Bridging Positivity and S-matrix Bootstrap Bounds Joan Elias Mir oa Andrea Guerrieribcd Mehmet Asm G um u sef aThe Abdus Salam ICTP Strada Costiera 11 34135 Trieste Italy

2025-04-27 0 0 6.69MB 61 页 10玖币
侵权投诉
Bridging Positivity and S-matrix Bootstrap Bounds
Joan Elias Mir´oa, Andrea Guerrierib,c,d, Mehmet Asım G¨um¨u¸se,f
aThe Abdus Salam ICTP, Strada Costiera 11, 34135, Trieste, Italy
bSchool of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel
cDipartimento di Fisica e Astronomia, Universita degli Studi di Padova, & Istituto Nazionale di
Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy.
dPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
eSISSA, Via Bonomea 265, I-34136 Trieste, Italy
fINFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
Abstract
The main objective of this work is to isolate Effective Field Theory scattering amplitudes
in the space of non-perturbative two-to-two amplitudes, using the S-matrix Bootstrap. We
do so by introducing the notion of Effective Field Theory cutoff in the S-matrix Bootstrap
approach. We introduce a number of novel numerical techniques and improvements both
for the primal and the linearized dual approach. We perform a detailed comparison of the
full unitarity bounds with those obtained using positivity and linearized unitarity. Moreover,
we discuss the notion of Spin-Zero and UV dominance along the boundary of the allowed
amplitude space by introducing suitable observables. Finally, we show that this construction
also leads to novel bounds on operators of dimension less than or equal to six.
January 18, 2023
arXiv:2210.01502v2 [hep-th] 17 Jan 2023
Contents
1 Introduction........................................... 2
2 The space of QFTs in 3 + 1-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Weakly coupled EFTs in 3 + 1 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Dualbounds........................................... 28
5 Dimension-sixoperators .................................... 36
6 Cutoff dependence, In medio stat virtus ........................... 38
7 Conclusions ........................................... 39
A Thenumericalsetup ...................................... 40
B Review of Mandelstam representation and partial waves . . . . . . . . . . . . . . . . . . 45
Cg0φ4theoryperturbatively................................... 47
D Subtracted dispersion relations for the O(n)theory ..................... 48
E Atoymodel........................................... 50
1 Introduction
Quantum Effective Field Theory is very much universal and has a wide range of application and
flexibility. Nevertheless, the principles of unitary evolution and causality imply constraints on the
space of feasible Effective Field Theories (EFTs), that is on EFTs with a consistent UV completion.
In other words, not anything goes and the coupling strengths of the interactions are subject to
inequality constraints.
A widely known example is the positivity bound: while a priori Wilson coefficients can take any
real value, the two-to-two forward scattering amplitude satisfies the positivity constraint ImM > 0,
implying that certain Wilson coefficients are positive [1] – see also studies in the context of the
chiral Lagrangian [2–4]. Several works have since then exploited positivity, leading to constraints
on renormalization group flows and the phenomenology of EFTs [5–17]. See also [18–35] for
interesting recent developments.
Another realisation of these principles is the recent version of the S-matrix Bootstrap [36–39].
In this approach unitarity is not linearised and it is treated non-perturbatively. These ideas have
been used in a number of theoretically and phenomenologically interesting theories such as two-
dimensional flux-tube effective field theories [40,41], four dimensional (pseudo)-Goldstone bosons
[42,43], Majorana fermion scattering [44], and higher dimensional supergravity [45,46].
In this work we obtain new S-matrix Bootstrap bounds on the space of two-to-two scattering
amplitudes in d= 4 spacetime. More concretely, we consider two examples: the two-to-two
scattering amplitude of a massive scalar singlet particle, and a massive scalar particle with internal
global O(n) symmetry. These amplitudes can be characterised by their Taylor expansion around
the crossing symmetric point (s, t, u)=4/3(1,1,1)m2in the centre of the Mandelstam triangle.
For the singlet theory, the first few terms of this expansion are given by
M(¯s, ¯
t, ¯u) = −c0+c2(¯s2+¯
t2+ ¯u2) + c3(¯s¯
t¯u) + O(¯s4,¯
t4,¯u4) ; (1.1)
2
while the s-channel amplitude of the O(n) scalar theory is
M(¯s|¯
t, ¯u) = −c0+cH¯s+O(¯s2,¯
t2,¯u2),(1.2)
where (¯s, ¯
t, ¯u) = (s, t, u)/m2−4/3(1,1,1). We will first show that unitarity, crossing, and analyt-
icity of the amplitude imply non-perturbative bounds on the ci’s. We will also characterise these
amplitudes with extremal values of the cicoefficients, and study a number of observables such as
UV/IR dominance or Low/High spin dominance.
For weakly coupled EFTs we may interpret the ci’s as Wilson coefficients of operators. For
instance consider the free O(n) scalar theory perturbed by the dimension-six operator ∆L=
gH∂µ(~
φ·~
φ)∂µ(~
φ·~
φ)/(4Λ2). Then, cH= 2 gHm2/Λ2+. . . at tree-level. This is a priori very sugges-
tive, because it is generally hard to set bounds on dimension-six operators using positivity methods,
or linearised unitarity on the imaginary part. This stems from the Froissart-Martin bound and
the fact that scattering amplitudes satisfy double subtracted dispersion relations. The dispersive
representation of the dimension six operators involve a real subtraction constant that cannot be
bounded unless we access the real part of the amplitude too. Nevertheless dimension-six operators
are of physical importance because they parametrise at leading order generic deviations from the
Standard Model predictions (barring the Weinberg operator for neutrino masses). Therefore it
is quite interesting that using the S-matrix Bootstrap one is able to bound these dimension-six
operators – as well as to characterise the amplitudes achieving such extremal values.
However the extremal values of the ci’s are often achieved by strongly coupled amplitudes and
therefore the weakly coupled EFT interpretation is not accurate. Namely |cH−2gHm2/Λ2|> O(1)
and as a consequence the bound on cHdoes not translate simply into a bound on gH. After finding
the bounds on the ci’s, one of our main objectives is precisely to amend this problem. That is we
will show that A) the S-matrix Bootstrap can output min/max values of Wilson coefficients for
theories that are described by a weakly coupled field theory for energies below a physical cutoff Λ,
and B) this construction provides min/max values of dimension-six operators as well.
In section 2we set the stage by determining precisely the space of amplitudes with maximal
civalues in the singlet case. In order to get this result we introduce a number of numerical
improvements that allow us to achieve a faster convergence of the Bootstrap algorithm. We also
compare in great detail the S-matrix Bootstrap bounds with a rigorous positivity approach. In
section 3we carve out the space of amplitudes with extremal ci’s that are weakly coupled in the
IR. In section 4we derive new dual bounds using linearised unitarity, and compare with the results
in section 3. In section 5we begin the exploration of the extremal values for the O(n) theory. In
section 6we address the role of the EFT cutoff in the bounds on the ci’s. Finally we conclude in
section 7.
2 The space of QFTs in 3+1-dimensions
We first study the space of QFTs in 3 + 1 dimensions that contain in the IR a single stable
scalar particle of mass m, even under field parity. 1We analyse a particular slice of this space
1This Z2-symmetry implies poles in the amplitude below threshold 0 <s<4m2are forbidden because of the
absence of the trilinear coupling. Our analysis could be easily generalised by relaxing this assumption.
3
by determining the possible values of the 2 →2 on-shell scattering amplitude with momenta
p1+p2→p3+p4.
p1
p2
p4
p3
M
As we shall see below this observable is very rich, containing a wealth of information about the
spectrum and properties of the theory.
It is possible to describe the 2 →2 scattering amplitude as a function of the three Mandelstam
invariants M(s, t, u). 2Due to crossing symmetry,Mis invariant under any permutation of its
arguments s↔t↔u. Momentum conservation s+t+u= 4m2further reduces the number of
independent variables to two: e.g. M(s, t)≡M(s, t, 4m2−s−t).
The amplitude M(s, t) is further constrained by the two particle sector of the unitarity con-
dition S†S1 where, as usual, the 2 →2 S-matrix is given by S=1+i(2π)4δ(4)(p1+p2−
p3−p4)M(s, t). It is useful to diagonalise unitarity by projecting onto partial waves f`(s)≡
1/(32π)R1
−1dx P`(x)M(s, t(s, x), u(s, x)) ,for `∈N, and x= 1 + 2t/(s−4m2). The unitarity
condition then takes the simpler form 2Im f`(s)>p(s−4m2)/s|f`(s)|2for s > 4m2.3The
inequality is saturated in the elastic region 4m26s < 16m2, due to the absence of multi-particle
processes.
We also assume maximal analyticity (or Mandelstam analyticity): the scattering amplitude is
an analytic function in the (s, t) complex planes everywhere except on the unitarity cuts s, t, u >
4m2, and possible bound state poles for 0 < s, t, u < 4m2.4In our setup, we assume the absence
of bound states below threshold 0 < s, t, u < 4m2, but generalising our results to include these
cases is possible. In Fig. 1we summarise the assumed analytic structure in the complex s-plane
at fixed −4m2< t∗<4m2. The right-hand cut starting at the two-particle threshold s= 4m2is
a consequence of unitarity. The left hand-cut is due to the physical u-channel process starting at
s=−t∗. The u-channel cut moves as we move t∗, and overlaps the s-channel cut for 4m2<s<−t∗
when t∗<−4m2.
All in all, the 2 →2 scattering amplitude is a function of two variables satisfying crossing-
symmetry, unitarity, and analyticity. These properties are summarised in Fig. 1. Any such
amplitude can be defined by its Taylor expansion around the crossing symmetric point
M(¯s, ¯
t, ¯u) = ∞
X
n,p,q=0
˜c(npq)¯sn¯
tp¯uq,(2.1)
with coefficients ˜c(npq)∈R, and with the shifted Mandelstam variables being m2¯x=x−4m2/3. In
this new variables the crossing symmetric point is at the origin (¯s, ¯
t, ¯u) = (0,0,0) and momentum
2Recall that the Mandelstam invariants are s= (p1+p2)2,t= (p1−p3)2and u= (p1−p4)2, where piare
Lorentz four-momenta.
3This is also consequence of real analyticity M(s∗, t) = M(s, t)∗. This is always true for the scattering processes
we are considering.
4The validity of Maximal analyticity of the scattering amplitude is a long-standing conjecture that has resisted
all the perturbative checks, see [47] for a review, and the references therein. In [48] it has been pointed out that even
for the simple case of the scattering of the lightest particles, its perturbative proof would require the cancellation
of an infinite number of intricate physical sheet Landau singularities.
4
XX
4m2
0
M(s,t*)
M(u,t*) ≡M(s,t*)
Crossing symmetry
Unitarity
2Imfℓ(s)≥s−4m2
s
|fℓ(s)|2
s-plane, t=t* fixed
−t*
Figure 1: Analytic properties of the amplitude M(s, t) in the complex s-plane for a fixed value of
−4m2< t∗<4m2. We denote in blue the crossing path continuing M(s, t∗) into M(u, t∗). The
black dashed vertical line passes through the s−ucrossing symmetric point (4m2−t∗)/2. Due
to real analyticity, the amplitude is real in between the two cuts and along the dashed line. The
right-hand cut is subject directly to the unitarity constraints.
conservation is given by ¯s+¯
t+ ¯u= 0. Given the values of all the coefficients {˜c(npq)}in (2.1), we
can reconstruct the whole amplitude M(s, t) by analytic continuation. Therefore, we parametrise
the space of amplitudes by the values of these Taylor coefficients.
The first few terms of (2.1) can be simply written as (1.1) after imposing ¯s+¯
t+ ¯u= 0, and a
straightforward linear redefinition of the coefficients {˜c(npq)}.
In perturbation theory the ci’s have a simple interpretation in terms of couplings or Wilson
coefficients in the EFT. For instance, consider the field theory Lagrangian
L[∂φ, φ] = 1
2(∂µφ)2−1
2m2φ2−1
24g0φ4+1
2
g2
Λ4[(∂µφ)2]2+1
3
g3
Λ6(∂µ∂ρφ)(∂ν∂µφ)(∂ρ∂νφ)φ+··· (2.2)
where Λ is the cutoff of the EFT, the dots ··· involve higher order corrections O(Λ−8) in the
derivative expansion and operators with more than four fields O(φ6). If the theory is weakly
coupled, i.e. pi, m Λ with gi.O(1), it is straightforward to compute the amplitude M(s, t)
from (2.2), leading to
c0=g0−4/3g22+. . . , (2.3)
c2=g22+. . . , (2.4)
c3=g33+. . . , (2.5)
where =m2/Λ2. The first term of these equations is based on standard field theory analysis:
the coupling ci=giiat tree level but it can be renormalised by O(g0) loops involving marginal
interactions. This explanation however is a bit too naive, which is demonstrated by the presence of
the second term 4/3g22in (2.3). This piece also arises from a tree-level correction because we are
5
摘要：

BridgingPositivityandS-matrixBootstrapBoundsJoanEliasMiroa,AndreaGuerrierib;c;d,MehmetAsmGumuse;faTheAbdusSalamICTP,StradaCostiera11,34135,Trieste,ItalybSchoolofPhysicsandAstronomy,TelAvivUniversity,RamatAviv69978,IsraelcDipartimentodiFisicaeAstronomia,UniversitadegliStudidiPadova,&IstitutoNazi...

展开>> 收起<<
Bridging Positivity and S-matrix Bootstrap Bounds Joan Elias Mir oa Andrea Guerrieribcd Mehmet Asm G um u sef aThe Abdus Salam ICTP Strada Costiera 11 34135 Trieste Italy.pdf

共61页,预览5页

还剩页未读， 继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

相关推荐

更多
立即下载
分类：图书资源 价格：10玖币 属性：61 页 大小：6.69MB 格式：PDF 时间：2025-04-27

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务

作者详情

相关内容

更多

热门标签

人际关系 动力学连接体 力的合成 配电装置 高考理综 全宋诗作者索引 公务员考试
/ 61
评分 收藏
分享
立即下载
客服
关注