi$ and $\phi$ have opposite behaviours. At a fixed
temperature $T=150$ MeV, $\rho_{00}^{h}>1/3$ for $J/\psi$, while
$\rho_{00}^{h}<1/3$ for $\phi$ mesons, indicating that $J/\psi$ ($\phi$)
mesons are more likely to be longitudinally (transversely) polarized. This
difference may arise from their different spectral functions because of
different masses. We also study their global spin alignment $\rho_{00}^{y}$,
i.e., the spin alignment measured along the direction of global OAM in heavy-
ion collisions. For both $J/\psi$ and $\phi$, the azimuthal angle dependence
can be fitted to an analytical form
$\rho_{00}^{y}-1/3=c_{1}+c_{2}\cos(2\varphi)$. For $J/\psi$ mesons, the
parameter $c_{1}$ decreases with increasing rapidity $Y$ and reaches a
negative value at $Y=1$, while for $\phi$ mesons the behaviour is opposite.
One can further expect that, in a more forward rapidity region $2.5<Y<4$,
$\rho_{00}^{y}$ for $J/\psi$ could also be smaller than 1/3 corresponding to
$\lambda_{\theta}>0$, which would agree qualitatively with the recent ALICE
experiment observations ALICE:2022dyy . This rapidity region is beyond the
scope of this paper because the related momentum is too large.

We emphasize that the global spin alignment for $\phi$ mesons given by the
soft-wall model in this paper quantitatively agrees with a previous prediction
obtained within the quark coalescence model Sheng:2023urn . In Ref.
Sheng:2023urn , the spin alignment is explained in terms of fluctuations of
the strong force field, with the rotation symmetry being broken by the motion
of vector meson and the magnitude of fluctuation being extracted by fitting
experiment results. The consistency between two models may imply that the
meson’s spin alignment is a non-perturbative property in the strongly
interacting matter. The validity of our model predictions for the azimuthal
angle dependence and rapidity dependence of $\rho_{00}^{y}$ for $J/\psi$ and
$\phi$ mesons in mid-rapidity region00 $|Y|<1$ would be tested in future
experiments.

## Acknowledgement

The authors thank Hai-Cang Ren for enlightening discussions. This work is
supported in part by the National Key Research and Development Program of
China under Contract No. 2022YFA1604900. This work is also partly supported by
the National Natural Science Foundation of China (NSFC) under Grants No.
12275104, No. 11735007. Si-wen Li is supported by the National Natural Science
Foundation of China (NSFC) under Grant No. 12005033 and the Fundamental
Research Funds for the Central Universities under Grant No. 3132023198.

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[a]Stefan Dittmaier

# Electroweak renormalization based on gauge-invariant vacuum expectation
values

Heidi Rzehak

###### Abstract

We briefly review a recently proposed scheme for a gauge-invariant treatment
of tadpole corrections in spontaneously broken gauge theories called Gauge-
Invariant Vacuum expectation value Scheme (GIVS). The tadpole scheme matters
in higher-order predictions of observables if not all free parameters are
fixed by renormalization conditions based on S-matrix elements, such as in
$\overline{\text{MS}}$ renormalization. In contrast to previously used tadpole
schemes, the GIVS unifies the properties of gauge invariance and perturbative
stability. The application of the GIVS to the Standard Model, for instance,
leads to very moderate electroweak corrections in the conversion of on-shell-
renormalized to $\overline{\text{MS}}$-renormalized masses. Moreover, in
models with extended Higgs sectors, the GIVS is less prone to perturbative
instabilities in the $\overline{\text{MS}}$ renormalization of Higgs mixing
angles than observed for the traditional gauge-independent tadpole treatment.
We illustrate this by considering the next-to-leading-order (electroweak and
QCD) corrections to the decay processes
$\text{h}/\text{H}\to\text{W}\text{W}/\text{Z}\text{Z}\to 4\,$fermions of the
CP-even neutral Higgs bosons h and H in a singlet Higgs extension of the
Standard Model and in the Two-Higgs-Doublet Model.

##  1 Introduction

Electroweak (EW) corrections in the Standard Model (SM) and its extensions are
an important ingredient in precision calculations for present and future
collider phenomenology. Renormalization, which is an important part of this
task, in particular includes the definition of vacuum expectation value (vev)
parameters, such as $v$ in the SM, in higher orders—an issue that is connected
to the treatment of tadpole loop contributions. In case, all model parameters
are fixed by renormalization conditions based on S-matrix elements, i.e. by
so-called on-shell (OS) conditions, higher-order predictions of observables do
not depend on the tadpole scheme. However, this is different, for instance, in
$\overline{\text{MS}}$ renormalization schemes, which are frequently employed
for mass parameters and mixing angles in extended Higgs sectors.

Explicit tadpole contributions are most conveniently removed via tadpole
counterterms which can be generated in two different ways in the Lagrangian:
in the course of parameter renormalization [1, 2], or alternatively via Higgs
field redefinitions [3]. The former, called Parameter Renormalized Tadpole
Scheme (PRTS) in the following, typically leads to small corrections
originating from tadpoles, but in general suffers from gauge dependences if
$\overline{\text{MS}}$ renormalization conditions are used for mass
parameters. The latter, usually called Fleischer–Jegerlehner Tadpole Scheme
(FJTS)111The FJTS [3] is equivalent to the $\beta_{t}$ scheme of Ref. [4]., is
free from gauge dependences, but is prone to very large corrections in
$\overline{\text{MS}}$ schemes, jeopardizing perturbative stability of
$\overline{\text{MS}}$ predictions. More details on tadpole renormalization
can, e.g., be found in Refs. [5, 6, 7, 8, 9, 10].

In Refs. [5, 6] we have proposed a new scheme for tadpole renormalization,
dubbed Gauge-Invariant Vacuum expectation value Scheme (GIVS), which is a
hybrid scheme of the two mentioned types, with the benefits of being gauge
independent and perturbatively stable.222Based on completely different
considerations about symmetry breaking in $R_{\xi}$ gauges, a similar idea for
a hybrid scheme has been mentioned in Sect. D of Ref. [10]. The GIVS is based
on the gauge-invariance property of Higgs fields, and the corresponding
parameters like $v$, in non-linear representations of Higgs multiplets. In
this article we briefly summarize the salient features of the GIVS and its
first applications within the SM [5], a Singlet Higgs Extension of the SM
(SESM) [6], and to the Two-Higgs-Doublet Model (THDM) [6].

##  2 Non-linear representations of Higgs sectors

One of the basic ideas underlying the GIVS is the use of a field basis in
which the Higgs fields developing non-vanishing vevs are gauge invariant. This
implies that explicit tadpole contributions induced by loop diagrams are gauge
independent. In such a field basis the would-be Goldstone-boson part of the
scalar Lagrangian is necessarily parametrized in a non-linear fashion. In the
following, we briefly sketch the structure of appropriate non-linear
representations for the SM, the SESM, and the THDM. For the full details, we
refer to Refs. [5, 6].

### Standard Model

Non-linear field representations are most conveniently formulated via matrix
fields. Denoting the usual, two-component SM Higgs doublet $\Phi$ and its
charge conjugate $\Phi^{\mathrm{c}}$, we form the $2\times 2$ matrix
$\mathbf{\Phi}=\left(\Phi^{\mathrm{c}},\Phi\right)$ and write it in some
“polar representation”

$\displaystyle\mathbf{\Phi}={\textstyle\frac{1}{\sqrt{2}}}(v+h)U(\text{\boldmath{$\zeta$}}),\qquad
U(\text{\boldmath{$\zeta$}})\equiv\exp\left(\frac{2\mathrm{i}\text{\boldmath{$\zeta$}}}{v}\right),\qquad\text{\boldmath{$\zeta$}}\equiv{\textstyle\frac{1}{2}}{\zeta_{j}\sigma_{j}},$
(1)

in which $h$ corresponds to the physical Higgs field, which is gauge
invariant, $\zeta_{j}$ ($j=1,2,3$) are real would-be Goldstone-boson fields
and $\sigma_{j}$ denotes the Pauli matrices. Since the matrix
$U(\text{\boldmath{$\zeta$}})$ is unitary, the gauge-invariant square
$\mathrm{tr}\bigl{[}\mathbf{\Phi}^{\dagger}\mathbf{\Phi}\bigr{]}=(v+h)^{2}$
does not involve would-be Goldestone-boson fields $\zeta_{j}$. In the SM Higgs
Lagrangian

$\displaystyle{\cal L}_{\text{H,SM}}$
$\displaystyle={\textstyle\frac{1}{2}}\text{tr}\left[(D_{\mu}\mathbf{\Phi})^{\dagger}(D^{\mu}\mathbf{\Phi})\right]-V_{\text{SM}}$
(2)

the first term is the kinetic part, which encodes the gauge interactions of
the scalar fields in the covariant derivative $D_{\mu}$. Owing to the non-
linear representation of $\mathbf{\Phi}$, this part is non-polynomial in
$\zeta_{j}$ and induces interaction vertices of arbitrarily many would-be
Goldstone-boson fields, but this is only a minor complication, and the usual
perturbative Feynman diagram calculus works as usual. Note that the SM Higgs
potential

$\displaystyle
V_{\text{SM}}=-{\textstyle\frac{1}{2}}\mu_{2}^{2}\mathrm{tr}\bigl{[}\mathbf{\Phi}^{{\dagger}}\mathbf{\Phi}\bigr{]}+{\textstyle\frac{1}{16}}{\lambda_{2}}\bigl{(}\mathrm{tr}\bigl{[}\mathbf{\Phi}^{{\dagger}}\mathbf{\Phi}\bigr{]}\bigr{)}^{2}=-{\textstyle\frac{1}{2}}\mu_{2}^{2}(v+h)^{2}+{\textstyle\frac{1}{16}}{\lambda_{2}}(v+h)^{4},$
(3)

with the free parameters $\mu_{2}^{2}$ and $\lambda_{2}$, is free from would-
be Goldstone-boson fields. With the of help the Nielsen identities [11], it is
easy to show that the one-point vertex function $\Gamma^{h}_{\mathrm{nl}}$ of
the gauge-invariant Higgs field $h$ is gauge independent; a simple one-loop
calculation confirms this.

### Singlet extension of the SM (SESM)

The SESM extends the SM by a real singlet scalar field $\sigma$, leading to a
second CP-even Higgs boson H. The SESM Higgs Lagrangian is given by

$\displaystyle{\cal L}_{\text{H,SESM}}$
$\displaystyle{}={\textstyle\frac{1}{2}}(\partial\sigma)^{2}+{\textstyle\frac{1}{2}}\text{tr}\left[(D_{\mu}\mathbf{\Phi})^{\dagger}(D^{\mu}\mathbf{\Phi})\right]-V_{\text{SESM}},$
(4)

with a non-polynomial kinetic part similar as in the SM and the gauge-
invariant Higgs potential

$\displaystyle V_{\text{SESM}}={}$
$\displaystyle-{\textstyle\frac{1}{2}}{\mu_{2}^{2}}\mathrm{tr}\bigl{[}\mathbf{\Phi}^{\dagger}\mathbf{\Phi}\bigr{]}-\mu_{1}^{2}\sigma^{2}+{\textstyle\frac{1}{16}}{\lambda_{2}}\big{(}\mathrm{tr}\bigl{[}\mathbf{\Phi}^{\dagger}\mathbf{\Phi}\bigr{]}\big{)}^{2}+\lambda_{1}\sigma^{4}+{\textstyle\frac{1}{2}}{\lambda_{12}}\mathrm{tr}\bigl{[}\mathbf{\Phi}^{\dagger}\mathbf{\Phi}\bigr{]}\,\sigma^{2},$
(5)

which again does not involve would-be Goldstone-boson fields $\zeta_{j}$. The
Higgs fields $h_{1}$ and $h_{2}$ corresponding to particle excitations in
$\sigma$ and $\mathbf{\Phi}$, respectively, are identified by

$\displaystyle\sigma=v_{1}+h_{1},\qquad\mathbf{\Phi}={\textstyle\frac{1}{\sqrt{2}}}(v_{2}+h_{2})U(\text{\boldmath{$\zeta$}}),$
(6)

with vev parameters $v_{1}$ and $v_{2}$, and $U(\text{\boldmath{$\zeta$}})$
denoting the same unitary would-be Goldstone-boson matrix as in the SM, where
$v=v_{2}$. The fields $(h_{1},h_{2})$ are rotated into a field basis of $h$
and $H$ corresponding to mass eigenstates,

$\displaystyle\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}=R(\alpha)\begin{pmatrix}H\\\ h\end{pmatrix},\qquad
R(\alpha)=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\
\sin\alpha&~{}\cos\alpha\end{pmatrix},$ (7)

where $\alpha$ is a real-valued mixing angle which is determined by the
parameters of the Higgs potential. In analogy to the SM case, the one-point
functions $\Gamma^{h}_{\mathrm{nl}}$ and $\Gamma^{H}_{\mathrm{nl}}$ of the
fields $h$ and $H$ are gauge independent in the non-linear representation.

### Two-Higgs-Doublet Model (THDM)

The THDM comprises two complex Higgs doublets $\mathbf{\Phi}_{n}$ ($n=1,2$),
which together contain eight real d.o.f.s. Denoting the vev parameters of
$\mathbf{\Phi}_{n}$ by $v_{n}$, we parametrize the $2\times 2$ matrix fields
according to

$\displaystyle\mathbf{\Phi}_{n}={\textstyle\frac{1}{\sqrt{2}}}\,U(\text{\boldmath{$\zeta$}})\,\bigl{[}(v_{n}+h_{n}){\rm
1\kern-1.19995pt\rule{0.3pt}{6.45831pt}\raisebox{0.0pt}{\rule{1.00006pt}{0.3pt}}}+\mathrm{i}c_{n}\sigma_{j}\rho_{j}\bigr{]},$
(8)

with the unitary would-be Goldstone-boson matrix
$U(\text{\boldmath{$\zeta$}})$ as in the SM with

$\displaystyle v\equiv\sqrt{v_{1}^{2}+v_{2}^{2}},\qquad
c_{1}=-\frac{v_{2}}{v}\equiv-\sin\beta,\qquad
c_{2}=\frac{v_{1}}{v}\equiv\cos\beta.$ (9)

The gauge-invariant fields $h_{1}$ and $h_{2}$ are CP even (under appropriate
assumptions on the Higgs potential) and are rotated into the fields $h$ and
$H$ as in (7) corresponding to neutral CP-even mass eigenstates h and H. The
three remaining fields $\rho_{j}$ describe a neutral CP-odd Higgs boson
$\text{A}_{0}$ via the gauge-invariant field $\rho_{3}$ and two charged Higgs
bosons $\text{H}^{\pm}$ via the fields
$\rho^{\pm}=(\rho_{2}\pm\mathrm{i}\rho_{1})/\sqrt{2}$. The proper
normalization of these fields fixes the constants $c_{n}$ as above. The two
Higgs mixing angles $\alpha$ and $\beta$ are frequently taken as basic input
parameters of the THDM. The Higgs Lagrangian of the THDM is given by

$\displaystyle{\cal L}_{\text{H,THDM}}$
$\displaystyle{}={\textstyle\frac{1}{2}}\text{tr}\left[(D_{\mu}\mathbf{\Phi}_{1})^{\dagger}(D^{\mu}\mathbf{\Phi}_{1})\right]+{\textstyle\frac{1}{2}}\text{tr}\left[(D_{\mu}\mathbf{\Phi}_{2})^{\dagger}(D^{\mu}\mathbf{\Phi}_{2})\right]-V_{\text{THDM}},$
(10)

in which the kinetic term is again non-polynomial in the would-be Goldstone-
boson fields $\zeta_{j}$ but polynomial in the fields $h_{n}$ and $\rho_{j}$
corresponding to physical Higgs bosons. In the non-linear Higgs representation
the gauge-invariant Higgs potential can be written as

$\displaystyle V_{\text{THDM}}={}$
$\displaystyle{\textstyle\frac{1}{2}}{m_{11}^{2}}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{1}\bigr{]}+{\textstyle\frac{1}{2}}{m_{22}^{2}}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{2}^{\dagger}\mathbf{\Phi}_{2}\bigr{]}-{m_{12}^{2}}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{2}\bigr{]}$
$\displaystyle{}+{\textstyle\frac{1}{8}}{\lambda_{1}}\big{(}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{1}\bigr{]}\big{)}^{2}+{\textstyle\frac{1}{8}}{\lambda_{2}}\big{(}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{2}^{\dagger}\mathbf{\Phi}_{2}\bigr{]}\big{)}^{2}+{\textstyle\frac{1}{4}}{\lambda_{3}}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{1}\bigr{]}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{2}^{\dagger}\mathbf{\Phi}_{2}\bigr{]}$
$\displaystyle{}+\lambda_{4}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{2}\Omega_{+}\bigr{]}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{2}\Omega_{-}\bigr{]}+{\textstyle\frac{1}{2}}{\lambda_{5}}\left[\big{(}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{2}\Omega_{+}\bigr{]}\big{)}^{2}+\big{(}\mathrm{tr}\bigl{[}\mathbf{\Phi}_{1}^{\dagger}\mathbf{\Phi}_{2}\Omega_{-}\bigr{]}\big{)}^{2}\right]$
(11)

with the two-dimensional projection operators
$\Omega_{\pm}=\frac{1}{2}(1\pm\sigma_{3})$, which select the original Higgs
doublet $\Phi_{n}$ or its charge conjugate $\Phi^{\mathrm{c}}_{n}$ from the
matrix field $\mathbf{\Phi}_{n}$. Obviously, the unitary Goldstone-boson
matrix $U(\text{\boldmath{$\zeta$}})$ again drops out in $V_{\text{THDM}}$. To
avoid flavour-changing neutral currents at tree level, we assume the
$\mathbb{Z}_{2}$ symmetry $\Phi_{1}\to-\Phi_{1}$ and $\Phi_{2}\to\Phi_{2}$
that is only softly broken by the $m_{12}^{2}$ term in $V_{\text{THDM}}$.
Moreover, all couplings in $V_{\text{THDM}}$ are assumed to be real in order
to conserve CP. Owing to the gauge invariance of the fields $h$ and $H$ the
one-point functions $\Gamma^{h}_{\mathrm{nl}}$ and $\Gamma^{H}_{\mathrm{nl}}$
are again gauge independent in the non-linear representation.

##  3 The GIVS in the SM

Before formulating the GIVS as introduced in Ref. [5] for the SM, we briefly
sketch the FJTS and PRTS for treating tadpoles in the course of renormalizing
the theory. The full renormalization procedure of the SM can be found in Ref.
[9]. Renormalization starts with the transformation that replaces the original
bare quantities in terms of renormalized ones and renormalization constants.
We mark bare parameters by subscripts “0” and bare fields by subscripts “B”.

In the following, we denote the physical Higgs-boson field of the usually
adopted linear Higgs representation by $v+\eta(x)$. One-particle-irreducible
Green function $\Gamma^{\dots}$, so-called vertex functions, involve tadpole
contributions if the splitting $v_{0}+\eta_{\mathrm{B}}(x)$ does not provide
an expansion of the effective Higgs potential about its true minimum (see for
instance App. C of Ref. [12]). Technically, it is desirable to eliminate such
tadpole contributions by appropriate parameter and field definitions. Choosing
$v_{0}$ such that $v_{0}^{2}=4\mu_{2,0}^{2}/\lambda_{2,0}$ at least to leading
order (LO) avoids tadpole contributions at tree level. In higher orders, the
explicit (unrenormalized) tadpole function $\Gamma^{\eta}$ can be cancelled
upon generating a tadpole counterterm $\delta t\,\eta$ in the counterterm
Lagrangian $\delta{\cal L}$. This is achieved by a tadpole renormalization
condition for the renormalized one-point function
$\Gamma_{{\mathrm{R}}}^{\eta}$ (in momentum space) of the physical Higgs
field,

$\displaystyle\Gamma_{{\mathrm{R}}}^{\eta}=\Gamma^{\eta}+\delta
t\overset{!}{=}0\quad\Rightarrow\quad\delta t=-\Gamma^{\eta}.$ (12)

Note that $\Gamma^{\eta}$ is a gauge-dependent quantity in contrast to its
counterpart $\Gamma^{h}_{\mathrm{nl}}$ in the non-linear representation. The
tadpole counterterm $\delta t\,\eta$ is generated in the Lagrangian by
appropriately choosing $v_{0}$ and, if needed, by a further redefinition of
the bare Higgs field $\eta_{\mathrm{B}}$. Inserting the field decomposition
$v_{0}+\eta_{\mathrm{B}}(x)$ into the bare SM Lagrangian ${\cal L}$, produces
a term $t_{0}\,\eta$ in ${\cal L}$ with

$\displaystyle
t_{0}={\textstyle\frac{1}{4}}v_{0}(4\mu_{2,0}^{2}-\lambda_{2,0}v_{0}^{2})$
(13)

at the one-loop level, where $t_{0}$ can be viewed as bare tadpole constant.
The tadpole schemes described below impose different conditions on $t_{0}$,
partially accompanied by appropriate field redefinitions of
$\eta_{\mathrm{B}}$, in order to generate the desired tadpole counterterm
$\delta th$ in the counterterm Lagrangian $\delta{\cal L}$.

In the FJTS the bare tadpole constant is consistently set to zero, $t_{0}=0$,
so that no tadpole counterterm is introduced via parameter redefinitions.
Instead, the tadpole counterterm is introduced by an additional field
redefinition

$\displaystyle\eta_{B}(x)$ $\displaystyle{}\;\to\;\eta_{B}(x)+\Delta
v^{\mathrm{FJTS}},\qquad\Delta v^{\mathrm{FJTS}}=-\frac{\delta
t^{\mathrm{FJTS}}}{M_{\text{H}}^{2}}=\frac{\Gamma^{\eta}}{M_{\text{H}}^{2}},$
(14)

in the bare Lagrangian. The field shift (14) distributes tadpole
renormalization constants to many counterterms in $\delta{\cal L}$ (see, e.g.,
App. A of Ref. [9]). Since the field shift (14) is a mere reparametrization of
the functional integral over the Higgs field, it does not alter predictions
for observables. Omitting the field shift would mean that explicit tadpole
diagrams had to be included, but the result would still be the same as in the
FJTS. In the FJTS, tadpole contributions correct for the fact that the
effective Higgs potential is not expanded about the location of its minimum,
but about the minimum of the potential in lowest order, which in the course of
renormalization receives further corrections. For this reason, renormalization
constants to mass parameters receive tadpole corrections in the FJTS, which
are rather large by experience. In OS renormalization schemes these
corrections cancel in predictions, but in other renormalization schemes such
as $\overline{\text{MS}}$ schemes this cancellation is only partial, and large
corrections typically remain. On the positive side, the FJTS respects gauge
invariance, i.e. the gauge independence of the parametrization of an
observable in terms of $v_{0}$ and the other bare parameters carries over to
the renormalized version of these parameters if the corresponding
renormalization constants do not introduce gauge dependences, which is for
instance the case in OS and $\overline{\text{MS}}$ schemes in the FJTS.

The idea behind the PRTS is to achieve an expansion of the Higgs field about
the true minimum of the renormalized effective Higgs potential (as obtained
from the effective action after renormalization) by appropriate relations
among the parameters of the theory. To this end, the bare parameter
$v_{0}=v+\delta v$ is renormalized in such a way that the renormalized
parameter $v$ is fixed by the renormalized W-boson mass $M_{\text{W}}$ and and
the SU(2) gauge coupling $g_{2}=e/s_{{\mathrm{w}}}$ according to
$v=2M_{\text{W}}/g_{2}$, where $s_{{\mathrm{w}}}=\sin\theta_{\mathrm{w}}$ is
the sinus of the weak mixing angle $\theta_{\mathrm{w}}$ and $e$ the electric
unit charge. The corresponding renormalization constant $\delta v$ is directly
fixed by the renormalization conditions on $e$, $M_{\text{W}}$, and
$s_{{\mathrm{w}}}^{2}={1-M_{\text{W}}^{2}/M_{\text{Z}}^{2}}$. In order to
guarantee the compensation of all tadpole contributions after renormalization,
the bare tadpole constant $t_{0}=t^{\mathrm{PRTS}}+\delta t^{\mathrm{PRTS}}$
is split into a renormalized value $t^{\mathrm{PRTS}}$ and a corresponding
renormalization constant $\delta t^{\mathrm{PRTS}}$ and we demand
$t^{\mathrm{PRTS}}=0$, so that

$\displaystyle\delta
t^{\mathrm{PRTS}}=v_{0}(\mu_{2,0}^{2}-{\textstyle\frac{1}{4}}\lambda_{2,0}v_{0}^{2})={v}(\mu_{2,0}^{2}-{\textstyle\frac{1}{4}}\lambda_{2,0}v^{2}-{\textstyle\frac{1}{2}}\lambda_{2,0}v\delta
v),$ (15)

where the second equality holds in one-loop approximation. Since the
renormalized parameter $v$, which is directly fixed by measurements, and the
original bare parameters $\mu_{2,0}^{2}$ and $\lambda_{2,0}$ are gauge
independent, the gauge dependence of $\delta t^{\mathrm{PRTS}}$ goes over to
$\delta v$, where it shows up as gauge dependence in the mass renormalization
constant $\delta M_{\text{W}}^{2}$. Trading the two bare parameters
$\mu_{2,0}^{2}$ and $\lambda_{2,0}$ of the Higgs sector for $v_{0}$ and
$M_{\text{H},0}$, the PRTS tadpole counterterms can be generated by the
replacements

$\displaystyle\lambda_{2,0}\;\to\;\lambda_{2,0}+\frac{2\delta
t^{\mathrm{PRTS}}}{v^{3}},\qquad\mu_{2,0}^{2}\;\to\;\mu_{2,0}^{2}+\frac{3\delta
t^{\mathrm{PRTS}}}{2v}$ (16)

in the bare Lagrangian with $t_{0}=0$, i.e. several vertex counterterms
receive contributions from $\delta t^{\mathrm{PRTS}}$ (see, e.g., App. A of
Ref. [9]). If $\overline{\text{MS}}$-renormalized mass parameters are used as
input, the gauge dependence of $\delta t^{\mathrm{PRTS}}$ enters the
parametrization of observables in the step where $\mu_{2,0}^{2}$ and
$\lambda_{2,0}$ are traded for $v_{0}$ and $M_{\text{H},0}$. Note that this
flaw of introducing gauge dependences does not invalidate the applicability of
the PRTS. On the positive side, the PRTS has the practical advantage over the
FJTS that contributions to mass renormalization constants are much smaller,
which, in particular, implies that conversions of renormalized mass parameters
between OS and $\overline{\text{MS}}$ renormalization schemes are typically
much smaller in the PRTS as compared to the FJTS.

The GIVS aims to unify the benefits of the FJTS and the PRTS: the gauge-
invariance property of the former and the perturbative stability of the
latter. To avoid potentially large corrections induced by tadpole loops as
inherent in the FJTS, the vev of the Higgs field is tied to the true minimum
of the effective Higgs potential. Gauge dependences are avoided by switching
to the non-linear Higgs representation (1) where the condition $v_{0}=v+\delta
v$ applies to the gauge-invariant component $v_{0}+h_{\mathrm{B}}(x)$. In
detail, in the non-linear Higgs representation the GIVS is identical to the
PRTS described above, i.e. the tadpole renormalization constant is fixed
according to

$\displaystyle\delta t^{\mathrm{GIVS}}_{\mathrm{nl}}=\delta
t^{\mathrm{PRTS}}_{\mathrm{nl}}=-\Gamma^{h}_{\mathrm{nl}},$ (17)

which is gauge independent as pointed out in the previous section. However,
actual next-to-leading-order (NLO) calculations are typically carried out in
the linear representation, and simply taking $\delta
t^{\mathrm{PRTS}}_{\mathrm{nl}}$ as tadpole renormalization constant there
does not fully cancel explicit tadpole loops. In order to fix this, we
calculate $\delta t^{\mathrm{GIVS}}$ in the linear representations from
$\delta t^{\mathrm{GIVS}}_{\mathrm{nl}}$ plus an extra term to restore $\delta
t=-\Gamma^{\eta}$ as demanded in (12):

$\displaystyle\delta t^{\mathrm{GIVS}}=\delta t^{\mathrm{GIVS}}_{1}+\delta
t^{\mathrm{GIVS}}_{2},\qquad\delta
t^{\mathrm{GIVS}}_{1}=-\Gamma^{h}_{\mathrm{nl}},\qquad\delta
t^{\mathrm{GIVS}}_{2}=\Gamma^{h}_{\mathrm{nl}}-\Gamma^{\eta}.$ (18)

The gauge-independent part $\delta t^{\mathrm{GIVS}}_{1}$ occurs in PRTS-like
tadpole contributions to counterterm which are generated as in (16) with
$\delta t^{\mathrm{GIVS}}_{1}$ playing the role of $\delta t^{\mathrm{PRTS}}$.
This part absorbs potentially large corrections to the location of the minimum
in the effective Higgs potential into renormalized input parameters. The
gauge-dependent part $\delta t^{\mathrm{GIVS}}_{2}$ occurs in FJTS-like
tadpole contributions to counterterm which are generated by the field shift

$\displaystyle\eta_{B}(x)\;\to\;\eta_{B}(x)+\Delta
v^{\mathrm{GIVS}},\qquad\Delta v^{\mathrm{GIVS}}=-\frac{\delta
t^{\mathrm{GIVS}}_{2}}{M_{\text{H}}^{2}}$ (19)

analogous to (14). In summary, knowing the tadpole counterterms of the FJTS
and the PRTS, as e.g., given in the SM Feynman rules of App. A of Ref. [9],
the generation of the one-loop GIVS tadpole counterterms is easily
accomplished by the substitutions

$\displaystyle\delta t^{\mathrm{PRTS}}\;\to\;\delta
t^{\mathrm{GIVS}}_{1},\qquad\delta t^{\mathrm{FJTS}}\;\to\;\delta
t^{\mathrm{GIVS}}_{2}.$ (20)

In Ref. [5] we have illustrated the perturbative stability of the GIVS by
considering the conversion of OS-renormalized masses of SM particles to
$\overline{\text{MS}}$ masses with the various tadpole schemes. Table 1 shows
the corresponding mass shifts induced by NLO EW corrections for the heaviest
particles in the SM.

<EMAIL_ADDRESS>M^{\overline{\text{MS}}{-}\mathrm{OS}}_{{\mathrm{EW}}}[\mathrm{GeV}]\hfil\lx@intercol\vrule\lx@intercol\vrule\lx<EMAIL_ADDRESS>M^{\overline{\text{MS}}{-}\mathrm{OS}}_{{\mathrm{EW}}}[\mathrm{GeV}]\hfil\lx<EMAIL_ADDRESS>&&\mathrm{FJTS}&\mathrm{PRTS}&\mathrm{GIVS}&&&\mathrm{FJTS}&\mathrm{PRTS}&\mathrm{GIVS}\\\
\hline\cr\mbox{W boson}&80.379&-2.22&0.82&0.74&\mbox{top
quark}&172.4&10.75&0.99&0.54\\\ \mbox{Z
boson}&91.1876&-0.77&1.25&1.14&\mbox{bottom quark}&4.93&-1.79&0.10&0.13\\\
\mbox{Higgs boson}&125.1&6.34&3.16&2.80&\mbox{$\tau$
lepton}&1.77686&-0.93&-0.028&-0.015\\\ \hline\cr\end{array}$

Table 1: On-shell masses $M^{\mathrm{OS}}$ of the heaviest SM particles and
differences $\Delta M^{\overline{\text{MS}}{-}\mathrm{OS}}_{{\mathrm{EW}}}$
between the $\overline{\text{MS}}$ mass $\overline{M}(\mu)$ and
$M^{\mathrm{OS}}$ induced by NLO EW corrections using the FJTS, PRTS, or GIVS
at the renormalization scale $\mu=M^{\mathrm{OS}}$ (taken from Ref. [5]).

The masses entering in $\Delta
M^{\overline{\text{MS}}{-}\mathrm{OS}}_{{\mathrm{EW}}}$ are chosen according
to the OS mass values given in Tab. 1, and for the PRTS the ’t Hooft–Feynman
gauge is chosen.

The values obtained in the PRTS and the GIVS are of comparable size while in
general the FJTS leads to larger differences between the OS and the
$\overline{\text{MS}}$ masses. As emphasized in the literature [13, 14, 15]
for the top quark before, the FJTS shift $\Delta
m^{\overline{\text{MS}}{-}\mathrm{OS}}_{\text{t},{\mathrm{EW}}}=10.75\,\text{GeV}$
in the conversion of fermion masses is much larger than the typical size of EW
corrections of the percent level. For the lighter fermions b and $\tau$, the
relative corrections $\Delta
M^{\overline{\text{MS}}{-}\mathrm{OS}}_{{\mathrm{EW}}}/M^{\mathrm{OS}}$ are
even larger than for the top quark in the FJTS, reaching up to $\sim 50\%$,
while the shifts in the PRTS and GIVS remain all moderate. Despite these large
corrections, the FJTS often is favoured in the literature in this context,
since it leads to a gauge-independent result in contrast to the PRTS. Note,
however, that these large EW one-loop corrections entail an enhancement of the
theoretical uncertainties due to missing higher-order corrections. The GIVS,
on the other hand, provides gauge-independent mass shifts that are moderate
and, thus, leads to smaller EW theory uncertainties, when those uncertainties
are estimated by the propagation of the known corrections to higher order as
typically done.

##  4 The GIVS in extended Higgs sectors

The generalization of the GIVS from the SM to extended Higgs sectors is fully
straightforward. The tadpole renormalization constant $\delta
t_{h_{n}}=-\Gamma^{h_{n}}$ of any Higgs field $v_{n}+h_{n}(x)$ that can
acquire a vev $v_{n}$ is generated in the GIVS from two parts as in (20). The
gauge-independent PRTS-like part $\delta
t^{\mathrm{GIVS}}_{h_{n},1}=-\Gamma^{h_{n}}_{\mathrm{nl}}$ is calculated in
the non-linear Higgs representation, and the gauge-dependent FJTS-like part
$\delta
t^{\mathrm{GIVS}}_{h_{n},2}=\Gamma^{h_{n}}_{\mathrm{nl}}-\Gamma^{h_{n}}$
accounts for the remaining contributions. For the SESM and THDM, the
application of the GIVS and its impact on the $\overline{\text{MS}}$
renormalization of the Higgs mixing angles are described in Ref. [6] in
detail. For the $\overline{\text{MS}}$ renormalization of the THDM mixing
angle $\beta$ we find that the PRTS in any $R_{\xi}$ gauge coincides with the
GIVS, a fact that puts existing PRTS results on a gauge-independent basis when
reinterpreted as GIVS results. We conjecture that this feature of the
$\overline{\text{MS}}$ renormalization of $\beta$ also carries over to
supersymmetric theories.

The renormalization of Higgs mixing angles as well as the weaknesses and
strengths of various schemes was discussed for the SESM and THDM already in
Refs. [7, 8, 16, 17, 12]. In particular, Ref. [12] analyzes existing and newly
suggested renormalization schemes wrt. the following criteria: (i) gauge
independence, (ii) symmetry wrt. mixing degrees of freedom, (iii) perturbative
stability, and (iv) smoothness for degenerate masses or extreme mixing angles.
While the suggested schemes based on field-theoretical symmetries or on OS
conditions widely meet these requirements, $\overline{\text{MS}}$
renormalization with FJTS tadpole treatment turned out to be particularly
prone to perturbative instabilities in specific parameter regions (large or
degenerate Higgs masses, extreme mixing angles). These features were
demonstrated in a comprehensive discussion of NLO predictions for various
Higgs-boson production and decay processes in Ref. [12], extending the earlier
discussions of Refs. [7, 8, 16, 17].

The results of Ref. [6] for the Higgs-boson decays
$\text{h}/\text{H}\to\text{W}\text{W}/\text{Z}\text{Z}\to 4f$ in the SESM and
THDM demonstrate that $\overline{\text{MS}}$ renormalization with GIVS tadpole
treatment mitigates perturbative instabilities significantly and produces
gauge-independent results very close to the gauge-dependent PRTS. Table 2
shows some results for the decays $\text{h}\to 4f$ in two THDM scenarios,
which illustrate that the scale uncertainty of the $\overline{\text{MS}}$ FJTS
results is not always reduced in the transition from LO to NLO. It should also
be mentioned that all $\overline{\text{MS}}$ variants run into problems with
perturbative stability in extreme parameter regions; for instance, the
$\overline{\text{MS}}$ schemes do not give reliable results for the
$\text{H}\to 4f$ decays of the heavy Higgs boson H in the THDM.

A1 A2 Ren. scheme tadpoles LO NLO LO NLO OS12 ($\alpha,\beta$) $0.89832(3)$
$0.96194(7)_{+0.1\%}^{-0.1\%}$ $0.87110(3)$ $0.92947(7)_{+0.1\%}^{-0.2\%}$
$\overline{\text{MS}}$ ($\alpha,\beta$) FJTS $0.89996(3)_{-7.4\%}^{+0.7\%}$
$0.96283(7)_{-0.2\%}^{+0.8\%}$ $0.88508(3)_{-10.0\%}^{+2.2\%}$
$0.93604(7)_{-11.0\%}^{+3.1\%}$ $\overline{\text{MS}}$ ($\alpha,\beta$) PRTS
$0.89035(3)_{+0.9\%}^{-2.8\%}$ $0.96103(7)_{+0.4\%}^{+1.2\%}$
$0.86130(3)_{+2.3\%}^{-6.1\%}$ $0.92784(7)_{+1.3\%}^{+1.3\%}$
$\overline{\text{MS}}$ ($\alpha,\beta$) GIVS $0.89082(3)_{+0.9\%}^{-2.7\%}$
$0.96106(7)_{+0.5\%}^{+1.2\%}$ $0.86249(3)_{+2.3\%}^{-5.8\%}$
$0.92808(7)_{+1.3\%}^{+1.3\%}$ $\overline{\text{MS}}$ ($\lambda_{3},\beta$)
FJTS $0.89246(3)_{+1.6\%}^{-15.1\%}$ $0.96108(7)_{+1.9\%}^{+17.3\%}$
$0.85590(3)_{+5.5\%}^{-29.8\%}$ $0.92723(7)_{+2.8\%}^{+18.3\%}$
$\overline{\text{MS}}$ ($\lambda_{3},\beta$) PRTS/GIVS
$0.89156(3)_{+1.7\%}^{-8.4\%}$ $0.96111(7)_{+2.1\%}^{+3.8\%}$
$0.85841(3)_{+5.0\%}^{-12.7\%}$ $0.92729(7)_{+2.6\%}^{+4.6\%}$

Table 2: LO and NLO decay widths $\Gamma^{\text{h}\to 4f}$[MeV] of the light
CP-even Higgs boson h for THDM scenarios A1 and A2 in different
renormalization schemes, with the on-shell scheme OS12 as input scheme (and
full conversion of the input parameters into the other schemes). The scale
variation (given in percent) corresponds to the scales $\mu=\mu_{0}/2$ and
$\mu=2\mu_{0}$, where $\mu_{0}$ is the average Higgs mass (for all details,
see Ref. [6]).

##  5 Conclusions

Recently we have proposed a new scheme for a gauge-invariant treatment of
tadpole corrections in spontaneously broken gauge theories called Gauge-
Invariant Vacuum expectation value Scheme (GIVS). The GIVS is a hybrid scheme
of the two most commonly used tadpole schemes, in which tadpole counterterms
are introduced either via parameter or via field renormalization
transformations. In the construction of tadpole counterterms, the GIVS makes
use of non-linear representations of Higgs sectors, in which fields that can
develop non-vanishing vacuum expectation values are gauge invariant, but
actual loop calculations are performed in the GIVS as usual. In contrast to
previously used tadpole schemes, the GIVS unifies the properties of gauge
independence and perturbative stability. We have illustrated these virtues by
discussing the conversion of on-shell-renormalized to
$\overline{\text{MS}}$-renormalized masses in the SM and the
$\overline{\text{MS}}$ renormalization of Higgs mixing angles in two
frequently considered models with extended Higgs sectors—a singlet Higgs
extension of the SM and the Two-Higgs-Doublet Model. We expect that the
generalization of the GIVS beyond the one-loop level works without problems.

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